NMR of Ordered Liquids

NMR of Ordered Liquids

NMR of Ordered Liquids

Edited by

E. Elliott Burnell University of British Columbia, Canada

and

Comelis A. de Lange University of Amsterdam, The Netherlands

Springer-Science+Business Media, B.Y.

A c.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-6305-2 ISBN 978-94-017-0221-8 (eBook) DOI 10.1007/978-94-017-0221-8

Printed on acid-free paper

All Rights Reserved © 2003 Springer Science+Business Media Dordrecht

Originally published by Kluwer Academic Publishers in 2003.

Softcover reprint of the hardcover 1 st edition 2003 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

This book is dedicated to our dear wives Bonnie and

Annette who put up with the many hours and late

evenings, not to mention the early mornings, spent

discussing and arguing about the various details of

this project

Contents

Contributing Authors xiii

Preface XXI

Introduction XXv

E.E. Burnell and CA. de Lange

Part I Basics

Basics ofNMR of molecules in uniaxial anisotropic environments 5 CA. de Lange and E.E. Burnell

1 Introduction 5 2 General Hamiltonian in uniaxial anisotropic liquids 6 3 The high-field approximation 9 4 Transformation to molecule-fixed axes 11 5 Orientation parameters 12 6 Molecular symmetry 15 7 Simple examples ofNMR spectra of orientationally ordered molecules 17 8 Orientational order induced by anisotropic solvents 21 9 Orientational order induced by strong electric fields 22 10 Orientational order induced by strong magnetic fields 23 11 Internal motion 24 12 Summary 26

2 Density matrix methods in NMR 27 M. Bloom, E.E. Burnell and CA. de Lange

1 Introduction 2 Brief introduction to the density matrix 3 PuisedNMR 4 Summary

3 Coherent averaging and correlation of anisotropic spin interactions in oriented

molecules Malgorzata Marjanska, Robert H. Havlin and Dimitris Sakellariou

1 Introduction 2 Spatial reorientation techniques

Vll

27 27 30 43

45

45 45

viii

4

3 4 5 6

Two-dimensional dynamic director correlations Spin manipulation based techniques Multidimensional techniques Conclusions

NMR OF ORDERED LIQUIDS

50 51 57 63

Multiple Quantum NMR Spectroscopy in Orientationally Ordered Fluids Leslie D. Field

67

1 NMR in orientationally ordered fluids 2 Spectral simplification by MQNMR 3 Excitation and detection of multiple quantum coherence 4 Selective deuteration for spectral simplification 5 Spectral analysis and simulation 6 Structural studies using MQNMR 7 Structural studies by 1 H MQNMR 8 Heteronuclear MQNMR 9 Other applications ofMQNMR in liquid crystalline solvents

5

67 68 69 76 77 77

80 84 84

Spectral Analysis of Orientationally Ordered Molecules Raymond T Syvitski

89

1 Introduction 2 Tools of the trade 3 Some examples of putting it together 4 Summary

89 90 96

103

Part II NMR of solute atoms and molecules

6 NMR of Noble Gases Dissolved in Liquid Crystals Jukka Jokisaari

109

1 Introduction 2 NMR properties of noble gases 3 Chemical shift 4 Quadrupole coupling 5 Relaxation 6 129Xe self-diffusion 7 Conclusions

7

109 109 110 121 127 130 133

NMR of partially ordered solutes with emphasis on structure determination 137 c.L. Khetrapal and G.A. Nagana Gowda

1 Introduction 137 2 Basic principles 138 3 Spectral analysis 141 4 Scope and limitations 142 5 Practical considerations for deriving precise molecular structural information 142 6 Aids for spectral analysis 145 7 Emerging developments and possible future directions 153 8 Conclusions 158

Contents IX

8 Observation and interpretation of residual dipolar couplings in biomolecules 163 Jean-Franr;ois Trempe and Kalle Gehring

1 Introduction 163 2 Theory 164 3 Measurement of residual dipolar couplings 166 4 Interpretation and applications of dipolar couplings and CSA in structural

biology 177 5 Summary 186

9 The search for high-resolution NMR methods for membrane peptide structure Christophe Fares and James H. Davis

191

1 Introduction: NMR of orientationally ordered systems 2 Theoretical background 3 Separated local field spectroscopy 4 High-resolution 1 H MAS NMR of small membrane proteins 5 Conclusions

191 193 197

209 210

Part III Theory, models, and simulations

10 Solutes as probes of simplified models of orientational order 221 E.E. Burnell and CA. de Lange

I Introduction 221 2 Obtaining a self-consistent set of solute orientational order parameters 224 3 Factors affecting solute orientational order 226 4 Orientational order of solutes in "magic mixtures" 230 5 Comparison of experimental and calculated orientational order in "pure"

liquid crystals and "magic mixtures" 234 6 Conclusions 239

11 Molecular Models of Orientational Order Alberta Ferrarini and Giorgio J Mora

241

1 Phenomenological models for short-range interactions 2 Modeling electrostatic interactions 3 Density functional theory 4 Conclusion Appendix: Polarization induced by a charge distribution in a dielectric

12 Molecular theory of orientational order Demetri J Photinos

1 Introduction 2 Order parameters, molecular structure and interactions 3 Approximation schemes for the potential of mean torque 4 Molecular models 5 Summary.

242 246 251 256 256

259

259 260 268 273 281

x NMR OF ORDERED LIQUIDS

13 Very Flexible Solutes: Alkyl Chains and Derivatives Edward T. Samulski

1 Introduction 2 Chronology ofNMR studies of flexible solutes 3 Models of flexible molecules 4 Conclusions

14 NMR Studies of Solutes in Liquid Crystals: Small Flexible Molecules Giorgio Celebre and Marcello Longeri

1 Introduction 2 Theoretical background 3 The conformational problem 4 Selected examples 5 Conclusions

15 Simulations of Orientational Order of Solutes in Liquid Crystals James M Polson

1 Introduction 2 Orientational distribution functions and mean-field potentials 3 Conformational behaviour of flexible solutes 4 Electrostatic interactions 5 Conclusions

Part IV Dynamic aspects and relaxation

16 Spin relaxation in orientationally ordered molecules Ronald Y. Dong

1 Introduction 2 Average Hamiltonian 3 Spin relaxation theory 4 Motional models 5 Applications of spin relaxation

17 Low-frequency NMR relaxometry of spatially constrained liquid crystals F Grinberg, M Vilfan and E. Anoardo

1 Introduction 2 Field-cycling relaxometry 3 Low-field proton relaxometry of confined liquid crystals 4 The dipolar-correlation effect 5 Deuteron NMR relaxometry of confined liquid crystals

18 NMR on macroscopically oriented lyotropic systems G. Oriidd and G. Lindblom

1 Introduction 2 Orientation dependent NMR interactions 3 Lipid translational diffusion 4 Preparation of macroscopically oriented lamellar systems

285

285 289 293 303

305

305 306 307 312 322

325

325 327 333 337 343

349

349 350 354 356 360

375

375 378 380 384 391

399

399 399 404 407

Contents

5 Examples

19 Dynamic NMR in liquid crystals and liquid crystalline solutions Zeev Luz

1 2 3 4 5

Introduction Dynamic proton NMR of solutes in nematic solvents Dynamic deuterium NMR spectra Dynamic carbon-13 MAS NMR Concluding remarks

Xl

411

419

419 420 428 439 447

Contributing Authors

Esteban Anoardo ([email protected]) was born in 1964 in Cordoba, Argentina. He studied Physics (graduate and Ph.D.) at the University of Cordoba, with specialization in Field-Cycling NMR applied to liquid crystal materials. During 1999-2000 he worked in Italy with Stelar srI for the development of a new Field-Cycling reI axometer. Later he moved to the University ofUlm (Germany) for a post-doctoral (AVH) research stay with Professor Rainer Kimmich. He is currently Professor of Physics at the University of Cordoba, with research interests in NMR relaxation and the associated instrumentation.

Myer Bloom ([email protected]) was born in 1928 in Montreal, Canada. He received B.Sc. and M.Sc. degrees in Physics from McGill University in 1949 and 1950 and completed his Ph.D. in Physics at the University of Illinois with Charlie Slichter in 1954. He is currently Emeritus Professor of Physics at the University of British Columbia, with research interests in the physics of biological systems. The main focus of his current research involves the role of the dominant poly-unsaturated lipids in animal brains in relation to the enlargement and evolution of the human brain that probably took place more than (approximately) 100,000 years ago.

E. Elliott Burnell ([email protected]) was born in 1943 in St. John's, Newfoundland. He received B.Sc. and M.Sc. degrees in Chemistry from Memorial University of Newfoundland in 1965 and 1967. He did his Ph.D. in Theoretical Chemistry in Bristol (UK) with Prof. A.D. Buckingham in 1970. He is currently Professor of Chemistry at the University of British Columbia, with research interests in NMR of orientationally ordered molecules and in intermolecular forces.

xiii

xiv NMR OF ORDERED LIQUIDS

Giorgio Celebre ([email protected]) was born in 1958 in Reggio Calabria, Italy. He graduated in Chemistry from the University of Calabria in 1984. He is currently Associate Professor of Physical Chemistry at the University of Calabria (Rende), with research interests in intermolecular forces and in conformational analysis, studied by Liquid Crystal NMR.

Jim Davis ([email protected]) was born in 1946 in Alexandria, MN, USA. He obtained Bachelors degrees in both Physics and Mathematics from Moorhead State University in 1969 and his Ph.D. from the University of Manitoba in Winnipeg in 1975. He then worked as an NRC post-doctoral fellow with Prof. Myer Bloom at the University of British Columbia before joining the faculty at the Department of Physics, University of Guelph in 1980. He has been using NMR for over 30 years.

Cornelis A. de Lange ([email protected]) was born in 1943 in Zaandam, The Netherlands. He graduated in experimental physics from the University of Amsterdam in 1966 and did his Ph.D. in Theoretical Chemistry in Bristol (UK) with Prof. A.D. Buckingham in 1969. He is currently Professor of Laser Spectroscopy at the University of Amsterdam, with research interests in atmospheric chemistry and NMR of oriented molecules.

Ronald Y. Dong ([email protected]) was born in 1942 in Shanghai, China. He graduated from Engineering Science at the University of Toronto in 1966, and obtained his Ph.D. in Experimental Physics at the University of British Columbia with Prof. Myer Bloom in 1969. He is currently a professor of Physics at Brandon University and an adjunct Physics professor at the University of Manitoba, Canada. Research interests include solid-state NMR, statistical theory, molecular dynamics, liquid crystals and biomolecules.

Contributing Authors

I

~., IJI'

xv

Christophe Fares ([email protected]) was born in 1972 in Montreal, Canada. He obtained a Bachelors of Science in Biochemistry from McGill University in 1994 and is expected to fi nish his Ph.D. in Biophysics at the University of Guelph under the supervision of Prof. lH. Davis and Prof. F.l Sharom in 2003. He is interested in pursuing research towards the application and development of solid state NMR methods on biological systems.

Alberta Ferrarini (albertaf@chfi .unipd.it) received the degree in Chemistry (1983) and the Ph.D. in Chemical Sciences (1989) from the University of Pad ova (Italy), under the supervision of Pier Luigi Nordio. She spent research periods in the groups of Gerd Kothe (University of Stuttgart, Germany) and Jack H. Freed (Cornell University, USA). In 1990 she became a research associate in the Physical Chemistry Department of the University of Padova, where since 2000 she has been an associate professor. Her research interests include the molecular interpretation of equilibrium and dynamic properties ofliquids and complex fuids (liquid crystals, membranes, colloids).

Leslie D. Field (L. [email protected]) was born in 1953 in New South Wales, Australia. He graduated with fi rst class honours in Chemistry in 1975 and undertook his Ph.D. in Organic Chemistry at the University of Sydney working with Professor Sev Sternhell. He spent postdoctoral periods studying with Professor George Olah at the University of Southern California (USA) and with Professor Jack Baldwin at the University of Oxford (UK) before returning to Australia to take up a post at the University of Sydney. He was Head of the School of Chemistry from 1996-2000 and he is currently Professor of Organic Chemistry with research interests in organometallic chemistry, catalysis and chemical applications ofNMR spectroscopy.

xvi NMR OF ORDERED LIQUIDS

Kalle Gehring ([email protected]) was born in 1958 and raised in Ann Arbor, Michigan. He has degrees from Brown University, the University of Michigan and the University of California (Berkeley). Following postdoctoral studies at Lawrence Berkeley Laboratory, the Ecole Polytechnique and the French CNRS, he took up a faculty position in Biochemistry at McGill University in 1994. His current research interests are structural genomics and residual dipolar couplings in NMR structure determination.

G.A. Nagana Gowda ([email protected]) was born in 1961 in Musandihal, India. He graduated from the University of Mysore in 1985 and completed his Ph.D. with Prof. C. L. Khetrapal in 1999 from the Bangalore University. He worked at the Indian Institute of Science till 2001 and presently he is working as Assistant Professor at the Center of Biomedical Magnetic Resonance, Sanjay Gandhi Post Graduate Institute of Medical Sciences, Lucknow, India. His Research Interests are in the area of Biomedical and Chemical Magnetic Resonance and NMR of Oriented Systems.

Farida A. Grinberg ([email protected]) was born in 1957 in Kazan, Russia. She graduated in experimental physics from the University of Kazan in 1979 and did her Ph.D. in NMR of polymers in 1987 in the same University. In 2002 she did her habilitation (second dissertation) work in slow molecular dynamics with Prof. R. Kimmich in the University ofUlm (Germany). She is currently a research scientist at the Max Planck Institute for Metals Research, Stuttgart, and a Privatdozentin at the University ofUlm. Her research interests include NMR of anisotropic liquids, diffusion studies, NMR imaging, nanostructured materials, and computer simulations.

Robert H. Havlin ([email protected]) was born in 1975 in Belleville, Illinois, USA. He graduated from the University of Illinois at Urbana-Champaign in 1997 with a B.S. in Chemistry where he did research with Prof. Eric Oldfi eld. Then he fi nished his Ph.D. in Physical Chemistry with Prof. Alexander Pines at the University of California, Berkeley where his research focused on switched angle spinning of liquid crystals and solid-state NMR dipolar recoupling methods.

Contributing Authors XVll

Jukka Jokisaari [email protected] ) was born in Kemi, Northern Finland. He graduated in experimental physics from the University of Oulu in 1968 and got his Ph.D. at the same university in 1974. He worked as a post-doctoral fellow with Professor Peter Diehl, University of Basel, Switzerland, during several periods between the late 70's and early 90's. He is currently Professor of Physics (atomic and molecular spectroscopy) at the University of Oulu, with research interests in NMR of noble gases in liquid crystals, determination of nuclear shielding, spin-spin coupling and quadrupole coupling tensors, applying NMR of solute molecules in liquid crystals, and determination of pore sizes in micro- and mesoporous materials applying 129Xe NMR.

C.L. Khetrapal ([email protected]) was born in 1937 in Sahival in Undivided India. He graduated from the University of Allahabad in 1959 and completed his Ph.D. with Prof. S.S. Dharmatti in 1965 from the Bombay University. He worked at the Tata Institute of Fundamental Research, Bombay as Reader, at the Raman Research Institute, Bangalore as Associate Professor, at the Indian Institute of Science as Professor, at the University of Allahabad as the Vice-Chancellor and currently he is Distinguished Professor at Sanjay Gandhi Post Graduate Institute of Medical Sciences, Lucknow, India. His Research Interests are in the area of Biomedical and Chemical Magnetic Resonance and NMR of Oriented Systems.

Goran Lindblom ([email protected]) was born in 1942 in Kalmar, Sweden. He graduated as Master in Chemical Engineering from the Lund Institute of Technology in 1969 and did his Ph.D. in Physical Chemistry, Lund University in 1974. He has been Professor in Physical Chemistry since 1981 at Umea University with research interests in Biophysical Chemistry, in particular of biological membranes and solid state NMR spectroscopy.

Marcello Longeri ([email protected]) was born in 1947 in Piombino, Italy. He graduated in Chemistry from the University of Pis a in 1972. He is currently Professor of Physical Chemistry at the University of Calabria (Rende), with research interests in NMR of Liquid Crystals.

xviii NMR OF ORDERED LIQUIDS

Demonstrating physical properties of thin-layer lyotropic liquid crystals (a soap fi 1m catenoid contained between two rings).

Zeev Luz ([email protected]) was born in 1932 in Munich, Germany and emigrated to Isreal (then Palestine) in 1934. He graduated with an M.Sc. in Physical Chemistry from the Hebrew University in Jerusalem in 1957, and obtained his Ph.D. from the Weizmann Institute of Science in 1961. He joined the scientifi c staff of the Weizmann Institute in 1964, and since 1997 has been Professor Emeritus in that Institute. Research interests include applications of magnetic resonance spectroscopy to study structure and dynamics in condensed phases.

Malgorzata Marjanska ([email protected]) was born in 1974 in Monki, Poland. She graduated in Chemistry from Loyola University of Chicago and did her Ph.D. in Physical Chemistry in the University of California, Berkeley with Prof. Alexander Pines in 2002. She is currently a post-doctoral fellow in the Center for Magnetic Resonance Research at the University of Minnesota working with Prof. Kamil Ugurbil.

Giorgio J. Moro (g.moro@chfi .unipd.it) has carried out research in physical and theoretical chemistry at Padova University, Italy, since 1987 as an Associate Professor and since November 2000 as a Full Professor of Physical Chemistry. His research activities focus on the theoretical study of condensed matter properties. Specifi c fi elds of interest are the simulation of spectroscopic observables, the effects of the solvent (and in particular of its local structure) on rotational and translational dynamics, conformational dynamics of chains and polymers, and stochastic models of kinetic processes. Liquid crystals represent the privileged systems for molecular dynamics studies and for the elaboration of original methods predicting order and structure on the basis of molecular organization. His work has resulted in more than 70 articles so far.

Contributing Authors xix

Greger Oradd ([email protected]) was born in 1961 in Asele, Sweden. He graduated in physics from Umea University in 1988 and did his Ph.D. in Physical Chemistry, Umea University in 1995. He is currently a Lecturer in Biophysical Chemistry at Umea University, with research interests in the pulsed fi eld gradient technique for self-diffusion measurements applied to ion conducting polymers and lipid bilayer systems of biological interest.

Demetri J. Photinos ([email protected]) was born in Cairo in 1947. He graduated in Physics from the University of Athens in 1970 and did his Ph.D. in Physics at Kent State University with Prof. D.S. Moroi in 1973. He is currently professor and chair of the Department of Materials Science at the University of Patras, with research interests in soft matter theory and computer simulations.

James M. Polson ([email protected]) was born in Windsor, Ontario, Canada. He received his B.Sc. and M.Sc. degrees in Physics from the University of Guelph in 1988 and 1990, respectively. He completed his Ph.D. in Physics with Profs. Elliott Burnell and Myer Bloom at the University of British Columbia in 1996. He is currently Assistant Professor of Physics at the University of Prince Edward Island, with research interests in the simulation of soft condensed matter systems.

Dimitris Sakellariou ([email protected]) was born in 1974 in Athens, Greece. He graduated in Physical Chemistry from the Ecole Normale Superieure de Lyon (France) in 1996. He did his Ph.D. in solid-state NMR in the Ecole Normale Superieure de Lyon with Prof. Lyndon Emsley in 2000. He is currently a post-doctoral fellow at the Lawrence Berkeley National Laboratory and University of California, Berkeley working with Prof. Alexander Pines. His current research interests focus on NMR of strongly oriented molecular systems.

xx NMR OF ORDERED LIQUIDS

Edward T. Samulski ([email protected]) was born in 1943 in Augusta, Georgia, USA. He graduated in textile chemistry from Clemson University in 1965 and did his Ph.D. in Physical Chemistry at Princeton University with Prof. A.V. Tobolsky in 1969. He is currently Cary C. Boshamer Professor of Chemistry at the University of North Carolina, Chapel Hill, with research interest in oriented soft matter.

Raymond T. Syvitski ([email protected])grewup in Thunder Bay, Canada on the western edge of Lake Superior. He graduated with a B.Sc. in chemistry from Lakehead University and received his Ph.D. (2000) in Anisotropic Intermolecular Forces in Liquid Crystals using NMR from the University of British Columbia in Vancouver, Canada. Ray is currently a Killam Postdoctoral Fellow at Dalhousie University in Halifax and is studying the structure of membrane bound proteins by solution and solid state NMR.

Jean-Fran~ois Trempe (jean. [email protected]) was born in 1978 in Montreal, Quebec, Canada. He obtained a B.Sc. degree in biochemistry from McGill University in 2000. Two years later, he completed a M.Sc. degree in biochemistry with Professor Kalle B. Gehring, where he studied the application of polymer-stabilized liquid crystals in biomolecular NMR. He is currently pursuing a D.Phil. in structural biology in the University of Oxford (UK).

Marija Vilfan ([email protected]) was born in Ljubljana and received her Ph.D. in physics in 1978 from the University of Ljubljana, Slovenia. She is currently research consultant at the Jozef Stefan Institute and associate professor of physics at the University of Ljubljana. NMR research of liquid crystals has been her main professional interest for more than 20 years. She is married and has two grown up children - both physicists by profession.

Preface

Since the fi rst successful detection of Nuclear Magnetic Resonance (NMR) in 1945, the importance of the technique has increased enormously through countless applications ofNMR to a huge variety of systems of physical, chemical, and biological interest. The impressive level of sophistication that NMR has achieved today is an excellent example of how the interplay between technological development and fundamental science continually rejuvenates an experimental method. The present book contains an overview of modem applications ofNMR to the most intriguing fourth state of matter which comprises anisotropic condensed fuid phases such as liquid crystals. Although the fi rst observation by Reinitzer of liquid crystals dates from 1888, the fascinating properties of these unusual, partially ordered liquids have continued to generate much scientifi c and technological interest and activity to the present day. As the contents of the present book show, the marriage between NMR and the study of anisotropic phases is generally a happy one. In this volume we hope to convey to the general reader some of the excitement that is felt by those active in the areas of both NMR and orientationally ordered liquids. We hope that the book may be a useful compendium of contributions that are of interest to experts in the fi eld, to those wishing to learn about this area of research, as well as to students who wish to obtain an overview about NMR and its applications to the study of ordered liquids. The title may seem presumptuous to some, as by no means do we even attempt to cover the entire range of ordered liquids that are known today. In fact, most of the ordered liquids discussed are liquid crystalline phases, with a large emphasis on the simplest of these, the nematic and smectic A uniaxial phases. In this context it should be noted that biological membranes are themselves uniaxial smectic A phases.

The philosophy behind the present book is that a full understanding of simpler systems may lead to a proper investigation of more complex ones. The applications discussed in the book span amazing ranges from various points of view. The molecular systems studied go all the way from molecular hydrogen to proteins, from the physical to the biological world. The highly detailed information obtained covers both time independent (spectroscopic) and time dependent (dynamic) facets of the realm of orientationally ordered molecules. The contributions included cover the entire spectrum between fundamental and technological science. Experimental results are discussed and interpreted in terms of varying approaches such as formal theory, phenomenological modeling, and computer simulations. This book does not and cannot solve all the

xxi

xxii NMR OF ORDERED LIQUIDS

outstanding questions and debates that make the present area of research such a lively one. However, we feel that the disparate range of ideas that sometimes arises from the pages of this book is a true reftction of how science works, and will help in defi ning future directions for the fi eld.

ELLIOTT BURNELL AND CORNELIS DE LANGE, FEBRUARY 2003

Introduction

E.E. Burnell and C.A. de Lange

Liquid crystals represent a fascinating state of matter that is intermediate between the liquid and the solid state. They are characterized by orientational, and sometimes positional order, while translational motion is often hindered but not prohibited. This so-called fourth state of matter has been known since 1888 [1], and nowadays comprises a staggering variety of molecules in the condensed phase that show partially ordered behaviour, ranging from relatively simple organic liquids to biological systems that can be designated as large by any standard. Liquid crystals can be broadly subdivided into thermotropics and lyotropics.

Thermotropics form homogeneous anisotropic liquid-crystalline phases over certain temperature ranges. With rod-shaped molecules we have a calimatic liquid crystal, and with disc-shaped a discotic. The ''simplest'' phase is the nematic, in which the molecules have orientational but no positional order, with the average direction of orientational order called the director. Cholesteric phases are formed from chiral nematics, resulting in a helical twist to the director throughout the sample. Smectic phases possess orientational plus translational order, where the molecules have a tendency to form planes. The smectic A phase has the director normal to the planes, whereas it is tilted in the smectic C phase. Depending on the arrangement of the molecules in the layers, there are many additional smectic phases.

Lyotropic liquid crystals depend on the heterogeneous mixing of at least two components, often water with an amphiphilic molecule such as a long-chain fatty acid salt or a phospholipid. The insolubility of water in the hydrocarbon, i.e. oil part, leads to many fascinating phases, including micelle (water surrounding clumps of lipids with their polar groups near the water), hexagonal (the micelles elongate and form hexagonally packed rods), lamellar (layers of lipid separated by layers of water, thus excellent models of biological membranes), and cubic. The vesicles (closed bilayers) formed by phospholipids are excellent models of biological cells and organelles.

All the above are characterized by molecular packing in domains in which the molecules have long-range orientational order. As this is not a book on liquid crystals, the reader is referred to excellent monographs on this subject for more detail [2,3]. In this monograph the focus will be on the simple uniaxial systems.

xxiii

XXIV NMR OF ORDERED LIQUIDS

What all the liquid-crystalline phases have in common is their liquid-like behaviour and their anisotropic intermolecular potential. The study of this anisotropic potential is a topic of great interest, both from the fundamental and the technological point of view. This book is about the investigation of these fascinating systems using the technique of nuclear magnetic resonance, NMR. Since its discovery in 1945 [4, 5], the NMR method has developed into a tool that is unequalled in the study of an abundance of physical, chemical and biological systems. The state-of-the-art arsenal of sophisticated pulse methods for manipulating nuclear spins, and the development of high-fi eld spectrometers with associated sensitivity and multinuclear capacity, make NMR a very attractive tool for the study ofthe liquid phase. Since the anisotropic NMR interactions usually dominate over the isotropic ones in partially ordered systems, NMR has developed into a unique tool for studying molecules that constitute liquid-crystal phases, as well as for monitoring small, well characterized solute molecules that serve to probe the anisotropic environment of these phases. The detailed information that is then obtainable from NMR studies is not available by any other physical method. For example, the spectral parameters obtained from the NMR of solutes in liquid crystals have in the past provided a wealth of molecular information, such as accurate molecular structures, anisotropies of chemical shielding and indirect spin-spin coupling tensors, quadrupole coupling tensors, information on intramolecular motions, and values of orientational order parameters. Many examples are to be found in this book. In addition, NMR is invaluable in the investigation of dynamics in molecular systems, and the several Chapters that are devoted to this aspect cover topics such as spin relaxation measurements, motional narrowing, and the direct measurement of molecular diffusion constants.

In the last two decades much research has been carried out with the aim of elucidating the physical and chemical mechanisms that lie at the root of the orientational order in liquid crystalline phases. Extensive studies on molecules dissolved as probes in the anisotropic environment, the use ofliquid-crystal mixtures with surprising properties, and the development of models, both theoretical and phenomenological, for the description of orientational order have all contributed to a signifi cant degree to our present level of understanding. In addition, computer simulation techniques have now developed to the point where solutes in an anisotropic environment can be treated at a satisfactory level. The combination of these experimental and theoretical approaches has proved to be instrumental in gaining a basic understanding of the factors that determine the behaviour of anisotropic liquids.

The book is divided into four parts. Part I provides a basic introduction to NMR spectroscopy and to some of the pulse techniques employed in the investigation of ordered liquids. Part II discusses applications ofNMR to studies ranging from atomic solutes dissolved in liquid crystalline solvents to the investigation of the structure of biological macromolecules - the methods developed for small solutes providing the basis for such investigations. In Part III we review the various models, theories and simulation approaches that have been used to investigate the interactions in ordered fuids at a molecular level. Part IV deals with some of the measurements that have been so successful in investigations of dynamics in ordered liquids.

INTRODUCTION xxv

With the current book we wish to achieve a number of aims. First, we believe that its publication is timely. As will become apparent from its contents, much understanding has been gained in recent years, but the discussion about fundamental issues is still heated and lively. We hope that this book in the years to come will help in focusing and guiding this scientifi c discussion in a way that benefi ts the entire community. This community is an active one, with many senior scientists playing a key role in developing novel ideas, with research students starting their careers in a highly stimulating area, and with young students being exposed to a competitive fi eld of science for the fi rst time. All these groups should benefi t from the book. With a signifi cant emphasis on introductory Chapters that outline the basic ideas necessary to understand the NMR of oriented liquids, the book provides useful material for specialized courses.

In editing a multi-authored book we, the editors, are very dependent on colleagues who are willing to dedicate much of their valuable time to writing Chapters that fi t into the overall framework of the book. With great pleasure we acknowledge the co-operation of all of them. They made our task a most enjoyable one.

References [I] Reinitzer, F. (1888), Monatsh. Chern., 9:421.

[2] de Gennes, P.G., and Prost, J. The Physics of Liquid Crystals, 2nd edition. Oxford University Press, Oxford, 1993.

[3] Chandrasekhar, S. Cambridge Monographs in Physics: Liquid Crystals. Cambridge University Press, Cambridge, 1977.

[4] Purcell, E.M., Torrey, H.C., and Pound, R.V. (1946), Phys. Rev., 69:37.

[5] Bloch, E, Hansen, w.w., and Packard, M. (1946), Phys. Rev., 70:474.

I

BASICS

Nuclear magnetic resonance, NMR, is a very diverse technique that has proven to be extremely useful in the investigation of a host of systems, ranging from homogeneous samples composed of atoms in the gas phase to heterogeneous samples of macroscopic proportions, such as used in whole-body magnetic resonance imaging, MRI. It is not the purpose of this book to review all such applications, but rather to concentrate on those aspects of NMR that are useful in the investigation of ordered liquids. This introductory part gives an overview of the basics of NMR, and provides background and definitions that are useful in general, and that are particularly helpful for reading the later parts of this book. Indeed, the book as a whole covers most of the important basic aspects of NMR.

The original, independent NMR experiments (in Boston and Stanford) followed different approaches, in that the Harvard group [1] used continuous-wave spectroscopy while the Stanford group [2] used radio frequency pulses and measured the signal in the time domain. Most NMR experiments are now performed in the time domain. However, the user often transforms the signal to the frequency domain and thinks in terms of spectroscopy. The basics of the spectroscopy relevant to investigations of partially ordered systems are presented in Chapter 1.

One extremely valuable tool that is used for the discussion of modern NMR experiments is the density matrix which provides the density or probability of the coherent superpositions of quantum states that are the key to NMR. Because of its importance in NMR, we devote Chapter 2 to an introduction to the spin density matrix. The understanding of the coherence transfer that is involved in multiple-quantum NMR (Chapters 4 and 5) and other sophisticated pulsed NMR experiments (Chapter 3) are readily described in terms of the density matrix. The density matrix is also invaluable in the description of NMR relaxation (Chapters 16 and 17) and motional narrowing (Chapter 19).

Modern-day NMR incorporates many complicated multi-pulse sequences, and Chapter 3 reviews some very sophisticated ones. In doing so, it outlines experiments for the simplification of the very complicated spectra that are often obtained from partially ordered liquids. The experimental techniques employed include modifications to magic angle spinning and spin decoupling experiments. Some of these experiments can be thought of as being at the forefront of modern-day NMR.

In order to use the couplings and chemical shifts that govern an NMR spectrum, it must first be analyzed. In the case of the 1 H NMR spectrum of orientationally oriented molecules, the spectrum rapidly increases in complexity with the number of protons in the molecule. Chapter 5 reviews some to the methods used for the analysis of such very complicated spectra. One important tool in this regard is multiple quantum NMR which is reviewed in Chapter 4. High-order mUltiple quantum NMR spectra in principle allow for the analysis of quite complicated spin systems.

References

[1] Purcell, E.M., Torrey, H.C., and Pound, R.V. (1946), Phys. Rev., 69:37.

[2] Bloch, E, Hansen, w.w., and Packard, M. (1946). Phys. Rev., 70:474.

3

Chapter 1

BASICS OF NMR OF MOLECULES IN UNIAXIAL ANISOTROPIC ENVIRONMENTS

C.A. de Lange Laboratory for Physical Chemistry. University of Amsterdam. Amsterdam. The Netherlands

E.E. Burnell Department of Chemistry. University of British Columbia. Vancouver, B. C. Canada

1. Introduction The NMR spectra of liquids are usually interpreted in terms of two types of scalar

quantities: the chemical shifts (a) and the indirect spin-spin couplings (J). Although a and J in "normal" NMR appear as scalars, in truth they represent tensorial properties of the molecule. The Brownian movement of the molecule and the resulting isotropic tumbling leads to a situation where only the isotropic part of the tensorial properties is expressed. The Hamiltonian of NMR in isotropic liquids is given by:

'H Bz L (1 iSO)I L JisoI- 1-= -- 'Y' -(1. ·z+ ..... 27T ' t', 'J ' J' i i<j

(1.1)

The magnetic field is along the laboratory-fixed Z-axis and the superscripts "iso" indicate that we are concerned with the isotropic parts of the (T- and J -tensors. The summation indices i and j run over all nuclear spins in the molecule. Since the Zeeman term has been divided by h, units are in Hz. For the description of all normal highresolution NMR spectra in isotropic liquids Eq. (1.1) is perfectly adequate.

NMR can also be used under conditions where orientational effects are present in liquids. In order to achieve orientational order, several methods can be employed:

(i) the use of liquid-crystalline solvents;

(ii) the use of strong electric fields;

(iii) the use of strong magnetic fields.

Although these ways of generating orientational order lead in principle to the same effects in NMR spectra, in practice the anisotropic effects in liquid crystals are much

5 E.E. Burnell and CA. de Lange (eds. J. NMR of Ordered Liquids. 5-26. © 2003 Kluwer Academic Publishers.

6 NMR OF ORDERED LIQUIDS

larger than when strong external fields are used. In fact, when liquid-crystal solvents are used, the anisotropic effects often dominate over the isotropic ones. In the following we shall limit ourselves to the so-called nematic liquid crystals which possess no translational order but show significant orientational order. These nematic phases are apolar with uniaxial symmetry. They often consist of long rod-shaped molecules which have the tendency to orient parallel to each other, either parallel or perpendicular to the magnetic field of the NMR spectrometer, depending on the macroscopic magnetic susceptibility anisotropy of the liquid crystal. These nematic liquid crystals are commonly used to provide an anisotropic environment for the study of solute molecules.

The first NMR experiments on solutes in nematic phases were carried out in 1963 [1]. Since then, the method has found broad applicability [2-5], as also illustrated by the contents of the present book. The first successful experiments on molecules oriented in strong electric fields were performed in 1963 [6], while observable orientational order arising from the use of very strong magnetic fields was not established experimentally until 1978 [7].

2. General Hamiltonian in uniaxial anisotropic liquids In this section the general Hamiltonian required to describe the NMR spectra of

molecules which are orientationally ordered in a uniaxial anisotropic environment is derived. In this description the second-order tensorial character of several molecular properties plays a central role. The Hamiltonian is constructed as a sum of a Zeeman term 'Hz which contains the chemical shielding, a term 'HJ which describes the indirect spin-spin couplings, a term 'HD which takes account of the direct interaction between magnetic dipoles, and finally a term 'HQ which describes quadrupolar interactions:

(1.2)

Although the physics clearly does not depend on the choice of axes, it is customary to define the Hamiltonian in terms of laboratory-fixed axes X, Y, Z. The reason is that the dominant term, the Zeeman term 'Hz, takes on a simple form in these axes. We shall consider the various terms in the Hamiltonian in more detail.

2.1 The Zeeman term The magnetic moment j1 of a nucleus is related to the spin operators via

h -j1 = "(-I. (1.3) 211"

Classically the interaction between a magnetic field jj and a magnetic moment j1leads to the interaction energy:

W=-j1.jj (1.4)

which in operator form for the magnetic field along the Z direction becomes:

h 'H = - "( 211" lz B z . (1.5)

Basics of NMR of molecules in uniaxial anisotropic environments 7

However, the field at the site of the nucleus is not quite identical to that of the external field B z, and the chemical shielding effect must be incorporated. After division by h to base our description on Hz as energy units we obtain:

1{z = - ~: L L ')'i(lza - ui,za)Ii,a. i a

(1.6)

Here 1 and (T are second-rank Cartesian tensors with components:

1 = and ( UXX uXy UXZ

(T = uy X Uyy Uy Z

UZX UZy UZZ ) (1.7)

The Cartesian tensor components are related to the laboratory-fixed axes X, Y, Z. The index a runs over X, Y, Z. Of course the use of Cartesian tensors is not essential, but a matter of free choice. Later in this Chapter we shall consider other formalisms that have been utilized, such as an expansion in real spherical harmonics, or a spherical tensor approach. Each of these formulations has its own advantages and drawbacks, but we shall not get involved in discussions about taste.

2.2 The indirect spin-spin coupling

The Hamiltonian for this contribution in Cartesian tensorial form is written as:

1{J = L L . Jij . lj = L L I i,aJij,a{3Ij,{3 (1.8) i<j i<j a,{3

where Jij is the second-rank tensor which describes the bilinear coupling between

spin vectors Land i;. The summation indices i and j number the nuclear spins, and a and (3label the laboratory-fixed axes X, Y, Z.

2.3 The direct dipole-dipole interaction

For the classical interaction energy between two magnetic dipoles J1i and J1j we have:

W .. - J1i· J1j _ 3(J1i . iij) (J1j . iij) ~ - 3 5 r ij r ij

(1.9)

with iij the vector from the spatial position of spin J1i to that of J1j with length r ij. In quantum-mechanical form (again after division by h):

(1.10)

The indices a and {3 again label the laboratory-fixed axes X, Y, Z, 6a{3 is the Kronecker delta function (1 if a = (3, Oif a =1= (3), and rij,a is thea componentofiij. Equivalently we can write:

1{D = L L Ii,aD~j,a{3Ij,{3 i<j a,{3

(1.11)

8

with

The convention usually employed in the literature is:

1 I

Dij,a{3 = "2 Dij ,a{3'

We shall therefore use:


(1.12)

(1.13)

L L 2Ii ,aDij,a{3h{3 (1.14) i<j a,{3

with

(1.15)

This second-rank Cartesian tensor which describes the direct dipole-dipole interaction is symmetric:

(1.16)

and traceless: (1.17)

The latter relationship can easily be proven from

(1.18)

2.4 The nuclear quadrupole interaction As derived in many textbooks, the quantum-mechanical nuclear quadrupole inter

action, after division by h to express the energy units as Hz, is given by the following spin Hamiltonian:

Here i labels all nuclei with I > !, and a and {3 run over the laboratory-fixed axes X, Y, Z. The eQi signify the nuclear quadrupole moments. The Vi,a{3 are the a{3 components of the the second-rank tensor that signifies the second-order derivatives ofthe potential energy with respect to position variables, calculated at the site of every nucleus:

(1.20)

This tensor is symmetric:

Vi,a{3 = Vi,{3a (1.21)


and, in the absence of free charges, traceless (Laplace's equation):

LVi,aa = O. (1.22)

The components of the electric field gradient tensor are defined as the negative of the Vi,a,B.

3. The high-field approximation In the overall Hamiltonian the Zeeman term dominates strongly over the other

contributions. For a 100 MHz NMR spectrometer the Zeeman term is of the order :::::: 108 Hz, whereas dipolar and quadrupolar interactions are rarely larger than:::::: 104

Hz. Indirect couplings are usually no larger than :::::: 102 Hz. This implies that offdiagonal matrix elements of 'HD + 'HQ + 'HJ that couple different values of

Iz = L Ii,z (1.23)

can be neglected to a very good approximation. The general tensorial form for the Hamiltonian derived in the previous section takes on a considerably simpler form under conditions where for a strong external magnetic field the Zeeman term dominates.

3.1 The Zeeman term 1i z When we consider the spin operators in Eq. (1.6), only Ii,z commutes with Iz. In

the high-field approximation Eq. (1.6) reduces to:

Bz,,"", 'Hz = - 211" ~ 'Yi(1- (Ji,zz)Ii,z.

i

3.2 The indirect spin-spin coupling term 1iJ

We use the well-known ladder operators:

Ij,+ = Ij,x + iIj,Y

Ij,- = Ij,x - ilj,Y.

(1.24)

(1.25)

(1.26)

Of all possible combinations of spin operators that occur in Eq. (1.8), only the combinations Ii,zIj,z, Ii,+Ij,_, and Ii,_Ij,+ commute with Iz. The only terms ofEq. (1.8) that survive are:

'HJ = ?=<. [Jij,zzIi,zhz + ~(Jij,XX + Jij,yy)(Ii,+Ij,- + Ii,_Ij,+)] . (1.27) t J

In order to obtain Eq. (1.27) an additional assumption has been made that the J -tensor is symmetric. However, a non-symmetrical J-tensor can always be expressed as the sum of a symmetric and an anti-symmetric part. It can be shown [2] that the influence of the anti-symmetric part of the J-tensor in the high-field approximation is negligible. The additional assumption is therefore hardly restrictive.


3.3 The direct dipole-dipole interaction 1-£ D

Starting from Eq. (1.15) we introduce spherical coordinates:

Tij,X = Tij sin ()ij cos ¢>ij Tij,Y = Tijsin()ijsin¢>ij

Tij,Z = Tij cos ()ij.

(1.28)

By employing the ladder operators defined in Eqs. (1.25) and (1.26), after some algebra Eq. (1.14) can be rewritten as:

with

1-lD = L I'i~j; (A + B + C + D + E + F) . . 411" Ti]· t<]

A = (1 - 3 cos2 (). ·)L zI· z t] t, ],

B = - ~(1 - 3cos2 (). ·)(L +1· _ + L _I· +) 4 t] t,], t,],

c = = -~sin() .. cos() .. e-i¢ij(L+1·z +Lz1·+) 2 t] t] t,], t,],

D = = -~ sin()·· cos()··e+i¢ij(L _I· z + L z1·_) 2 t] t] t,], t,],

E=

F=

3 . 2 () -i2""··J I = --SIn ··e 'i"J.+.+ 4 t] t,],

3 . 2() +i2"'··J I = -- SIn i]·e 'i"J i - ]._. 4 ' ,

(1.29)

(1.30)

(1.31)

(1.32)

(1.33)

(1.34)

0.35)

Just as with the indirect spin-spin coupling only terms with 1i,zhz, 1i,+h- and 1i,-h+ commute with Jz. This means that only the so-called secular terms A and B survive in the Hamiltonian in the high-field approximation:

Finally, with

we obtain:

I'i/'j h ( 3 2 () ) = -- 1- cos i· 811"2T~. ]

t]

1-iD = L2Dij,zz [Ii,zhz - ~(Ii'+Ij,- + 1i,-h+)] . i<j

(1.37)

(1.38)

In a uniaxial nematic liquid crystal the terms C-F do not playa role even if the high-field approximation did not hold. For a uniaxial phase with cylindrical symmetry


around the laboratory-fixed Z-axis all angles ¢ij are equally probable. The ensemble averages over e±i</>ij and e±i2</>ij are zero, since

(1.39)

The terms C-F do playa significant role in nuclear spin relaxation processes.

3.4 The nuclear quadrupole interaction 1iQ

Of all the combinations of spin operators that occur in Eq. (1.19) only Ii,zIi,z, Ii,+Ii,- and Ii,_Ii,+ commute with /z. Eq. (1.19) therefore reduces to:

'H 1 '" eQi Q = h ~ 6Ii (2Ii - 1)

•

x ~ [2Vi,zzI~z + ~(Vi,xx + Vi,yy) (Ii,+Ii,- + Ii,_Ii,+)] .

(l.40)

Here the symmetry of the field gradient tensor has been used. By employing both Eq. (1.22) and the expression

we obtain: 1 '" eQi 2 2

'HQ = h.w 4!.(2f. _ 1) Vi,zz(3Ii,z - Id· i"

An abbreviation which is often used in the literature entails:

eQiVi,zz qi,ZZ = h

leading to an alternative expression for Eq. (1.42):

'1J '" qi,ZZ (12 12) nQ = .w 4J.(2f. _ 1) 3 i,Z - i .

i"

4. Transformation to molecule-fixed axes

(1.41)

(1.42)

(1.43)

(1.44)

So far we have developed a description in laboratory-fixed axes X, Y, Z. This is a logical choice for the Zeeman term which depends on the magnetic field of the NMR spectrometer. However, the various molecular properties such as chemical shielding tensors, spin-spin interactions and quadrupolar interactions are more conveniently formulated in a molecule-fixed axes system x, y, z. This necessitates a transformation from laboratory-fixed to molecule-fixed axes for such molecular properties. Of all the tensor components that occur in the high-field approximation, only the ZZcomponents playa key role. For an arbitrary second-rank tensor A the transformation from laboratory-fixed axes X, Y, Z to molecule-fixed axes x, y, z goes as follows:

Azz = Lcos(;lz,acos(;lz,,l3Acl<,l3 a,,l3

(1.45)


where the summation indices a, f3 run over x, y, z. The quantity cos OZ,a is the cosine of the angle between the laboratory-fixed Z-axis and the molecule-fixed a-axis. Eq. (1.45) can be rewritten:

121 Azz = a(Axx+Ayy + Azz) + aL'2(3cosOz,aCOSOZ,f3-8af3)Aaf3. (1.46)

a,f3

The quantity

A (1.47)

signifies the trace of the tensor and is independent of the axes system chosen. The tensor transformation of the direct dipolar and quadrupolar interactions will take a simple form because these tensors have zero trace both in laboratory-fixed and molecule-fixed axes.

5. Orientation parameters Since the molecular tumbling causes the molecular axes to move continually with

respect to the laboratory-fixed axes, the angles between both frames of axes change all the time. In order to calculate an ensemble average, averaging over all tumbling angles must be performed. In this averaging process it is important to know if the molecular tumbling is isotropic or anisotropic. It is customary to define the following ensemble averages:

1 Saf3 = '2 < 3 cos OZ,a cos OZ,f3 - 8af3 > (1.48)

where Saf3 are the elements of the so-called Saupe orientation tensor [8] The angular brackets indicate the ensemble average. From Eq. (1.48) it is easy to derive that this tensor is symmetric and traceless:

(1.49)

(1.50)

Introducing spherical coordinates where the Z-axis makes an angle 0 with the z-axis, and the projection of the Z-axis onto the xy-plane makes an angle ¢ with the positive

Basics ofNMR of molecules in uniaxial anisotropic environments

x-axis, we obtain:

Syy

~ < 3 sin 2 0 cos2 4> - 1 >

~ < 3sin2 0sin2 4> - 1 >

1 2" < 3 cos2 0 - 1 >

~ < sin 2 0 sin 4> cos 4> >

~ < sin 0 cos 0 cos 4> >

~ < sin 0 cos 0 sin 4> > .

It can easily be established that:

1 <

2 Sxx, Syy, Szz < 1

3 Sxy, Sxz, Syz

3 < < 4' 4

13

(1.51)

(1.52)

Moreover, for isotropic tumbling, evaluation of the angular brackets in Eq. (1.51) leads to zero for all elements of the S-tensor. For instance:

121r d4> l1r sin OdO (~ sin2 0 cos2 4> - ~) = O. (1.53)

For anisotropic tumbling Saj3 =1= 0 in general. We shall now relate the molecular properties occurring in the Hamiltonian to molecule

fixed axes. For the chemical shielding occurring in the Zeeman term this leads to:

1 2 C7i,ZZ = 3(C7i,xx + C7i,yy + C7i,zz) + 3 L Saj3C7i,aj3.

a,j3

(1.54)

The contribution to the chemical shielding that remains on isotropic tumbling is:

Hence:

C7~so t

iso C7i,ZZ - C7i C7r:niso

t

The Zeeman term of Eq. (1.24) can now be written as:

'l.1z __ B Z "'" (1 iso _ ",a. niso)l. ,~ --2 ~'Yi - C7i V t t,Z'

7r . z

(1.55)

(1.56)

(1.57)


For the indirect spin-spin interaction we write:

Jf;o = ~(Jij,XX + Jij,YY + Jij,ZZ) = ~(Jij,XX + Jij,yy + Jij,zz). (1.58)

In similar fashion as with the chemical shielding:

Jij,ZZ = Jl;o + ~ L So.(3Jij,o.(3 = 0.(3

From Eq. (1.59) it follows that:

~(1 .. xx + 1·· yy) 4 tJ, tJ, = ~J~~o _ ~J~.niso

2 tJ 4 tJ .

Eq. (1.27) can now be rewritten as:

riJ = L Jl;o [Ii,zhz + ~(h+h- + Ii,-h+)] i<j

or alternatively as:

+ L Jijniso [Ii,zhz -l(Ii,+h- + Ii,-h+)] i<j

(1.59)

(1.60)

(1.61)

riJ = L Jl;o! .1j + L Jijniso [Ii,zhz -l(Ii,+h- + Ii,-h+)]. (1.62) i<j i<j

For the traceless D-tensor which describes the direct spin-spin interaction:

D Daniso 2 ""' S D ij,ZZ = ij = 3" L..J 0.(3 ij,o.(3· 0.(3

(1.63)

From Eq. (1.15) we obtain:

(1.64)

The contribution to the Hamiltonian of the direct spin-spin interaction is therefore:

riD = L 2Dijniso [Ii,zhz -l(Ii,+h- + Ii,-h+)] . i<j

For the traceless tensor which describes the quadrupolar interaction we obtain:

(1.65)

(1.66)

Basics of NMR of molecule s in uniaxial anisotropic environments

and for the contribution to the total Hamiltonian:

qaniso ~ i 2 2

HQ = L...J 41-(21- _ 1) (3Ii,z - Id· i ~ ~

In summary, for the total Hamiltonian we obtain:

H = - B2z L l'i(1 - O":so - O"iniso)Ii,Z 7r .

• + LJfrI:· ij

i<j

+ ~<. (2Ditso + Jijniso) [Ii,zhz -l(Ii,+h- + Ii,-h+)] ~ J

qaniso ~ i 2 2 + L...J 4Ii(2Ii _ 1) (3Ii,z - Ii)'

~

15

(1.67)

(1.68)

The Hamiltonian of Eq. (1.68) is the key result which describes the NMR spectra of molecules in uniaxial anisotropic environments. For isotropic tumbling all anisotropic terms are zero and the Hamiltonian reduces to the familiarform of Eq. (1.1).

6. Molecular symmetry In general the traceless, symmetric S-tensor possesses 5 independent elements.

Since the orientation parameters depend on the two spherical angles () and ¢, it is clear that they must be related to the spherical harmonics Vim ((), ¢) or to the Wigner rotation

matrices D~ o((), ¢). In tum, the Wigner rotation matrices are related to the spherical harmonics a~cording to [9]:

(1.69)

A probability function P( (), ¢) which is the probability per unit solid angle that the magnetic field is at polar angles () and ¢ with respect to the molecular frame has also been introduced in the literature [10]. This probability function can be expanded into real spherical harmonics:

1 P((), ¢) = - + cxPx + CyPy + czPz + C3zLr2D3z2_r2

47r

+ Cx Ly2DxLy2 + CxyDxy + cxzDxz + CyzDyz + ..... . (1.70)

When the components of the second-rank tensors defined in the laboratory frame are averaged with P ((), ¢ ), the second-rank orientational order in anisotropic liquids is determined by the motional constants c3zLr2, Cx2_y2, cxy , Cxz , and Cyz.

The interrelationships between Snyder's c-parameters [10], Saupe's S-parameters [11,12], and Wigner rotation matrices [9] are given in the following conversion table.


Table 1.1. Interrelationships between Snyder's c-parameters, Saupe's S-parameters and Wigner rotation matrices

C3z 2 _r2

Cx 2_y2

cxy cx •

cyz

(5)t Szz

(5/3)!(Sxx - Syy) 1

2(5/3) 2 Sxy 2(5/3)! Sx. 2(5/3)! Syz

(5)! < D50 > ) 1 2 2 (5/2 ~ < D20 + D_20 >

i(5/2)! < D~o - D:20 > -(5/2)! < Dio - D:lO > -i(5/2)! < Dio + D:lO >

In order to describe the orientational order of a molecule dissolved in an anisotropic solvent, we have to define our molecule-fixed axes x, y, z in the molecule. So far we have not discussed how to choose these axes, and in principle the choice is immaterial. However, when the molecular spin system possesses a degree of symmetry, it is convenient to choose the molecule-fixed axes in such a way that the orientation tensor is diagonal in these axes as much as possible. For example, if the molecule possesses axial symmetry, it is customary to choose the z-axis along the symmetry axis, leading to only one independent orientation tensor element Szz, with

Szz - 2Sxx = - 2Syy

Sxy = Sxz = Syx = O. (1.71)

In the following table the independent elements of the S-tensor are given for every point group.

Table 1.2. Independent non-zero elements of the Saupe order matrix for the various point groups

Point group

C1, Ci

C2 , C2h, Cs C2v , D2, D2h

Cn, Cnh, Cnv (n = 3 - 6) Cooh, Dooh D2d D n , Dnd, Dnh (n = 3 - 5) D6, D6h S4, S6 Kh, 0, Oh, T, Td

Szz, Sxx - Syy, Sxy, Sxz, Sy. Szz, Sxx - Syy, Sxy Szz, Sxx - Syy Szz Szz Sz. Szz Szz Szz all S"'i3 = 0


7. Simple examples of NMR spectra of orientationally ordered molecules

7.1 Two spins I = ~,the AB spectrum

The total Hamiltonian for two spins ~ is given by Eq. (1.68):

'1.J = B Z [ (1 iso aniso)I + (1 iso ,-raniso)I ] IL - 211" rA -O"A - O"A A,Z rB -O"B - VB B,Z

+ Jiso ~ . iB (1.72)

+ (2DaniSo + Janiso) [IA,zIB,Z - ~(IA,+IB,- + IA,_IB,+)] .

Using the spin functions a and f3 as a basis, the relevant matrix elements are:

a f3 < ¢>mIIA,zl¢>n > = ~ ( ! ~~ ) (1.73)

aa af3 f3a f3f3 aa

U 0 0

n < ¢>ml~ . iBl¢>n > af3 1 1 (1.74) -14 21 f3a 2 -4 f3f3 0 0

1 < ¢>mIIA,zIB,z - 4(IA,+IB,_+IA,_IB,+)I¢>n >

aa af3 f3a f3f3 aa

U 0 0

n (1.75) af3 1 1 -1 -1 f3a -4 -4 f3f3 0 0

With

VA = Bz (1 iso -rA- -O"A 211"

_ O"'Aniso) (1.76)

and a similar definition for VB we obtain a 4 x 4 secular problem which blocks out into two 1 x 1 problems and one 2 x 2 problem. After diagonalization the energy levels are:

(1.77)


with c = ~[(I/A - I/B)2 + {JiSO _ ~(2DaniSo + JaniSo)}2J%

2 2 . (1.78)

The wave functions are:

with

'1/11 £lA£lB

'1/12 £lAf3B cosw + f3A£lB sinw

'1/13 = £lAf3Bsinw - f3A£lBcosW

'1/14 = f3Af3B

C sin 2w = ~JiSO _ ~(2DaniSo + Janiso).

(1.79)

(1.80)

If for the sake of argument we neglect the usually small Janiso, and assume a negative Daniso with !2Daniso! > !Jiso!, the four allowed NMR transitions are given in Table 1.3 in order of increasing frequency, and the spectrum is displayed in Fig. 1.1. Following the discussion of the AB spectrum we shall consider two limiting cases.

Table 1.3. Transition frequencies and intensities for an AB spin system orientationally ordered in an anisotropic solvent.

Transitiona

~v(4, 2) ~v(3, 1) ~v(4, 3) ~v(2, 1)

Frequency

_l(vA + VB) - C + 1Ji80 + 1(2Dani80 + r ni80 ) -1(vA + VB) - C _ 1Ji80 _ !(2DaniSo + r ni80 ) _! (VA + VB) + C + 1JiSO + !(2Dani8o + r ni80 ) -1(vA + VB) + C -If so _1(2Dani80 + r ni8o )

a The levels between which the transitions take place are labeled L:l.v(i,j) = Vj - Vi.

b The intensity of transition L:l.v(i,j) is proportional to! < ,pj!IA,X + IB,X!,pi > !2.

!2e1 izDaniso+Janiso +Jiso ! .. II ..

-(~ +~ )/2

Figure 1.1. AB spectrum.

Intensitl

1 + sin2w 1 - sin2w 1 - sin2w 1 + sin2w


7.1.1 The A2 spin system. Both nuclei have the same chemical shift, leading to sin 2w = 1. The NMR spectrum reduces to a doublet with splitting

~ 12Daniso + Janiso I. (1.81)

I ~(2Daniso +jisO ) I

Figure 1.2. A2 spectrum.

7.1.2 The AX spin system. If (VA - VB) is large compared to Daniso and Janiso, Eq. (1.78) reduces to

1 C = "2 (VA - VB). (1.82)

In this approximation the NMR spectrum reduces to four lines of equal intensities, with doublet spIittings

(1.83)

Figure 1.3. AX spectrum.


7.2 Linear molecule that contains one spin I = 1

The quadrupolar interaction for one spin 1= 1 has the form (Eqs. (1.43) and (1.68»:

1 2 2 'HQ = '4qzz (3Iz - I ). (1.84)

The Zeeman term has the form 'Hz = vIz (1.85)

with

Bz (1 iso aaniso). v = -,- -a -211"

(1.86)

Both 'HQ and 'Hz have only diagonal matrix elements in the usual basis of the I = 1 spin functions, leading to an NMR spectrum consisting of a doublet with a doublet splitting which is expressed in a number of ways in the literature:

(1.87)

2B .. ..

-t)

Figure 1.4. Spectrum of uncoupled I = 1 spin system

7.3 D2 molecule with two spins I = 1

The molecule D2 contains two spins I = 1. Because of nuclear spin statistics we distinguish ortho- and para-D2' The possible transition frequencies and intensities can be calculated analytically and are presented in Table 1.4. From the Table and from the experimental spectrum (see Fig. 10.3 of Chapter 10 and [13]) it is clear that the NMR spectrum shows transitions of unequal intensities separated by 13JI. Since this indirect J-coupling arises predominantly from the Fermi contact interaction, its sign is positive. The sign of the S zz order parameter can be derived immediately from the order in which the transitions separated by 13JI appear in the spectrum. This unusual situation offers interesting possibilities when D2 is employed as a probe molecule in nematic solvents (see Chapter 10).


Table 1.4. Transition frequencies and intensities for ortho- and para-D2 orientationally ordered in an anisotropic solvent. (Reprinted with permission from [13]).

Transitiona Frequenc/ IntensityC

ortho-D2 81 - 82 1/ - B - 3D 1 80 - 81 I/-p-R (Y - 0)2[2(1 + y2)]-1 88 -81 I/-P+R (Y 0 + 1)2[2(1 + y2)]-1

8_1 - 80 I/+P+R (Y - 0)2[2(1 + y2Wl 8_1 - 88 I/+P-R (Y 0 + 1)2[2(1 + y 2W l

8_2 - 8_1 1/+ B +3D 1 para-D2 Ao -AI 1/+ B - 3D 1

~ A-I - Ao 1/ - B +3D 2"

aThe levels between which the transitions take place are labeled as symmetric S or antisyrnrnetric A with respect to permutation symmetry. They are further classified according to the eigenvalues of the Z component of the total angular momentum.

bR = !(4B2 +9J2 + 12D2 - 4BJ -8BD-12JD)!. In shorthand notation: B:; BW 2 ) andD:; DDD. cy = (R - B + P + D)[V2(J - D)]-I.

7.4 Larger molecules with more spins The complexity of NMR spectra of orientationally ordered molecules increases

rapidly with the number of nuclear spins. If the number of spins is ~ 10 or larger, the number of allowed NMR transitions becomes very large, the intensity for many of these transitions becomes very low, and, especially for molecules with low symmetry, the problem of overlapping transitions becomes severe. In practice this often leads to very broad, unresolved NMR features. As will become apparent from various Chapters in this book, the problem of spectral complexity and congestion in the case of somewhat larger solute molecules dissolved in anisotropic solvents is a serious one (see Chapters 4, 5, and 7). Sophisticated spectral analysis methods and multiple quantum techniques (see Chapters 2, 4, and 5) are relatively recent approaches to overcoming such difficulties.

8. Orientational order induced by anisotropic solvents The easiest way to create a relatively high degree of orientational order is by dis

solving solute molecules into anisotropic solvents such as nematic liquid crystals. Typically the molecules that make up a nematic liquid crystal contain ~ 20 protons, leading to extremely broad featureless NMR spectra of such oriented liquids. The NMR spectra of the smaller solute molecules are usually better resolved and appear as structured spectra on the broad liquid-crystal background. From an analysis of such NMR spectra much information about solute molecular properties (e.g., molecu-


lar geometries, anisotropies in chemical shielding, anisotropies in indirect couplings, signs of indirect couplings, quadrupolar constants) can be obtained. The degree of orientational order measured for various well characterized solutes provides important keys to understanding the mechanisms that lie at the root of the processes leading to orientational order (see Part III). The NMR study of the dynamics of liquid-crystal molecules and of solutes in anisotropic solvents has led to a deeper understanding of the relaxation processes that are particular to partially oriented liquids (see Part IV). Many of the Chapters in this book are concerned with exactly these problems.

The NMR spectra of complex solutes (containing many protons) in normal anisotropic solvents that cause a high degree of orientational order are essentially impossible to analyse. In these cases selected anisotropic solvents that lead to a low degree of orientation can be employed instead. Under such conditions the NMR spectra are dominated by the normal isotropic terms in the Hamiltonian. However, the small residual dipolar couplings appear to be a rich source of structural information which would not be available otherwise. The novel and important application of this method to the case of biomolecules is discussed in Chapter 8.

9. Orientational order induced by strong electric fields If a liquid consisting of molecules with a permanent electric dipole moment is

subjected to a strong electric field, the interaction Hamiltonian between the static electric field E and the electric dipole moment [lei is given by:

'It = -[l.E = -I-lel,ZEZCOse (1.88)

with the electric field along the laboratory-fixed Z-axis. Here e is the angle between the vector of the electric dipole moment and the electric field. The competition between the orienting influence of the electric field and the temperature-dependent molecular movement in the liquid is obtained from Boltzmann statistics:

I-'el ZEZ cos 0

1 f01l" (3 cos2 e - l)e . kT sin OdO 2"\3cos2 0-1)E = l-'elZEZ.cosO

2 f 01l" e ! kT sin OdO (1.89)

= 1 + 3 [I-lelk~Ez] -2 _ 3 [I-lelk~Ez] -1 coth [I-lelk~z ] .

For I-lel,zEz/kT « 1 the coth can be expanded into a Taylor series, and neglecting terms of order E~ and higher we obtain:

~ 13 0 2 0 -1) = ~ [I-lel,zEz] 2

2 \ c s E 15 kT (1.90)

The orientational order increases with the square of the electric field. In Fig. 1.5 the 2H NMR spectrum of fully deuterated nitrobenzene is shown as a function of electric field strength. The traceless quadrupole interaction tensor does not lead to splittings in the isotropic case, but when partial orientation is induced every transition becomes a doublet.

Basics ofNMR of molecules in uniaxial anisotropic environments 23

Figure 1.5. 2H NMR spectra of nitrobenzene-ds in electric fields of 0, 34, 46, and 62 kV I cm (from top to bottom). (Reproduced with permission from [14]. Copyright 1984 American Chemical Society).

10. Orientational order induced by strong magnetic fields If a liquid that possesses an anisotropic magnetic susceptibility tensor XO'.(3 is placed

in a strong magnetic field B, the following interacting should be considered:

(1.91)

With B along the Z direction of the laboratory-fixed frame we obtain:

1i

0.92)

Here XII and X.L are the components of the X tensor parallel and perpendicular to the molecular z-axis. The anisotropy in the magnetic susceptibility is given by:

.6.X = XII - X.L· (1.93)

24

(45)

2Hz ..


(1,3,6.8) ("~~,10)

1HZ

12,7) ~9~1 70 0 2 6 0 3

5 7

9 10

7~2 6 5 4 3

(1,8) (3,6) (9,10} (2,7)

Figure 1.6. 2H NMR spectra of pyrene-dlO and phenanthrene-dlO in a magnetic field of 11.7 Tesla. (Reproduced with permission from [14]. Copyright 1984 American Chemical Society).

The degree of orientational order can again be obtained using Boltzmann statistics. Under the neglect of terms of order Bi and higher we obtain:

1 ( 2 > 1 B1D..x 2" 3 cos B-1 B = 15 kT' (1.94)

The orientational order increases with Bl As an example we show in Fig. 1.6 the 2H NMR spectra of pyrene-dlO and phenanthrene-dlO in which the quadrupolar splittings due to the magnetic-field induced orientational order can be observed. Because these molecules have lower than axial symmetry additional order parameters must be calculated for a detailed interpretation.

11. Internal motion So far we have developed the theory for "rigid" molecules. Of course real molecules

are far from rigid, since they undergo either small-amplitude vibrational motions or large-amplitude conformational changes. Clearly, it is important to discuss the consequences of such internal motions for our NMR experiments. We shall discuss the situation for the dipole-dipole interaction which can be expressed as the temperature and quantum average of an operator Dij,zz where i and j label the two nuclei that are coupled:

h'Yi'Yj ~ Dij,zz(Qm, D) = - 4 2 ~.(Q ) LJ cos Bij,k(Qm) cos Bij,I(Qm)Skl(D). (1.95)

7r TtJ m k I ,

The Bij,k indicates the angle between the ij-direction and the molecular k-axis. The Qm indicate the internal vibrational coordinates, the D stands for the angles associated


with reorientational motion. The observable couplings are obtained by averaging over all these motions. For small-amplitude vibrational motions the averages over internal and reorientational motions can be performed separately, usually to quite a good degree of approximation:

/ L COS Oij,k(Qm) ~os Oij,I(Qm)Skl(O)) = \ k I rij(Qm)

, (1.96)

L / cos Bij,k(Q;;) cos Bij,I(Qm)) (Skl(O)) . k,l \ r ij (Qm)

In this approximation the reorientational and internal motions are decoupled, and the averaging over internal vibrational motions can be performed separately. This is the approach normally taken when the NMR experiment aims at obtaining accurate molecular geometries of solute molecules [15-18].

For large-amplitude conformational changes the situation is more complicated. In the early days ofNMR spectroscopy of oriented molecules it was commonly assumed that the interconversion between various conformers was very fast and that the solute undergoing these large-amplitude motions could be considered as an "average" molecule, often with an "effective" symmetry higher than that of the separate conformers. This point of view has proved to be untenable [19,20]. For a proper description of the observed dipolar couplings the following expression holds as an extension of Eq. (1.63):

(Dij,zz) = ~ Lpn L S~{3Dij,Ot{3' n Ot,{3

(1.97)

Here pn stands for the probability of conformer n, the Dij,Ot{3 signify the dipolar couplings associated with conformer n, and the S~{3 describe the orientation tensor of conformer n. It should be noted that from the NMR experiment alone there is no possible way of separating the products of the conformer probability pn and the conformer order tensor S~{3' The study of molecules that undergo conformational change is a topic discussed in several Chapters of the book (see Chapters 12, 13, 14, 16 and 19).

The separation of internal small-amplitude vibrational motions and overall reorientational motions discussed above is normally adequate, but fails in one particular case. When we consider molecules with very high symmetry, such as tetrahedral solutes (e.g., CH4, CD4), such molecules should show no reorientational order, and hence no dipolar couplings in anisotropic environments. Nevertheless, methane and its deuterated isotopomers all show dipolar (and where applicable quadrupolar) couplings, albeit of small magnitudes. These observations can be explained in detail by assuming that there is a correlation between internal and reorientational motions which does not allow a complete decoupling. Although the underlying theory for vibrationreorientation coupling shows interesting physics [21-24], we shall not dwell on this topic in the context of the present Chapter.


12. Summary In this Chapter the theory underlying NMR spectra of molecules that possess orien

tational order is developed. Orientational order can arise when molecules are dissolved in an anisotropic environment, and the theory described here pertains to a situation of uniaxial symmetry of the surrounding medium. Alternatively, orientational order can be achieved when appropriate molecules in the liquid phase are subjected to high electric or magnetic fields. The tensorial character of the chemical shielding, dipolar interactions, indirect couplings, and quadrupole couplings is emphasized, and the consequences for NMR spectroscopy are discussed. The tensors and the relevant tensor transformations are described in a Cartesian framework. However, different notations based on real spherical harmonics or the spherical tensor formalism abound in the literature. In the present Chapter the results of the various descriptions are compared.

References [1] Saupe, A., and Englert, G. (1963), Phys. Rev. Letters, 11:462. [2] Buckingham, A.D., and McLauchlan, K.A. (1967), Progress in Nuclear Magnetic Resonance

Spectroscopy, Pergamon Press, Oxford, 2:63. [3] Diehl, P., and Khetrapal, C.L. (1969), NMR, Basic Principles and Progress (P. Diehl, E. Fluck and

R. Kosfeld, Eds.), Springer-Verlag, Berlin, 1:1. [4] Khetrapal, C.L., Kunwar, A.C., Tracey, A.G., and Diehl, P. (1975), NMR, Basic Principles and

Progress, Springer, Berlin, 9. [5] Emsley, J. w., and Lindon, J.C. NMR Spectroscopy using Liquid Crystal Solvents. Pergamon Press,

Oxford,1975.

[6] Buckingham, A.D., and McLauchlan, K.A. (1963), Proc. Chem. Soc., page 144.

[7] Lohman, J.A.B., and Maclean, C. (1978), Chem. Phys., 35:269. [8] Saupe, A. (1964), Z Naturforsch. A, 19:161. [9] Zare, R.N. Angular Momentum. Wiley, New York, 1988.

[10] Snyder, L.C. (1965), J. Chem. Phys., 43:4041. [11] Maier, w., and Saupe, A. (1959), z. Naturforsch. A, 14:882. [12] Maier, w., and Saupe, A. (1960), z. Naturforsch. A, 15:287. [13] Bwnell, E.E., de Lange, C.A., and Snijders, J.G. (1982), Phys. Rev., A25:2339. [14] van Zijl, P.C.M., Ruessink, B.H., Bulthuis, J., and Maclean, C. (1984), Accounts Chem. Res.,

17:172. [15] Bulthuis, J., and Maclean, C. (1971), J. Magn. Reson., 4:148.

[16] Lucas, NJ.D. (1971), Mol. Phys., 22:147.

[17] Lucas, N.J.D. (1971), Mol. Phys., 22:233.

[18] Lucas, NJ.D. (1972), Mol. Phys., 23:825. [19] Bwnell, E.E., and de Lange, C.A. (1980), Chem. Phys. Letters, 76:268.

[20] Bwnell, E.E., de Lange, C.A., and Mouritsen, O.G. (1982), J. Magn. Reson., 50:188.

[21] Snijders, J.G., de Lange, C.A., and BwneIl, E.E. (1982), J. Chem. Phys., 77:5386.

[22] Snijders, J.G., de Lange, C.A., and BwneIl, E.E. (1983), J. Chem. Phys., 79:2964.

[23] Snijders, J.G., de Lange, C.A., and BwneIl, E.E. (1983), Israel J. Chem., 23:269.

[24] Lounila, J., and Diehl, P. (1984), J. Magn. Reson., 56:254.

Chapter 2

DENSITY MATRIX METHODS IN NMR

M.Bloom Department of Physics, University of British Columbia, Vancouver, B. c., Canada

E.E. Burnell Department of Chemistry, University of British Columbia, Vancouver, B. c., Canada

C.A. de Lange Laboratory for Physical Chemistry, University of Amsterdam, Amsterdam, The Netherlands

1. Introduction The previous Chapter 1 reviews the frequency domain NMR spectroscopy of ori

entationally ordered molecules. However, most modem NMR experiments involve accumulation of signal in the time domain, followed by Fourier transformation in order to obtain the frequency spectrum. As Abragam realized during the writing of his classic book [1], time-domain NMR is most conveniently handled using the statistical density matrix. In this Chapter we present a brief introduction to the spin-part p of the density matrix, followed by examples designed to give the reader insight into the use of p in various aspects of NMR. First, we treat an ensemble of uncoupled spin I = ~ particles, followed by a discussion of an ensemble of uncoupled spin I = 1 particles. The latter is the simplest system that allows full insight into the power of the density matrix approach. The excellent book by Goldman [2] treats the system consisting of two coupled I = ~ spins, and the reader is referred there, and to several other books, for further reading [3-6].

2. Brief introduction to the density matrix Recall that any state vector 'Ij; of a spin system can be written as a linear combination

of a complete set of orthonormal, time independent basis states <Pn, which in Dirac notation is

n

27 E.E. Burnell and c.A. de Lange (eds.), NMR of Ordered Liquids, 27-44. © 2003 Kluwer Academic Publishers.

(2.1)


where the coefficients Cn are in general time-dependent, complex numbers, and where we use the notation In >= l<Pn >. For state vector 'l/J, the observable 0 associated with operator 0 is given by the equation

0= <0> = < 'l/J101'l/J >

= LLcqc; <plOlq > p q (2.2)

= LLCqC;Opq. p q

The Opq =< plOlq > are time-independent matrix elements that are readily calculated.

It is often appropriate to represent the system under investigation by an ensemble of particles (spins) in various states 'l/Ji of the spin Hamiltonian. Hence, the quantity observed in some "measurement" of 0 will be the statistical average over all state vectors 'l/Ji of the ensemble

o = < 0 > = L L cqc; Opq. p q

We define the spin density matrix

Pqp = cqc; = < qlplp > .

(2.3)

(2.4)

Each element Pqp is the density or probability of the coherent superposition of the basis states <pp and <pq. The observable quantity is then

p q

(2.5) p q

Tr{pO}

where we use bold symbols to denote matrices. The time dependence of the system is governed by the SchrOdinger equation

- ~ EN = it. l• i at 'f/

(2.6)

which can be written as the quantum-mechanical form of the Liouville theorem [2,4]:

dp i dt = Ii [p, 1-£]. (2.7)

The term in square brackets is the commutator

[p,1-£] = p1-£ - 1-£p (2.8)

Density matrix methods in NMR 29

and 1£ has matrix elements 'Hpq =< plHlq >. In NMR it is often convenient to use a time-independent Hamiltonian, and to put the time dependence into the coefficients Cp, i.e. into the density matrix. If 1£ is time independent, the solutions ofEq. (2.7) in matrix form are

(2.9)

We refer to Chapters 16 and 17 for a discussion of NMR relaxation which is more complicated as it involves a time-dependent Hamiltonian.

2.1 Comments on spectroscopy To gain insight into the density matrix and its use in describing NMR spectroscopy,

it is useful to examine p in a basis in which 1£ is diagonal, i.e. the basis states used, </>n, are taken to be the eigenfunctions of the spin Hamiltonian. In this basis the time evolution of the matrix elements of p(t) in Eq. (2.9) is readily shown (by expanding the exponentials) to be

ppq(O)e*(Eq-Ep)t

Ppq(O) (cos Wqpt + isinwqpt) (2.10)

where Wqp = Eq-n EP . This equation is interesting from the NMR point of view. It

indicates that each element Ppq will evolve in time at angular frequency Wqp = Eci/ p. In other words, Eq. (2.10) describes the precession of the coherent superposition of time-independent eigenstates </>p and </>q. The population or density of this coherence is given by the absolute value Ippql. The normal NMR spectrum (discussed in Chapter 1) arises from those elements with Wpq :::::: Wo, i.e. the superpositions of eigenstates that differ by the flip of a single spin. We think of each element Ppq as precessing in time about the "Hamiltonian direction". Here we associate a pseudo direction with the Hamiltonian. For the case of the Zeeman Hamiltonian this direction is the magnetic field direction taken as the real space-fixed direction Z. However, there is no real vectorial direction for the quadrupolar, dipolar or indirect spin-spin Hamiltonians. For the many-spin systems discussed in the various Chapters of this book, the Hamiltonian involves many Zeeman, quadrupolar and coupling terms. However, when we refer to the "Hamiltonian direction" we mean the precession axis associated with Eq. (2.10). In this sense NMR spectroscopy involves precession of elements of p about some axis that we call the "Hamiltonian direction". These ideas apply equally to Hamiltonians that include only static fields as well as to those that involve rotating radio frequency rjfields, as long as we can transform to a rotating reference frame in which the Hamiltonian is static (the usual case).

2.2 Properties of p

Because p is a statistical average density of a quantum system, it must satisfy several properties. Because quantum observables are real quantities, p must be Hermitian with


diagonal elements Ppp ~ O. In most physical systems studied, it is appropriate to think of p as a probability density which requires Tr{p} = 1. In addition, for the special case of an equilibrium system which requires p(t) = p(O), the off-diagonal terms in Eq. (2.10) must be zero, i.e. at equilibrium Ppq = 0 if Eq i= Ep.

For an m dimensional system, i.e. a system with m independent orthonormal Hilbert space basis states, p has m 2 - 1 independent elements (the trace condition is the extra element). For example, for I = ! there are two basis states, and p has four terms, three being independent. For I = 1, there are three basis states and nine terms (eight independent). For two coupled spin! particles, there are 16 terms (15 independent). In the treatment that follows, we shall want to think of a multi-dimensional Liouville space, with a component associated with each independent element of p. We can describe NMR in terms of populations of these independent elements of p. We have already seen that, if we represent p in a diagonal basis of 1-£, spectroscopy involves elements of p precessing about the "Hamiltonian direction", and for each element this precession can be expressed in a three-dimensional sub space of the overall (m 2 - 1) dimensional space. These ideas can be applied equally to both rotating reference fields as well as to large constant fields.

3. Pulsed NMR 3.1 N identical non-interacting spin I = ~ particles

We wish to develop an intuition for pulsed NMR. We start with the simplest possible system in order to develop a terminology that will be useful in the understanding of more complex spin systems. We start with the calculation of magnetization and precession for an ensemble of non-interacting spin I = ~ particles in a magnetic field of magnitude Bo that lies along the space-fixed Z direction. The spin Hamiltonian then includes only the Zeeman term of Chapter 1 which we write for convenience as

(2.11)

where Wo = -,Bo. Note that Hamiltonians are written in energy units in this Chapter. Each spin has a magnetization or a magnetic moment vector, /-L = ,n < I >. The net magnetization vector M is then the magnetization averaged over all spins times the number of spins, N:

M = < IV! > = N < jL > = N,n< I >. (2.12)

Eq. (2.7) is then

: = iwo[p, Iz]. (2.13)

In order to gain insight into the mathematics, it is instructive to introduce matrices. For I = 1 we use the two Hilbert space basis states In > and 1,8 > for m = + 1 and

Density matrix methods in NMR

1(3 > = (~). The Pauli matrices for the spin operators in this basis are

1 (0 1) Ix = 2" 1 0

1 (0 -i) Iy = 2" i 0

where the matrix elements are calculated according to equations such as:

(IX)pq = < plixlq > .

The density matrix defined in terms of this Hilbert space is

p = (PU P12). P21 P22

31

(2.14)

(2.15)

(2.16)

(2.17)

For NMR a more convenient way to write p is as a complete set of spin operators in Liouville space. For example, for spin I = ! we can write p in terms of the complete

set of spin operators Ix, Iy, Iz, and 1 = ! (~ ~) which is one half the unit

matrix. These spin operators are closely related to NMR observables. These Liouville space basis operators are orthogonal as

(2.18)

where fJa {3 is the Kronecker delta function (1 if a = (3, 0 if a i- (3). The density matrix is then

p = aal + axlx + ayly + azlz. (2.19)

The important point about this representation is that the coefficients ai are observables of an NMR experiment. We set aa = 1 to meet the requirement Tr{p} = 1. We can also write the matrix for any quantum-mechanical operator Q in the same basis, i.e.

Q = qal + qXIX + qyly + qzlz. (2.20)

We now use Eq. (2.5) and matrix multiplication to calculate the observable

where Ii includes the unit matrix. For example, the Z magnetization is readily calculated by setting Q = Iz (i.e. qo = qx = qy = 0; qZ = 1 in Eq. (2.20», and we obtain lz = !az. The ai are, therefore, the density or population of the i magnetization, and Eq. (2.21) allows us to decompose p into the various contributions. Note that


there are no additional observables for non-interacting spin ! particles because any observable can be represented as a superposition of linear terms only. For example, Iz2 = 1; Iz3 = Iz; etc.

We now use this formalism with Eq. (2.13) to calculate the time evolution of p under the influence of the Zeeman Hamiltonian Eq. (2.11):

: == p = iwo[p, IzJ. (2.22)

Expanding p in terms of Eq. (2.19), using a == !!ft, and noting that the matrices Ii are constant in time, we obtain

0,0 1 + axlx+ayly + azlz = (2.23)

iwo {ao[1, IzJ + ax [Ix,IzJ + ay[Iy, IzJ + az[Iz,Iz]}.

Using the commutation relations

Eq. (2.23) becomes

[1,IzJ = 0

[Ix, IzJ = -ily

[Iy, IzJ = ilx

[Iz,IzJ = 0

0,01 + axlx + ayly + azlz = iwo {-iaXly + iaylx}. (2.24)

We are interested in the time dependence of the NMR observables. To obtain equations that are useful for this purpose, we multiply each side of Eq. (2.24) by Q equal to one of the 1, Ix, Iy, or Iz matrices (i.e. we set one qi = 1 in Eq. (2.20) and set the other three qi = 0). Since these are time-independent quantities, they commute with the ai. Hence, we can take the trace of both sides of the resulting equation. This involves taking the trace of each term in the same manner as was done in Eq. (2.21). The general result for each coefficient ap can be written

a; = iwoLaqTr{Ip[Iq,Iz]}. q

(2.25)

Thus we obtain four equations, one associated with the time dependence of each of the observables I, Ix, Iy, and /z:

0,0 =0

ax = -WOay

ay (2.26)

=woax

az =0.

The three independent ai in these equations are the densities or probabilities of the X, Y and Z magnetizations that can be measured in an NMR experiment. The equations


show that the Z magnetization (and the unit matrix) are constant with respect to time, whereas the X and Y magnetizations are time-dependent and coupled. We can now form the following linear combinations of the two coupled differential equations:

ax + iay ax - iay

+ iwo(ax + iay) - iwo(ax - iay).

This procedure immediately decouples the equations and we find the solutions:

ax(t) + iay(t) = (ax(O) + iay(O))eiwot

ax(t) - iay(t) = (ax(O) - iay(O))e-iwot .

By separating the real and imaginary parts we obtain:

ax(t) = ax(O) coswot - ay(O) sinwot

ay(t) ay(O) coswot + ax(O) sinwot.

(2.27)

(2.28)

(2.29)

These are identical to the solutions of the normal Bloch equations, and describe Larmor precession of magnetization around the field B z at angular frequency Wo. Hence, this quantum-statistical treatment of the spin system leads to a description of the magnetization that is identical to that for a classical magnetization. The a coefficients of the three independent elements of the density matrix, ax, ay, and az, are proportional to the components of the magnetization vector which precesses about the magnetic field Bo that lies along the Z direction. The equations are identical to those obtained for precession of a classical magnet in a magnetic field.

It is worth emphasizing the remarkable significance of the precession equations just obtained. If we have some spin magnetization (Mx say) and wish to know qualitatively how it will evolve, we need only work out the commutator of the magnetization operator with the Hamiltonian. We then discover a new magnetization (My in our current example with it = lU..ioiz ), and its commutator with 1-£ gives us yet another magnetization that will develop. In the current example, this new magnetization is Mx. Hence in this case we find that Mx evolves into My which evolves back into Mx. The loop is closed, and we obtain the precession equations for the magnetization about the Hamiltonian (magnetic field) direction. Of course, this is exactly what is expected from Eq. (2.10) because 1-£ is diagonal in this case. If we use a basis in which 1-£ is not diagonal, we expect coupling among more than two equations.

3.1.1 Rotating frame. In thinking about NMR it is useful to transform into a reference frame that is "rotating" about Z at angular frequency w = 27Tl/, where l/ is normally chosen to be the detection if of the spectrometer which is also normally the if of the applied pulses. In this frame the precession frequency in Eq. (2.29) becomes D.w = Wo - w. We have transformed the effective magnetic field seen by the spins from Bz to Beff = -D.w/'y. If we set w = wo, i.e. if we set the spectrometer "on resonance", then the right hand sides of Eqs. (2.26) are all zero, and ax(t) = ax(O) and ay (t) = ay (0) are constant with respect to time, where the prime denotes an axis


in the rotating frame. The result is that there is no precession of magnetization in this on-resonance rotating frame. In what follows we shall discuss the effects of pulses and of quadrupolar terms in the Hamiltonian. The mathematics is simplified by working in the on-resonance rotating frame where we can forget about precession due to the main magnetic field B z.

3.1.2 Pulses. We apply an ifpulse along the laboratory-fixed X axis of strength BI = -~ cos(wt + J). For now we set the pulse phase J = O. We decompose the pulse into two counterrotating components, and discard the component that precesses in the direction opposite to Wo as it is too far off-resonance to affect the magnetization by much. Note that the discarded counterrotating component of BI does cause a small shift, known as the Bloch-Siegert shift [7], which we shall ignore for w~ « w6. Hence, the Hamiltonian associated with the pulse is (in the laboratory frame):

ilpulse A A

-li- = WI (Ix coswt + ily sinwt).

In the rotating frame this Hamiltonian is time independent, being ilrot;ru!se Thus, the effective Hamiltonian in the rotating frame is:

ilrot AI AI T = flwlz + wIlx ·

(2.30)

(2.31)

Note that the phase (J = 0) we have chosen for ilpulse leads to an X' pulse. If we now

play the same game as above in section 3.1 (for il = liwoiz ) and use Eqs. (2.5, 2.7, 2.18, and 2.19) to work out the precession equations for p, we find that its elements precess around

Beff = k(Bo +~) + i~l (2.32)

at frequency

(2.33)

where i, j, k are unit vectors along axes X', Y', Z' in the rotating frame. For simplicity in what follows, we shall drop the primes on the rotating frame axes and the subscript "rot" for the rotating frame Hamiltonian.

The algebra above is complicated by the mixing caused by the presence of two terms in the Hamiltonian which is no longer diagonal in the basis chosen. The algebra is much simpler if we are on-resonance, i.e. (w = wo), when Eq. (2.31) takes the

especially simple form ilhot = wdx. We now find that

ay(t)

az(t)

ay(O) cos wIt - az(O) sinwit

az(O) cos wIt + ay(O) sinwit. (2.34)


Again we have precession, this time of Z or Y magnetization about the "Hamiltonian" or X direction. We could have anticipated this result from examination of commutators.

In order to keep the algebra simple we here assume, as is often the case for discussion of basics, that pulses are sufficiently short and strong that we can neglect precession due to terms other than those that describe the pulse in 'Hrot . In the current example this is equivalent to working on resonance during the pulse.

3.1.3 Spherical tensor representation - a different basis for p. In section 3.1 the problem of N non-interacting spins I = ! was treated by expanding the density matrix p in terms of the four orthogonal basis functions 1, Ix, I y, and I z. This approach led to a set of differential equations that were partly coupled. By taking appropriate linear combinations of the coupled equations (see Eq. 2.27), decoupling resulted and a solution could be obtained immediately (see Eq. 2.28). This suggests that, by taking a different basis set for the expansion of p, the decoupling of the resulting differential equations may be achieved right away. The basis set for the problem of N non-interacting spins I = ! must again consist of four independent functions for which we now select the unit tensor and three spherical tensors of rank 1. We shall not dwell upon the unit matrix which transforms like a scalar. Under coordinate transformation the spherical tensors of rank 1 transform among themselves like vectors in three-dimensional space. We expand p in this new basis set which will prove to be extremely convenient:

where

We note that

1

P = ao,ol + L al{tTl,p. {t=-l

T1,o = Iz = ~ (1 0) 2 0-1

which leads to the orthogonality condition

(2.35)

(2.36)

(2.37)

(2.38)

(2.39)

(2.40)


Recalling Eq. (2.5), the observables are calculated as

Tl,,,,, = < 1'1,,,,, > = Tr{pTl,,,,,} = L a11-',Tr{Tl,,",,, Tl,,,,,} 1-"

(-1)1-' = -2-a1- w

It is easy to prove that

[Iz,I±l = ± I±. (2.42)

Then, for the Zeeman Hamiltonian it = /u,;oiz , Eq. (2.7) gives

(2.43)

with solution

(2.44)

There are no coupled equations to solve. Indeed, we note that for the JL = ±1 terms we obtain a result identical to that of Eqs. (2.28 and 2.29). Also, note the connection with Eq. (2.10) which applies because 'H. is diagonal in the chosen basis. The I± terms, Eqs. (2.37, 2.38), are the elements P12 and P21 in this basis. The values of al±l-' are the density or population of the coherences between states la > and 1,8 >. These are the coherences that precess about Z and are detected as Ix and Iy directly in an NMR experiment.

3.2 N identical spin I = 1 particles with nuclear quadrupole interaction

From Chapter 1 the Hamiltonian for a spin I = 1 particle in a magnetic field Bo can be written

it A WQ A2 - = woh + -(3Iz - 2) n 3

(2.45)

where wQ = 3Q: z (see Eq. 1.44). The NMR spectrum is then a doublet with lines at angular frequencies Wo ± wQ.

Since for spin I = 1 there are three independent spin functions (usually denoted II, m >, with m ranging from -I to +1), the spin density matrix p now has nine terms in Liouville space which can be represented by the unit matrix and eight additional independent ones. As before, any orthogonal basis set will do, and the choice is one of convenience alone. In this case we take basis spin functions that under rotation transform like the s (1), p (3) and d (5) real spherical harmonic solutions of the angular part of the hydrogen atom problem. The s function transforms as a scalar, is a constant,


and not interesting. The three p functions transform among themselves as vectors in three-dimensional space. The five d functions also transform among themselves. In NMR our pulses are along a specific direction in the rotating frame, and we often refer to magnetizations pointing along certain directions in this frame. Hence, it would seem convenient to choose a basis that is consistent with this Cartesian aspect of our NMR experiment. As for the spin I = ! case, we choose Ix, I y, and I z as the three basis matrices that transform as the real hydrogen p angular wave functions. Note that we could equally well have chosen a spherical tensor representation (1+, 1_ and Iz), but the basis used here is more suitable for the present discussion.

The eight independent matrices that transform as the hydrogen p and d wave functions are:

Txz

Tx

1Y~~IY~~O ~'

Tz ~ ~Iz ~ ~ 0 ~ ~ ~(3Iz2-2) ~ ~ (~

( 0

1 2 2 1 -(Ix -Iy) =- 0 J2 J2 1

1 1 ( 0 J2(Iz I x + IxIz) ="2 ~

Tyz ~ ~(Izly + /ylz) ~ ~ (! ~: 1 1 (0 0

Txy = j(\(IxIy + IyIx) = j(\ 0 0 v2 v2 i 0

(2.46)

(2.47)

(2.48)

(2.49)

(2.50)

(2.51)

(2.52)

(2.53)

It is easy to show that the Ti matrices are orthogonal in the sense of Eq. (2.18), and this time they are normalized

(2.54)


The following equations and definitions from quantum mechanics are useful for checking the matrices in Eqs. (2.46 - 2.53)

iz II,m > = mII,m>

i+ II,m > J(I - m)(I + m + 1) II, m + 1 > L II,m> = J(I + m)(I - m + 1) II, m - 1 > (2.55)

i+ = ix + ily

L = ix - ily .

This matrix representation is useful for the determination of the commutators needed to calculate the time evolution of the density matrix. In Table 2.1 we present a commutator table for I = 1. For spin I > 1 basis sets can be constructed in a straightforward manner along very similar lines.

3.3 An example

It is instructive to follow the evolution of the magnetization through an NMR experiment, and we choose one that involves three pulses. In particular, we choose the leener-Broekaert [8] three-pulse experiment of Fig. 2.1 because it demonstrates the power of the density matrix approach. We have already seen for spin I = ! that the !:::..W term causes precession about the effective magnetic field in the rotating frame, and the same result is obtained for the I = 1 case. In order to keep the algebra simple, we go on resonance where the j z term disappears in it and the rotating frame Hamiltonian

(a) (b) (c) (d) (e) (f) (g)

Figure 2.1. The Ieener-Broekaert [8] pulse sequence which is chosen to demonstrate the utility of the density matrix for understanding a pulsed NMR experiment. Top, pulse sequence: (a) equilibrium magnetization M along Z; (b) I pulse along X in the rotating frame; (c) evolution under HQ; (d) ~ pulse along Y in the rotating frame; (e) evolution underHQ; (f) ~ pulse along Y in the rotating frame; (g)

evolution under HQ. Bottom: cartoon of NMR signal. The signal at (g) is known as the Ieener-Broekaert echo.


Table 2.1. Commutator table for spin I = 1 using a Cartesian basis for pa

Ix Iy Iz Txz Tyz TZ 2 -iB Txy

Ix 0 iIz -iIy -iTxy A+B -iV3Tyz -iTyz iTxz Iy -iIz 0 iIx -A+B iTxy iV3Txz -iTxz -iTyz Iz ily -iIx 0 iTyz -iTxz 0 2iTxy

Txz iTxy A-B -iTyz 0 ~Iz -i::t{-ly ~ly Tyz -A-B -iTxy iTxz -lIz 0 . ../3 I ~Ix

·J- I ~2 x

TZ 2 iV3Tyz -iV3Txz 0 . ../3 I 0 0 IT y -IT x -iB iTyz iTxz -2iTxy 2'ly 2~Ix 0 0 Txy -iTxz iTyz 2B ~Ix 2'ly 0 -iIz

aTo save space we write A == iv'3Tz 2 and B == iTx2_ y2.

becomes

(2.56)

We start with an equilibrium magnetization M == M = Mo lying along the Z direction which from Eq. (2.12) is M = N"(liIz (see Figs. 2.1a and 2.2). In other words, the coefficient az is the only non-zero contribution to p (other than ao which we can safely ignore), or p = azTz = ~Iz. As we are only interested in relative

values of magnetizations, we arbitrarily set az = 1 and write p(O) = Tz. As for the spin I = ! case, an X pulse with if = nw1ix (for this discussion we

take"( > 0 and therefore Wl < 0) causes precession (Eq. (2.34) or Table 2.1) of Tz about the X axis to Ty to -Tz to -Ty ... (Figs. 2.lb and 2.3). If the pulse is of duration -Wl t = 'IT /2, its effect is to rotate M by 'IT /2 from the Z to the Y direction in the rotating frame. Recall that we commonly assume in pulsed NMR that all rf pulses are sufficiently short and strong that during the pulse the time evolution of the

z M

J------y

X (a)

Figure 2.2. Equilibrium magnetization M lying along Z before pulses are applied.

-2B 2; Ix

~ly 0

iIz 0


~ / y

X (b)

Figure 2.3. Precession due to an X pulse; all diagrams are drawn to indicate precession for'Y > 0 and therefore Wi < 0 and Wo < O.

density matrix is determined by the pulse terms in the Hamiltonian alone, while the time evolution between pulses is governed solely by the other terms in the Hamiltonian.

We note that the effect of the first pulse is to convert Z magnetization (i.e. zeroquantum coherence), into X or Y magnetization which involves coherence between states that differ by total b..M = ±1 (i.e. one-quantum coherence). In high field, precession about the total spin Hamiltonian (Eq. (1.68) of Chapter 1) does not change the order of coherence. Hence, starting from equilibrium Tz magnetization, we cannot populate all elements of p with a single pulse (assuming no precession during the pulse), because the pulse can only rotate Tz into Tx or Ty. As we shall see below, precession of X or Y magnetization about the spin Hamiltonian generates magnetizations that can populate many elements of p with a second pulse. We note that spin-lattice relaxation processes (see Chapters 16 and 17) can also change orders of coherence.

Let us assume that the first pulse is a 7f /2 pulse such that the density matrix immediately after the pulse is p = Ty. This Ty magnetization undergoes a time evolution

according to Eq. (2.7), with the appropriate Hamiltonian it = jI/i;.;;Q TZ 2 given by

Eq. (2.56). Analysis following the procedures used in section 3.1 yields the general result for each coefficient ap :

Ctp = ~ L aq Tr {Tp [Tq , 1-t]} . q

(2.57)

From the commutator table we see that in the presence of the Hamiltonian ;-iwQ T Z2,

Ty will evolve into Txz, from which in tum -Ty will be generated. The result is two coupled equations similar to those in Eqs. (2.26 to 2.29):

ay(t)

axz(t)

ay(O) coswQt - axz(O) sinwQt

= axz(O) coswQt + ay(O) sinwQt (2.58)

where ay(O) = 1 and axz(O) = o. Thus, taking wQ > 0, we have precession of magnetization about the "Hamiltonian axis" TZ 2 from the "axis" Ty to Txz to -Ty to -Txz ... (Figs. 2.1c and 2.4). These "axes" are orthogonal "directions" in our


L / y

zz (c)

Figure 2.4. On-resonance precession due to the quadrupoJar Hamiltonian; the precession direction is indicated for WQ > O.

eight-dimensional Liouville space. The basis functions are chosen in such a way that the Ty magnetization evolves between only two "directions" in the eight-dimensional space (Ty and Txz) under the influence of a pure quadrupolar Hamiltonian in the high-field approximation. The coil of an NMR spectrometer can only detect Tx or Ty magnetization, and hence the signal appears to "vanish" when it becomes Tx z magnetization or coherence. Some authors call Tx z "anti-phase" magnetization, and attempt to picture it as two opposing vectors pointing along + X and - X axes. However, the power of the current picture is that the magnetization is always represented as a "direction" or "vector" (lying somewhere in our eight-dimensional Liouville space). This picture is especially useful when the basis chosen for p is such that in the time evolution (or "precession") of the magnetization, only a small number of basis "directions" is involved.

Let us assume that our magnetization precesses for a time t such that wQt = 7r /2, when according to Eq. (2.58) P = Txz. Let us now apply a pulse that is 7r /2 shifted in phase from the first pulse, i.e. we apply a pulse along the Y axis in the rotating frame, with the Hamiltonian during the pulse being it = tiw1iy . We can again work through the algebra. However, examination of the commutator table shows that the pulse will generate {v'3 TZ2 - TXLY2} which will then regenerate -Txz etc. Again, we have "precession", but this time one of the "axes" is not a basis "direction", but rather a linear combination of the two "axes" TZ 2 and T X 2_Y2. In a manner of speaking, these "directions" define a "plane" for the magnetization. Working through the algebra we find that the precession equations are

axz(t)

1 '2(aA - aB)(t)

axz(O) cos 2wlt + ~(aA - aB)(O) sin2wlt

1 . '2(aA - aB)(O) cos 2wlt - axz(O) sm 2wlt

(2.59)

where aA = v'3az2 and aB = aX2_y2 (see Figs. 2.1d and 2.5). We note two things. First, the rotation due to the pulse is twice as fast as earlier, and a 7r / 4 pulse causes a 7r /2 rotation. Second, the pulse transfers f':l.M = ±1 coherence into f':l.M = 0 or zeroquantum coherence T Z2, and into f':l.M = ±2 or two-quantum coherence T XL y2 .


(A-B)/i

~---'--- xz

Y (d)

Figure 2.5. Precession due to a Y pulse. A = i,j3Tz 2 and B = iTxL y2

In general, the second pulse is able to lead to population of many elements of p, and the idea of multiple-quantum NMR (see Chapter 4) is based on this concept. In the multiple-quantum experiments described elsewhere in this book (see Chapters 4 and 5) there are many more coupling terms in ii, and the experiment is not performed on resonance. Hence the equations are far more complicated, but the present example is an excellent one for giving an intuitive picture of the experiment. An analogous treatment often applied to the discussion of a system of coupled spins ~ in the weak coupling limit involves using product spin! functions as a basis. In this limit one can consider

ii operating individually on each spin. For a two-spin system the magnetizations discussed here are applicable. The book by Goldman [2] gives an excellent description of the system of two coupled spins !.

If the second, Y pulse is of duration WI t = 1f /4, after the pulse we have magne-.. vI3 d a X2 - y2 Th fi T . hId T . . tlzatlOns TaTz2 an - 2 . erst, Z2, commutes WIt zan Z2, z.e. It

commutes with 1{., and is "constant" in time. It is often referred to as quadrupolar order. The second, or double-quantum term T X2 _ y2, will now evolve under the Hamil

tonian t Z2. Again we can work out the precession equations following the procedure outlined in section 3.1. We see from the commutator table that TX2_Y2 commutes with TZ 2. In this case there is no precession (see Figs. 2.1e and 2.6). However, if

(A-B)/i

M

J-----XY

ZZ (e)

Figure 2.6. On-resonance precession due to the quadrupoJar Hamiltonian. A = i,j3Tz 2 and B = iTx 2_y2.


we are off resonance such that there is the term Awiz in the Hamiltonian, the commutator table tells us that (if Aw < 0) -TXLy2 will precess into Txy which will precess into TXLy2 etc. Working out the algebra gives the precession frequency of this double-quantum coherence as 2Aw:

= aXLY2(0) cos2Awt - axy(O) sin2Awt

axy(O) cos 2Awt + aXLy2 (0) sin 2Awt. (2.60)

In general, an N -quantum coherence precesses at N Aw, and this fact allows separation of the various multiple-quantum orders in NMR experiments (see Chapters 4 and 5).

If after a certain time, tl say, we apply a third pulse of duration 7r / 4 along Y, then the algebra and commutator table tell us that T Z2 will precess into - Tx z, T XL y2 into Txz, and Txy into Tyz (see Figs. 2.1f and 2.7). These coherences will then precess

under it = Jj'tiM.JQ TZ2 into Ty and -Tx after a time wQt. As Tx and Ty are the

measured signal of an NMR experiment, an echo will be formed (see Figs. 2.1g and 2.8). In the case that the double-quantum coherence has relaxed or dephased, the echo is a measure of the T Z2 remaining. When the echo amplitude is obtained as a function of tl (the time between the second and third pulses), this allows the experimental determination of the spin-lattice relaxation time, T1Q, associated with the quadrupolar order. In addition, Fourier transformation of the echo amplitude as a function of tl gives the spectrum of precessions (in this case of T Z2 and T XL y2, Txy) during this time, and thus in general gives the multiple-quantum spectrum (see Chapter 4).

4. Summary In this Chapter we have presented the power of density matrix methods in the inter

pretation and understanding of NMR spectroscopy experiments. While the example discussed in section 3.3 is specific for a very particular experiment performed on resonance for non-interacting spin I = 1 particles, the concepts are very general and easily extended to more complicated spin systems, such as systems of coupled spin I = ! nuclei. We chose the simple system for discussion because it allowed us to

(A-B)/i

M

I~···· -xz

y (0

Figure 2.7. Precession due to a Y pUlse. A = i /3T z 2 and B = iT x 2 _ Y 2 •


xz

;1~i ;;;::W'--Q-t--M- Y

(g)

Figure 2.8. On-resonance precession due to the quadrupolar Hamiltonian.

follow and understand quite a sophisticated experiment in complete detail. This example highlights the intuition that can be gained from the density matrix approach to NMR spectroscopy in one or more dimensions. It also demonstrates the power of commutators in gaining insight into which magnetizations evolve in the presence of various pulses and Hamiltonians.

In this Chapter we use "vectors" in a spin space of appropriate dimensions to represent magnetizations. This approach gives insight into experiments where the "observable" X and Y magnetizations seem magically to vanish, only to reappear at a later time; of course the magnetizations are simply precessing into one of the "unobservable" dimensions of our density matrix space. We think that this vectorial description gives excellent intuition for NMR experiments, and for the evolution of the magnetization that is key to such experiments.

Acknowledgments The authors are grateful to Dentry's (Vancouver) and Brouwerij 't IJ (Amsterdam)

for providing an atmosphere conducive to scientific discourse.

References [1] Abragam, A. The Principles of Nuclear Magnetism. Clarendon, Oxford, 1961. [2] Goldman, M. Quantum Description of High-Resolution NMR in Liquids. Oxford University Press,

Oxford, 1988.

[3] Ernst, R. R., Bodenhausen, G., and Wokaun, A. Principles of Nuclear Magnetic Resonance in One and Two Dimensions. Clarendon Press, Oxford, 1987.

[4] Munowitz, M. Coherence and NMR. Wiley, New York, 1988.

[5] Slichter, C. P. Principles of Magnetic Resonance. Springer-Verlag, New York, 3 rd edition, 1990.

[6] Bloom, Myer. Physics of NMR Spectroscopy in Biology and Medicine, Maraviglia, B., Editor, page 121. Elsevier, Amsterdam, 1988.

[7] Bloch, F., and Siegert, A. (1940), Phys. Rev., 57:522. [8] Jeener, J., and Broekaert, P. (1967), Phys. Rev., 157:232.

Chapter 3

COHERENT AVERAGING AND CORRELATION OF ANISOTROPIC SPIN INTERACTIONS IN ORIENTED MOLECULES

Malgorzata Marjanska, Robert H. Havlin, and Dimitris Sakellariou Materials Sciences Division. Lawrence Berkeley National Laboratory and

Department of Chemistry. University of California. Berkeley. California. U.S.A.

1. Introduction Nuclear magnetic resonance (NMR) spectroscopy has been widely used to study

liquid crystalline phases and molecules dissolved in these oriented phases. However, the observed spectra are complex and difficult to assign. This complexity arises from the existence of unaveraged intramolecular dipolar couplings, Dij , the magnitudes of which exceed typical chemical shift differences, leading to second-order spectra. Also, the spectral complexity increases rapidly with the size of the molecular spin system, or with a lowering of the symmetry. The second-order spectra can be difficult to analyze, but they possess a wealth of information about the structure, conformation, and orientational order of the molecules. Various techniques have been designed to simplify the second-order spectra to facilitate analysis.

In this Chapter, several novel techniques that aid in the analysis of the liquid crystal spectra are presented. A description of techniques that use spatial reorientation, such as variable angle spinning (VAS) and switched angle spinning (SAS), are outlined. Following this is a discussion of techniques that simplify the spectra through the use of spin space manipulations, such as heteronuclear and homonuclear decoupling. Finally, multidimensional experiments that use both heteronuclear and homonuclear decoupling as building blocks are presented.

2. Spatial reorientation techniques Spatial reorientation techniques have been employed to manipulate and control the

averaging of anisotropic interactions beyond what the static liquid crystalline environment naturally provides [1-3]. By manipulating the director experimentally, the natural spatial averaging ofliquid crystals can be utilized to further average interactions or even provide decoupling similar to the well-known magic angle spinning (MAS)

45 E.£. Burnell and CA. de Lange (eds.). NMR of Ordered Liquids. 45-65. © 2003 Kluwer Academic Publishers.


in solids [4]. This section describes some of the recent work that has utilized director manipulations to aid in the analysis and assignment of anisotropic interactions in liquid crystals.

2.1 Director manipulations 2.1.1 Spinning. Initial investigations demonstrated that the liquid crystal director can be manipulated using sample spinning [5]. During sample spinning, the magnetic aligning forces become time dependent, and the director aligns in such a way as to minimize the potential energy per rotation cycle. Below a threshold rotation frequency (wr / B5 '" 4.8 Hz T-2), the director aligns according to the rotation frequency and the characteristic hydrodynamics of the sample [1, 6]. However, above this critical spinning frequency, the director of the liquid crystals aligns to minimize the potential energy due to the magnetic force. This energy has the form [6]:

(3.1)

where () is the angle between the spinning axis and the magnetic field, Bo, ", is the angle between the spinning axis and the director, and ~X is the diamagnetic susceptibility anisotropy (~X = XII - X.l)· From inspection of Eq. (3.1), it can be seen that the director of a sample with a positive value of ~X will align parallel to the spinning axis (", = 0°) for () between 0° and 54.7°, and perpendicular to the spinning axis (", = 90°)for() between 54.7° and 90°. Fornegative values of ~X, the director exhibits the opposite behavior. Therefore, the alignment of the director can be reoriented by changing the axis of rotation.

The degree to which the liquid crystals follow the behavior described above depends upon the magnitude of the magnetic potential energy (Eq. (3.1). Therefore, as the angle of the spinning axis approaches the magic angle, the preference for alignment is reduced until at the magic angle there is no preferential alignment. This leads to a random distribution of directors. Under these conditions, the spectrum resembles a solid-state spectrum of a random powder; thus, spinning at exactly the magic angle is avoided. However, when close to the magic angle, the ordered phase can be maintained such that the director remains parallel or perpendicular to the spinning axis.

2.1.2 Other methods. Several alternatives to the use of spinning for director reorientation have been investigated. These alternatives are sought to avoid the drawbacks of the dependence of the aligning mechanism on the angle of the spinning axis. For example, in samples with a long director reorientation time, the sample can be rotated in the field to create a non-equilibrium director alignment [7]. However, this technique is largely dependent on the orienting characteristics of the particular liquid crystal and typically only works for polymeric liquid crystals or those with a high viscosity. Other studies have shown that an alignment can be "trapped" through the use of a polymer that prevents reorientation when the sample is rotated from the aligned state [8-10]. Electric fields have also been employed to induce an ordering

Coherent averaging and correlation of anisotropic spin interactions in oriented molecules 47

that can be switched on and off [11]. These types of dynamic director experiments retain the averaging properties of the liquid crystal, with the added benefit of allowing manipulations of the magnetic interactions, as described below.

2.2 Scaling interactions

Although the static anisotropic interactions such as dipolar couplings, chemical shift anisotropies, and quadrupolar couplings are initially scaled by the averaging properties of the liquid crystal (for a detailed description refer to [12]), changing the director orientation with respect to the magnetic field can further scale these interactions. Previous studies have demonstrated that changing the orientation of the director can scale anisotropic interactions in both liquid crystals and solutes dissolved in the liquid crystalline phase [6]. The interactions are scaled by the factor (3cos2 () - 1)/2 «() is the angle between the spinning axis and the magnetic field, Eo) when the director is aligned parallel to the spinning axis. This scaling has been observed for dipolar couplings [6], chemical shift anisotropies (CSA) [1], and quadrupolar couplings [13,14].

2.2.1 Dipolar coupling. The commonly used benefit of techniques that reorient the director is the scaling of dipolar couplings [2]. These techniques do not have the disadvantages of similar techniques that use spin manipulations such as scaling of the chemical shift and sample heating. During spinning, the observed dipolar coupling, .6.?r (between two nuclei whose chemical shielding difference is much larger than this coupling), scales with the director orientation in the following way:

A obs _ 2D .. (3 cos2 () - 1) J.. (3.2) Uij - tJ 2 + tJ

where b.?r is the observed splitting, Dij is the dipolar coupling in the liquid crystal phase in the absence of spinning, and Jij is the isotropic J coupling. Typically, only the isotropic component of the J coupling needs to be included since the anisotropic J coupling is usually small in light nuclei such as 1 H; however, the anisotropic J coupling would also scale with (3 cos2 () - 1)/2 [15]. The scaling properties of the dipolar interaction can prove particularly useful in the interpretation of second-order spectra. At spinning angles where the spectrum is reduced to first-order, the dipolar couplings can be measured directly from the splittings. Fig. 3.1A demonstrates how the dipolar couplings are scaled to their isotropic (liquid-like) values at the magic angle such that only the J coupling is evident. The dependence of the first-order dipolar splitting as a function of (3 cos2 () - 1) /2 plotted in Fig. 3.1B reveals that the linear relation passes through zero for two of the coupling pairs; this is a result of the dipolar coupling and the J coupling having opposite signs. Thus by fitting the linear dependence of the observed first-order coupling, the static dipolar couplings in liquid crystals can be determined.

2.2.2 Chemical shift anisotropy (CSA). The observed chemical shift also varies with the director angle. As shown in Fig. 3.1A the central frequency of all the peaks varies due to the scaling of the CSA by (3 cos2 () - 1)/2. This dependence

48

A. pure liquid

in liquid crystals under MAS

in liquid crystals spinning at 25°

__ -"~ ___ .--J 1.-___ -' I--

I 10

I o

kHz

I -10


B. Splitting Dependence

100-r-----------,

o

-100

.g,~ -200

<I -300

-400

-500

-1000 000

.'(1;,0 -1500 I

jg 00 -2000 '800

-2500

-3000

0.6 0.8

Shift Dependence

-3500-l--~-~-~-~-_I 0.0 0.2 0.4 0.6 0.8 1.0

(3cos2e-1 )/2

Figure 3.1. (A) 19F spectra of neat C2F31 (top), and of C2F31 dissolved in a nematic liquid crystal spinning at 2 kHz, (B) The dependence of the dipolar splitting (Eq_ (3.2» (top) and of the shift (Eq_ (3.3» (bottom) as a function of (3 cos2 8 - 1}/2. (Reproduced with permission from [16]. Copyright 2002 Academic Press)_


of the shift as a function of (3 cos2 B-1) /2 plotted in Fig. 3.1B illustrates how the averaged value of the CSA changes with the alignment director. Although the CSA is frequently an anisotropic tensor under static conditions, the averaging from the liquid crystal allows observation of only the components parallel to the director; thus a sharp resonance is observed at a frequency 8c~s:

.l:obs _ .l:aniso (3 cos2 B-1) .l:iso ucs - ucs 2 + ucs (3.3)

where 8cJsiso represents the anisotropic component of the CSA that remains after motional averaging in the liquid crystalline phase. The full CSA tensor cannot be determined from the linear dependence of Eq. (3.3) because only the averaged anisotropic shift, 8c~so, is accessible, and any information about the asymmetry of the CSA has been lost. Despite this, quantitative determinations of the averaged anisotropic component of the CSA can be obtained by fitting Eq. (3.3) for various values of B. Many studies have successfully separated 8CSiso and 8~s using the linear dependence of the observed chemical shift (Eq. (3.3» on the angle of the spinning axis [17]. Knowledge of 8CSiso not only allows for the prediction of the observed resonance frequency, but also provides structural insight typically associated with the CSA. In addition, 8~s remains an invaluable tool for assignment.

2.2.3 Quadrupolar coupling. The study of quadrupolar couplings in liquid crystals with varying director orientations also benefits from the (3 cos2 B-1) /2 scaling. Several studies have utilized the linear dependence of the quadrupolar splitting on (3 cos2 B -1)/2 to characterize the behavior of the liquid crystalline phase [14, 18]. The quadrupolar splitting, boQbs , scales as:

A obs aniso (3 cos2 B-1) i...J.Q = q 2 (3.4)

where qaniso is the averaged quadrupolar interaction [12]. The scaling of the quadrupolar interaction has received less attention than that of the dipolar interaction due to the lack of a strong correlation between the quadrupolar splitting and static structural properties. However, the study of the dynamics of ions, 2H, and 14N could potentially utilize the variable director method to assist in spectral analysis.

2.2.4 Perpendicular phase considerations. In the preceding discussion on anisotropic interactions, it was assumed that the directors align at a single angle with respect to the magnetic field. However, the directors may also align in a plane perpendicular to the axis of rotation. In this case, a distribution of (3, the angle between the director and the magnetic field, about the spinning axis is obtained, and spinning sidebands reminiscent of those that appear in solid-state spectra are observed. These spinning sidebands occur at even multiples of the spinning frequency instead of at every integer multiple of the spinning frequency. This can be understood by analyzing


the time-dependence of the angle f3 [18]:

(3cos2 f3(t) -1) 2

1 = 4" [(3 cos2 0 - 1)(3 COS2 1] - 1)

+ 3 sin(20) sin(21]) cos(wrt + 'ljJ)

+ 3 sin2(O) sin2(1]) cos(2wrt + 2'ljJ)]

(3.5)

where Wr is the spinning frequency, 0 is the angle between the spinning axis and the magnetic field, 1] is the angle that the director makes with the spinning axis, and 'ljJ is the initial phase. Thus when 1] = 90°, only the cos(2wrt) term contributes and gives sidebands at 2wr . Although information about anisotropic interactions is encoded within the sidebands, a quantitative determination requires numerical fitting. This method of analysis is not difficult for a simple sample, but it becomes more difficult when samples with multiple inequivalent spins are considered.

3. Two-dimensional dynamic director correlations The techniques described in the previous section work well for simple spin sys

tems; however, in complex spin systems, assignment becomes difficult even for a first-order spectrum. Additionally, it can be challenging to assign resonances due to the variable peak positions and splittings for samples with large CSAs or quadrupolar couplings. Thus, spreading out the information into another dimension and correlating the anisotropic dimension with the isotropic dimension can be a very powerful tool for spectral analysis.

Perhaps the simplest experimental design to effectively utilize the isotropic dimension is a two-dimensional dynamic director experiment. In this experiment, anisotropic interactions evolve at one spinning angle and the isotropic spectrum is observed after hopping to the magic angle. A correlation can be obtained between two different director orientations that provides anisotropic information through cross-peaks in the two-dimensional spectrum.

A two-dimensional correlation can help to identify the coupling pairs [16]. With an appropriately designed pulse sequence, dipolar cross-peaks exist between dipole coupled spins with their identity maintained by the isotropic dimension. With an initial angle of evolution adjusted such that the spectrum is made first-order, followed by switching the spinning axis angle to the magic angle for observation of the isotropic dimension, a two-dimensional dynamic director correlation reveals the dipolar couplings in the form of clearly separated cross-peaks.

A two-dimensional dynamic director experiment can also reveal information about the CSA. A recent study correlated the CSA values obtained at two different angles in 13C natural abundance polymer liquid crystal samples [17]. By evolving first at 0° and then reorienting the non-spinning sample to 90° for acquisition, both (8~s and 8CSiso) can be obtained. At 0°, the resonance frequency is (8~s + 8CSiso), while the frequency at 90° is (8~s - 8c~sO /2). The cross peaks at these frequencies help to improve spectral resolution by expanding what normally is the ID signal into a second


dimension. Knowledge of the cross-peak coordinates allows the direct determination of 8iso and 8aniso cs cs·

Although little has been reported on the switched angle correlation of the quadrupo-lar interaction [14], there is great potential for assigning quadrupolar couplings based upon isotropic chemical shifts in a similar fashion as described for dipolar couplings.

Much of the potential of two-dimensional dynamic director reorientation has yet to be realized; presumably this is due to some of the experimental challenges that are not easily overcome on commercially available spectrometers and probes. Nevertheless, the averaging obtained using the reoriented director method exceeds any other available method and warrants future exploration and implementation.

4. Spin manipulation based techniques

4.1 Heteronuclear dipolar decoupling Ideally, heteronuclear decoupling completely removes interactions between abun

dant (e.g. 1 H) and dilute (e.g. 13C) spins. The ability to perform efficient heteronuclear decoupling is of prime importance in NMR since the sensitivity of rare nuclei such as 13C and 15N is directly linked to resolution enhancement. The decoupling techniques have different requirements depending on the nature of the sample (e.g. liquids or solids). Liquid crystals have special requirements, and it is not always appropriate simply to adapt existing liquid or solid state decoupling schemes.

In liquid crystalline samples, the carbon spectrum is typically broad and usually not well-resolved due to the presence of the 13C_1 H dipolar and J couplings. Additionally, the dispersion and anisotropy of the proton chemical shift, exacerbated by unaveraged homonuclear dipolar interactions, constitute the main contribution to broadening of the spectra of liquid crystals and other oriented media [19]. The aim of heteronuclear proton decoupling is to average out these interactions, to provide narrow resonances of carbon or other rare nuclei. The most common method of heteronuclear decoupling is on-resonance continuous wave (CW) rf irradiation. In order to increase the bandwidth of the decoupling, the highest technically possible rf power is used (greater than the size of the dipolar couplings and proton offsets). Such brute-force techniques when applied to liquid crystals can lead to temperature gradients inside the sample and cause a perturbation of the order parameter. This can have a significant effect on the spectral resolution [20]. To avoid this problem, low power decoupling ("-' 20 kHz) and a low duty cycles (less than 1 %) are typically used for high-resolution liquid crystal NMR.

Beyond CW decoupling, one approach to liquid crystal decoupling borrows from the decoupling schemes of the liquid state. After the introduction of noise decoupling [21], many attempts were made to modulate the if irradiation in order to perform a more efficient low power heteronuclear decoupling [22,23]. The goal of low power broadband decoupling was partially achieved in the liquid state by heteronuclear decoupling schemes such as MLEV [24-27], WALTZ [28], GARP [29,30] and more recently, by schemes based on adiabatic frequency sweeps [31-36]; (for a review on heteronuclear decoupling in liquids see [30]). However, these techniques that were


developed for isotropic solutions do not perform well in liquid crystals [37] because they do not take into account the presence of proton-proton and proton-carbon dipolar interactions. The liquid-state decoupling schemes are based on the ability to apply broadband 7r-pulse rotations to the protons (independent of the resonance offset and the if amplitude). However, in the presence of proton-proton dipolar couplings, the 7r rotations are no longer perfect and the performance is degraded.

Another approach is to borrow decoupling schemes from the solid-state, where dipolar couplings are explicitly averaged away during the pulse sequence using coherent averaging theory [38]. The two-pulse phase modulation (TPPM) technique introduced by Bennett et al. [39] narrows the lines significantly compared to CW decoupling. Other frequency and phase-modulated schemes are presented in the literature, and are interpreted in terms of a second averaging effect [40,41]. However, decoupling techniques used for solids usually benefit from magic angle spinning (MAS) which averages all anisotropic interactions to the zeroth order. In general, liquid crystalline samples are not spun at the magic angle. Thus the decoupling method for liquid crystals has to consider a static Hamiltonian containing all the anisotropic interactions, or more precisely, their averaged projections along the director. Therefore, all heteronuclear decoupling sequences devised for rotating solids using the reintroduction of proton-proton dipolar couplings for enhancing the self-decoupling effect [42] or using rotor-symmetrized pulse sequences [43,44] are not appropriate.

Decoupling schemes for liquid crystals can also be designed by drawing from both liquid and solid state techniques. The early attempts were based on phase alternated pulses (such as the ALPHA sequence) [37]. Then sequences called COMARO (for composite magic-angle rotation) were introduced based on the extension of Waugh's decoupling theory [45, 46] to coupled multi-spin systems [47]. These sequences exhibited a broadband performance with respect to resonance offset in deuterium decoupling [48].

More recent developments in liquid crystal decoupling pulse design use small phase angle rapid cycling (SPARC) [49] or small phase incremental alternation (SPINAL) [50]. The design of both sequences was inspired by TPPM which represents the current state of the art in solid-state decoupling, and ALPHA which formerly gave the best performance [37].

The SPARC-16 pulse sequence is based on a 16-step phase cycling of the two-step p P TPPM decoupling method for solids. The sequence contains 16 elements where the width of the square pulse P is 180° ± 30° and the phase shift is ±(100-12°).

SPARe -16 = PPPP PPPP PPPP PPPP. (3.6)

A comparison between the efficiency of different decoupling methods is shown in Fig 3.2 as a function of decoupling power. This study shows that the decoupling efficiency is directly related to the amplitude of the decoupling field. While GARP and WALTZ-16 are very efficient for isotropic liquids, they do not decouple the benchmark liquid crystal 4-n-pentyl-4'-cyanobiphenyl (5CB) efficiently. The COMARO and COMARO-2 sequences [48] yield sharp peaks for every carbon, but not as sharp


GARP

WALTZ-16

~~~ ALPHA-~~~

~~ Waugh-3

~ XY-32~~~

CW

~~~ COMAR~~~

COMAR~~~

TPPM~~~

SPARC~~~ ~:o l~O 1.;0 .'e .'0 ~ l~O J.~O l~ O .~ 4~ ~ ... ...

Figure 3.2_ 13C spectra of 4-n-pentyl-4'-cyanobiphenyl (SeB) at 9.4 T and ambient temperature for various pulse sequences. The same decoupling power was used for the spectra in each column: 13 kHz, 19 kHz and 22 kHz for the first, second and third columns respectively. (Reproduced with permission from [49]. Copyright 1998 Academic Press).


as the peaks obtained using TPPM. Of the decoupling methods compared in Fig. 3.2, the SPARC-16 pulse sequence proved to have the best performance.

The SPINAL pulse sequence has a stepwise variation of the phase. If P lO represents a square pulse of nominal flip angle 1650 and phase 100 , then:

Q = PlOP-lOP15P-15P20P-20P15P-15 (3.7) Q = P-lOPlOP-15P15P-20P20P-15P15 (3.8)

SPINAL -16 = QQ (3.9) SPINAL - 32 = QQQQ (3.10) SPINAL - 64 QQQQ QQQQ. (3.11)

The performance of the SPINAL-64 sequence [50] is compared to WALTZ, TPPM and SPARe in Fig 3.3.

The best on-resonance performance is obtained with SPINAL. The better performance of SPINAL over TPPM was explained theoretically using a Fourier decomposition of the irradiation scheme. For TPPM, a single frequency is present due to

WALTZ-16

TPPM

SPARC-16

SPINAL-54

ii i I I I 200 160 120 80 .0 0

ppm

Figure 3.3. 13C spectra of 5CB at 9.4 T and ambient temperature for various pulse sequences. The decoupling power was 18 kHz. (Reproduced with permission from [50]. Copyright 2000 Academic Press).


the simple two-phase alternation. SPINAL contains more frequencies, covering a much wider spectrum. According to Fung et al. [50], this leads to a more broadband saturation of the proton spectrum and to better decoupling.

At present, it seems that there is no clear theoretical framework that satisfactorily describes the current state of the art heteronuclear decoupling pulse sequences used for liquid crystalline samples, and is able to predict unambiguously more efficient and robust pulse sequences. On the other hand, a simple model [51] proved to be able to reproduce the experimental results of TPPM and could constitute a first step towards the development of new sequences. Also, decoupling methods based on both numerical and theoretical methods such as iterative maps potentially could be important in the development of better decoupling sequences for liquid crystals.

4.2 Homonuclear dipolar decoupling Homonuclear decoupling techniques use multipulse sequences to remove the dipole

dipole couplings between abundant spins such as 1 H. These techniques are used for the determination of heteronuclear couplings and chemical shift in samples with homonuclear dipolar couplings. The decoupling efficiency depends on many factors including the coupling system and the applied decoupling sequence. The primary goal of the homonuclear decoupling sequence is to remove the homonuclear dipole coupling terms in the Hamiltonian to the highest possible order while retaining the chemical shift. The efficient homonuclear decoupling with highest scaling factor is crucial for obtaining well-resolved spectra. The chemical shift and all heteronuclear dipolar couplings are scaled by the scaling factor of the decoupling sequence. Many homonuclear decoupling techniques have been developed for the solid-state. Much time and effort has been devoted towards the study and application of these techniques to liquid crystals as well as to the design of new decoupling pulse sequences unique to liquid crystals. In the discussion that follows, only decoupling sequences that have been applied to liquid crystalline studies are presented. In principle, other solid-state decoupling techniques could be used as well.

The first sequence able to remove the homonuclear couplings was introduced by Lee and Goldburg [52]. In their method (also called "CW magic-angle decoupling"), a strong rf field is applied off-resonance so that in the rotating frame the spins experience an effective field at an angle of 54.7° with respect to the Z'-axis. The pulsed analog of Lee-Goldburg decoupling was introduced by Waugh and co-workers [38, 53]. This four pulse sequence, called WAHUHA, became a building block for many other homonuclear decoupling sequences, and several improvements have been proposed in order to provide more efficient decoupling. Some of the improved pulse sequences are: MREV-8 [54,55] which corrects for rf inhomogeneity; BR-24 and BR-52 [56] which remove the second-order average Hamiltonian without reintroducing any of the dipolar terms which vanish for MREV-8; windowless BLEW-48 [57], in which during continuous rf irradiation the phases of 1r /2 pulses are switched; and TREV-8 [58] which involves the use of time-reversal spin echos. Lee-Goldburg decoupling has been improved by switching the frequency of rf irradiation (FSLG) and the phase by 1r


every time a 27T rotation about the effective field is completed [59,60]. The frequency switching also can be accomplished by a series of pulses with well-defined phases [61,62]. Recently, the magic sandwich high order truncation (MSHOT-3) and high order truncation (BHOT-4) sequences [63] were introduced which eliminate higher order (fourth) dipolar coupling terms based on z-rotational decoupling. The scaling factor for these two sequences is the lowest of all the above mentioned decoupling methods, 0.186.

Fung has compared the efficiencies of homonuclear decoupling methods in two studies [61,64]. In the first study, the performances of Lee-Goldburg off-resonance CW irradiation, WAHUHA, MREV-8, BR-52 and BR-24, BLEW-48, and TREV-8 have been compared in monodeuterobenzene dissolved in the liquid crystal ZLI 1242 [64]. For each decoupling method, deuterium spectra were recorded with a IH decoupling power of 14 kHz. The "CW magic-angle decoupling method" proved to be very sensitive to resonance offsets and usually gave asymmetric peaks. For the windowed sequences such as WAHUHA, MREV-8, BR-52, BR-24 and TREV-8, the ratio r =(window width) / (7T /2 pulse width) was 4.0. For WAHUHA and TREV-8, the efficiency decreased when r was reduced, and for BR-24 the opposite was true. BLEW-48 and BR-24 with r = 1 showed the best decoupling performance, and further experiments were done to study the effects of resonance offsets. BLEW-48 was found to be less sensitive to resonance offsets and gave the best performance overall. In the second study, the performances of MREV-8 (the most widely used sequence), BLEW-48 (the best sequence from previous study), and of new pulse sequences such as FSLG-2, MSHOT-3, and BHOT-4 were studied using benzene dissolved in two types of liquid crystals, TNC 1291 and ZLI 1167. For each decoupling method, 13C spectra were recorded with IH decoupling powers of 11 kHz for TNC 1291 and 10 kHz for ZLI 1167. All the decoupling methods except for BHOT-4 produced first-order spectra. On the basis of results obtained for each of the decoupling methods, it was concluded that BLEW-48 gave the best resolution for 13C spectra of benzene dissolved in liquid crystals. BLEW-48 also performed very well when it was included in a two-dimensional proton-detected local-field experiment (for the description of this experiment see next section). However, the long cycle time, moderate scaling factor, and lack of windows can be a drawback for some applications. The scaling factor influences the obtainable resolution, and the lack of windows does not permit the use of BLEW-48 during acquisition.

A new class of pulse sequences has been designed especially for oriented systems. These sequences reduce the dipolar splittings through spin manipulations. In the previous sections, the variable angle spinning technique was shown to reduce the dipolar splittings using spatial averaging. However, these techniques work only for uniaxial systems such as nematics. The goal of the new method is to produce a first order spectrum using multiple pulses, which means that the dipolar couplings are coherently reduced so that the values of the scalar and dipolar couplings are small compared to the chemical shift differences. Some of the new pulse sequences are Metronome-12 (MET-12) [65], Zig-Zag-32 (ZZ-32), and Flip-Flop-16 (FF-16) [66]. Due to introduction of many errors when these pulse sequences are placed during


acquisition, the performance of these pulse sequences was studied in a two-dimensional fashion using the standard COSY pulse sequence, where the pulse train was applied during the tl period [67,68]. The second-order I9p spectrum in W2 was correlated with the I9p_IH dipolar coupled first-order spectrum in WI. The effectiveness ofMREV-8 and FF-16 was demonstrated on 1,3-dichloro-4-fluorobenzene dissolved in ZLI 1167. It was concluded that both sequences are successful in producing first-order spectra in the WI dimension.

5. Multidimensional techniques Both heteronuclear and homonuc1ear decoupling sequences have been used as build

ing blocks in multidimensional experiments used for liquid crystals. This includes very powerful two-dimensional techniques such as separated local field spectroscopy and proton-detected local-field spectroscopy as well as three-dimensional techniques. Many of the liquid crystalline techniques were first developed for solid state NMR.

5.1 Local field spectroscopy

5.1.1 Separated local field spectroscopy. One such technique is separated local field (SLF) spectroscopy [69-72]. As shown in Pig. 3.4, the information about I3C_1 H dipolar couplings is obtained in the tl dimension under I H_I H homonuclear decoupling, and the I3C chemical shift information is obtained in the t2 dimension under IH heteronuclear decoupling. A slice in the WI dimension at a particular I3C resonance in the direct dimension shows a first-order spectrum that contains 2m lines, where m is the number of coupled protons. The structure of this "multiplet" is due to repeated splitting by I3C_1 H couplings. Hence, the 13C_1 H dipolar couplings can be calculated directly from the splitting pattern, taking into consideration the characteristic scaling factor of the homonuc1ear decoupling sequence. The accuracy to which the dipolar couplings can be obtained depends upon the efficiency of the homonuclear decoupling sequence.

In the first experiments performed on oriented systems, the proton-coupled carbon-13 spectrum of p-methoxybenzylidene-p-n-butylaniIine (MBBA) with the director aligned along the magnetic field was obtained with off-resonance CW decoupling [73]. The linewidths that were observed in W2 were broad and the couplings determined from the WI dimension were not very accurate due to the presence of residual strong I H_I H couplings. Better resolution and linewidths can potentially be obtained by simply using increased decoupling power. However, as stated earlier, high decoupling powers can introduce thermal gradients [20] in liquid crystalline samples that lead to a distribution of directors and further broadening of the lines.

5.1.2 RHODIUM. Pung and co-workers pioneered a technique which greatly improved the quality of the carbon spectra without the need for higher decoupling power. Their technique, RHODIUM [76] (Removal of HOmonuclear DIpolar coupling and the Use of off-Magic-angle spinning) is a combination of SLP spectroscopy and


,aoo

~ Broadband BLEW·4a II BLEw.4a l Broad band L 900 ,aoo

13C ~/J~" ~ V V --..::TO

I, 12

~I 2 00

=t=~ 180

r 160 4

~\ 3' 140

:::~ 3 2' i 4' 2

~ 120 ~ .., ... ... 100 .<:

~/ ..

..... 80 lIS

u ...

±~ ~

60 tl

v Ii 40 a

~; =iJc:=!

~ 2 0 Ol

0

4 0 00 Hz

Figure 3.4. The structure of 7CB. The pulse sequence used for the SLF experiment. 2D l3C spectra of 7CB obtained by RHODIUM method. The spectra in the WI dimension are presented on the left side and the spectrum in the W2 dimension is presented on the right side, The spectra were obtained with variable angle spinning at 45,6° and a decoupling power between 20 and 25 kHz, (Reproduced from [74] by permission of the Royal Society of Chemistry).


i 8 ..... .... .... ..<: III

.-t CS 0

i ..<: 0

::? ....

t,

15 ~-~,

20

~ c> 0=0-25

30 ~ 8 ~ 35 <! ~ I>

120 ..0.

125 _0_ 130 ---.... 135

140 .. 150

~o·

I I I I I

Broad band

__ ~A 4'

~ .

£~ '- MJ\ 3'

--~ l A ____ __ ~A l'

1000 500 0 ·500 ·1000

lH_13C local dipolar field (Hz)

Figure 3.5. The structure of 5CB. The pulse sequence used for the PDLF experiment. 2D contour plot of PDLF·VAS spectra of 5CB (left side) and the spectra in WI dimension (right side). The spectra were obtained with variable angle spinning at 48.2° and a decoupling power between 30 and 35 kHz. (Reprinted with permission from [75]. Copyright 1996 American Chemical Society).


variable-angle spinning (VAS). RHODIUM relies on the fact that the dipolar and other anisotropic interactions are scaled by (3 cos2 e - 1) /2 under rapid spinning conditions as discussed earlier [1]. A second-order spectrum can therefore be simplified to firstorder using this technique. The reduction of dipolar couplings allows the local field experiments to be performed under mild rf conditions.

The RHODIUM method has been used in systematic studies of the orientational ordering of many nematic liquid crystals [74,76-87]. In all these experiments, the BLEW-48 decoupling sequence was used with the decoupling power varying between 20 and 28 kHz. The spinning angle, e, for liquid crystals with ~X > 0 varied from 37.9° to 48.6°, and for liquid crystals with ~X < 0 was 66.1 0. Fig. 3.4 shows a typical 13C spectrum obtained using this technique. It can be seen that for 4-n-heptyl-4'-cyanobiphenyl (7CB), all seven peaks in the aliphatic chain and all nine peaks in the mesogenic core are resolved in the W2 dimension. In the WI dimension, first-order spectra with I3C_I H splittings are observed. It can be seen that the methyl group exhibits a quartet and all methylene groups exhibit triplets. Each of the quaternary carbons is split into a triplet by the ortho protons, and each of the protonated carbons exhibits a doublet of doublets from the directly bonded protons and the ortho protons. The couplings can be obtained from line-shape analysis of the first-order spectra obtained in WI, and from these a C-H bond order parameter can be defined.

5.1.3 Proton-detected local-field spectroscopy. A variation of local field spectroscopy called proton-detected local-field (PDLF) spectroscopy, based on work by Weitekamp et al. [88] and Caravatti et al. [89], was first applied towards the simplification of the heteronuclear dipolar spectra of polymers [90] and oriented systems [91], and offered a substantial gain in resolution and sensitivity over SLF spectroscopy. This technique yields high resolution 13C_1 H couplings by detecting the dipolar local field produced by the I3C spins at the site of each proton in the indirect dimension of a two-dimensional experiment (Fig. 3.5). In the presence of multiple-pulse homonuclear decoupling, each proton "sees" the local field of a single I3C spin, effectively isolated due to the low natural abundance (1.1 %) of the I3C nuclei. The ideal spectrum contains a maximum of m doublets for a 13C nucleus coupled to m protons. The splitting of each doublet is proportional to the strength of the heteronuclear I3C_IH coupling. Fig. 3.6 shows simulations of SLF and PDLF spectra for benzene dissolved in nematic liquid crystals and for SCB. The number of lines increases dramatically for the SLF spectrum which causes a decrease in overall sensitivity. A 10-fold increase in sensitivity is observed in the PDLF spectrum over the SLF spectrum. The smaller dipolar couplings add to the linewidth in the SLF experiment and form a broad peak close to zero frequency in the PDLF experiment. The PDLF technique has been used to obtain high resolution spectra of membrane samples [92,93], solutes such as benzene in the nematic phase [91], and membrane constituents such as cholesterol [94].

The PDLF technique has also been combined with variable angle spinning. A comparison between PDLF with and without variable angle spinning [75] was conducted using the well-characterized liquid crystalline compound SCB. It was concluded that for mechanically orientable samples such as nematic liquid crystals, high resolution


A.

SLF

PDLF ill iii I I I i I I

-4.2 -3.2 -2.2 -1.2 -0.2 0.8 1.8 2.8 3.8

kHz

B.

SLF

PDLF

00 I I I I I I I I I

-1.5 -1.0 -0.5 0 0.5 1.0 1.5

kHz

Figure 3.6. Theoretical comparison of the relative performance of SLF and PDLF experiments for (A) benzene and (B) 5CB. (Reprinted with permission from [75]. Copyright 1996 American Chemical Society).

spectra could be obtained under relatively mild rf conditions. Fig. 3.5 shows an example of the spectral simplicity and resolution that can be obtained using the PDLF spectroscopy combined with VAS. The zero frequency peaks in the WI dimension are due to unresolved long-rage couplings.

5.1.4 3D techniques. PDLF spectroscopy also has become a building block for reduced-three-dimensional [95] and three-dimensional [96] experiments. In order to assign long-range dipolar couplings, for example, between two non-bonded I3C

and 1 H nuclei, the I3C_I H dipolar couplings must be correlated with both I3C and 1 H chemical shifts. During the first evolution period, the protons evolve under heteronuclear dipolar couplings with the homonuclear couplings and chemical shifts removed. During the second evolution period, the protons evolve only under the chemical shift interaction with both the homonuclear and the heteronuclear couplings removed. Finally, after cross polarization from 1 H to I3C, 13C is detected in the direct dimension under heteronuclear 1 H decoupling. This technique yields a pairwise dipolar field


spectrum [75]. For each carbon-proton pair in the W3-W2 plane, the WI line contains a single splitting which represents the averaged dipolar coupling between the two nuclei. This experiment allows for the measurement of not only long-range splittings but also of short- and medium-range splittings. The reduced-three-dimensional experiment has the same evolution periods as the three-dimensional experiment, but the first two evolution periods are incremented simultaneously, rather than independently. This simultaneous incrementation produces a single frequency dimension after Fourier transformation [95]. In each 13C cross section, the I3C_I H dipolar splittings are displaced according to the IH chemical shift (Fig. 3.7) .

15 .Oeo .' ... ~ 20 ~ 0000> •

.. ' 25

...... ~ Ii 30 g: ~ .•. S'. '.:.:------»k 35- , ~ i"~"" ~ Ij.f ....

40-

\ .. ~ .Q In

r-i I I I I

" -1000 0 1000 2000

U ....

.. ~. 9 .". .Q 120 u U "

<:' .. ~ M . '

I; . I ... 130 ""'0;-00 ... • . ".;::6:>. , .. ..-... ... :

...... ~ 140 "

41>- . . . . 150 '$' o=- .... . ~

I I I I I I I -1000 0 1000 2000 - 1000 0 1000

C-H dipolar coupling Frequency (Hz)

& IH chemical shift (Hz)

Figure 3.7. Reduced-3D spectrum of 5CB. (Reprinted with permission from [95]. Copyright 1996 American Chemical Society).


6. Conclusions This Chapter has reviewed different techniques that are used to simplify NMR spec

tra in order to obtain structural, orientational, and conformational information about liquid crystals or solutes dissolved in liquid crystals. The ability to simplify the spectra comes from the experimental manipulation of the spatial degrees of freedom through sample spinning, or of the spin degrees of freedom through tailored mUltiple-pulse sequences. The combination of these techniques in multidimensional NMR spectroscopy of liquid crystals provides a very powerful tool for obtaining structural and dynamical information. In the future, these techniques could simplify significantly the study of complex molecules such as peptides, proteins and other species of biological interest, upon orienting of these molecules in liquid crystalline media.

Acknowledgments

We thank Professor Alexander Pines for his constant guidance and encouragement and for his advice regarding this Chapter. The authors would also like to thank Jeffry Urban and Jamie Walls for reading the final version of the manuscript. This work was supported by the Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences Division, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.

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Chapter 4

MULTIPLE QUANTUM NMR SPECTROSCOPY IN ORIENTATIONALLY ORDERED FLUIDS

Leslie D. Field School of Chemistry, University of Sydney, Sydney NSW, Australia

1. NMR in orientationally ordered fluids Most compounds that form a liquid crystalline phase are able to dissolve a small

amount of solute compound without destroying the structure of the mesophase. Since the first reports of using liquid crystals as solvents for NMR spectroscopy, high resolution NMR in liquid crystalline solution has developed into an important area [1]. The amount of solute that can be dissolved in a nematic mesophase depends on the mesophase and the solute, but solute concentrations of 5% w / ware common and concentrations of 10 - 15% w / w or higher are sometimes achievable without disrupting the liquid crystalline properties of the solvent.

The maximum number of lines appearing in the one-quantum NMR spectrum of a compound partially oriented in a nematic phase can be calculated using Eq. (4.1), with sample calculations shown in Table 4.1.

Number of transitions (2N)!

(4.1) (N - 1)!(N + 1)!

where N is the number of interacting spins I = 1/2 in the spin system. Symmetry in the spin system decreases the number of different transitions. For a

completely unsymmetrical six-spin system, one would expect 792 transitions in the IH one-quantum spectrum. A more symmetrical molecule such as benzene, C6H6, exhibits only 72 transitions in the one-quantum proton spectrum.

Table 4.1. Maximum number of transitions in the one-quantum spectrum of a partially oriented compound as a function of the number of spin Yz nuclei.

Number of spins 2 3 4 5 6 7 8 9 10

Transitions 4 15 56 210 792 3003 1.1 x 104 4.4 X 104 1.7 X 105

67 E.E. Burnell and CA. de Lange (eds.), NMR of Ordered Liquids, 67-87. © 2003 Kluwer Academic Publishers.


In higher spin systems, it becomes impossible to resolve or assign individual transitions for an iterative computer analysis. Severe overlap between transitions in unsymmetrical spin systems containing more than about 7 spins makes analysis of onequantum spectra virtually impossible, unless there are elements of symmetry in the spin system to simplify the observed spectrum.

2. Spectral simplification by MQNMR Multiple quantum NMR (MQNMR) [2-7] can considerably reduce the complexity

of the spectrum of a solute dissolved in a liquid crystalline solvent [8]. The main benefit of MQNMR lies in the fact that there are fewer transitions in the high-order mUltiple quantum spectra.

The number of M-quantum transitions in a system of N nuclei (1 = !) without symmetry is given by Eq. (4.2). Representative values are summarised in Table 4.2.

Number of M -quantum transitions (2N)!

(4.2) (N - M)!(N + M)!

in an N -spin system

Table 4.2. The number of transitions in the M -quantum spectra as a function of the number (N) of interacting nuclei (I = t)·

M N, number of interacting spins (I = !) 2 4 8 12 16 20 24

1 4 56 1.1 x 104 2.6 X 106 5.7 X 108 1.3 X 1011 3.1 X 1013

2 28 8.0 x 103 2.0 X 106 4.7 X 108 1.1 X 1011 2.7 X 1013

4 1 1.8 X 103 7.4 X 105 2.3 X 108 6.3 X 1010 1.7 X 1013

8 1 1.1 X 104 1.1 X 107 5.6 X 109 2.3 X 1012

12 1 3.6 x 104 7.7 X 107 7.0 X 1010

16 1 9.1 x 104 3.8 X 108

20 1 2.0 X 105

24 1 N-3 56 5.6 x 102 2.0 X 103 5.0 X 103 9.9 X 103 1.7 X 104

N-2 28 1.2 x 102 2.8 X 102 5.0 X 102 7.8 X 102 1.1 X 103

N-1 4 8 16 24 32 40 48

In the N -quantum spectrum of an N -spin system, there will always be only one transition. The (N - 1 )-quantum spectrum of an N -spin system will contain 2N transitions. For large N, the N - 1 and N - 2 quantum spectra will contain relatively few transitions and will therefore be much less complex and more easily analysed than the one-quantum spectrum. The frequencies of the resonances in the MQNMR spectra are governed by exactly the same parameters as those in conventional single-quantum NMR spectra (chemical shieldings, scalar, dipolar and quadrupolar couplings). In principle, the N, N - 1 and N - 2 quantum spectra contain sufficient transitions to solve the spectrum completely for all of the spectral parameters for most spin systems.

Multiple Quantum NMR Spectroscopy in Orientationally Ordered Fluids 69

In many cases, the analysis of the high-order MQNMR spectra has been used to provide initial spectral parameters that have then been used as a starting point for the more complex analysis of the single-quantum spectrum. The one-quantum spectrum can typically be obtained with higher digital resolution. The resonances are typically sharper, providing spectral parameters that are inherently more accurate.

3. Excitation and detection of multiple quantum coherence

MQNMR spectra are typically obtained by two-dimensional NMR methods. MQ coherences (MQC) are not directly observable. However, MQ coherences generated in a spin system can be converted to observable single quantum coherences and MQNMR spectra can be obtained by using an appropriate 2D NMR experiment.

In the early 1980s, Pines et al. [9, 10] demonstrated elegantly that multiple quantum NMR spectra could be used to simplify the complex NMR spectra that arise from large spin systems in orientationally ordered molecules. In a seminal study using the side-chain deuterated liquid crystaI4-cyano-4'- pentylbiphenyl (5CB-dll), Sinton and Pines [9] obtained excellent 5Q, 6Q and 7Q spectra of the eight-proton spin system and successfully extracted dipolar couplings from the MQ spectra. The intramolecular dipolar couplings were then used to model the structure and conformation of the biphenyl subunit, including an estimate of the torsional angle between the phenyl rings.

H H H H

C5D .. **CN H H H H

5CB-d11

3.1 Non-selective excitation of MQC

The most general method [11] for generating MQ coherence in a homonuclear coupled spin system is by means of two non-selective IT /2 pulses separated by a fixed time interval (T). The value of T should be greater than (or of the same order as) the reciprocal of the dipolar couplings in the spin system and this simple two-pulse sequence generates all orders of MQC in the spin system. Coherences are not excited uniformly and the efficiency with which the various orders ofMQC are excited depends specifically on the couplings and chemical shifts of the nuclei in the spin system and the choice of the interval T. Broadband excitation techniques have been proposed [12-14] where the value of T in the preparation sequence is varied in either pseudo-random or systematic fashion to achieve a more uniform excitation in the multiple-quantum domain. Weitekamp et al. [15) devised an experimental search procedure to optimize


Figure 4.1. Basic three-pulse sequence for generating MQNMR spectra

the delays in the preparation period of the MQ excitation sequence. Wimperis [16] used average Hamiltonian theory to propose broadband excitation sequences to achieve uniform excitation ofMQ coherence over a range of couplings in the spin system. Once generated, MQC must be converted to single-quantum coherence (SQC) for detection and this is usually achieved by a third 1f /2 pulse in the sequence.

If tl, the time for which MQ coherences are permitted to evolve, is incremented systematically in a 2D NMR experiment, 2D transformation against tl and t2 will give a 2D spectrum with single-quantum transitions in hand MQ spectra in fl. MQNMR spectra are then typically extracted as fl projections from the 2D spectrum. If the phase of all three pulses in the simple sequence in Fig. 4.1 is maintained constant, and the phase of the detector is kept constant, then the resulting MQ spectrum contains all orders of coherences. The various orders can be separated by judicious choice of the transmitter offset. MQ coherences evolve at frequencies given by the sums of the chemical shifts of the nuclei contributing to the coherence. Hence, spectra of order M are offset by approximately M.6.w, where 6.w is the average chemical shift of nuclei in the spin system with respect to the transmitter offset. By carefully choosing the transmitter offset, the various orders of MQ coherence can be separated with no overlap (Fig. 4.2). In principle, this permits all orders of coherence to be obtained simultaneously. In practice, the spectral width required to accommodate all MQ spectra without overlap leads to a digital resolution that is typically too low to permit reliable spectral analysis.

A technique that is sometimes used is Time Proportional Phase Incrementation (TPPI). If the phase of the excitation radiation is incremented by b..¢ = b..W.b..tl each time that tl is incremented by b..tl in the 2D NMR experiment, the N-quantum spectrum in the 2D NMR spectrum is offset by a frequency N. b..w from the transmitter. MQ orders are then separated in frequency by b..w in the same fashion as if a frequency offset had been employed.

3.2 Order selective detection A number of methods have been developed to enable MQ spectra of various orders

to be observed selectively. Phase cycling in the basic three-pUlse sequence (Fig. 4.3) permits observation of the spectrum of any particular order of coherence. Orderselective detection of the spectrum of order N is achieved by co adding 4N FIDs at


CI

HX;x-::?' I H (1) H ~ I

H

It :t j ••

40 . t It. . • • f ..

~ t . ·ri I t~ t . t :~ 20

l"~ t: t ~ 10

11 0 . t~ ~I 00 (kHz) •

~ 14i t't 10

t . 'f fill t~t t!1 ;. 20 .. . .. -40 "

'~i i . 1 • . Ii

~ . . j ~ •

4 2 0 -2 -4 12 (kHz)

Figure 4.2. MQNMR spectra of m-chloroiodobenzene (1) oriented in liquid crystalline solution from a 2D NMR experiment with non-selective excitation and detection (Fig. 4.1). The transmitter offset is set to provide dispersion between the MQ spectra.


Tt/2 1[/2 Tt/2

~ l' ~ t1 ~ t2

~ID 4J1 4J2 4J3 4J4

1 Tt/2N Tt/2N 0 Tt/2 2 Tt/N 1[/N 0 Tt

3 3Tt/2N 3Tt/2N 0 3Tt/2

4N 0 0 0 0

Figure 4.3. Phase cycle required for selective detection of MQ spectra.

each value of tl with the phase of the first two pulses incremented from 271'/ 4N to 271' in steps of 271'/ 4N while the receiver phase is incremented by 71'/2 at each step [17-20]. Fig. 4.4 provides an example of the selective detection of the three-quantum spectrum of the four-spin system.

Magnetic field gradients can be utilized for order-selective detection of the spectrum of any desired order N [21]. A magnetic field gradient pulse of amplitude g is applied at the end of the MQ evolution period, prior to the mixing pulse of the MQ pulse sequence (Fig. 4.5). A gradient of amplitude N g immediately after the mixing pulse selects only signals from the N-quantum spectrum [22-26]. Changing the ratio of gl to g2 selects spectra of various coherence orders. Fig. 4.6 shows a selection of MQ spectra for 2,4-dimethylfuran, (2) (an eight-proton spin system) obtained using the sequence in Fig. 4.5 as a function of the gl : g2 ratio [27]. In order to select the desired order of coherence, the magnetic field gradients must be matched accurately in the correct ratio.

Since the field gradient sequence does not rely on perfect signal cancellation over a long cycle of FIDs, spectra obtained using this method are typically cleaner than those obtained using phase cycling for MQ filtration. Moreover, since there is no need to acquire multiple FIDs in an extended phase cycle to achieve MQ filtration, spectra can be obtained more rapidly.

Multiple quantum coherence transfer echoes and their phase properties have also been used to selectively detect desired orders of MQ spectra. An additional (fixed) period d is inserted into the MQ pulse sequence, prior to the mixing pulse (Fig. 4.7). As a consequence, MQ coherences evolve beyond the normal h period, and coherence transfer echoes of the N -quantum coherences reach a maximum in the detection period at a time N x d [22, 28, 29]. The spectra of different orders of coherence can be obtained by selecting the appropriate window in the detection period.


3.3 Order selective detection using three-dimensional NMR

Syvitski et al. [30] have reported a 3D sequence that separates the spectra of different coherence orders. The sequence employs a non-selective 2D excitation sequence (as in Fig. 4.1), but systematically increments the phase of the first two pulses in the excitation sequence by b.¢ for each value of tl' The 3D interferogram is then a function of tl, ¢ and t2' Fourier transformation against these parameters [4, 17,31] produces MQ spectra of various orders, cleanly separated in the "pseudo-frequency" dimension resulting from transformation against ¢. There is considerable time saving in obtaining MQ spectra simultaneously in a single 3D acquisition compared to the time required to obtain individual MQ spectra selectively using phase cycling.

3.4 Selective excitation of MQC Non-selective excitation of MQC excites all possible MQ coherences. However,

most of the spectral intensity remains in the single quantum and lower order multiple quantum spectra [32], but the intensities in the simpler high-order coherences are comparatively weak. There have been a number of approaches to selectively excite only a few orders of coherence and to more effectively focus the available intensity into one order or into a set of selected orders. Order-selective excitation has been achieved by

<l- F2 --l>

Figure 4.4. 3Q IH NMR spectrum of m-chloroiodobenzene (1) orientationally ordered in liquid crystalline solution. The II projection on the left is the MQ spectrum. (Reproduced with permission from [20]).


1[/2 1[/2

~ t ~ Figure 4.5. Pulse sequence required for selective detection of coherence of order M using magnetic field gradients.

the use of phase-cycled sequences of rf pulses [33-38] and composite pulse trains [39]. Pines et al. [33-38] established that it was possible to construct Nk-quantum selective excitation sequences (k = 0, ±1, ±2, .... ) by repeating an appropriate sequence with a phase increment of 27l' / N between repetitions (Fig. 4.8).

Specific pulse trains [Pl ... . Pq] employed in each section of the sequence were found to be effective in generating selected orders of MQC. Sequences involving several thousand cycled pulses have been explored [33-38]. Such extended repeating pulse sequences with a high transmitter duty cycle have a tendency to cause heating and temperature gradients in the sample and this has limited the application of these sequences to liquid crystalline solutions where temperature stability is not critical. The length of the excitation sequence that can be employed is also limited by relaxation times

CH3'Jt--iH

A~ H 0 CH3

(2)

7

6 g2/g1

5

i i

24 16 8 o -8 ·16kHz

Figure 4.6. IH Multiple Quantum spectra of 2,4·dimethylfuran (2) obtained using the sequence in Fig. 4.5 as a function of the 91 : 92 ratio.


Figure 4.7. Pulse sequence for selective detection of coherence of order N by capturing coherence transfer echoes.

in the spin system. The preferred phase-cycled excitation sequences generate MQC and compensate for errors that arise through pulse imperfections. Pulse sequences that are selective for desired orders of coherence in spin systems of given topology and symmetry have also been developed [40,41]. In relatively simple spin systems where resonances of some nuclei are well separated, Rendell and Burnell [42,43] have demonstrated the feasibility of DANTE frequency-selective pulses to selectively excite specific nuclei in the spin system. This approach has been employed to simplify and to aid the analysis of MQ spectra.

3.5 MQ filters to simplify lQ spectra

Avent [44] has demonstrated that the NMR spectra of solutes orientationally ordered in liquid crystalline solvents can be considerably simplified by selecting transitions between spin eigenstates belonging to a given symmetry class. The allowed transitions in the Nand N - 1 quantum spectra of an N -spin system must be between eigenstates belonging to the totally symmetric representation of the spin permutation group. By creating N or N - 1 quantum coherences, and then converting them into observable 1 Q coherences, only the subset of lQ transitions between totally symmetric eigenstates is observed. Carravetta et al. [45] extended this concept to larger spin systems. Iterative analysis of the MQ-filtered single-quantum spectra of spin systems of up to eight spins (1 = ~) was used to determine dipolar coupling constants of molecules orientationally ordered in liquid crystalline solution.

~=O ~=21tIN ~=41tIN ~=2(N-1 )1tIN

G1"'P~ G1"'P~ G1'''P~ -.------.-- ~1".p~

Figure 4.8. General fonn of the pulse sequence employed for order-selective excitation [33-38].


10.0

i --+ - ~ 4· ~ - ..... I f ··t ~ !

• I .. • ; t ; .. ., • • l OT - - · · ., .. •

- . . · · I I .. ., : : .. . . I I ! • · , . t .. t 't ' , :--.. .. ~ • t ~

~ If>'"! t • H------..,-------+-------,----- - -t 10.0 86.0 93.8

60T

10 1.6

F,eqyency /kHz

109.4

70T

117.2

Figure 4.9. 6Q and 7Q spectra of 1,1,1,6,6,6-d6-hexane orientationally ordered in a liquid crystalline solvent. (Reproduced with permission from [52]).

4. Selective deuteration for spectral simplification The proton NMR spectra of solutes dissolved in liquid crystalline solvents can be

simplified by selective, partial deuteration followed by deuterium decoupling [46,47] . However, deuteration is generally synthetically challenging, time consuming and, in many cases, simply impractical or impossible. In addition, deuterium substitution is known to cause small but significant changes to orientation and structure [48].

Drobny [7] reported the MQ spectra of n-hexane-d6 (deuterated in the terminal methyl groups) dissolved in a nematic solvent. Analysis of the high-quantum spectra (6Q and 7Q) gave the dipolar coupling and these were fitted to various models for the alkyl chain conformations.

Pines et al. [49] introduced the concept of combining both selective deuteration and MQNMR to obtain dipolar coupling constants in larger spin systems. In a mixture of randomly deuterated benzenes, it was possible to extract all coupling constants by extracting sub-spectra containing 2, 3, 4 and 5 protons. This principle was extended to the study of randomly deuterated alkanes [49-53] where the analysis of doublequantum filtered COSY spectra, INADEQUATE spectra and mUltiple quantum spectra permitted the measurement of all interproton dipolar couplings. A detailed analysis of dipolar couplings obtained for alkanes from C6 to CIO [53] indicated a trans-gauche energy difference in the range 2.1-2.7 kJ / mol, consistent with other measurements of alkane conformational probabilities in the condensed phase. Analyses of the 6Q and 7Q spectra of hexane-d6. deuterated in the terminal methyl groups, were used to establish the relative signs of the dipolar couplings (Fig. 4.9).


5. Spectral analysis and simulation For a given spin system, the energy levels can be calculated from the dipolar cou

pling constants (Dij ), the scalar coupling constants (Jij ), the chemical shifts (O"i), and the quadrupolar coupling constants (qi). The spin eigenstates can be labeled according to their Zeeman quantum number (M) and irreducible symmetry representation. The frequencies of MQ transitions can be calculated from energy level differences in the spin system. N -quantum transitions occur between spin eigenstates of the same symmetry where the difference in the Zeeman quantum number (b.M) between states is equal to N. A number of groups have devised algorithms that enable the frequencies in experimental MQ spectra to be fitted iteratively to calculated frequencies using modifications of the computer programmes LAOCOON [54,55] or LEQUOR [56].

It should be noted that these approaches calculate the frequencies of the MQ transitions without attempting to predict transition intensities. For the purposes of spectral analysis this is usually adequate. The transition intensities actually observed in an MQNMR experiment are a complex function of the specific pulse sequence used to generate the spectrum and the spin system itself. An estimate of the intensities of MQ transitions can be obtained by a complete simulation of the effect of a given pulse sequence on a given spin system. However, this approach has yet to be used in the analysis of MQNMR spectra.

The transition intensities in MQ spectra derived from the simple three-pulse sequence (Fig. 4.1) have been estimated using a statistical approach where all symmetryallowed transitions were assumed to be excited equally. Averaging over a range of T

values, the integrated intensity per order decreases as the observed order of coherence increases. However, since the number of transitions decreases, the average intensity per transition increases as the observed order of coherence increases [32]. The estimation of transition intensities in MQ spectra has been extended to excitation sequences involving order-selective composite pulse excitation and the calculations provide a reliable means for theoretically optimising the efficiency of the excitation sequence [57].

The analysis of MQNMR spectra has been used frequently to provide good initial estimates of parameters to permit the analysis of 1 Q spectra where these are well resolved. Celebre et al. [58] have automated this procedure and have described a general protocol for sequentially analysing MQ spectra followed by the analysis of the lQ spectrum to yield accurate spectral parameters.

6. Structural studies using MQNMR MQNMR, in principle, provides a means of analysing the spectra of larger spin

systems than is possible by the analysis of single-quantum spectra. The analysis of MQNMR spectra of partially ordered species yields the dipolar coupling constants (Dij ) between each pair of nuclei (i and j) in the spin system. The Dij are inversely


proportional to the cube of the internuclear distance rij and are given by

D h,l Sij ij = - 411"2 r?

~J

(4.3)

where ,H is the proton magnetogyric ratio, and Sij is the order parameter of the internuclear direction between i and j (see for example [59]). The diagonal order parameters Sxx, Syy and Szz can have any value between -0.5 and 1.0, whereas the off-diagonal order parameters Sxy, Syz and Sxz vary between -0.75 and +0.75 [60]. The relative positions of the nuclei in the spin system can be determined provided that sufficient independent Dij values can be measured. Information about the molecular structure, as well as the molecular order matrix, can thus be deduced.

For a system of coupled I = ~ nuclei, the transition frequencies in the mUltiple quantum spectra are determined by the dipolar coupling constants, the scalar coupling constants, and the chemical shifts of the nuclei. In theory for an N -spin system, the spectra of order N - 1 and N - 2 contain sufficient transitions to measure all dipolar coupling constants and chemical shifts. For additional accuracy and confidence, the N - 3 quantum spectrum can be analysed to provide redundancy and more reliable estimates of the Dij .

6.1 Estimating order parameters by MQNMR The total spectral width of any MQ spectrum is dominated by the values of the

dipolar couplings in the spin system. The dipolar couplings in tum are a function of the order parameters and the internuclear distances (Eq. (4.3)). The iterative analysis of most spectra of orientationally ordered molecules typically begins with an approximate geometry in conjunction with a trial and error assessment of the molecular order tensor to provide a visual match to spectral features. Model compounds or molecular modeling can be used to provide a reasonable molecular geometry. It is, however, much more difficult to obtain reasonable estimates of the orientation parameters.

Field et al. [61] have developed a general procedure for estimating the Saupe order matrix for spin systems orientationally ordered in liquid crystal media by iteratively fitting the experimental and simulated spectral widths of the high-order multiple quantum spectra.

For the eight-spin system indene (3), orientationally ordered in a liquid crystalline solution, the observed spectral widths of the 5Q, 6Q and 7Q spectra were measured as 30428 Hz, 23185 Hz and 14940 Hz, respectively (Fig. 4.10).

The proton spin system of indene has Cs symmetry. H7 and Hs are out of the yz-plane, and three independent order parameters (Syy, Szz and Syz) are required to define the molecular orientational order. Approximate atomic coordinates for the protons of indene were obtained from a molecular modeling programme [62] and the widths of simulated MQ spectra were obtained by varying the order parameters Syy and Szz over all possible values (with Syz fixed arbitrarily at 0). There is a continuum of combinations of Syy and Szz that can give a 6Q spectrum whose width is 23185 Hz; likewise there is a continuum of combinations of Syy and Szz that can give a 7Q


H4

H3 Z

(3) H6 J-v H2

H1 X

7Q

14940~: I L~I • ;111 , w.--i i i i i i

3S 30 25 20 15 10 101:

60

23185 Hz

~ ~

i i 3S 5 IOIz

5Q

30428 Hz ,

:~ ~ , ,

30 25 10 S 0 KIlt

Figure 4.10. 5Q, 6Q and 7Q Spectra of indene (2) partially oriented in liquid crystalline solution. (Reproduced with permission from [61]).

spectrum whose width is 14940 Hz. However, there are very few combinations of Byy

and Szz that can simultaneously fit the observed spectral widths of both the 6Q and 7Q spectra. This immediately gives reasonable estimates of the possible order parameters that can provide a match to the experimental results. A computer algorithm was developed to search for estimates of the orientation parameters based on the spectral widths of the MQ spectra.


7. Structural studies by lH MQNMR Until now there are few published studies where the geometry of a spin system has

been established using MQNMR. These studies include the following: Bromobenzene, C6H5Br, (4), is a five-spin, rigid aromatic system. The 4Q spec

trum was obtained and analyzed [58] to provide spectral parameters that were used as starting parameters for an automated analysis of the 1 Q spectrum.

H H

H-Q-Br (4)

H H

Phenylacetylene, CSH6, (5), is a six-spin, rigid aromatic system. The 5Q spectrum was obtained and analysed [58] to provide spectral parameters that were used as starting parameters for an automated analysis ofthe lQ spectrum.

Naphthoquinone, ClOH6, (6), is a six-proton, rigid aromatic spin system. The 5Q spectrum was obtained and analysed [58] to provide spectral parameters that were used as starting parameters for an automated analysis of the 1 Q spectrum.

H o H H

(6)

H H

H o


1,3-Dichloro-2-ethenylbenzene, CgH6Ch, (7), is a six-spin system in an aromatic molecule with an unsaturated two-carbon side chain where there is restricted rotation between the aryl and the olefinic groups. The high-order MQ spectra were obtained and analysed [42] to provide spectral parameters that were used as starting parameters for the analysis ofthe lQ spectrum. The structure was refined with a model incorporating torsional motion between the planes of the aromatic and olefinic groups. The best solution was found for a dihedral angle of approximately 45°.

KH H~H (7)

H CI

I-Bromonaphthalene, ClOH7Br, (8), is a seven-proton, rigid aromatic spin system. The 5Q and 6Q spectra were obtained and analysed [20] simultaneously to give all dipolar couplings, and these were used directly to estimate the molecular geometry.

H Br

HWI~H (8) H ~ ...:::::: H

H H

Benzyl bromide, C7 H 7 Br, (9), is a seven-spin system in an aromatic molecule with a one-carbon side chain. There is a low barrier of rotation of the CH2Br group with respect to the five-spin, rigid aromatic ring. The 5Q and 6Q spectra [63] were analysed to provide spectral parameters that were used as starting parameters for an automated anal ysis of the 1 Q spectrum.

5=<:vH

H~Br (9)

H H


3-Phenyl-l-propyne, CgHs, (10), is an eight-spin system in an aromatic molecule with a three-carbon side chain. The 4Q, SQ, 6Q and 7Q spectra [63] were obtained and analysed to provide spectral parameters that were used as starting parameters for an automated analysis of the 1 Q spectrum.

H~ ~JtH (10) C~

H H ~C '\

H

4-Cyano-4' -pentylbiphenyl ClsHsDllN, (11), (SCB-dll ), is an eight-proton aromatic spin system. The SQ, 6Q and 7Q spectra were obtained and analysed to provide dipolar couplings that were used to estimate the molecular geometry [9,10]. The structure of the molecule was modeled to give an estimate of the torsional angle (30 ± 2°).

eN (11 )

H H H

Indene, CgHs, (3), is an eight-spin system comprising an aromatic ring with a fused five-membered ring. The spin system has Cs symmetry and is essentially rigid. MQ coherence was generated non-selectively [61] and MQ spectra were detected selectively using magnetic field gradients. The 6Q, 7Q and 8Q spectra were analysed to provide dipolar couplings that were used to estimate the molecular geometry.

H H

:~H (3) H H H


Biphenyl, C12HlO, (12), is a ten-spin system with D2 symmetry. The eight-quantum IH NMR spectrum was analysed [64] to provide all dipolar coupling constants and chemical shifts, which were then used as starting parameters for the analysis of the 1 Q spectrum. The molecular geometry was obtained using vibrationally corrected dipolar couplings. It was concluded that the molecule has an equilibrium dihedral angle of approximately 37° .

H H H H

H~H (12)

H H H H

Butane, C4HlO, (13), is a flexible alkane with a low barrier to rotation about each of the C-C bonds. This is a ten-proton spin system. The analysis [65] of the 7Q and 8Q spectra provided spectral parameters that were used as starting parameters for the analysis of the lQ spectrum. The analysis of the lQ spectrum provided dipolar couplings similar to those derived from the MQ spectra, but of higher accuracy. The structure of the molecule was modeled to give an estimate of the trans-gauche energy difference in the range 2.1-3.0 kJ / mol.

p-Xylene, CSHlO, (14), m-Xylene, CSHlO, (15), a-Xylene, CSHlO' (16), are all ten-spin systems in aromatic molecules with two one-carbon side chains. The highorder MQ spectra were analysed [66, 67] to provide starting parameters for the analysis of the 1 Q spectrum. There is a low barrier of rotation of the CH3 groups with respect to the four-spin, rigid aromatic spin system.

H H

CH,-i ~CH3 H H (14)

CH3 H

H*C~ H H (15)

H CH3

Hi ~CH3 H H (16)


8. Heteronuclear MQNMR Heteronuclear Multiple Quantum Coherence, (HMQC) (coherence involving spins

of different type (e.g. IH /2H, IH /13C, IH /15N) has been used to study molecules in liquid crystal solvents. Minoretti et al. [68] demonstrated that there is a significant sensitivity advantage when the MQ coherences of a relatively insensitive nucleus (such as 2H) can be detected indirectly by observing 1 H nuclei coupled to the heteronuclear spin system.

Castiglione et al. [69] have used HMQC as an aid to the automatic analysis of spectra from molecules orientationally ordered in liquid crystalline solution.

Sandstrom et al. [70] have used heteronuclear MQ coherence to correlate shifts in 13C and 2H spectra of fully deuterated 4-cyano-4'-pentylbiphenyl (5CB) orientationally ordered in a magnetic field. The carbon-deuterium correlation permitted the assignment of the 2H spectrum. Deuterium quadrupole splittings were then used to estimate the molecular order matrix.

9. Other applications of MQNMR in liquid crystalline solvents

9.1 Diffusion in anisotropic media using MQNMR The normal NMR spin-echo measurement of diffusion can be extended by using

MQNMR. The use of MQNMR relies on the fact that N -quantum coherences dephase N times more quickly than do single quantum coherences in an inhomogeneous magnetic field. Hence, higher quantum coherences are more sensitive to diffusion. In typical spin-echo experiments molecules are position-labeled in a magnetic field gradient after an initial rf pulse. Diffusion is then monitored by measuring the amplitude of the spin echo following a second pulse.

Martin et al. [25] measured the diffusion of methylene chloride (CH2Ch) in liquid crystalline solvents by following the decay of the double quantum coherence. Zax et al. [26] measured the diffusion of benzene in a nematic liquid crystal by using a pulse sequence to non-selectively excite MQC, allowed this MQC to evolve in the presence of a magnetic field gradient, and then selectively detected a coherence transfer echo. The echo decay for each order of coherence was fitted to a modified Stejskal-Tanner equation [71] verifying that the rate of decay is proportional to N2. They concluded that the N -quantum coherences are much more sensitive than are single-quantum methods in measuring diffusion. This makes MQ diffusion measurements attractive in viscous media (or for the measurement of relatively slow diffusion rates) where diffusion measurements by single-quantum methods would require prohibitively large field gradients.

9.2 MQ relaxation in anisotropic media

Molecular reorientation processes in anisotropic media, such as liquid crystals, are described by the spectral densities of the motion of a molecule, Jq(w), and are


usually estimated from a range of relaxation measurements. In quadrupolar systems the relaxation of zero, double, triple, and higher-order coherences has been used to estimate the spectral densities of motion for small deuterated organic molecules (including CDCl3 [72], D-C=C-C=N [73], CD2Cl2 [74-76], and CD3-C=N [77-79]) dissolved in liquid crystals. The relaxation of the 2Q and 3Q resonances of 23Na (I = 3/2) and the 4Q and 5Q resonances of 170 (1 = 5/2) has also been measured in lyotropic liquid crystals [80].

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NMR Spectroscopy Using Liquid Crystal Solvents, Pergamon Press, 1975; (b) J. W. Emsley (Ed), Nuclear Magnetic Resonance of Liquid Crystals, D. Reidel Publishing Co., 1985.

[2] Ernst, R. R., and Wokaun, A. Principles of Magnetic Resonance in One and Two Dirnensions. Oxford University Press, London, 1980.

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[4] Weitekamp, D. P. (1983), Adv. Magn. Reson., 11:111.

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[6] Slichter, C.P. Principles of Magnetic Resonance. Springer-Verlag, BerlinlNew York, yd edition, 1990.

[7] Drobny, G. P. (1985), Ann. Rev. Phys. Chern., 36:451.

[8] Field, L. D. ,Encyclopedia of NMR, eds. D. M. Grant and R. K. Harris, page 3172. Wiley, Chichester, 1996.

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[l3] Braunschweiler, L., Bodenhausen, G., and Ernst, R. R. (1983), Mol. Phys., 48:535.

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[52] Gochin, M., Pines, A., Rosen, M. E., Rucker, S. P." and Schmidt, C. (1990), Mol. Phys., 69:671.

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[57] Warren, W. S., Murdoch, J. B., and Pines, A. (1984),1. Magn. Res., 60:236.

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[59] Diehl, P., and Khetrapal, C. L. NMR, Basic Principles and Progress, P. Diehl, E. Fluck and R. Kosfeld eds., volume 1, page 1. Springer-Verlag, 1969.

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[61] Field, L. D., Pierens, G. K., and Ramadan, S. (2002), J. Magn. Res., 156:64.

[62] HyperChem TM Realease 4.5 for SGI, Molecular Modeling System, Hypercube, Inc.

[63] Castiglione, E, Celebre, G., De Luca, G., and Longeri, M. (2000), J. Magn. Res., 142:216.

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[67] Syvitsky, R. T., and Burnell, E. E. (2000),1. Magn. Res., 144:58.

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[69] Castiglione, E, Celebre, G., De Luca, G., and Longeri, M. (2001), Liquid Crystals, 28: 1403.

[70] Sandstrom, D., and Zimmerman, H. (2000), J. Phys. Chem. B., 104:1490.

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[74] Poupko, R., Void, R. R., and Void, R. L. (1979), J. Magn. Res., 34:67.

Multiple Quantum NMR Spectroscopy in Orientationally Ordered Fluids

[75] Bodenhausen, G., Void, R R, and Void, R L. (1980), J. Magn. Res., 37:93.

[76] Void, R R., Void, R L., Poupko, R., and Bodenhausen, G. (1980), J. Magn. Res., 38:141.

[77] Jaffe, D., Void, R R, and Void, R L. (1982), J. Magn. Res., 46:475. [78] Jaffe, D., Void, R R, and Void, R L. (1982), J. Magn. Res., 46:496.

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87

Chapter 5

SPECTRAL ANALYSIS OF ORIENTATIONALLY ORDERED MOLECULES

Raymond T. Syvitski College of Pharmacy, Dalhousie University, Halifax, NS, Canada

1. Introduction The NMR spectra of solute molecules dissolved and orientation ally ordered in

liquid-crystalline solvents can provide a wealth of valuable information; directional properties such as anisotropies in indirect spin-spin coupling constants and in chemical shifts (which can be used for studies of nuclear shielding tensors) can be determined [1-3]. Other pertinent information that can be extracted are the magnitudes and absolute signs of the dipolar, quadrupolar and indirect spin-spin coupling constants [1-3].

The dipolar couplings have a strong dependence on internuclear distances and, for flexible molecules, are influenced by the probability of each conformer. Unlike microwave or electron diffraction, proton NMR spectroscopy can be used to study molecules in the condensed phase and is one of the few techniques available for the accurate determination of relative proton positions for molecules and rotational potential barriers for flexible molecules (see Chapters 13 and 14).

The average second rank orientational order parameters that can be extracted from dipolar or quadrupolar couplings provide information about the anisotropic intermolecular forces that are responsible for orientational ordering. Thus, order parameters can be used to examine statistical theories of liquid crystals (see Part III). Liquid crystal molecules tend to be very flexible and exist in a large number of conformations. For a theory or model to be successful, the probability of each conformer must be properly considered. Often it is easier and more informative to dissolve small solutes which probe these anisotropic forces (see Chapters 10, 14 and 15). Solutes can be chosen so as to emphasize specific anisotropic interactions. For example, there is some debate as to the relative importance of long-range electrostatic interactions and short-range repulsive forces (which are related to the size and shape of the molecule) for orientational ordering in liquid crystals. There is also some debate over which long-range interactions are most important. Thus, molecules of similar size and shape have been utilized to distinguish between steric and electronic effects on the intermolecular potential. The anisotropic short-range interactions are similar for similarly substituted

89 E.E. Burnell and c.A. de Lange (eds.), NMR afOrdered Liquids, 89-104. © 2003 Kluwer Academic Publishers.


molecules, but the long-range interactions (such as those due to the multi pole moments and dispersion forces) are different. By comparing the order parameters among solutes, effects of long-range interactions can be examined (see Chapter 10, 14 and 15).

It is possible to determine very detailed and precise molecular information from the NMR spectra of orientationally ordered molecules. However, herein lies the problem: in order to determine accurate structures and / or rotational potential barriers, or to examine statistical theories of anisotropic intermolecular potentials, we require precise and accurate dipolar couplings; further, for statistical theories we require couplings for a number of solutes under identical conditions [4-6]. Typically, solutes are dissolved in different sample tubes and results are scaled to account for variation in the solvent orientational order that results from different sample conditions. However, the scaling of order parameters is fraught with uncertainty (see Chapter 10). Thus, in some cases many solutes are co-dissolved in the same sample tube, which complicates the NMR spectrum and analysis [7,8].

It is possible to obtain order parameters from analysis of deuterium NMR spectra. Deuterium NMR has the advantage of producing relatively simple spectra: typically a doublet for every deuteron, unless there are strongly dipolar coupled protons within a few angstroms of the deuteron. Unfortunately, the inherently broad lines and reSUlting inaccurate frequencies inhibit the use of deuterated molecules for accurate measurements of quadrupolar and dipolar couplings. Ultimately, we are required to analyze high-resolution proton NMR spectra. If we are examining statistical theories, it may be the proton NMR spectrum of a number of solutes co-dissolved in the same sample tube. Unfortunately such spectra, that by necessity are very strongly coupled, tend to be exceedingly complex. Moreover, in the case of more than one solute, we generally cannot distinguish which peaks are associated with which solute (see the following sections). Even for a single simple solute, analysis may be a formidable task, and it requires a great deal of know-how and experience to extract experimental spectral parameters from the analysis of one-dimensional high-resolution spectra. In practice, anisotropic spectra can be analyzed only if a good set of starting parameters is known. For example, the spectrum of orientationally ordered 1,3-dichloro-2ethenylbenzene could not be analyzed using conventional methods. NMR "tricks" were required to obtain estimates of some dipolar couplings which were then used to predict a good set of starting parameters [9]. Therefore, when analyzing the spectrum of a single solute containing many spins, or of many co-dissolved solutes (some of which themselves may have very complicated NMR spectra), we require novel "tools" and strategies to overcome inherent difficulties.

2. Tools of the trade There are a number of "tools" or aids that have proven to be invaluable when ana

lyzing spectra of orientationally ordered molecules. These tools can be, for example, NMR pulse sequences that "simplify" spectra, computer programmes or graphical user interfaces that aid in the analysis of the spectra, or theories / models that are

Spectral Analysis of Orientationally Ordered Molecules 91

used to predict initial starting parameters. Nevertheless, the minimum essential "tool" is a computer programme that can take as input chemical shifts, indirect spin-spin couplings, direct spin-spin couplings, and if necessary quadrupolar couplings. The programme must be able to calculate a set of transitions that can be assigned to experimental frequencies; the transitions are calculated from the appropriate selection rules for 1D high-resolution or multiple quantum (see other Chapters in Part I) spectra and from the spin Hamiltonian

'H= - '"" vT z + '"" '"" [(J" + 2D· ·)1· zl· z ~ t t, ~ ~ tJ tJ t, J,

i i j>i

+ ~(J .. - D· ·)(1· +1· _ + I· _I· +)] 2 tJ tJ t, J, t, J,

(5.1)

where Iz, 1+ and 1_ are spin operators, Vi is the resonance frequency of nucleus i, and Jij and Dij are the indirect and direct coupling constants between nuclei i and j in the same molecule. Using a least-squares regression algorithm, the programme must be capable of iterating spectral parameters by minimizing the square of the differences between the calculated transitions and the experimental frequencies assigned to these transitions. Transitions are repeatedly assigned and I or unassigned (either by manually typing in the frequencies and assignments, or by way of a graphical user interface) and parameters adjusted until a reasonable fit to the experimental spectrum is obtained; there are basic programmes, such as LEQUOR [2], DANSOM [10] and DAISY [11], available for this purpose. More sophisticated programmes can "automatically" analyze spectra. This is similar to the manual method of analysis except the computer automatically assigns I unassigns experimental frequencies to calculated transitions and performs the iteration until a satisfactory fit is obtained. This automatic analysis has been demonstrated for single, relatively simple solutes dissolved in a liquid crystal [12-14], but not for multiple co-dissolved solutes. Recently these programmes have shown promise since they are capable of distinguishing artifacts in MQ spectra [15] (Fig. 5.1).

A number of research groups [12-15, 17, 18] have taken these programmes and made other very useful modifications that add tremendous flexibility to the set of "tools." Typically, Dij are independently adjusted within the least-squares routine. However, since dipolar couplings are related to structural and order parameters (for rigid molecules, Eq. (5.2)), the set of independent adjustable parameters can be considerably reduced. For example, the molecule o-xylene has ten independent Dij parameters which require adjustment, but only two independent order parameters. For essentially inflexible molecules, Dij can be expressed (in SI units) as

D .. - _ /1-on/Yi'Yj '"" S (cos OCt cos 0/3) tJ - 8 2 L..J Ct/3 3

7r r·· Ct/3 tJ

(5.2)

where the angular brackets indicate a statistical average over all intramolecular motions, O! and f3 are the molecular fixed x, y, z axes, Tij is the internuclear distance


between nuclei i and j, and (}o: and (}(3 are the angles between the internuclear vector and the molecular 0: and f3 axes. 80:(3 is equal to !(3 cos (}o:,z cos (}(3,Z - 80:(3), where the angle brackets indicate a statistical average over all orientations of the molecule, and (}o:,z and (){3,Z are the angles between the magnetic field direction Z and molecular 0: and f3 axes. For flexible molecules the dependence of Dij couplings on geometry is more complex, since the conformational equilibrium must be considered. Direct couplings between nuclei belonging to the same rigid subunit can be calculated in terms of an 80:{3 for that rigid fragment, further reducing the total number of independent parameters.

With the least-squares routine, 80:(3, structural parameters (either x, y, z coordinates [15,17,18] or the more intuitive bond angles and lengths [4]), and / or the Dij can be adjusted independently. The Dij are calculated from 80:(3 and structural parameters. If a Dij is to be adjusted independently, it is not calculated but allowed to vary freely. Thus, the dependence of the Dij on 80:(3 is removed; this is useful for molecules that have internal rotations, where the potential barrier is uncertain, or if specific structural parameters are not well known. Derivatives of the line positions with respect to the Dij are calculated analytically. Line positions are complicated functions of 80:(3 and structural parameters. Thus, their derivatives are not obtained analytically, but are calculated using finite difference and structural data from other studies. It should be noted that line positions are very sensitive to minor changes in structural

a

c

..

b

d

. .

Hz

Figure 5.1. The 300 MHz IH spectra of benzyl bromide dissolved in the nematic liquid crystal ZLI 1132 [16]: (a) 6Q refocused; (b) 6Q; (c) 5Q refocused; and (d) 5Q. The real lines are marked with an * in (a), (b) and (c) to distinguish them from artifact lines. In (d) only lines assigned in the automatic iteration process are marked with an *. (Reprinted from [15] with permission from Academic Press).

Spectral Analysis ofOrientationally Ordered Molecules 93

parameters. Reasonably good estimates of proton positions are required. These leastsquares programmes are most useful for MQ spectra where small changes in spectral parameters do not affect the spectrum to a large degree, unlike the situation for 1 Q high-resolution spectra for which slight changes can be very significant.

In order to begin the iteration process, a reasonably good set of input spectral parameters is required. Another tool at our disposal is the ability to predict relatively precise spectral parameters. Typically, the isotropic chemical shifts and indirect spinspin couplings are accurate enough for initial estimates. However, the dipolar couplings can be tricky. There now exists a vast collection of data on various solutes dissolved in numerous liquid crystals, and it is possible to use calculated dipolar couplings based on a "best guess" of molecular order parameters from these previous studies. This does not always prove to be effective, especially if the molecule or liquid crystal has never been utilized previously. Models and / or theories can be used to predict order parameters of an arbitrary molecule (see Part III), but not necessarily an arbitrary liquid crystal. On the other hand, one liquid crystal mixture, 55wt% Merck ZLI 1132 in EBBA (N-(p-ethoxybenzylidene)-p'-n-butylaniline), has been well studied and the models that predict orientational ordering in this liquid crystal are very good (see Fig. 10.10 in Chapter 10 and in [19]). Typically, estimates of order parameters from these models have been accurate enough to begin the iteration process.

It is possible that the particular molecule of interest has not been studied in the liquid crystal being investigated. In this case theories or models may not be useful. One can now utilize a variety of NMR "tools." If the synthetic resources are available, estimates of spectral parameters can be determined from nuclear quadrupolar couplings from deuterium spectra [20] and in favourable cases, nuclear dipolar couplings from the proton spectra of randomly or specifically deuterated [21-26] or 13C labeled [12, 13] molecules. However, such isotopically labeled compounds are not readily available and the synthesis of a wide variety of such molecules is very time consuming and can be problematic [27]. Using natural abundance 13C it is possible to estimate directly the magnitude of large 1 H_l H dipolar couplings with an inverse detect 13C_ IH Heteronuclear Single Quantum Correlation (Fig. 5.2) or Heteronuclear Multiple Quantum Correlation experiment [13]. However, this is a 2D experiment that uses the insensitive 13C nuclei for detection, and due to the inherently broad lines, small couplings may be poorly defined; this may pose a problem when starting analysis of the ID high-resolution proton spectrum.

Homonuclear two- [28-30] or three- [31] dimensional multiple-quantum (MQ) NMR spectroscopy (Chapter 4) can be quite useful since it is possible to obtain reasonably accurate couplings and chemical shifts from spectra that are "simplified" by having considerably fewer lines than the one-dimensional lQ high-resolution spectrum [9,17,18,32-34]. Additionally, for samples that contain multiple solutes MQ NMR can be used to separate solutes. For example, in a sample containing the two solutes a-xylene and a-chlorotoluene, only a-xylene can contribute to MQ spectra of order higher than 7Q.

The use of MQ spectra is not entirely straightforward; the intensities depend on both spectral and experimental parameters in such a complicated manner that they are


not normally used in the spectral analysis. In addition, due to the sparseness of lines care must be taken so that an incorrect and therefore meaningless "fit" is not obtained. Nevertheless, the analysis tends to be far easier than for the normal 1 Q high-resolution spectra, and often only minor adjustments to parameters determined from MQ spectra are required in order to obtain a "fit" to the lQ high-resolution spectrum (see [9,17, 18,34]).

Sometimes even MQ spectra are too complicated to analyze. To further simplify the spectra the first two pulses of the basic three-pulse sequence (see Chapter 4) can be made frequency-selective which in favourable cases causes some MQ transitions to be suppressed. This has been demonstrated for 1,3-dichloro-2-ethenylbenzene [9]. The lQ high-resolution spectrum (Fig. 5.3) has two regions that are well isolated from the main portion of the spectrum and contain transitions only for the aromatic protons. By exciting these regions, the 3Q spectrum was greatly simplified and contained information for the couplings between aromatic and vinyl protons only (Fig. 5.4); the strong couplings between the aromatic protons were suppressed. From the analysis of the 3Q spectrum the 1 Q high-resolution spectrum was easily fit.

A 3D multiple-quantum NMR technique has significant advantages over the 2D technique: all MQ orders are acquired simultaneously and under the exact same conditions with the same digital resolution, and in the same amount of time that is required to collect a single MQ order with the 2D technique. The 2D technique cancels out unwanted MQ orders by co-adding spectra that have been acquired with a constant

Hz -~l

DlSt,D.s J,S + 2D" I

'000 I I III~ t I BC2

I , 'I I

6000 J2l + 2023

8000

F6 I 10000 Fl

Br-Cr-F5

I 12000

,J IrS+201l

II H3-Cr-Br I uooo

Ii ~ (iC I

16000 II II 1,1

18000 J,,+2D tJ

5000 0 -5000 Hz

F2

Figure 5.2. The 500 MHz HSQC spectrum of 1.I-diftuoro-l.2-dibromoethane orientationally ordered in the nematic liquid crystal ZLI 1132. The frequency separations used to determine the dipolar couplings are marked along with the two chemical shift positions of the 13C'S. (Reprinted from [13] with permission from Elsevier Scientific).


H(4)

H(l)

4000 3000 2000 1000 frequency/Hz

Figure 5.3. The 400 MHz 1 Q high-resolution NMR spectrum of l,3-dichloro-2-ethenylbenzene partially oriented in EBBA. The outermost multiplets on either side of the spectrum arise from the three protons H(l), H(2) and H(6) attached to the aromatic ring. (Reprinted from [9] with permission from Academic Press).

phase increment of the first two pulses with respect to the third pulse. For the 3D version, instead of co-adding spectra, each acquisition is stored in a 3D interferogram where the acquired signal is a function oftt. the phase increment, and the acquisition time t2. After the 3D Fourier transform, the typical 2D MQ-NMR spectra occur at "pseudo frequencies" [31]. As a further modification it is possible to combine the frequency-selective and 3D techniques. This could be useful as a separation technique if there are multiple solutes with one solute having a set of transitions that are isolated from those of other solutes in the spectrum.

For samples that contain more than one solute, the lQ high-resolution spectra typically contain a forest of peaks. Distinguishing which peaks come from which solute, especially when there are overlapping frequencies, can be very difficult. In certain cases (see below), after the spectrum from one solute has been analyzed, the calculated frequencies and intensities are accurate enough to use the calculated spectrum to "subtract off" the experimental one, leaving only frequencies from the other solutes visible [8].


c

b

t a 3Q

o 10000 20000 30000 Frequency/Hz.

Figure 5.4. (a) The 400 MHz MQ NMR spectrum of 1 ,3-dichloro-2-ethenylbenzene partially oriented in EBBA. (b) The 400 MHz selective excitation MQ NMR spectrum acquired using a frequency-selective MQ pulse sequence. (c) An expanded view of the 3Q region noted in (b). (Reprinted from [9] with pennission from Academic Press).

3. Some examples of putting it together As a simple example of mUltiple solutes, the lQ high-resolution spectrum of

propyne, acetonitrile and 1,3,5-trichlorobenzene co-dissolved in the same sample tube [35] is shown in Fig. 5.5. The spectrum of 1,3,5-trichlorobenzene is a triplet with a splitting of 13DHHI, and thus DHH could be determined without the aid of a leastsquares fitting routine. For the spectrum of acetonitrile, there is a triplet with a splitting of 13DHHI and two sets of doublets centred around each of the lines of the triplet which are due to the two H - -13C couplings. The splittings are 12D H13e + J H13e I, and were required in order to determine the absolute sign of DHH. The spectrum of


*

*

-5000 o 5000 10'

Frequency

Figure 5.5. The experimental high-resolution spectrum (top) of propyne, acetonitrile and 1,3,5-trichlorobenzene dissolved in ZLI 1132. The main triplet from acetonitrile has been truncated. The calculated spectra of acetonitrile and propyne are in the middle and on the bottom. The two sets of doublets from the coupling to the two 13C nuclei are visible in the experimental and calculated (indicated with 0) spectra of acetonitrile and are required in order to determine the absolute sign of D H H. Resonances marked with * are from 1,3,5-trichlorobenzene. Resonances marked with + and the many low-intensity resonances around the centre of the spectrum are from unknown impurities. In the experimental spectrum, the rOlling base line which is approximately 50 kHz wide is from the liquid crystal molecules.

propyne is slightly more complex. Since acetonitrile and propyne are molecules of similar size and shape, the initial guess of the methyl D H H for propyne was taken directly from the DHH of acetonitrile, and the DHH between the methyl and single proton was determined from the structure. Propyne is a four-spin system and could not be easily analyzed by simply examining the spectrum, so the D H H 's were determined using the least-squares routine. However, the spectrum is quite trivial to analyze and only the proton-proton dipolar couplings were required. The absolute signs of the DHH'S were found by setting the initial value of the J coupling to the isotropic value -3.551 Hz [36].

A much more difficult and involved example is that of the multiple co-dissolved solutes o-xylene, o-chlorotoluene, o-dichlorobenzene and 1,3,5-trichlorobenzene (TCB) in ZLI 1132 [8] (Fig. 5.6). The spectra of the individual molecules o-xylene and o-chlorotoluene are complex. Even though initial estimates for spectral parameters


...u.w~~oiIIIIU"IIIIIJ~IilIIiI ..... (~Cal~·~MJ .. ..1.,'lL .... j.JIt~JjJ~"' c

A - c I " \ .. L "a.IJjJ.1~11'~ D

.-.J1w..II"-!"I.LAIoIhL"t'"!!l ...... I. ...... ,,,L ..... L_~ .... l ..... ,j_jUI'--...lJIlhlLl .. J_"_(JC:~(Calc.), ,.. \I. I. l...lwL'1! llliJLI .. uJJ E

D-E

acC)

I (calc.) G __ ~~U-~-L~~__________ ~ Cl ____ ~ ________ ll_-J~~~ __

-5000 o Frequency

F-G d P P

I I 5000

* I'I II

I I

-3000 -2000 -1000 o Frequency

Figure 5.6. Strategy for spectral analysis: full spectra are displayed on the left and expansions on the right. A: experimentallQ spectrum. B: predicted a-xylene spectrum from the parameters determined by analysis of the 8Q spectrum (Fig. S.7B). C: calculated from the fit to the IQ high-resolution spectrum of a-xylene. Note that there are only minor differences between spectra B and C. D: difference spectrum between A and C. The negative residuals in D are due to slight differences between line shapes of calculated and experimental spectra. E: calculated spectrum of O-chlorotoluene. F: difference spectrum between D and E. G: calculated a-dichlorobenzene spectrum. H: difference spectrum between F and G. Note that, when calculated spectra are subtracted from experimental ones, resonances from the other molecules are readily visible. Resonances marked with * are from TCB. Resonances indicated with * are from impurities, and the resonance indicated with <> is from the partially protonated acetone used for field-frequency lock. The calculated 1 :2: I triplet of TCB is not displayed. (Reprinted from [8] with permission from Academic Press).


were obtained from previous studies, they were not close enough to begin analysis of the 1 Q high-resolution spectrum, and frequencies from anyone solute could not be readily identified. In this case, the proton 8Q NMR spectrum was acquired, not only to simplify the analysis, but as a method to filter out transitions that belonged only to o-xylene. The other molecules had less than 7 nuclear spins and thus could not contribute to the 8Q spectrum.

Typically, for complex solutes the N, N - 1 and N - 2 MQ spectra (where N is the highest MQ order) are analyzed first to obtain accurate estimates of spectral parameters. The 1 Q high-resolution spectrum is then analyzed to obtain more accurate spectral parameters. For 0-xylene analysis began with the 8Q spectrum, Fig. 5.7 A, using a least-squares routine to adjust the three resonance frequencies and the two 80/(3, until a reasonable fit was obtained (Fig. 5.7C). Adjusting 5 parameters instead of 13 makes analysis easier, but the procedure still remains non-trivial. Slight deviations in spectral parameters could cause difficulties when analyzing the lQ high-resolution spectrum, especially with the added complexity of other solutes. Therefore, to obtain accurate spectral parameters, a fit of the Dij to the resonance frequencies was carried out readily (Fig. 5.7B).

The high-resolution spectrum predicted from the analysis of the 8Q spectrum of o-xylene is displayed in Fig. 5.6B. By comparing the experimental Fig. 5.6A with the predicted one, there are many resonances which can be immediately assigned, even

A

I I I I ) I I , I I I I I I I

o 5000 10. 1.5x 10· 2.5x10· Frequency

Figure 5.7. A: experimental +8Q spectrum. Only resonances from o-xylene are observed. For an Nspin-~ system, N is the highest attainable MQ order. B: calculated +8Q spectrum of a-xylene obtained by adjusting frequencies and Dij . C: calculated +8Q spectrum of a-xylene obtained by adjusting frequencies and So{3'


in the presence of resonances from the other molecules. The precise fit to the 1 Q high-resolution spectrum of Fig. 5.6A is then very easy. Figs. 5.6B and C compare the 1 Q high-resolution spectra that are calculated from the parameters obtained from the fit to the experimental SQ spectrum and from the independent fit to the experimental 1Q high-resolution spectrum.

After analysis of the high-resolution spectrum of o-xylene the resulting fitted spectrum was subtracted from the experimental one, and resonances from the other solutes could be identified (Fig. 5.6D). The initial dipolar couplings for o-chlorotoluene were calculated from the order parameters of o-xylene. The off-diagonal order parameter was set to zero. In the spectrum of o-chlorotoluene there is a group of resonances to high frequency from the main portion of the spectrum (see Fig. 5.6D and E). The fine structure is due to the Dij between methyl and ring protons, and by assigning some of these resonances certain Dij could be determined roughly. This aided the identification of resonances in the main portion of the spectrum. Once a few resonances within the main portion of the spectrum were correctly assigned, the spectrum was analyzed quickly.

Again, after the high-resolution spectrum of o-chlorotoluene was fitted and subtracted from the experimental one, resonances from o-dichlorobenzene were easily identified (see Fig. 5.6F and G). In Fig. 5.6H only a few resonances remain after the fitted 0-xylene, o-chlorotoluene and o-dichlorobenzene spectra are subtracted from the experimental one. The remaining resonances correspond to TCB, acetone-d5 (from the lock) and an unknown impurity.

One final example of a single, but very complex solute is butane dissolved in the liquid crystal mixture of 55wt% Merck ZLI 11321 EBBA [34]. The 1Q high-resolution 1 H NMR spectrum of butane consists of a thick mass of lines spanning a frequency range of 10 kHz with essentially no notable features, and sits on the broad liquidcrystal 1 H spectrum (Fig. 5.S). A horizontally expanded region of the spectrum is shown in the lower half of Fig. 5.9. It is apparent that while the spectral line density is high, overlap is not so severe as to make it impossible to determine the frequencies of most of the lines. Thus, a fit of the experimental spectrum is possible in principle. However, the complexity of the spectrum makes it extremely difficult to do so without very accurate initial estimates of the coupling constants and chemical shifts. Small deviations from the true values of these parameters would alter the line frequencies and intensities enough to generate a spectrum with significantly different fine structure from that of the experimental spectrum.

In the previous example of the multiple co-dissolved solutes the initial parameters were taken from previous studies. The strategy used for the butane spectrum utilized the CI Model described in Chapter 10, with parameters optimized for the liquid crystal mixture of 55wt% Merck ZLI 11321 EBBA according to the results of an earlier study [37] to predict molecular order parameters and thus dipolar coupling constants. A value for the methyl-group rotational potential of 3.0 kJ 1 mol was used to generate initial conformer probabilities. The predicted spectral parameters were not adequate to begin analysis of the 1 Q high-resolution spectrum. However, a trial spectrum based on the predicted dipolar couplings, isotropic chemical shifts, and indirect coupling


I I -4 o 2 4

Frequency 1kHz

Figure 5.B. Experimental (bottom) and simulated (top) spectra of partially oriented butane in the liquid crystal mixture of 55wt% Merck ZLI 1132 I EBBA. (Reprinted from [34] with permission from the American Institute of Physics).

I I I I I I I I I I I I I

-0.5 -0.4 -0.3 -0.2 -0.1 o Frequency 1kHz

Figure 5.9. Expanded region of Fig. 5.8. (Reprinted from [34] with permission from the American Institute of Physics).


I I J I -6 -4 -2 0 2 4 6

Frequency 1kHz

Figure 5.10. Experimental and simulated 7Q spectra of partially oriented butane. For the simulated spectrum, the line intensities have been arbitrarily set equal, since the intensity of each MQ transition is a complicated function of the preparation time in the pulse sequence and the parameters in the spin Hamiltonian. (Reprinted from [34] with permission from the American Institute of Physics).

constants provided an adequate starting point to fit simultaneously the 7Q and 8Q spectra. Figs. 5.10 and 5.11 show the frequencies of all the lines calculated in the fitto the MQ spectra. A large number of these lines in the experimental 7Q spectrum have very weak intensities and are barely discernible. if at all. from the noise. The trial lQ

-6 -4 -2 0 2 4 6 Frequency 1kHz

Figure 5.11. Experimental and simulated 8Q spectra of partially oriented butane. (Reprinted from [34] with permission from the American Institute of Physics).


high-resolution spectrum that was predicted from the fit to the MQ spectra proved to be an excellent starting point for the fit of the experimental 1 Q high-resolution spectrum. Assignment of spectral lines was tedious, but trivial. A lQ high-resolution spectrum simulated using the fitted dipolar coupling constants, chemical shift difference, and J-couplings is shown in Fig. 5.8, together with the experimental one. The high quality of the fit is more evident in the expanded plot of a region of the spectrum shown in Fig. 5.9.

4. Summary We have numerous "tools" at our disposal when analyzing NMR spectra of ori

entationally ordered molecules. These tools include analysis programmes, graphical user interfaces, NMR pulse programmes, and models and theories that predict starting parameters. Anyone of these can be effective when determining spectral parameters. However, combining the tools can take a virtually impossible task and turn it into a challenge that is achievable.

Acknowledgments I would like to thank E. Elliott Burnell, the Natural Sciences and Engineering Re

search Council of Canada for financial support and the E. Merck Company of Darmstadt, Germany for their gift of Merck ZLI 1132.

References [1] Buckingham, A.D., and McLauchlan, K.A. Progress in Nuclear Magnetic Resonance Spectroscopy,

volume 2. Pergamon Press, Oxford, 1967, p 63. [2] Diehl, P., and Khetrapal, C.L. NMR Basic Principles and Progress, volume 1. Springer-Verlag,

Berlin, 1969.

[3] Emsley, J. w., and Lindon, J.C. NMR Spectroscopy using Liquid Crystal Solvents. Pergamon Press, Oxford, 1975.

[4] Syvitski, R.T., and Burnell, E.E. (1997), Chem. Phys. Letters, 281:199. [5] Syvitski, R. T., and Burnell, E. E. (2000), J. Chem. Phys., 113:3452. [6] Syvitski, R. T., Pau, Monita Y-M., and Burnell, E. E. (2002), J. Chem. Phys., 117:376. [7] Syvitski, R.T., and Burnell, E.E. (1999), Can. J. Chem., 77: 1761. [8] Syvitski, R. T., and Burnell, E. E. (2000), J. Mag. Res., 144:58. [9] Rendell, J.C.T., and Burnell, E.E. (1995), J. Magn. Reson., A 112:1.

[10] Stephenson, D. S., and Binsch, G. (1980), Org. Mag. Res., 14:226. [11] Hagele, G., Engelhard, M., and Boenigk, W. Simulation und automatisierte Analyse von Kernres-

onanzspektren. VCH, Weinheim, 1987. [12] Castiglione, F., Celebre, G., De Luca, G., and Longeri, M. (2001), Liquid Crystals, 28: 1403. [13] Vivekanandan, S., and Suryaprakash, N. (2001), Chem. Phys. Lett., 338:247. [14] Takeuchi, H., Inoue, K., Ando, Y., and Konaka, S. (2000), Chem. Lett., 11:1300. [15] Castiglione, F., Celebre, G., De Luca, G., and Longeri, M. (2000), J. Mag. Res., 142:216. [16] Lounila, J., and Jokisaari, J. (1982), Prog. NMR Spectrosc., 15:249. [17] Chandrakumar, T., Polson, J.M., and Burnell, E.E. (1996), J. Magn. Reson., A 118:264.

[18] Polson, J.M., and Burnell, E.E. (1994), J. Magn. Reson., A 106:223. [19] Burnell, E.E., and de Lange, C.A. (1998), Chem. Rev., 98:2359.


[20] Dong, R.Y. Nuclear Magnetic Resonance of Liquid Crystals. Springer-Verlag, New York, 2nd

edition, 1997. [21] Higgs, T.P., and MacKay, A.L. (1977), Chern. Phys. Lipids, 20:105. [22] Emsley, J.w., Luckhurst, G.R., and Stockley, c.Pl. (1981), Mol. Phys., 44:565. [23] Emsley, J.w., Fung, B.M., Heaton, N.J., and Luckhurst, G.R. (1987), J. Chem. Phys., 87:3099. [24] Janik, B., Samulski, E.T., and Toriurni, H. (1987),1. Phys. Chem., 91:1842. [25] Delikatny, E.J., and Burnell, E.E. (1989), Mol. Phys., 67:757. [26] Rosen, M. E., Rucker, S.P., Schmidt, C., and Pines, A. (1993), J. Phys. Chem., 97:3858. [27] Khetrapal, C.L. (1995), Int. J. of Modem Phys., 9:2573. [28] Drobny, G., Pines, A., Sinton, S. w., Weitekamp, D. P., and Wemmer, D. (1979), Faraday Symp.

Chem. Soc., 13:33. [29] Sinton, S. w., and Pines, A. (1980), Chem. Phys. Lett., 76:263. [30] Warren, W. S., and Pines, A. (1981), J. Am. Chem. Soc.,103:1613. [31] Syvitski, R.T., Burlinson, N., Burnell, E.E., and Jeener, J. (2002), J. Magn. Res., 155:251. [32] Bodenhausen, G. (1981), Prog. NMR Spectrosc., 14:137. [33] Slichter, C. P. Principles of Magnetic Resonance. Springer-Verlag, New York, 3rd edition, 1990. [34] Polson, J.M., and Burnell, E.E. (1995), J. Chem. Phys., 103:6891. [35] Syvitski, Raymond T. Probing anisotropic intermolecular forces in nematic liquid crystals using

NMR and computer simulations. PhD thesis, University of British Columbia, 2000. WWW address:http://www.chem.ubc.ca/facultylburneJVgroup/syvitskilthesislindex.htrnl.

[36] Sykora, S., J. Vogt, H. Bosiger, and Diehl, P. (1979), J. Mag. Res., 36:53. [37] Zimmerman, D. S., and Burnell, E.E. (1993), Mol. Phys., 78:687.

II

NMR OF SOLUTE ATOMS AND MOLECULES

Since the discovery in 1963 that small solutes such as benzene dissolved in nematic liquid crystals show well-resolved NMR spectra dominated by anisotropic interactions [1], the method has been exploited and perfected. Such studies can produce detailed information about intramolecular interactions in the solute species, as well as on the interaction between the solute and the intermolecular anisotropic potential provided by the liquid-crystal solvent.

The simplest solutes are clearly atoms. Chapter 6 reviews the investigations of atomic solutes possessing nuclei with non-zero nuclear spin. Atoms would seem to be ideal and robust probes for exploring details of the anisotropic potential in liquid crystals, and of course this is true to a large extent. However, for many-electron atoms, the liquid-crystal environment affects and perturbs the atomic electronic wave functions in a most interesting manner. In addition, it is shown that atomic solutes provide a sensitive and unique means of detecting phase transitions in anisotropic liquids.

For the study of molecular properties of solutes dissolved in liquid crystals the NMR method is not without limitations. Much research has been carried out over the past 40 years to overcome the restrictions associated with the study of orientationally ordered solutes. One important stimulus has always been to extend and modify the method in order to apply it to larger solutes than originally deemed possible. In Chapter 7 an overview is presented of the developments that have led to where we are today. In addition to the present scope and limitations ofNMR of solutes in anisotropic solvents, possible future directions are also discussed. Clearly, the possibility of obtaining accurate liquid-phase structures from the use of dipolar couplings measured from the NMR of orientationally ordered solutes constitutes one of the important successes of the method.

High-resolution NMR employing isotropic liquid solvents has for many years contributed greatly to our understanding of protein structure in solution. In recent years much effort has gone into the weak alignment of proteins in order to obtain information on anisotropic interactions. Chapter 8 gives an overview of the significant progress that has been made in employing novel anisotropic solvents that lead to quite small solute anisotropic interactions, in particular the dipolar couplings. Under these conditions the protein NMR spectra are well-resolved and dominated by the familiar isotropic interactions, with the residual dipolar couplings containing valuable novel structural information. This method of structure determination of orientationally ordered proteins has led to a lot of excitement and shows great promise for the future.

The native environments for many proteins are the membranes of biological cells which show liquid-crystalline behaviour. The NMR spectra of membrane-bound proteins are usually dominated by large dipolar couplings that lead to broad, structureless NMR spectra, as expected for solutes containing a large number of magnetic nuclei. In the case of membrane-bound proteins, the magnitude of the dipolar couplings can be decreased by the use of model membranes that exist in structures exhibiting small orientational order. In this spirit Chapter 9 concentrates on two specific experiments, viz. separated local field (SLF) and 1 H magic angle spinning (MAS) methods. These additions to the NMR arsenal have proven to be very useful for the elucidation of structures of membrane-bound proteins.


108

References [1] Saupe, A., and Englert, G. (1963), Phys. Rev. Letters, 11:462.


Chapter 6

NMR OF NOBLE GASES DISSOLVED IN LIQUID CRYSTALS

Jukka Jokisaari NMR Research Group, Department of Physical Sciences, University of Oulu, Finland

1. Introduction Interestingly, the first results of the 129Xe NMR experiments were published as early

as 1951 when Proctor and Yu [1] reported the magnetic moment of the isotope. One can regard the experiment by Loewenstein and Brenman in 1978 [2] as the starting point of the application of xenon NMR to the investigation of liquid-crystalline systems. They were the first, to the author's knowledge, to record the 131 Xe NMR spectrum of xenon in a liquid-crystalline environment, a lyotropic liquid crystal composed of polY-J-benzyl-L-glutamate (PBLG) and chloroform. This work will be discussed in section 4. A huge growth of interest in 129Xe NMR spectroscopy emerged in the 1980s when its use was suggested for deriving information on pore size in microporous materials, such as zeolites and molecular sieves. The first 129Xe NMR experiments on xenon in a thermotropic liquid crystal were performed in 1988 by Bayle et al. [3]. Several review articles have been published On Xe NMR, the most comprehensive being that by Ratcliffe [4] which covers various applications. The review by Bonardet et al. concentrates on 129Xe NMR of porous solids [5], and the one by Jokisaari deals with NMR of noble gases in isotropic liquids and liquid crystals [6].

2. NMR properties of noble gases There are five stable noble gas isotopes, 3He, 21 Ne, 83Kr, 129Xe and 131 Xe, that

possess a non-zero spin quantum number and thus are NMR active. The properties related to NMR are shown in Table 6.1.

From an experimental point of view it is clear why 129Xe has become the most popular of the noble gases; its relative receptivity is much larger than that of the other noble gas isotopes, and it is a spin-1I2 nucleus which implies that the NMR spectrum consists of a single resonance line. On the other hand, 3He also has spin 112 and the receptivity can be enhanced by using enriched gas. Still, it is less popular in NMR spectroscopy, the reason being the small chemical shift range. This will be discussed in section 3. The use of Xe gas pressures of a few atm makes feasible the detection of

109 E.E. Burnell and c.A. de Lange (eds. j, NMR of Ordered Liquids, 109-135. © 2003 Kluwer Academic Publishers.


Table 6.1. NMR properties of noble gases.

Natural Gyromagnetic NMR Quadrupole abundance ratio Relative freq. moment

Nucleus Spin (%) (107 radT- 1s-1 ) receptivitya (MHz)b (10-31 m 2)

3He 112 0.00013 -20.380 0.00336 304.710 21Ne 312 0.257 -2.113 0.0359 31.577 101.55(75)C 83Kr 912 11.55 -1.033 1.24 15.391 253(5)d 129Xe 112 26.44 -7.441 32.3 110.642 131Xe 312 21.18 2.206 3.37 32.798 -120(20)d

a Relative to l3C. b At 9.4 T. C Ref. [7]. d Ref. [8].

the 129Xe resonance with a single scan. This is advantageous because the spin-lattice relaxation time of this Xe isotope in liquid-crystalline phases is tens of seconds [9-11]. The remaining three isotopes are quadrupolar. Their applicability is restricted because of their small Larmor frequencies which, especially in the case of 83Kr, often leads to problems with acoustic ringing. However, their NMR spectra reveal information that is not available with the spin-lI2 isotopes. Their receptivity can be enhanced by using isotopically enriched gases.

An additional feature that favours the use of 129Xe (and 3He) in NMR studies of various materials is the possibility of increasing its receptivity, i.e. polarization, by optical pumping (OP) [12,13]. This procedure may increase the polarization by up to 1Q5-fold as compared to thermal polarization. OP 129Xe has not yet been applied to liquid crystal research.

3. Chemical shift 3.1 Experimental observations

The shift detected in the NMR resonance frequency of a noble gas nucleus arises exclusively from environmental effects. Therefore, it is somewhat misleading to call the shift a chemical shift. However, this is a common habit and is adopted here too. On the other hand, terms such as "shielding" and "shielding change" are also used to refer to "chemical shift".

Generally, the total shielding (relative to a reference shielding, defining the chemical shift, 8 = -(a - a 0» of a nucleus in an atom or molecule is comprised of several contributions

a - (70 = ab + aa + aw + aaniso· (6.1)

In Eq. (6.1), ao is the shielding of a reference substance, ab is due to bulk susceptibility, aa arises from the magnetic anisotropy of neighbouring solvent molecules, aw is

NMR of Noble Gases Dissolved in Liquid Crystals 111

a consequence of van der Waals interactions, and the last term (j aniso comes from an anisotropic shielding tensor. For liquid-crystalline samples, both (jb and (j a are anisotropic. In noble gas NMR, the shielding of the gas at zero pressure is used as a reference. The shielding of 129Xe in pure gas depends upon gas density according to

(6.2)

where (j 0 is the shielding in vacuo and p is the density (given in amagat) [14]. The virial coefficients (ji (i = 1,2,3) arise from two-, three- and four-body interactions and are temperature dependent. At low densities the last two terms in Eq. (6.2) are negligible, and consequently, the shielding varies linearly with density. In practice, gas pressures of 1-5 atm are applied and the (j 0 value is obtained by using the known values of the second virial coefficient (j1 and the density.

In isotropic solutions, the nuclear shieldings (or more precisely the gas-to-solution shifts defined as (j m = (j - (j 0 - (jb) of the various noble gas isotopes are linearly related [6]. However, this is not necessarily valid for the shielding values measured in liquid-crystalline solutions because of the anisotropic contribution, (j aniso, shown in Eq. (6.1). The anisotropic shielding tensor arises in part from the anisotropic forces that deform the originally spherical electron cloud of a noble gas atom. The anisotropy of the shielding tensor (together with the discrete density change) leads to an abrupt jump in the shielding at the isotropic-nematic phase transition. This is illustrated in Fig. 6.1.

Fig. 6.1 displays the cases in which the liquid crystal (LC) director is aligned itself either along the external magnetic field, Bo, or perpendicular to it. In each case, the 129Xe shielding decreases linearly with decreasing temperature in the isotropic phase. The slopes of the straight lines in the isotropic states are typically from 0.15 to 0.25 ppm K-1. Difference in behaviour is detected at the isotropic - nematic phase transition: when the liquid crystal director is along Bo, the shielding decreases abruptly by several ppm (Fig. 6.1 top), whereas when the director is perpendicular to Bo the shielding increases abruptly (Fig. 6.1 middle). Fig. 6.1 bottom in tum displays a case in which the liquid crystal director rotates from the parallel to the perpendicular direction because of the change of sign (which follows from the change in sample temperature) of the anisotropy of the liquid crystal volume diamagnetic susceptibility tensor. Such an experiment for 129Xe was first reported by Bayle et aZ. [3]. These observations indicate clearly that 129Xe NMR of xenon in liquid-crystalline solutions gives information on the orientation of the liquid crystal director, and consequently, reveals the sign of the anisotropy of the diamagnetic susceptibility tensor.

Particularly interesting behaviour of 129Xe shielding as a function of temperature can be detected in the liquid crystals that possess several mesophases. An illustrative example, xenon in NCB 84 (l-butyl-c-4-( 4' -octylbiphenyl-4-yl )-r-I-cyclohexanecarbonitrile), is shown in Fig. 6.2 [16]. This particular liquid crystal has the following phases [17]:

112

a :: .... A . . v

! ...

-" -.. -t. -..

- 1


:uo l2$ uo J.J.5 340 J4.S ~ 3S$ s-o H5

T/K

-02"'=--=:500:--:_::!;-c'~":-';:"~'1O;;-;-!'2>~l>O!::-=3"~"'c!::.-:"'''&:--'300~'''±--~''O~''' T/K

a -.,~ .. ..

T/K

Figure 6.1. 129Xe shielding (i.e. negative chemical shift) of xenon in: (top) EBBA, (black circles, xenon pressure ca. 1 atm, open circles, xenon pressure ca. 5 atm); (middle) ZLI 1167 (black circles), 87.5:12.5 (open circles), and 78.0:22.0 (open squares) mixtures of ZLI 1167 and EBBA; and (bottom) critical 77.0:23.0 mixture of ZLI 1167 and EBBA. The large jump of the shielding at 309 K is due to the 90° rotation of the liquid crystal director. The point (x) is obtained by linear extrapolation from the isotropic phase, whereas the point (0) is calculated from a = ~(all + 2a .i). (Reprinted with permission from [15]. Copyright (1990) Taylor&Francis).

As in previous cases, the shielding is linearly dependent upon temperature in the isotropic phase, the slope being 0.183 ppmK- l . When moving to the nematic phase from the isotropic phase, the shielding decreases suddenly by 4.9 ppm. At the nematic

NMR of Noble Gases Dissolved in Liquid Crystals

·180

·182

IS ·184 'Co ~ ~

I: b

·181

I .188

b

~ ·180 .. 1 .192

~ ·184

·'9&

.. .. ... ....

.... _ .... _._ ................. .. . .

. . . . . ... .. .. .......... ... .... . ..,.-

Temperature T (K)

... ..... ....

113

Figure 6.2. 129Xe shielding (i.e. negative chemical shift relative to low pressure gas) ofxenon in NCB 84. (Reprinted with pennission from [16]. Copyright (1992) American Institute of Physics).

to smectic A transition no abrupt jump can be detected. However, the slope and curvature change remarkably allowing for the detection of the phase transition. This seems to be a common feature of such transitions [15,18]. The smectic A - smectic C transition has only a minor effect on the shielding behaviour; a slight change in the slope is detectable at 339 K. The lower temperature transition from the smectic C to the smectic G causes an appreciable change in the slope, whereas the transition from the smectic G to the solid exhibits a large jump of 9.6 ppm toward more shielded values.

Bharatam and Bowers [10] applied 129Xe NMR to the mixture (LC-l) of 60CB (4-cyano-4'-n-hexyloxybiphenyl; 27.16 wt%) and 80CB (4-cyano-4'-n-octyloxybiphenyl; 72.84 wt%), and to the mixture (LC-2) of 7CB (4-n-heptyl-4'-cyanobiphenyl; 73 wt%), 80CB (18 wt%) and 5CT (4-n-pentyl-4'-cyanoterphenyl; 9 wt%). Both LC-l and LC-2 display a reentrant nematic phase, the phase transitions being the following:

LC-l: reentrant nematic - 304 K - smectic A - 319 K - nematic - 351 K - isotropic, LC-2: reentrant nematic - 284 K - smectic A - 306 K - nematic - 340 K - isotropic.

The 129Xe NMR spectra recorded at various temperatures yield a chemical shift jump of 9 ppm at the isotropic - nematic phase transition and a clear change in the slope at the smectic A - reentrant nematic phase transition, as shown in Fig. 6.3. However, the phase transition nematic - smectic A is not apparent. Obviously this finding is an indication that no redistribution of xenon atoms takes place during the formation of the smectic phase, contrary to what is observed for the respective transitions in, for

114

0

10

20

30

I I .. : .: •

u .... =-~ = t!!l •

• •

134~KI

350

I I I

u :c ca ~ Z

• •


• ... I ••

u :c ~ e rIl

'. ;e : ., I • I • I

1 z

I Q,I

~

I •

1306 KI1284 KI ••• I I ••

:: e. I I I I I I

300

T(K) 250

Figure 6.3. 129Xe chemical shift (relative to the value in the isotropic phase) as a function of temperature in LC-2. (Reprinted with permission from [10]. Copyright (1999) American Chemical Society).

example, NCB 84, HAB and FELIX-R&D liquid crystals. All the phase transitions of LC-l and LC-2 are clearly visible via spin-lattice and spin-spin relaxation time measurements, as discussed in section 5.

The above examples indicate that the 129Xe shielding is indeed very sensitive to phase transitions, making 129Xe NMR a useful tool for the determination of phase diagrams. This has been applied to the binary mixtures of ZLI 1132 (mixture of trans-4-n-alkyl-( 4-cyanophenyl)-cyclohexanes) and EBBA (n-p-ethoxybenzylidene)p-n-butylaniline) liquid crystals [19].

The temperature dependence of the 21 Ne shielding is very similar to that of 129Xe. For example, the abrupt shielding decrease at the isotropic - nematic phase transition of 5CB (4'-n-pentyl-4-biphenyl carbonitrile) is ca. 1 ppm, while the total shielding range is about 2 ppm in the temperature range 296-313 K [20].

As Table 6.1 indicates, the relative receptivity of 3He is very low. However, NMR experiments become feasible with 3He-enriched gas; if the degree of enrichment is


2.8

2.7

S Po. 2.6 Po.

- 2.5 ........

CD

iF 2.4 ........ b'

:2.3 .. b ~ 2.2 r-

........ 2.1

2.0 0.80

I

0.85

• ••• •••••

•

I

••

•

I

0.90 0.95 1.00 T/Tm

.. .. .. ..

115

-

-

-

1.05

Figure 6.4. 3He shielding relative to zero-pressure gas in the ZLI 1167 (open circles) and 5CB (closed circles) liquid crystals as a function of reduced temperature. (Reprinted with permission from [22]. Copyright (1994) Gordon and Breach).

99.95 at%, the relative receptivity is ca. 2500. Fig. 6.4 shows the 3He shielding as a function of reduced temperature, T /TNJ, in the 5CB and ZLI 1167 liquid crystals. Comparison of the results for ZLI 1167 to the 129Xe results (shown in Fig. 6.1 middle) reveals several differences. First, 3He in ZLI 1167 is more shielded than in zeropressure gas, which is opposite to the behaviour detected for 129Xe. Secondly, the total chemical shift range of 3He is about 0.6 ppm, while that of 129Xe is ca. 11 ppm over the temperature ranges studied. Thirdly, the shielding jumps at the nematic -isotropic phase transition are clearly observable in both cases, but the slopes of the curves are opposite in sign. Fourthly, the 3He NMR spectra allow detection of the smectic A - nematic phase transition in ZLI 1167. Obviously, helium does not disturb the liquid crystal as much as xenon which is much larger in size. 21 Ne NMR spectra also yield the smectic A -nematic transition, but not as clearly as shown in Fig. 6.4 [21] (see also section 4). The different behaviour of 129Xe and 3He in ZLI 1167 is a result of the different physical mechanisms that are responsible for the shielding; for 3He the shielding is dominated by the terms (fa and (fb, whereas the dominant contributions to the 129Xe shielding are (faniso and shifts arising from the liquid crystal density (see Eq. (6.1».

3He NMR of helium dissolved in the liquid crystal NCB 84 appears even more effective in detecting phase transitions than 129Xe NMR [22,23]. Fig. 6.5 shows the anisotropic part, «(fa +(fb)aniso, of the total shielding of 3He as a function of


-0.2 r-:--,....--..---or--....---,--...----T---T---.--,

-0.25

e -0.3

~ ....... -0.35

~ ~ .g -0.4 a 1 + .. -0.45 b

'-"

-D.S

-0.55 c

2.7 r----------. 2.6 ~ ................. .

~ 12$

1~1 : i:: / . ...:1

t·I::. 21 •••••••••••• 370 375 380 385 390 395 400 40S 410 415

T [K]

* · +++ : .. : · '-- ... . · ............... . ~:............... : ...... .

N

310 320 330 340 350 360 T [KJ

370 380 390

Figure 6.5. Anisotropic part of (ua + Ub) for 3He in NCB 84. (Reprinted with permission from [23]).

temperature. The anisotropic contribution is defined as [23]

( ) aniso S [ 11 isol 2(471" )pNAA ]p.( e ) (j a + (jb ~ - LC 3n Xd + '3 3'" - 9 M uXd 2 cos Bn . (6.3)

The symbols in Eq. (6.3) have the following meaning: SLC is the orientational order parameter of the liquid crystal, n is the number of the nearest liquid crystal molecules, X~80 and D.Xd are the isotropic diamagnetic volume susceptibility and the anisotropy of the susceptibility tensor, 1 is the shape factor (dependent upon the sample dimensions), 9 is a factor that takes into account the finite height of the sample (g = 0 for an infinitely long cylinder parallel to Bo), p is the density, N A is Avogadro's number, M is the molar mass, and P2 is the second-order Legendre polynomial, with e Bn being the angle between Bo and the liquid crystal director, n.

3He reveals a feature of the liquid crystal NCB 84 that is not detected through 129Xe. In the smectic C phase the 3He spectrum is a superposition of two components, a narrow one and a broad one. As Fig. 6.5 indicates, the components shift in opposite directions when the temperature is changed. An explanation for the presence of the broad component may be the existence of small domains which possess different orientational order relative to the external magnetic field, and consequently the 3He nuclei in these domains resonate with varying frequencies. The domains are built up during the transition from the smectic A to the smectic C phase. The narrow component in tum is interpreted as arising from the sample regions that are fully oriented [23].


21S

210

20S

200

i S 195 to

190

185

180

260 280 300 320 340 360 380 400

T(K)

117

Figure 6.6. 129Xe chemical shift relative to free xenon gas in pure forrnamide (FA, open squares) and in three crAB I FA systems: CTAB 78 wt% and FA 22 wt% (black triangles); crAB 60 wt% and FA 40 wt% (open triangles); and crAB 25 wt% and FA 75 wt% (black circles). (Reprinted with permission from [24]. Copyright (1996) Society for Applied Spectroscopy).

In principle, similar noble gas NMR experiments as performed in thermotropic liquid crystals can be applied to lyotropic liquid crystals. However, such experiments are scarce, although the first 131 Xe experiments were carried out in the lyotropic solution PBLG / CDCh [2]. The reason for this may be the low solubility of noble gases in water-based lyotropics. An example of a non-water based lyotropic phase is the system obtained by dissolving cetyltrimethylamrnonium bromide (CTAB) in formamide (FA). This system is known to form micelles at low CTAB concentration (~ 20 wt%) and hexagonal (He.), cubic (QaJ and lamellar (La:) mesophases at higher concentrations and elevated temperatures. The 129Xe chemical shifts of xenon in three CTAB / FA systems as a function of temperature are shown in Fig. 6.6. The results for 129Xe in pure formamide are also shown. In each case a clear phase transition can be detected, the low temperature phase being a phase in which liquid and solid coexist in equilibrium [24]. In the solid / liquid phase the 129Xe (as well as 131 Xe) NMR spectra display two resonance peaks; the high-frequency peak follows the behaviour shown in Fig. 6.6, while the low frequency peak appears at the position of free xenon gas (see Fig. 6.7). The appearance of the latter peak is a consequence of the formation of


200 150 100 50 o pp

Figure 6.7. 131Xe NMR spectra of xenon in CTAB(25 wt%) I FA(75 wt%) sample at various temperatures. The top spectrum was recorded at 282 K and the bottom spectrum at 388 K. Temperature steps between successive spectra are 11.8 K. The two topmost spectra indicate a resonance at 0 ppm which arises from relatively large pores in the solid phase. The 131 Xe shielding changes by ca. 9 ppm at the isotropic - solid !liquid phase transition. (Reprinted with permission from [24]. Copyright (1996) Society for Applied Spectroscopy).

relatively large pores filled with xenon gas during the crystallization process. The pore size has to be large because, if it were small, the peak should have shifted toward high frequency (small shielding) as compared to free xenon. Experiments in zeolites and molecular sieves have indicated that (roughly) the smaller the pore size, the smaller the shielding constant of 129Xe [5].

3.2 Theoretical

As described in the previous section, the liquid crystal solvent can modify the shielding of a solute atom in two ways: first, indirectly by changing the local magnetic field (contributions (Ta and (Tb in Eq. (6.1)); and secondly, directly by distorting the electronic cloud of the atom (contributions (TW and (Taniso in Eq. (6.1)). Ylihautala et ai. discuss the contribution of attractive van der Waals interactions to the shielding of


noble gas atoms in liquid-crystalline environments [25]. The effect of an electric field (E, magnitude E) on the shielding can be represented as a series expansion

(6.4)

where 0'(1) and 0'(2) are the first and second shielding hyperpolarizability tensors, respectively, and the effects of the electric field gradients have been omitted. Assuming that higher than second-order shielding hyperpolarizabilities are insignificant, and noticing that for an atom 0'(1) vanishes, the shielding tensor element along the external magnetic field, i.e. along the Z axis ofthe laboratory frame, becomes

1 (2) 2 2 (2) 2 O'zz = 2"[O'isoE + 3~0' P2(COS()EB)E]. (6.5)

In Eq (65) notations 0'~2) = 1(0'(2) + 20'(2)) and ~0'(2) = 0'(2) - 0'(2) have been used • • tSO 3 II 1. II 1. '

and BEB is the angle between the momentary electric field and the external magnetic field. The direction and magnitude of the electric field at the position of a solute atom fluctuate. In NMR experiments, the average < O'zz > is observed:

(6.6)

In the case that the magnitude and direction of the electric field are independent, the second average in Eq. (6.6) reduces to (P2(cos(BEB)) (E2), where (P2(cosBEB)) can be considered as the orientational order parameter of the electric field relative to the external magnetic field.

The above theory was applied to the 129Xe and 21 Ne shielding behaviour in EBBA, corrected for bulk susceptibility effects. Comparison of the experimental shielding and shielding anisotropy results with the theoretical predictions of electric effects suggest that the shielding of noble gases induced by a liquid crystal environment is mainly due to attractive van der Waals interactions, although repulsive interactions should not be disregarded.

In another study, Ylihautala et al. developed a statistical mechanical approach to the evaluation of the gas-to-solution shifts, am, of a noble gas atom in a nematic liquid crystal [26]. Let us define the coordinate systems as follows: (X,Y,Z) is the laboratory frame, (a,b,c) is the frame at the position of the solute atom, and (Xi,Yi>Zi) is the frame located in solvent molecule i. The orientation of the solute atom shielding tensor relative to the laboratory frame depends on the coordinates of all solvent molecules. It is convenient to choose shorthand notations so that e includes the positional and orientational coordinates of the solute, and ,n the coordinates Xl, x2, ... ,xn , corresponding to e, for the solvent molecules 1,2,3, ... n.

In NMR experiments, an average of the shielding tensor, <(7">, is observed [26]:

(6.7)


where p(n+1)(~,(n) is the (n+ I)-particle distribution function. It gives the probability for the configuration (~, (n) and is determined by Boltzmann statistics. Following the treatment of [26], the (n + I)-particle (including the solute atom and n solvent molecules) distribution function is written as a product of (n + 1) single particle distribution functions and a (n + 1 )-particle correlation function. Skipping the details, it appears that the nuclear shielding constant of a noble gas atom observed in an NMR experiment in a nematic liquid crystal environment depends upon the product of the rank li orientational order parameters:

((JZZ) = 2: 1/J(h, ... ,In)(PIt)(Pz2 )··· (PzJ, (6.8) It , ... ,In

where the coefficients 1/J(h, ... , In) are defined by combinations of the various configurational and orientational averages. The application of Eq. (6.8) is in practice impossible because of the large number of variables. A simplification may be derived with the assumption of pairwise additivity of the noble gas atom shielding perturbation. In this approximation,

(6.9)

Here p is the liquid crystal number density. The result of Eq. (6.9) is the same as that derived from different starting points in [16] which was based on two assumptions: (a) the shielding perturbation is directly proportional to the density; and (b) the anisotropy of the perturbation is directly proportional to the orientational order parameter (P2)

= SLC. In the phenomenological theory of [16] it was assumed that the shielding constant is temperature dependent via the temperature dependence of the density and orientational order parameters. However, the shielding constant may also experience an indirect temperature effect, i.e. ((J) and (b.(J) are temperature dependent. There are arguments for these two factors to be slowly varying functions of temperature. Therefore, they can be approximated by linear functions of temperature. Consequently, when temperature dependence is taken into account, the pairwise additive shielding can be represented in the form [26]:

2 ((Jzz) = p(T){(Jo[l- E(T - To)] + 3b.(J[1- b.E(T - To)]SLC(T)}, (6.10)

where the temperature dependence of the density can be approximated by

p(T) = Po[l - a(T - To)] (6.11)

in the isotropic phase, and by

p(T) = Po{[l - a(T - To)] + b.p} Po

(6.12)

in the nematic phase of the liquid crystal. The term b.p/Po takes into account the possible density jump at the isotropic - nematic phase transition. The reference temperature To is chosen to be equal to TNI.


The above theory (using the pairwise approximation) has been applied to 129Xe shielding behaviour in EBBA [26], and in the so-called critical mixture of EBBA (23 wt%) and ZLI 1167 (77 wt%) [11]. In the former case, partially deuterated EBBA (4-ethoxybenzylidene-2,6-dideutero-4' -n-butylaniline) was used. Hence, the orientational order parameters, SLC, at various temperatures could be determined with the aid of the 2H quadrupole splittings and the D DH dipolar couplings obtainable from the 2H NMR spectra. Fitting the experimental points to the Haller function [27]

SLc(T) = (1 - y TT Y NI

(6.13)

results in y=0.9988 and z=0.182, and a continuous function of temperature. Furthermore, the temperature dependence of the number density could be derived from the work by Bahadur and Chandra [28]. Consequently, when fitting Eq. (6.10) to the experimental data there remain two adjustable parameters «(]"o and E) for the isotropic phase and four «(]" 0, E, !1(]" and !1E) for the nematic phase. Choosing the reference temperature To to be equal to the isotropic - nematic phase transition temperature, TN I = 347 K, and assuming the (]" 0 and E to be the same in the isotropic and nematic phases, there remain four adjustable parameters which get the following values: (]" 0 = -197.2 ppm cm3g-1,!1(]" = -38.2 cm3g-1, E = 2.3.10-4 K- 1 and!1E = -63.6.10-4 K-1.

The higher value of the thermal expansion coefficient a = 8.09.10-4 of EBBA as compared to the E value indicates that the temperature dependence of the 129Xe shielding is dominated by the temperature dependence of the density, i.e. the Xe-EBBA pair correlation function contributes much less than the density. However, the situation is reversed for the shielding anisotropy, i.e. the temperature dependence of !1(]" arises almost totally from the solute-solvent pair correlation function. For the critical mixture of ZLI 1167 and EBBA, the !1E is negative as it is for pure EBBA, but its magnitude is approximately one order smaller [11]. Consequently, in this case the variation of !1(]" with temperature arises from the variation in density.

4. Quadrupole coupling In an anisotropic environment the electric quadrupole moment of a quadrupolar

noble gas nucleus couples with the electric field gradient (efg) at the nuclear site. Consequently, the total Hamilton operator (consisting of the Zeeman, shielding and quadrupole interactions) of the nucleus in the laboratory frame is (in frequency units)

'lJ 'YBo( )J qzz (J 2 2) fL = - 27f 1 - (]" Z + 4J(2J _ 1) 3 z - I (6.14)

where the symbols have their usual meaning and qzz is the quadrupole coupling constant defined as

eQ < Vzz > qzz =

h (6.15)

In Eq. (6.15), eQ (e is the positive elementary charge) is the electric quadrupole moment and < Vzz > the average value of the negative of the total efg at the nuclear site in


Table 6.2. Sternheimer antishielding factors, 100, of the quadrupolar noble gas nuclei.

Nucleus 100

21Ne -9.145 83Kr -79.98 131Xe -168.5

the direction of the external magnetic field. From the above, one can conclude that the NMR spectrum of a quadrupolar noble gas nucleus in a liquid crystal consists of 2I equidistant peaks; the spectra of the spin-312 nuclei, 21 Ne and 131 Xe, are triplets with relative theoretical intensities of 3:4:3, while the spectrum of 83Kr is a nonet with intensity ratios of 9:16:21:24:25:24:21:16:9. Examples are shown in Fig. 6.8. The quadrupole splitting, 21BI, (distance between two successive peaks)

3 21BI = 21(21 _ 1) IqZZP2(COSOBn) I (6.16)

is often large for the 131 Xe nucleus (several hundred kHz). Therefore, even small temperature fluctuations and gradients (the efg is proportional to the orientational order parameter of a liquid crystal [29]) may lead to substantial broadening of the satellite transitions (see Fig. 6.8 middle). Furthermore, the intensities of the outermost peaks of a broad spectrum may be reduced by the narrow width of the power spectrum of the exciting pulse. An example of a small 131 Xe quadrupole coupling is that measured by Loewenstein and Brenman for xenon dissolved in the mixture of PBLG and dchloroform [2]. The quadrupole splitting, 2lBxel, is only 2675 ± 50 Hz, indicating small orientational order of the medium and possibly a small total efg.

The quadrupole splitting is enhanced by the Stemheimer antishielding effect which is a measure of the susceptibility of the atom to the distortion caused by the environment [30). The Sternheimer antishielding factors of the quadrupolar noble gas nuclei are shown in Table 6.2. The total efg at the nuclear site is (1- "foo) times the efg created by the surroundings. The quadrupole splitting is scaled by the spin factor 3/21(21 - 1) (see Eq. (6.16); it is 1/2 for the spin-3/2 nuclei and 1/24 for the spin-912 nucleus), which helps in detecting 83Kr. However, acoustic ringing may present problems, as mentioned above. An additional disadvantage in the case of 83Kr is that the signal intensity is distributed among nine resonance lines [31].

The first successful 131Xe NMR experiments for natural xenon dissolved in a thermotropic liquid crystal were performed in a mixture of ZLI 1132 (55 wt%) and EBBA (45 wt%) in which the efg created by the liquid crystal molecules has been shown to be vanishingly small [32,33). However, it appeared that the 131 Xe quadrupole coupling,


t iii i Iii iii iii I Iii i' I' iii Iii' I Iii i, j iii i I 4 3 2 j 0 -1 -2 -3 -4

2000 1000 o -1000 -2000 Hz

123

Figure 6.8. (Top) 21Ne NMR spectrum of isotopically enriched neon (95 at%) in the liquid crystal ZLI 1167. 21BNei '" 3 kHz. (Reprinted with permission from [21]. Copyright (1991) Elsevier). (Middle) l3lXe NMR spectrum of natural xenon in the liquid crystal ZLI 1167. 21Bxel '" 28 kHz. (Bottom) 83Kr NMR spectrum of isotopically enriched krypton (75 at%) in the liquid crystal ZLI 1167. 21BKri '" 10 kHz. (Unpublished spectrum from this laboratory.)

iqll,Xei = 4iBxei where the subscript II refers to the efg along the liquid crystal director, is large (200-250 kHz in this particular mixture depending upon temperature), and about twice as large as in pure ZLI 1132 (100-115 kHz) [34]. Consequently, a natural conclusion is that there must be another source for the efg in this particular mixture. Generally, the total efg experienced by a noble gas nucleus in a liquid crystal is a superposition of at least two different efg contributions. This conclusion is nicely supported by 21 Ne experiments at various temperatures of isotopically enriched neon in ZLI 1132, EBBA, and three mixtures thereof. The results are shown in Fig. 6.9.

The series of 21 Ne NMR experiments allows the following conclusions. First, the absolute value, iqll,Nei, of the quadrupole coupling constant increases rapidly with increasing concentration of EBBA. Second, the temperature dependence of i qll ,N e i is different in different liquid crystal solutions, and does not behave as the orientational order parameter, revealing the fact that the total efg is due to at least two contributions.


9 34 (e) Ce)

7 32

5 30

3 28 .85 .90 .95 1.00

7 26

i (b)

e > 5 24 .. - .85 .90 .95 1. 00 '0 .:::..

~18 3 23

!i ~ Cd) :S l - 1 21

.85 .90 .95 1 . 00

7 19 (a)

5 17

J 15

13 .90 .95 1.00 .85 .90 .95 1.00

,. TO

Figure 6.9. Electric field gradients experienced by 21Ne in: (a) ZLI 1132; (b) ZLI 1132 (78 wt%) I EBBA (22 wt%); (c) ZLI 1132 (55 wt%) IEBBA (45 wt%); (d) ZLI 1132 (35 wt%) I EBBA (65 wt%); and (d) EBBA. (Reprinted with pennission from [35]. Copyright (1993) Taylor&Prancis).

Similar behaviour as shown in Fig. 6.9 for neon is also observed for 83Kr and 131 Xe [35].

A phenomenological model for the temperature dependence of the quadrupole coupling (analogous to the temperature dependence of the shielding, see section 3.2), based on an efg from the permanent dipoles of the liquid crystal molecules (referred to as external efg) on the one hand, and from the deformation of the spherical electronic


cloud on the other hand leads to the function [16]

Bnem(T) = ~[1- a(T - To))(~Bd + ~B~xt 1 T + ~Bext-T ) SLc(T) P2 (COS()Bn)

NI

125

(6.17)

which is valid for nematic phases. In Eq. (6.17), a is the isobaric thermal expansion coefficient of the liquid crystal, the term ~Bd arises from the deformation of the electron cloud of the atom, and ~Bext = ~B~xt + ~B~xtT/TNI is due to the permanent dipoles of the liquid crystal molecules, and is assumed to depend linearly upon temperature. SLc(T) is the orientational order parameter which in the analysis of experimental data can be modeled by the Haller function in Eq. (6.13). The solid lines in Fig. 6.9 are results ofleast-squares fits to the simplified form ofEq. (6.17):

(Flfot) = (1 -,oo)(A~ ext + B~xtTT )(1 - y TT Y P2(COS()Bn)

, NI NI (6.18)

where (F,fot ) is the average total efg at the nuclear site in the direction of the liquid

crystal director, and A~ ext (= const X (~Bd + ~B~xt) in the present approximation) includes the contributions from the electron cloud deformation and external efg (the temperature independent part), whereas B~xt (= const x ~B~xt) represents the temperature dependent part of the external efg [35]. The data analyses were carried out as joint fits utilizing the quadrupole couplings of 83Kr and 131 Xe when available, and neglected the temperature dependence of the density of the liquid crystals used. In some cases only 21 Ne results were used [35]. Fitting parameters are collected in Table 6.3. Munster performed 21Ne NMR experiments for neon dissolved in 5CB (4'-npentyl-4-bipheny1carbonitrile), 7CB (4'-n-heptyl-4-biphenyl carbonitrile), and 7PCH (trans-4' -n-heptyl-( 4-cyanophenyl)cyclohexane). He applied Eq. (6.18) to analyse the results assuming the density to remain constant [20]. The behaviour of the quadrupole splitting as a function of temperature is similar to that shown in Fig. 6.9: the splitting first increases with decreasing temperature, and reaches a maximum in each liquid crystal at about T /TN I = 0.97, after which it decreases monotonically. This means that the factors A~ ext and B~xt are of opposite sign. Their values are given in Table 6.3. '

NMR spectra of quadrupolar noble gas nuclei do not yield signs of quadrupole coupling constants. However, the above data analysis allows determination of relative signs of the coefficients A~ ext and B~xt; they possess opposite signs in ZLI 1132, and the same signs in EBBA. The sign combinations in Table 6.3 were deduced from the study of the molecule dideuterium in similar environments [32,33]. However, the experiments do not allow the separation of deformational and external contributions to the total efg. According to the results derived from the behaviour of the dideuterium molecule the external efg should be vanishingly small in the 55 :45 mixture of ZLI 1132 and EBBA. The 21 Ne experiments show that the coefficient A~ ext goes through zero when the concentration of EBB A increases, and the zero value appears near the 55:45 mixture. The parameter B~xt has the same sign independent of mixture composition,


Table 6.3. A~.ext and B~xt values for 21 Ne in ZLI 1132, EBBA, and their mixtures, and in three other liquid crystals.

Liquid crystal a A~.ext 11018 Vm- 2 B~xt 11018 Vm- 2

ZLI 1132 4.91 ± 0.25 -6.05 ± 0.29 ZLI 1132(78) 1 EBBA(22) 1.78 ± 0.18 -2.95 ± 0.22 ZLI 1132(55) 1 EBBA(45) 0.15 ± 0.13 -1.55 ± 0.15 ZLI 1132(35) 1 EBBA(65) -1.18 ±0.76 -2.31 ± 0.81 EBBA -1.71 ± 0.07 -3.72 ± 0.09

5CB 4.89 -6.33 7CB 5.93 -7.45 7PCH 4.00 -5.03

a The numbers in parentheses indicate the amount (in wt%) of the liquid crystal in the mixture. The data

for the first five liquid crystals are from [35], while those of the three last liquid crystals are from [20].

No error estimates are given in the latter work.

but its absolute value is minimum at the composition where the AI d,ext factor vanishes. This obviously means that the temperature-independent terms b:..Bd and AB~xt in Eq. (6.17) cancel each other at this composition . .6.B~xt must be of similar magnitude as AB;xt, but of opposite sign, i.e. it is positive. This also implies that ABd is negative.

Eq. (6.17) is valid only for nematic phases. The extension of the validity of the equation to the smectic A phase requires consideration of a density wave, and thus introduction of new order parameters as described by McMillan [36]. The quadrupole splitting becomes

Bnem+sm(T) =~[1- a(T - To)] (b:..Bd + AB~xt + AB;xtT:I)

x [SLc(T) + 2CO'l(T)T1(T)]P2(COS(}Bn), (6.19)

where a1 (T) is the mixed translational-orientational order parameter, T1 (T) is the translational order parameter in the smectic phase, and C is a temperature-independent constant which measures the positional distribution of atoms with respect to the uniform distribution [16]. The analysis of the experimental 21 Ne quadrupole coupling results (combined with the 129Xe chemical shift results) in the liquid crystals NCB 84 by application ofEq. (6.19) leads to good agreement between calculated and experimental quadrupolar couplings, as shown in Fig. 6.10. The order parameter a1(T) can be modeled with an expression similar to that for SLc(T) (see Eq. (6.13», whereas a


1

-.. to'!

:5 0 '-"

CQ 0

tID -1

= ~ ~

6 -2 CJ

~ - -3 0 e 1 -4 ::s Cf

-5 0.'87 0.9 0.93· 0.96 0.99

Reduced temperature T /TNI

Figure 6.10. 21Ne quadrupole splitting, 2BNe (in kHz), of neon in the nematic and smectic A phases of the liquid crystal NCB 84. The solid lines result from the least-squares fit to Eq. (6.19). Four points in the neighbourhood of the phase transition were omitted in the fit. (Reprinted with permission from [16]. Copyright (1992) American Institute of Physics).

more flexible function must be used for 71 (T) [16]:

T T 71(T) = (1 + x-T )(1 - Y-T Y

NI NI (6.20)

Applying certain constraints, all three order parameters, S Le, 0'1 and 71, can be derived from the analysis.

5. Relaxation 5.1 129Xe relaxation

The relaxation mechanisms of 129Xe in solution are not known in every detail. However, there is experimental evidence that in protonated solvents the dominant relaxation mechanism is the dipole-dipole interaction between 129Xe and the protons of solvent molecules [37-40]. In typical isotropic solvents, the 129Xe T1 values range


I 3 I

~ I ~ ~ 1.4

-trt :c :c .... ; ..

l to:!

Q,I ~ 2.8 Z

1.2 I

~ 2.6 - = -I 1 ~ I ~ U

2.4 U

~ ~

13O~KI ~ .... &:I.

I: 0.8 ~ I: • po(

I • po(

...-... = I ...-... ~

~

'!~i 2.2 -~ '--" 0.6 '--" I: I I I: - I -I

2 I I

0.4

1351 KI 1319 KI 1.8

0.2 I I

1.6

0 2.8 3 3.2 3.4

1000 jT(rl)

Figure 6.11. In(T1 ) (squares, axis at right) and In(T2) (circles, axis at left) as a function of 1000jT for 129Xe in the liquid crystal LC-l. Solid lines represent linear regression fits to the results in each phase. (Reprinted with permission from [10]. Copyright (1999) American Chemical Society).

from ,....,70 s to ,...., 1000 s [41]. In liquid crystal solutions, Tl is some tens of seconds. The most comprehensive investigation of the 129Xe spin-lattice (T1) and spin-spin (T2) relaxation times over the various mesophases was performed by Bharatam and Bowers [10] in the mixtures LC-l and LC-2 (for the compositions of the liquid crystals, see section 3). The dependencies ofln(T1) and In(T2) on the inverse temperature, lOOO/T, are shown in Figs. 6.11 and 6.12.

The figures show a clear change in the slopes of the straight lines when passing through phase transitions. Even the nematic - smectic A transition is nicely observable, which was not the case for the chemical shift results. The situation is very similar to the behaviour of the 129Xe self-diffusion coefficient along the external magnetic field, as will be discussed in the next section. The difference in the Tl and T2 values at all temperatures of the LC-l and LC-2liquid crystals is interpreted as an indication


3 5

(J (J (J

:= := :=

! ~ (U

2.5 ~ 4.5 Z tI) Z .. I:

• i 2

~ 4 N

N ~ I I U U (J ... 128~KI ...l ...l =- • = Q

Q 1.5 b 3.5· .... ..... = {I} ---- joooI ..... N -I. to- ~ ~ ---- ~ Q Q 1 3 ...... ......

~ 0.5 1340KI 1306KI : . 2.5 , , , , , , , , ,

0 2 2.8 3 3.2 3.4 3.6

1000 jT(rl)

Figure 6.12. In(Tl) (squares, axis at right) and In(T2) (circles, axis at left) as a function oflOOOjT for 129Xe in the liquid crystal LC-2. Solid lines represent linear regression fits to the results in each phase. (Reprinted with permission from [10]. Copyright (1996) American Chemical Society).

of the dominance of slow motional modes. The change of the slope of the straight lines of Figs. 6.11 and 6.12 indicates the change of the activation energy, Ea. In LC-l, Ea increases on going from the isotropic to the nematic phase. In addition, Ea is about three times smaller in the nematic phase (14.6 and 15.5 kJ I mol as derived from Tl and T2) than it is in the reentrant nematic phase (40.8 and 48.2 kJ I mol). However, in LC-2 the activation energies in the nematic and reentrant nematic phases are practically equal (32-36 kJ I mol). The large activation energy in the reentrant nematic phase of LC-l is most likely due to the presence of microcrystalites in the reentrant nematic phase [10]. Activation energies of similar magnitude were derived from the temperature dependence of the 129Xe Tl in the nematic phases of ZLI 1132, and the mixture ZLI 1132 (55 wt%) I EBBA (45 wt%) [9].


5.2 Relaxation of quadrupolar noble gas nuclei

The relaxation of quadrupolar nuclei is predominantly due to the interaction between the nuclear electric quadrupole moment and the fluctuating e!g at the nuclear site. As discussed in the previous section, the e!g may originate from at least two different sources: the e!g created by the liquid crystal molecules, and the e!g caused by the deformation of the electron distribution. The resonance lines of the quadrupolar nuclei are naturally broad (as shown for example in Fig. 6.7), and consequently the spin-spin relaxation time, T2, can be estimated from the linewidth at half height. For example, the 131 Xe T2 values of xenon in lyotropic and nematic liquid crystals are 1-10 ms [9, 24]. The 83Kr T1 values, obtained by the inversion-recovery method, range from ,,-,6 ms to ",23 ms in the isotropic phases of the liquid crystals ZLI 1132, ZLI 1167, EBBA and EBBA(45 wt%) I ZLI 1132(55 wt%), the activation energy being ",30 kJ I mol [42]. No relaxation results for the quadrupolar noble gases have been reported in the mesophases of liquid crystals, which is understandable from the experimental point of view.

6. 129Xe self-diffusion The self-diffusion tensor, D, of 129Xe is a valuable quantity (in addition to the

shielding tensor) when studying the structural and orientational properties of liquid crystals. A widely applied NMR technique in self-diffusion experiments is the pulsed gradient spin echo (PGSE) method [43,44]. The 129Xe spin-lattice relaxation time measurements in isotropic solutions have revealed thermal convection flows when using 10 mm sample tubes in experiments without sample spinning [38]. This is due to the commonly used temperature regulation scheme in which a stream of heated or cooled air flows around the sample tube in the probe. At elevated temperatures, this may cause thermal convection within the sample which may lead to additional echo amplitude decay and faster apparent self-diffusion than is really the case. In order to avoid convection problems, a technique based on double spin echo (DSE) pulse sequence was introduced [18]. In the DSE experiment the first moment of the gradient sequence is zero, and consequently most of the convection artifacts are eliminated. The method has been applied to the determination of the 129Xe self-diffusion tensor in the ferroelectric FELIX-R&D liquid crystal (consisting of phenylpyrimidene derivatives; from Hoechst AG, Germany) [18], and in the critical mixture ofZLI 1167 (77 wt%) and EBBA (23 wt%) [11]. In both cases, 129Xe enriched (70 at%) gas was used allowing for the experiments to be performed in reasonable time.

The pure FELIX-R&D possesses the following phases: C - 6°C - Sc* - 54°CSA -59 °C-N* -68°C-I. Fig. 6.13 displays the stack plot of the 129Xe NMR spectra at various temperatures. The shielding (chemical shift) behaviour observable at the isotropic - nematic phase transition is typical for a uniformly oriented nematic phase with its director along Bo, i.e. the anisotropy of the diamagnetic susceptibility tensor, ~Xd, is positive. The chiral nematic phases with positive diamagnetic anisotropy orient in low magnetic fields so that their chiral axis is aligned perpendicular to Bo [45]. The


310-

3'~ ~

Sc· : ~~A

;U.v·

325-~ ..,. \--' SA :

~~n c--

b ...... 335-~g r-N-

~ -"=

~

340 ;: :: I- -

t-= ~d.~ . t:--

~ ~

-194 I -1~2 I -1~0 I -lks I -1~6 I -1~4 I

-182

I~e shielding I ppm

Figure 6.13. The 129Xe spectrum of xenon in the FELIX-R&D liquid crystal at various temperatures at 11.746 T. (Reprinted with permission from [18]. Copyright (2001) Taylor&Francis).

129Xe shielding behaviour at the isotropic - nematic transition of the FELIX-R&D liquid crystal most likely is an indication of the unwinding of the helix. Consequently, the nematic phase is a non-chiral phase with positive diamagnetic anisotropy. The increase of the shielding at the nematic - smectic A transition is due to the redistribution of xenon atoms during the layer formation [16, 18].

The variation of the 129Xe self-diffusion coefficient as a function of temperature, shown in Fig. 6.14, can be described by the Arrhenius equation

Ea) D = Doexp(- RT ' (6.21)

132

-....


4

4

isotropic

nematic

smectic A ,

2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 Reciprocal temperature J 1 O'K·1

Figure 6.14. 129Xe self-diffusion coefficients in the parallel direction, DII (black circles), and in the perpendicular direction, D 1. (black triangles), with respect to the external magnetic field in the ferroelectric FELIX-R&D liquid crystal. The solid lines are from fits to the Arrhenius Eq. (6.21). (Reprinted with permission from [18]. Copyright (2001) Taylor&Francis).

where Do is the pre-exponential factor, Ea the activation energy, and R the gas constant. Eq. (6.21) is valid also in the case of an anisotropic diffusion tensor, i.e. for the elements of the tensor. Fig. 6.14 shows the results of least-squares fits to the Arrhenius equation. The phase transitions are clearly visible in the behaviour of the diffusion coefficient D II . In contrast, the shielding behaviour does not reveal the smectic A - smectic C* transition at all. The slopes of the straight lines in Fig. 6.14 change at the phase transitions, indicating that the diffusion (along the external magnetic field) activation energies vary in the different mesophases. Moreover, the diffusion coefficients in the perpendicular direction to Bo are larger than those in the parallel direction. In the nematic phase the anisotropy of the diffusion tensor, defined as D.l / D II , is ca. 1.6 and increases up to ca. 6 in the smectic phases. The faster diffusion rate in the perpendicular direction can be explained by the layer structure of the smectic phases, and additionally by the non-uniform distribution of xenon atoms. During the formation of a smectic A phase xenon atoms are expelled towards the smectic interlayer space where the density is smaller and diffusion faster along the layer surface than in and through the aromatic core region. Another reason for the restricted diffusion along the director is the finding that the chiral chain in ferroelectric liquid crystals is bent with respect to the molecular long axis [46]. This conclusion is in full agreement with that drawn from chemical shift behaviour.


-v, oj: 0

~ ., >< ~ ... 0

c ., 'u I:: ... 8 (J

c .2 III

:5 ~ "il CI)

IO

~ 1... .....

cridcal point I

t ........... ............ , :;1,'

isotropic phase

... I

-. -->:-: ~-:::.~> ....... . .. '. r ':-i ....... ). nematic phase;

nliBo

nematic phase; n LBo

I+---r-~--~--~--T---~--~~---T--~--~ 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35

r-111()l K·I

133

Figure 6.15. 129Xe self-diffusion coefficient in the direction of Do in the isotropic phase and in the nematic phases with the director parallel and perpendicular to Do in the critical mixture of ZLI 1167 and EBBA. (Reprinted with permission from [11]. Copyright (2001) The Owner Societies).

The 129Xe self-diffusion experiments performed for xenon in the ZLI 1132 liquid crystal applying the PGSE technique did not yield any anisotropy in the diffusion tensor [47]. The diffusion coefficient at room temperature in the direction of the external field, DII' (and in the direction of the director; ZLI 1132 orients along the external field) appeared to be (2.0 ± 0.3) . 10-10 m2s-1 which is 2-3 times smaller than the values measured for xenon in the nematic phase of the ferroelectric FELIX-R&D. Contrary to what was observed in ZLI 1132, a slightly anisotropic self-diffusion tensor was measured for xenon in the critical mixture of ZLI 1167 (77 wt%) and EBBA (23 wt%), D l/ DII = 0.78 [11]. This is an interesting liquid crystal mixture because the orientation of the director can be changed from the parallel to the perpendicular direction with respect to Bo by changing the sample temperature (see Fig. 6.1 bottom). Consequently, such a situation allows for the determination of the diffusion coefficients DII and D .1

parallel and perpendicular to the director simply by applying the DSE method with the Z gradient only. Fig. 6.15 shows the experimental points, Arrhenius fits, and extrapolations.

7. Conclusions NMR spectroscopy of noble gases dissolved in thermotropic and lyotropic liquid

crystals is a powerful means for deriving information on the properties of liquid crystals. In particular, due to the large polarizability of xenon the 129Xe chemical shift is sensitive to phase transitions, orientational order parameters, and liquid crystal director


orientation. The chemical shift of 3He is dominated by local magnetic fields and by bulk susceptibility effects. Quadrupolar noble gases may be exploited for probing both the electric field gradients created by the liquid crystal environment and the deformation of the atomic electron cloud in the anisotropic solvent. Combining experimental results of spin-1I2 and quadrupolar nuclei, and applying theoretical models allows for the derivation of translational and mixed translational-orientational order parameters, in addition to conventional second-rank orientational order parameters of smectic A phases. In principle, the temperature dependence of the tilt angle in smectic C phases is also obtainable from 129Xe chemical shift results, although such determinations have not been reported so far.

Acknowledgments

The author is grateful to the Academy of Finland for financial support (grant 43979).

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41:115. [6] Jokisaari, J. (1994), Progr. NMR Spectrosc., 26:1. [7] Sundholm, D., and Olsen, J. (1992), J. Phys. Chern., 96:627. [8] Raghavan, P. (1989), At. Data Nucl. Data Tables, 42:189. [9] Diehl, P., and Jokisaari, J. (1990), J. Magn. Reson., 88:660.

[10] Bharatarn, J., and Bowers, c.R. (1999), J. Phys. Chern. B,103:2510. [11] Ruohonen, J., and Jokisaari, J. (2001), Phys. Chern. Chern. Phys., 3:3208. [12] Happer, W, Miron, E., Schaefer, S., Schreiber, D., van Wijngaarden, WA., and Zeng, X. (1984),

Phys. Rev. A, 29:3092. [13] Brunner, E. (1999), Concepts in Magn. Reson. 11;313. [14] Jameson, A.K., Jameson, C.J., and Gutowsky, H.S. (1970), J. Chern. Phys., 53:2310. [15] Jokisaari, J., and Diehl, P. (1990), Liq. Cryst., 7:739. [16] Lounila, J., MUnster, 0., Jokisaari, J., and Diehl, P. (1992), J. Chern. Phys., 97:8977. [17] Kiefer, R., and Baur, G. (1990),Liq. Cryst., 7:815. [18] Ruohonen, J., Ylihautala, M., and Jokisaari, J. (2001), Molec. Phys., 99:711. [19] Jokisaari, J., Diehl, P., and MUnster, O. (1990), Mol. Cryst. Liq. Cryst., 188:189.

[20] MUnster, O. PhD Thesis, Department of Physics, University of Basel, Switzerland, 1993. [21] Ingman, P., Jokisaari, J., Pulkkinen, 0., Diehl, P., and MUnster, O. (1991), Chern. Phys. Lett.,

182:253. [22] Seydoux, R., MUnster, 0., and Diehl, P. (1994), Mol. Cryst. Liq. Cryst., 250:99. [23] Seydoux, R. PhD Thesis, Department of Physics, University of Basel, Switzerland, 1994. [24] Ylihautala, M., Ingman, P., Jokisaari, J., and Diehl, P. (1996), Appl. Spectrosc., 50:1435. [25] Ylihautala, M., Lounila, J., and Jokisaari, J. (1999), Chern. Phys. Lett., 301:153. [26] Ylihautala, M., Lounila, J., and Jokisaari, J. (1999), J. Chern. Phys., 110:6381.

[27] Haller, I. (1975), Progr. Solid State Chern., 10: 103.

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[30] Stemheimer, R.M. (1954), Phys. Rev., 95:736.

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135

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Chapter 7

NMR OF PARTIALLY ORDERED SOLUTES WITH EMPHASIS ON STRUCTURE DETERMINATION

C.L. Khetrapal and G.A. Nagana Gowda San jay Gandhi Post Graduate Institute of Medical Sciences, Lucknow, India

1. Introduction The 1963 observation of Englert and Saupe [1] that the 1 H NMR spectrum of benzene

in a nematic liquid crystalline phase has a complex appearance compared to a single line in an isotropic medium (Fig. 7.1) has led to the discovery of a new method for the determination of molecular structure. It provides the only direct method for the precise determination of molecular geometries in the liquid phase and is the latest addition to the earlier existing list of methods such as X-ray, neutron and electron diffractions and microwave spectroscopy.

During the initial period of nearly three decades from the discovery, the method was essentially used for the study of molecular structure and to explore the scope and limitations of the method. Due to its inherent utility and enormous applicability, several monographs and reviews on the subject have appeared during this period. The entire literature to date can be traced back from a recent report published by the Royal Society of Chemistry [2]. It was visualized already during the first decade after its discovery that, although the technique provides valuable information on the structure of small molecules, its utility for larger systems such as bio-molecules would be highly restricted unless some novel developments took place. This is due to the rapid increase of spectral complexity with the number of interacting nuclei, as is obvious from Fig. 7.2 [3]. Efforts, therefore, switched from assessing the scope of the method to the development of the theoretical and experimental procedures needed to apply the technique to larger molecules. Considerable success has been achieved in this direction and the use of dipolar couplings has now become a most valuable tool, particularly for bio-molecular structural studies. After a short summary of the basic principles involved in the interpretation of NMR spectra of orientationally ordered solutes, the current status and possible future directions of the field will be reviewed in this Chapter.

137 E.E. Burnell and c.A. de Lange (eds.), NMR afOrdered Liquids, 137-161. © 2003 Kluwer Academic Publishers.


_'000 Hz---<

II, ,

I .I lU II n I I -

Figure 7.1. Proton NMR spectrum of benzene oriented in a nematic phase.

2. Basic principles 2.1 The Hamiltonian

The Hamiltonian describing high-resolution NMR spectra of orientation ally ordered molecules with spin I = ~ nuclei in diamagnetic materials differs from that of the "isotropic" case in that additional terms due to direct dipole-dipole couplings and anisotropies of indirect spin-spin couplings and of chemical shifts are present. The anisotropies of indirect spin-spin couplings have the same functional dependence as the direct dipole-dipole couplings and hence cannot be experimentally separated from the direct couplings themselves.

The Hamiltonian (1{) for such systems can, therefore, be written as follows [4]:

'l..J _ ~(1 _ a~so _ a~niso)/) J. + ~(Jis.o + 2D·· + J~.niso)I· I· I L ~ t t 0 t,Z ~ tJ tJ tJ t,Z J,Z

i<j

+~ L,(JijO - Dij - ~J0niso)(Ii,+Ij,_ + Ii,_Ij,+) (7.1) i<j

In this equation, the first term gives the Zeeman interaction of nuclei in the magnetic field and (1- O'jSO) and J};o represent chemical shifts and indirect spin-spin couplings as observed in normal high-resolution NMR of isotropic media, and are one-third of traces of the corresponding second-rank tensors. The quantity ainiso is the anisotropy of the chemical shielding, Jipiso is the anisotropy of the indirect spin-spin coupling,

and Dij is the direct dipole-dipole coupling. For nuclear spin I > ! a quadrupolar term 'HQ must be added to the Hamiltonian.

NMR of partially ordered solutes with emphasis on structure determination 139

~-----...I LCH~'2 1 CH3CN _JA...------Jl---_____ _

H, /0, ,.el3 ,C-C,

H H

1 Hf=-C-CHz

L 1 CHj: !!!CH

i1 l u~l ____ ~ILJ~, ____ ~ * A 4x t)

0 Figure 7.2. Increase of spectral complexity as a function of increase of number of interacting nuclei. Proton NMR spectra of various molecules oriented in liquid crystal solvents. (Courtesy Z. Luz)

2.2 Anisotropic interactions and their relation to order parameters and internuclear distances

The direct dipolar coupling Dij between nuclei i and j is given by:

(7.2)

where h is Planck's constant, ()ij is the angle between the magnetic field direction and the axis connecting nuclei i and j separated by a distance rij, and Ii and Ij are the


magnetogyric ratios of the nuclei i and j. The average is taken over all inter- and intra-molecular motions. If i and j belong to the same rigid part of the molecule, Eq. (7.2) reduces to:

(7.3)

where Sij defines the degree of order of axis ij. Eq. (7.3) represents a very simplified picture since it neglects influences of all types

of intramolecular motions, including vibrations and conformational changes. Some of the complications arising from this simplified picture are discussed in section 5. The S-values of different axes in the molecule are interdependent, and the average secondrank orientational order of a rigid molecule is given by a symmetric and traceless matrix Spq with five independent elements. If x, y, z are molecule-fixed Cartesian axes, and Bx, By, Bz are the angles between these axes and the magnetic field direction, the tensor elements Spq are given by:

(7.4)

where p, q = x, y, z and Opq is the Kronecker delta with Opq = 1 for p = q and = 0 otherwise. Eq. (7.4) prescribes the range of the diagonal elements of the S tensor to be between -0.5 and + 1. Absolute signs of dipolar couplings can therefore be determined when the magnitude of a diagonal S tensor element is greater than 0.5.

The matrix elements Spq can be transformed to a new molecule-fixed Cartesian axis system a, b, c according to:

Saa = L cos Q pa cos QqaSpq

p,q

(7.5)

where Q pa and Q qa are the angles between the original p, q axes and the new a axis. Molecules with sufficiently high symmetry (tetrahedral, octahedral, cubic, etc.), such as methane, tetramethyl silane, fullerene (C60), and ions such as Na+, Li+, Cl-, BFt and NHt ' are expected not to orient in an anisotropic environment. The explanations of these effects range from a distortion of the electronic clouds in the case of monatomic ions and noble gases (see Chapter 6 of this book) to the effects of interaction between reorientational motion and asymmetric vibrational modes in the high-symmetry molecules [5]. To our knowledge, fullerene and SF6 are the only molecules for which no small splittings have been observed when dissolved in an anisotropic environment [6,7].

It is convenient to relate the observed chemical shielding anisotropy ainiso to elements of the shielding tensor defined in molecule-fixed axes through the transformation:

a~iso t (7.6)


Note that the shielding tensor is not necessarily symmetric. A similar relationship holds for Jttso (see Chapter 1 of this book).

When the spin system contains nuclei with spin 1 > !, an additional term 'HQ must be added to the spin Hamiltonian. This term leads to quadrupolar splittings /:).V in the NMR spectra of orientationally ordered solutes that are given by:

1 fj.v = 21(21 _ 1) Qc[3Szz + "l(Sxx - Syy)] (7.7)

where "l is the asymmetry parameter of the electric field gradient tensor (defined in terms of second-order spatial derivatives ofthe potential energy V, see Chapter 1) and is given by:

(7.8)

Q c = eQ[.. is the nuclear quadrupole coupling constant where eQ is the nuclear quadrupole moment.

3. Spectral analysis As pointed out in section 1, the spectra of orientationally ordered molecules, partic

ularly when dissolved in commonly used rod-like nematic thermotropic liquid crystals with a high degree of orientational order, become rapidly complex with increasing number of interacting nuclei. This is due to the fact that spins are in general strongly coupled in such cases. On the one hand, such a situation puts an upper limit on the number of interacting nuclei for which the spectral analysis can be carried out conveniently; on the other hand, it poses a challenge for researchers to develop ways of enhancing the method towards investigations of larger spin systems such as bio-molecules.

The standard procedures that are followed for the analysis of the high-resolution NMR spectra of isotropic systems are used in orientationally ordered systems as well, by appropriately taking into account the contributions of anisotropic interactions. The standard computer programmes, such as LAOCOONOR [8] or LEQUOR [9], are often used. Efforts have also been made to develop "automatic" procedures for analysing NMR spectra [10-18]. The "automatic" programmes avoid the assigning of experimental to calculated transitions. The use of the automated WIN-DAISY programme has been demonstrated and applied to the 1 Hand 19F spectra of 3-methyl-3-cyanocYclopropene and bis-trifluoromethyl mercury dissolved in liquid crystals [19]. The original algorithms for analyses of complex NMR spectra of molecules dissolved in nematic solvents have been improved by providing a wider choice of smoothing functions, by incorporating a principal component regression method, and by allowing the possibility of using molecular coordinates and order parameters as the fitting variables [20]. The efficiency of this procedure has been tested on sixteen molecules with spectra of increasing complexity. It was shown to be valuable for analysing the spectra of "rigid" solutes provided that a reasonable estimate of the molecular geometry was available. However, its utility decreases significantly when the guess starting


parameters are far from their real values, or when the system has a large number of independent spectral parameters. Several multi-dimensional techniques have been developed for the analysis of the spectra of specific complex systems. Many spectral editing techniques have also been developed.

4. Scope and limitations The method of dissolving solutes in liquid crystals has been extensively used to de

termine molecular geometries, anisotropies of indirect spin-spin couplings and chemical shifts, and quadrupole coupling constants. Relative signs of direct and indirect couplings have also been determined. If the sign of the order parameter, and hence of the dipolar coupling, is known, signs of indirect spin-spin couplings can be determined. Alternatively, if the signs of indirect spin-spin couplings are known, those of dipole-dipole couplings can be obtained.

It must be emphasized that the NMR spectra of orientationally ordered molecules provide only relative internuclear distances. This is obvious from Eq. (7.3) which indicates that the only observable is ~, which does not permit a separate determination rij

of Sij and r i/. Prior knowledge of at least one internuclear distance is required in order to determine absolute values of internuclear distances, as well as order parameters. The absolute sign of a diagonal element of the order matrix is positive when its magnitude is greater than 0.5. This fixes the sign of the corresponding dipolar coupling to be negative (see Eq. (7.3». This situation has been observed in benzo (l,2-C: 4,3-C')bisisothiazole [21] oriented in the nematic phase of N-(p-ethoxybenzylidene)-p' -nbutyl aniline (EBBA).

5. Practical considerations for deriving precise molecular structural information

The following practical considerations are essential before using the parameters obtained from the analysis of NMR spectra of solutes in liquid crystals for precise structure determinations.

5.1 Influence of molecular vibrations

Since dipolar couplings are averages over all intramolecular motions, the effects of molecular vibrations should be taken into account in order to compare geometrical data derived from various physical techniques such as X-ray, electron and neutron diffractions, and microwave and NMR spectroscopies. The need for vibrational corrections was realized right from the early days of NMR of orientationally ordered molecules [22], and several advances in this direction have been made subsequently [23-33]. A simple approach is to consider the stretching in-plane and out-of-plane motions of only the light atoms such as hydrogen, and to neglect the motions of the heavy atoms in the molecule. The direct dipolar coupling can then be expanded in a Taylor series about the equilibrium geometry, and terms higher than quadratic are neglected. The


quadratic terms can be conveniently approximated from harmonic force fields which are often available in the literature. The computation of linear anharmonic terms is more complicated. However, it has been pointed out that the average r a structure, derived by applying only harmonic corrections, can be compared directly with the geometrical data obtained from other techniques. A computer programme (called VIBR [29]) which uses literature values of valence force fields is available for computing vibrationally corrected dipolar couplings from which the r a structure can be derived.

5.2 Interdependence of the average order parameter and :i-r ij

As pointed out earlier, Eq. (7.3) is based upon the fact that average molecular orientational order and intramolecular motions do not vary in a correlated manner. Although this assumption is reasonable in most of the cases, changes of molecular order with intramolecular motion in a correlated manner have been shown to be important in some cases [5,33-40].

The anomalous temperature variation of the order parameter of methyl iodide [41], the apparent solvent and temperature dependence of the molecular geometry of acetylene [42], methanol [43,44], and methyl fluoride [45], and the abnormally large apparent value of the chemical shift anisotropies of methanol and tetramethyl silane [46,47] cannot be explained on the basis of simple considerations. The concept of multiplesite theory was originally introduced in order to rationalize some of these results. In this theory it has been suggested that the molecules exchange between two or more sites with slightly different geometries, and with orientation parameters of opposite signs. Subsequently, the effects of interactions between molecular rotation and intramolecular motions of solute molecules on observed dipolar couplings in cases such as benzene, acetylene, and methyl fluoride have been emphasized [39,40,48].

Large amplitude intramolecular motions, such as conformational changes, present special problems (see Chapters 13 and 14 of this book). For example, the conformational dependence of the ordering matrix of cyclopentene has been investigated, and attempts have been made to understand the results in terms of the theory for the potential of mean torque in uniaxial liquid crystals [49,50].

Situations as discussed above should serve as a warning of potential problems involved in the determination of molecular geometries from NMR studies of orientationally ordered species. These problems are especially serious for solutes with a low degree of orientational order, and for highly flexible molecules which interconvert between two or more conformers [51,52]. It may be advantageous to perform experiments under conditions of varying molecular orientational order. The problems alluded to in this section attain special significance in view of the current widespread applications of the technique to bio-molecular structural studies where weak or residual dipolar couplings are utilized (see also Chapter 8 of this book).


5.3 Anisotropy of indirect spin-spin couplings As mentioned in section 2, due to the identical orientational dependence of the

direct dipolar couplings and the anisotropic part of the indirect spin-spin couplings, the couplings observed in the spectra of orientationally ordered molecules are in fact equal to the sum of the direct dipolar and the anisotropic contributions of the indirect spin-spin couplings. Hence, in general, the anisotropic parts of the indirect spinspin couplings must be subtracted from the experimental couplings in order to obtain the direct dipolar couplings required for structural and conformational studies. In practice, anisotropic contributions to indirect spin-spin couplings have been shown to be negligible for HH couplings. Therefore, the HH direct couplings as obtained from experiments are used straightaway for structural studies. However, for other nuclei such as 19p, there are significant contributions from the anisotropy of the indirect couplings. Hence, studies of such systems need these effects to be considered before attempting to derive geometrical information [53]. Although no systematic and reliable method is yet available for the evaluation of anisotropic contributions to J couplings, the usefulness of theoretical ab initio methods has been demonstrated [54,55].

5.4 Size of the molecule and derivation of structural information

The fundamental Eq. (7.3) that relates molecular geometry and order parameters indicates that the number of independent dipolar couplings in a system should not be less than the number of independent order and geometrical parameters so as to determine the molecular geometry without an assumption other than that of a scaling distance. If the number of dipolar couplings derived equals the "sum" of the number of independent order and geometrical parameters, one can solve the simultaneous equations connecting the various parameters. However, if the number of independent dipolar couplings is less than the "sum" defined above, one has to reduce the number of independent parameters, for example by making assumptions about the molecular geometry. The minimum number of such assumptions equals the number by which the set of dipolar couplings falls short of this "sum". When the number of dipolar couplings exceeds the "sum", an iterative procedure which computes the geometry and order parameters by a weighted least squares fitting method is used. The programme SHAPE [56] is most commonly used for such a purpose.

Unless the molecule is linear, no new geometrical information can be derived for a system with fewer than four interacting nuclei. This fixes the lower limit of the number of interacting nuclei required for the derivation of structural information. The upper limit is set by experimental problems and the practical difficulties of analysing NMR spectra. Therefore, it appears that the NMR technique provides structural details only for molecules in which the number of interacting nuclei falls within a limited range. Attempts have been made to devise suitable experimental and theoretical procedures to broaden the scope of the method. Such efforts will be evaluated critically in the following section.


Means of obtaining additional geometric details in systems that normally do not provide sufficient information from a single NMR experiment have been explored and successfully employed [57--64]. It was found to be quite useful to perform experiments under conditions for which the asymmetry of the order matrix was varied, and to utilize 13C and 15N satellites in proton NMR spectra. Also, studying the resonances of nuclei other than the protons is helpful. Examples of successful experiments include studies ofN,N-dimethyl uracil [63], substituted thiophenes [60], and N-methyl imidazole [62], to mention a few.

6. Aids for spectral analysis 6.1 Multinuclear and multi pulse techniques

As in isotropic systems, multinuclear and mUltipulse techniques have been employed in orientationally ordered systems to aid in the spectral analysis, to enhance sensitivity, and to obtain information that is otherwise difficult to get. For relevant references, the reader is referred to [2] and references therein. The reader is also referred to Chapters 4 and 5 of this book.

6.2 Specific deuteration followed by high-power deuteron decoupling

The method of measuring proton NMR of specifically deuterated species in conjunction with high-power deuteron decoupling was proposed as early as 1973 [65]. If the extent of the deuteration is large such that the NMR spectrum is dominated by molecules that contain up to only two protons, the spectral analysis becomes trivial and the derived dipolar couplings can be used for the determination of the molecular structure. The utility of such a technique has been demonstrated in the analysis of the proton NMR spectra of cyclohexane [65] and cydooctane [66]. The spectra of cyclohexane with and without specific deuteration are shown in Fig. 7.3. It is evident from the figure that the proton spectrum of the 98 % deuterated species shows only molecules that contain only one or two protons. The HH dipolar couplings corresponding to species with both protons on the same carbon atom, and those separated by 3, 4 and 5 bonds in the same molecule can be derived in a straightforward manner from the observed doublet splittings. These dipolar couplings can then be used to obtain structural information. If one requires more precise information (free from the isotopic effects), one can use the dipolar couplings obtained in the above fashion as starting parameters for the analysis of the proton spectrum measured for the fully protonated solute. Since specific deuteration is not trivial, this method, although valuable in principle, has not found widespread application.

6.3 Use of deuterium spectra

Deuterium spectra may sometimes be used to aid the analysis of complex proton spectra. The deuterium spectra are dominated by quadrupole-split doublets which


2

~-...,.-...... -3

___ -----.- 5

Figure 7.3. Proton NMR spectra of cyc10hexane oriented in a nematic phase. Trace 1 is the spectrum of ordinary cyc1ohexane; the markers on trace 2 are 100Hz apart and provide the frequency scale for trace 1. Trace 3 is spectrum of cyc1ohexane-d11 . Trace 4 is the spectrum of the same sample of cyc1ohexane-d11

with deuterium decoupling (at reduced sensitivity, and the markers are 100 Hz apart). Trace 5 is the same as trace 4, but at a sensitivity 32 times larger than in trace 4. Temperature, 80°C; spectrometer frequency, 100 MHz (reproduced from [65] with permission).

provide estimates of order parameters if the values of the quadrupole coupling constants are known. The order parameters so derived can be used to estimate HH dipolar couplings to be used as starting parameters for the analysis of the more complex proton spectrum. Typical proton and deuterium NMR spectra of ethanol and CH3CD20H dissolved in a nematic phase [67] are shown in Fig.7.4A and B.

Despite the difficulties in preparing deuterated compounds, deuterium NMR ofbioand model-membrane systems has been extensively used for the study of orientational order and mobility of the fatty acyl chains in these systems.

6.4 Use of smectic liquid crystals A sample that is cooled from a higher temperature nematic phase into a smectic A

phase yields an ordered phase with its director no longer influenced by the magnetic field. When the angle a between the director and magnetic field directions is varied, the anisotropic interactions scale as (~ cos2 a - !). When a is close to the "magic angle", first-order spectra that are easily analysable are expected. The principle of the method has been demonstrated for 1,1,I-trifluorotrichloroethane in the smectic phases of p-(p-2-n-propoxyethoxybenzylideneamino)acetophenone (Fig. 7.5) [68], and fluoroform in p-n-pentoxybenzylidene-p-aminoacetophenone. The usefulness of this method is limited by the large linewidths caused by the relatively high viscosities of smectic phases.

NMR o/partially ordered solutes with emphasis on structure determination 147

A

SOOHz

Figure 7.4. (A) Proton NMR spectrum of CH3CH20H dissolved in a nematic liquid crystal. (B) Deuterium NMR spectrum of CH3CD20H dissolved in a nematic liquid crystal. (C) Deuterium NMR spectrum of CH3CD20H dissolved in CHCla. (Reprinted from [67] with permission from the copyright owner).

45° 65° eoo

Figure 7.5. 19p NMR spectra of l,l,l-trifluorotrichloroethane oriented in the smectic phase of p-(p-2-n-propoxyethoxybenzylideneamino)acetophenone. The spectra were recorded at several angles with respect to the initial direction (0°) along which the sample was oriented. Solute concentration, 20 mole %; temperature 50°C. (Reproduced from [68] with permission).


500Hz

E-O.o

E-4.S

E-S.O

E in IN/an

Figure 7.6. Influence of electric fields on the NMR spectrwn of cis-l ,2-dicholoroethylene dissolved in a nematic phase [69]. Concentration = 22 mole %, temperature = 27°C, spectrometer frequency = 60 MHz.

6.S Use of electric field in conjunction with magnetic field

If an electric field is applied to a sample in the magnetic field of an NMR spectrometer, the optic axis of the liquid crystal rotates into the direction of the electric field provided that the electric field strength is above a critical value. As in the case of smectic phases, the anisotropic interactions vary as (3cos 2a - 1)/2, where a is the angle between the electric and the magnetic fields. This method can also be used to scale the dipolar couplings. Its first application was demonstrated as early as 1969 [69] for a special case in which an electric field was applied perpendicular to the magnetic field (Fig. 7.6). This method has not been exploited, probably due to experimental difficulties.

NMR o/partially ordered solutes with emphasis on structure determination 149

------.~-3Z"C

1 500Hz '

Figure 7.7. 31p NMR spectra of trimethyl phosphite oriented in a 0.3:1 (by weight) mixture of EBBA:ZLI 1167. At 33°C. the two spectra corresponding to those at 32 and 34°C coexist.

6.6 Use of liquid crystal mixtures

It has been observed that, in a mixture of liquid crystals with opposite diamagnetic anisotropies, a director orientation change of 90° (associated with a scaling of anisotropic interactions by a factor -2) occurs at a critical temperature and concentration where macroscopic diamagnetic anisotropy vanishes [70]. Under appropriate conditions the coexistence of two orientations of the director, one perpendicular and the other parallel to the direction of the magnetic field, can be observed [71]. A typical spectrum is shown in Fig. 7.7 [72]. Such experiments have led to some novel applications of the NMR of orientation ally ordered molecules. Examples are the precise determination of molecular geometries, the determination of diamagnetic anisotropies of liquid crystals, the determination of chemical shift anisotropies without the use of a reference or a change of experimental conditions, the separate determination of dipolar and indirect spin-spin couplings, and the discovery of a unique procedure for the assignment of spectral lines in 13C spectra of oriented systems. A brief review of the subject is available [73]. However, it has been pointed out in experiments on molecules such as methane and tetramethylsilane in anisotropic media that the determination of proton shielding anisotropies is severely hampered by the presence of anisotropic "local" solvent effects [74-76]. The difficulties are less severe for other nuclei where the chemical shielding anisotropies are usually larger.

Director dynamics in mixed liquid crystals of opposite diamagnetic anisotropies near the critical point have recently been studied with sample spinning as a function of spinning speed and the angle that the spinning axis makes with the direction of


the magnetic field [77]. It was observed that a switch over of the director orientation occurs as a function of spinning speed and angle. Such experiments may serve as aids for spectral analyses.

6.7 Multiple quantum spectroscopy The use of multiple quantum spectroscopy as an aid in the analysis of normal

single-quantum spectra has proved to be useful because spectral complexity rapidly decreases with increasing multiple quantum order [78-92]. In a molecule with N interacting nuclei, the N - 1 and N - 2 quanta spectra in principle provide complete spectral information. Utility of the method was proposed and first demonstrated [79] for benzene and n-hexane. Various techniques for obtaining multiple quantum spectra have been developed.

Of the many published examples of multiple quantum spectroscopy of orientationally ordered systems (see Chapter 4), we quote a few. A study of 2,3-dimethylmaleic anhydride has been used to establish that the motion of the two methyl groups is not correlated [85]. Application to the pure nematic liquid crystal p-cyano-p'-n-pentyldu-biphenyl has shown that the dihedral angle between the phenyl planes is 32 ± 10

[83]. The technique has also been applied to biphenyl (Fig 7.8) [93]. The eightquantum spectrum was analysed first to obtain approximate dipolar couplings and the chemical shifts which in tum were used to derive precise spectral parameters from analysis of the single-quantum spectrum. The vibrationally corrected dipolar couplings were used to obtain geometrical parameters. The eqUilibrium dihedral angle of 370 found in the nematic phase is less than the value determined in the gas phase.

Multiple quantum methods have been employed to analyze the single-quantum spectrum of butane dissolved in a liquid crystal. Information on the conformer probabilities and trans - gauche energy differences were derived [94].

A double-quantum 13C NMR technique has been used to determine 13C_13C dipolar couplings in natural abundance in the nematic phase of (4-n-pentyl-4'-cyanobiphenyl) [95]. These dipolar couplings were interpreted in terms of the structure and orientational order of the cyanobiphenyl fragment.

An automatic method for the analysis of multiple quantum spectra of orientation ally ordered molecules has been described [96], and the parameters derived have been used for analysis of the single-quantum spectrum. Its utility and limitations were demonstrated for the proton spectra of bromobenzene, ethylbenzene and napthaquinone dissolved in nematic solvents.

Despite experimental difficulties and the large amount of spectrometer time required to obtain the necessary information, mUltiple quantum NMR methods applied to complex orientationally ordered molecules are expected to become increasingly important (see Chapter 4).

6.8 Near magic angle spinning experiments

A convenient method for systematically scaling anisotropic interactions in orientationally ordered systems is provided by spinning the sample about an axis that makes

NMR o/partially ordered solutes with emphasis on structure determination

I I

-10

10 15 20 Frequency/kHz

A B --

I I I I -5 0

Frequency 5

1kHz

I I 10

151

Figure 7.8. Calculated (bottom trace) and experimental (middle trace) single-quantum spectra of partially oriented biphenyl. Top trace corresponds to the eight-quantum spectrum of the same compound. (Reproduced from [93] with permission).

152

Fa Ff \}::=:={I / \

Fa Br

-


H

•

-

__ ~ ____ -rl~ ____ -~---__ ~~ ____ ----

Figure 7.9. 19F NMR spectra of bromotrifluoroethylene oriented in a nematic solvent as a function of the angle between the axis of rotation and the magnetic field. Spinning rate 70 Hz. (Reproduced from [97] with permission).

an angle a to the magnetic field direction. When a equals "magic angle" (54.7°), the anisotropic interactions vanish. When spinning close to the "magic angle", strongly coupled spectra can be reduced to weakly coupled ones, as has been demonstrated for the 19F spectrum of bromotriftuoroethylene orientationally ordered in a nematic solvent (Fig. 7.9) [97].

However, the utility of the method is limited by interference of lines from the liquid crystal medium. The use of fully deuterated liquid crystals as solvents may be helpful in such cases. As mentioned in section 6.6, the use of magic angle and near magic angle sample spinning has been widely employed to study the director dynamics of liquid crystals. Studies involving the use of side-band intensities observed in the spectra of orientationally ordered systems spinning at the magic angle are valuable for investigations of bio-molecular structure and mobility of fatty acyl chains in bio- and model-membrane systems [98].


6.9 Zero field NMR If NMR spectra of orientionally ordered molecules could be obtained at zero mag

netic field, they would be relatively easy to interpret since they depend only on two types of anisotropic parameters, viz. dipolar and quadrupolar couplings. The underlying theory and experimental procedures for obtaining such spectra have been discussed in the literature [99-101]. Further developments are essential before this approach can be employed routinely.

7. Emerging developments and possible future directions It is evident that the major hurdles to the widespread use of NMR spectroscopy of

orientationally ordered systems for the determination of molecular geometries stem from two factors: (a) the solubility of the compound of interest in an appropriate liquid crystal solvent; and (b) the rapid increase in complexity of the NMR spectrum with increasing number of interacting nuclei. Efforts of enhancing the scope of the method have concentrated on overcoming both these problems, and considerable success has been achieved. This is essentially because of the following three developments: (1) the use of natural abundance 2H NMR; (2) the use of high magnetic fields for partially orienting molecules; and (3) the discovery of new liquid crystals with low order parameters.

7.1 The use of natural abundance 2H NMR

2H isotopic substitution as an aid in the analysis of complex proton spectra has been discussed in section 6. It was pointed out that it could not be employed routinely due to difficulties associated with the synthesis of isotopically enriched compounds. However, with the development of high-field, high-sensitivity NMR instrumentation, it is now possible to obtain natural abundance 2H NMR spectra of liquid crystals [102], as well as of molecules dissolved therein [103], in a reasonable time. The 2H quadrupole coupling constants obtained from such experiments provide information on order parameters that can be used as a starting point for analyses of more complicated proton spectra. Small differences between the quadrupole splittings observed in natural abundance 2H NMR spectra and those observed in specifically deuterated samples have been attributed to deuterium isotope effects [104]. The existence of such isotope effects has been known for a long time [105, 106].

Proton decoupled natural abundance deuterium NMR has been used to study chiral molecules orientionally ordered in organic solutions of poly(,),-benzyl-L-glutamate) (PBLG) [107]. Chiral discrimination is observed through measurements of deuterium quadrupole splitting differences (Fig. 7.10).

7.2 Use of high magnetic fields for partially orienting molecules

The small orientional effects induced by high magnetic fields were first observed in isotropic solutions as early as 1986 (Fig. 7.11) [108]. These effects are proportional

154

4doublelS .-cI (o,m)

zs. zoo n< 101 Ha

00


•

CDH z

00

•

Figure 7.10. Natural abundance 2 H-{ 1 H} NMR spectrum of (± )-phenyl alcohol dissolved in PBLG at 320 K. The peaks arising from deuterons in the ortho, meta and para positions of the benzyl group are defined by the symbols ~-d (0, m) and ~-d (p). In the insert, eight lines associated with the ortho and meta deuterons of the R and S benzyl groups are shown. (Reproduced with permission from [107]).

to the square of the magnetic field and to the anisotropy of the molecular magnetic susceptibility tensor. Such experiments are of particular interest because they eliminate the need to use a liquid crystal solvent as an orienting medium. This is particularly valuable in bio-molecular structural studies since the information that can be obtained from anisotropic couplings can be combined with that obtained from Nuclear Overhauser enhancement (NOB) and scalar coupling measurements in order to obtain refined structures in liquid phase (see Chapter 8).

The availability of high-field spectrometers (17.6 T) has enabled the study of the magnetic field orientation of duplex and quadruplex DNA [109]. The observation of small magnetic field induced splittings in nucleic acids and proteins in solution appears more feasible via heteronuclear couplings than by 2H NMR. Magnetic field induced orientation of the diamagnetic protein 15N emiched human ubiquitin has been observed by measuring 1 JNH at three different magnetic field strengths, and using two different techniques [110]. The results indicate two contributions to the magnetic field dependence of the splittings: an orientation independent contribution caused by a dynamic frequency shift; and an orientation dependent shift reSUlting from 15N_1H dipolar couplings. The apparent field dependence of one-bond J-couplings between Cll<_Hll< for the 13e labeled protein ubiquitin has been attributed to residual dipolar couplings due to weak orientation of the protein in the magnetic field [111].


Figure 7.11. Nonnal and resolution enhanced 2H NMR spectra of nitrobenzene-d5 dissolved in ether at 14.09 Tesla. (Reproduced from [108] with permission).

The solvent dependence of H-H dipolar couplings in strong magnetic fields has been used to characterize aromatic-aromatic interactions of benzene, hexafluorobenzene, naphthalene, and some mono-substituted benzenes [112]. The solvent dependence of the dipolar couplings allows the determination of the association thermodynamics of these aromatic compounds. The results show that benzene, naphthalene and some mono substituted benzenes tend to stack parallel to hexaftuorobenzene. For benzenenaphthalene complexes, evidence for a T-type structure was indicated. These results can be understood on the basis of quadrupole-quadrupole interactions, where it should be realized that the sign of the molecular quadrupole of hexaftuorobenzene is opposite to that of the other aromatics.

The fine structure that has been observed in the 500 and 750 MHz 1 H NMR spectra of the hydrated fullerenes C60H2 and C60H4 was attributed to the magnetic field induced orientation of these molecules [113, 114]. At 750 MHz a splitting of 0.30 Hz results from the residual dipole-dipole interaction between the two magnetically and chemically equivalent protons in C60H2. A corresponding splitting of 0.74 Hz was observed for C60H4.

7.3 The discovery of liquid crystals with low order parameters

Although the potential use of liquid crystals with a low degree of orientational order for obtaining first order spectra of solute molecules was envisaged over two decades ago [115], major interest in this area has developed over the last few years. It is obvious that such experiments will be increasingly important in NMR investigations of the molecular structure of bio-molecules. Historically, the use of lyotropic liquid crystals that provide weakly coupled spectra of solute molecules dates back to the 1960's [116]. However, a real breakthrough in the field was realized when these meth-

156 NMR OF ORDERED UQUIDS

ods were extended to weak molecular alignments obtained using different orienting media. One example is the use of lipid bilayer fragments, called "bicelles" [117], produced from mixtures of long-chain phospholipids such as Di-Myristoyl Phosphatidyl Choline (DMPC) and Di-Hexanoyl Phosphatidyl Choline (DHPC) in aqueous buffer [118]. One can dilute such a bicelle solution to produce a cooperatively oriented homogeneous medium with spaces between the bicelles that are large enough to accommodate soluble proteins. The orientational order can thus be tuned to yield favourable residual dipolar splittings by adjusting the concentration of the bicelles in the mixture [119]. Another type ofbicelle which aligns with its major axis parallel to the external magnetic field can be made from I-Dodecanoyl-2-(Biphenyl-4-acetyl)-Sn-glycero-3-Phospho Choline (DB PC) and 1,2-Di-Hexanoyl-Sn-glycero-3-Phospho Choline [120]. The suitability for biomolecular structural studies was tested by recording an HMQC spectrum of a trisaccharide. Distances and angles derived from 1 H_15N, 1 HCt_13CCt and l3CCt_13C' dipolar couplings of human ubiquitin dissolved in a bicelle medium agree well with crystal structure data [121]. Highly accurate one-bond N-H, C-H, C-C, C-N, and two-bond C-H dipolar couplings have been measured in l3C /15N enriched ubiquitin orientationally ordered in bicelles [122]. Vibrationally corrected bond lengths have been obtained. Using the C'-N bond length of 1.329 A as the scaling distance, the CCt-C', N-H and CCt_HCt bond lengths have been calculated. While the CCt-C' distance of 1.526 A is found to be in agreement with crystal structure studies, the N-H and CCt_HCt distances are considerably longer. This is attributed to fast librations of these bonds resulting in a reduction of the corresponding observed dipolar couplings.

In recent years, the use of residual dipolar couplings for the NMR determination of the structure of complex bio-molecules has undergone rapid growth, as evidenced by several recent review articles [2,123-127]. In a review with 84 references the elucidation of protein structure and dynamics through field-induced dipolar couplings is presented [123]. Another review on the NMR of nucleic acids and proteins with 256 references has a substantial portion devoted to the measurement of residual dipolar couplings in samples where the solute is partially oriented, either by a magnetic field, or an orienting medium [124]. A survey of the use of various methods, including the use of residual dipolar couplings in biological systems, is also available [127].

A three dimensional experiment for the measurement of proton-proton dipolar couplings in partially oriented proteins has been described [128]. Resonance assignments and local structure refinements have been obtained, and protein folding has been studied. A method to obtain the sign of the residual dipolar coupling in the case of methyl protons (which is normally not possible due to absence of J couplings) has been proposed [129]. The pulse scheme employed creates magnetization via the one-bond 13C_l H coupling of the methyl group, such that the sign of the cross peak reflects relative signs of DHH and 1 J13CH'

An alternative method for weakly orienting biological macromolecules that is based on anisotropically compressed and stretched polyacrylamide hydrogels [130, 131] has been proposed and used [132,133]. The structures of peptide fragments of HIV-l envelope protein gp41 (residues 228-304), when bound to micelles and bicelles, have

NMR of panially ordered solutes with emphasis on structure determination

Protons (1.4)

Isotropic in MTAI

157

Figure 7.12. Proton NMR spectra of cis, cis-mucononitrile: (top trace) in ZLI 1167; (central trace) 5.4 % in MTAI; and (bottom trace) isotropic phase in MTAI.

been compared. It has been demonstrated that detergent micelles can induce significant strain in attached helical peptides. This confirms that small bicelles are well suited for studying the detailed peptide structure in a biologically relevant environment. Unambiguous identification of the weak curvature observed in the detergent micelle environment is generally not possible by conventional NMR, using NOEs and J couplings, but it can be determined quantitatively by residual dipolar couplings. Considering the usefulness of lyotropic media for obtaining NMR spectra of systems with low orientations, it has been considered worthwhile to explore the existence of thermotropic liquid crystals that can provide weakly coupled spectra. To this purpose, neat phases of quaternary ammonium halides based on trioctylamine have been studied [134, 135]. They exhibit liquid crystalline behaviour within certain temperature ranges, and provide weakly coupled proton NMR spectra of solute molecules. The spectrum of cis, cis-mucononitrile is shown in Fig. 7.12 as an example. As is evident from the top trace, the proton spectrum of cis, cis-mucononitrile in the liquid crystal


ZLI 1167 is strongly coupled, and it is difficult to assign lines to specific protons. The spectrum in the liquid crystalline phase of N-methyl- N,N,N-trioctadecylammonium iodide (MTAI) (middle trace) shows essentially two groups of lines corresponding to protons 2,3 and 1,4, as shown in the figure centred nearly around the isotropic positions of these protons. The splittings within the two groups of lines provide (J + 2D) 12

and (J + 2Dh3, whereas (J + 2Dh4 falls within the line width. This illustrates the possible use of such phases for obtaining first order spectra. As a by-product of such investigations, it has been observed that solutes such as dimethyl sulphoxide and methyl iodide induce another phase identified as a smectic phase within certain temperature and concentration ranges.

8. Conclusions Three recent developments, viz., the observation of natural abundance deuterium

NMR spectra of orientationally ordered molecules, the use of molecular orientation by high magnetic fields, and the discovery of liquid crystals with a low degree of orientation, have revolutionalized the application of NMR to structural studies of complex bio-molecules. These developments have resulted in an explosive growth of activity in this area. However, when exploiting the small residual dipolar couplings that are observed, either by magnetic field induced orientation, or that result from the use of weakly oriented anisotropic media, considerable caution is required [136]. Some of the possible pitfalls have been discussed in the present Chapter.

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Chapter 8

OBSERVATION AND INTERPRETATION OF RESIDUAL DIPOLAR COUPLINGS IN BIOMOLECULES

Jean-Fran~ois Trempe and Kalle Gehring Biochemistry Department, McGill University, Montreal, Quebec, Canada

1. Introduction Residual dipolar couplings were introduced a few years ago as a new type of con

straint in biomolecular structure determination. Thereafter, the field has literally exploded, with over a hundred and fifty articles published since 1995. The method is increasingly popular because of the relative ease with which spectral measurements can be made compared to NOE analysis, and the high amount of information about structure and dynamics of macromolecules that can be extracted.

Previously, high-resolution structure determinations offolded polypeptides by NMR relied heavily on the measurement of small proton-proton distances from NOE spectra, as well as on dihedral angle restraints from J-coupling measurements [1]. Anisotropic interactions such as the dipolar coupling and chemical shift anisotropy normally vanish in solution due to isotropic molecular tumbling. Partial orientational ordering through direct magnetic ordering of the substrate [2-4] or dissolution of the biomolecules into a partially ordered liquid crystalline medium [5,6] can bring back those interactions. Weak orientational order is normally sought in order for the NMR relaxation properties to be as close as possible to those in the isotropic liquid state, as well as to obtain firstorder spectra.

Incorporation of anisotropic interactions into biomolecular structure determination generally proceeds through the following five steps:

1 Choice of the proper orienting system for the biomolecule under study.

2 Measurement of residual dipolar couplings and chemical shift anisotropies using appropriate pulse sequences.

3 Preliminary analysis of the anisotropic interactions to extract the order parameters needed for the structure calculation.

4 Calculation of a three-dimensional structure that is consistent with the experimental results.

163 E.E. Burnell and c.A. de Lange (eds.), NMR of Ordered Liquids, 163-190. © 2003 Kluwer Academic Publishers.


5 Analysis of the internal dynamics, if required.

This Chapter describes the physical basis of these weak anisotropic interactions and their applications, in the context of the five steps enumerated above. First, a short theoretical background will be given. In the following section, the various techniques employed to induce weak partial ordering ofbiomolecules in solution and to measure residual dipolar couplings and chemical shift anisotropies will be summarized. In the final section, the interpretation of anisotropic interactions in terms of structure and dynamics of a macromolecule will be reviewed. Readers interested in a more detailed description of the latter topics are referred to recent review articles [7-10].

2. Theory The theory ofNMR of orientationally ordered molecules in an anisotropic environ

ment is reviewed in Chapter 1. Here we summarize the aspects that are relevant to the interpretation of the residual dipolar couplings and chemical shift anisotropies that are measured in biomolecules with weak orientational order. The description is simplified by the fact that the induced degree of orientational order is sufficiently low that the resulting NMR spectra can be described in the high-field weak-coupling limit which leads to first-order spectra.

Under these circumstances, splittings observed between two magnetically inequivalent spin 1/2 nuclei are given by Jliso + 2Daniso + Janisol. For the light nuclei commonly found in biological systems, the term Janiso is safely neglected. From Eqs. (1.63 and 1.64) of Chapter 1, the residual dipolar coupling between nuclei p and q that reside in the same molecule can be written (in Hz) as

Daniso IPlqh 2 pq - 47r2 3"

x I: / (3 cos Opq,o. co~ Opq,(3 - 00.(3) x (3 cos Oz,o. cos OZ,(3 - 00.(3)) (8.1)

(3 \ 2~ 2 0.,

where a and {3 are the molecular-fixed x, y, z axes. In addition, rpq is the internuclear distance, IP and Iq are the gyromagnetic ratios of the two nuclei, Opq,o. is the angle between the internuclear direction and the axis a, and 0 z,o. is the angle between the axis a and the laboratory-fixed Z magnetic field vector. The angular brackets indicate averaging over both inter- and intramolecular motions.

In most cases applicable here, for example the one-bond and two-bond heteronuclear dipolar couplings measured in proteins and nucleic acids, the two nuclei of interest reside in the same "rigid" fragment of the molecule. This fragment can be a pair of directly bonded atoms, a peptide plane or rigid sugar ring, a secondary structure element, or even a whole folded protein domain. In these cases, the effects of smallamplitude vibrations and librations can be treated such that averages over intra- and intermolecular motions are evaluated separately. Under these circumstances, averaging

Observation and interpretation o/residual dipolar couplings in biomolecules 165

over intermolecular motions leads to the familiar Saupe order matrix,

Sa{3 = \ ~ cos (lz,a cos (lz,{3 - ~Oa{3 ) (8.2)

and the motional average of the term < rpq3 > can be written as r;q~eff (see Table 8.1 in section 3 for values of r pq,eff). For large-amplitude conformational motions, the above separation is not valid for dipolar couplings between nuclei that reside in different "rigid" subunits of the interconverting molecule. Such complications are dealt with in Chapters 12, 13, 14, and 19. However, this is rarely the case for interactions normally measured in biological macromolecules, and (with the exception of some long-range IH_IH couplings) a rigid fragment can always be defined for an interaction.

Assuming that internal and reorientational motions can be separated, and that the nuclei p and q reside in the same rigid subunit of the macromolecule, in the principal frame of the order tensor, Eq. (8.1) reduces to

A "(p"(qh - rdc 2 3

47r r pq,eff

X [Szz (~cos2 (I -~) + ~ (Sxx - Syy) sin2 (I cos 2¢] (8.3)

where (I and ¢ are the polar angles in this frame, as depicted in Fig. 8.1. The "generalized order parameter" Ardc (often denoted S) which can be estimated from relaxation measurements is frequently introduced to account for rapid angular fluctuations of the internuclear vector (in which case r;~eff must be taken to account for bond stretching vibrations only). In the biomolecular literature, the residual dipolar coupling is defined as 2D;~iso which is equivalent to the D~q of Eqs. (1.11)-(1.13) of Chapter 1 of this book. In this context, the dipolar coupling equation is often written:

D~q((I,¢) = 2D;~iso((I,¢) = Da [(3cos2(1-1)+~RSin2(1cos2¢] (8.4)

where Da and R are the magnitude and rhombicity of the dipolar coupling tensor, defined in Hz as [11]:

D A "(p"(qh S a = - rdc 4 2 3 zz

7r r pq,eff (8.5)

The symbol A is frequently used in the biological NMR literature to represent the Saupe molecular (or rigid fragment) order tensor of Eq. (8.2).

Similarly, the chemical shielding of a nucleus p can be written as

with

iso 1 ( ) O'p = 3" O'p,xx + O'p,YY + O'p,zz . (8.7)


z

q

~~-"---y

x

Figure 8. I. Angles for the description of anisotropic interactions in orientationally ordered molecules. In the principal axes of the order matrix, only the polar angles «(), ¢) are required to calculate the dipolar coupling between p and q, using Eq. (8.3).

The terms (jp,iOt are the angles between the ith axis of the chemical shielding tensor and the axis a of the order tensor. It is important to write both the order tensor and the chemical shielding tensor in the same molecule-fixed axis system. The principal axis systems of these two tensors are, in general, different, and the angles between them are a priori unknown.

3. Measurement of residual dipolar couplings It is evident from theory that the observation of anisotropic NMR interactions re

quires that the molecule under study has orientational order. As we will see in the present section, there are two general ways to achieve this. Thereafter, NMR measurements of induced residual dipolar couplings will be discussed.

3.1 Magnetic ordering of biomolecules The technique has its roots in the development of ultra-high field NMR instrumenta

tion, where molecules with relatively low magnetic susceptibility become sufficiently orientationally ordered to express dipolar couplings [2, 12]. Molecules with all electron spins paired generally possess an overall magnetic susceptibility which is considerably smaller than that for molecules with unpaired spin. Therefore, it is not surprising that the first experiments on orientational ordering in a magnetic field were performed using paramagnetic complexes and ions that possess unpaired electron spin. The degree of orientation for diamagnetic molecules is much smaller, but has been observed for several polyaromatics.

The first study of a magnetically ordered protein was done by Tolman et al. using paramagnetic iron-bound myoglobin [3]. Molecules of biological interest with paired spin, such as nucleic acids that possess relatively large magnetic susceptibility anisotropies due to the stacking of aromatic bases, also exhibit orientational order in strong magnetic fields [13].

Observation and interpretation of residual dipolar couplings in biomolecules 167

In general, the magnetic susceptibility of a molecule is anisotropic and is described by a second-rank tensor. The interaction between a magnetic field and this anisotropic susceptibility leads to a small degree of orientational ordering of the molecule (see also Chapter 1). The energy of this interaction is given by

1 - -E = --Bo . X· Bo 2

(8.8)

where Eo is the magnetic field vector, and X is the anisotropic magnetic susceptibility tensor.

An expression for the orientational probability distribution function can be derived from Eq. (8.8). It is thus possible to write the elements of the Saupe order tensor in terms of the magnetic susceptibility tensor. The principal axis system of the order tensor must coincide with that of the magnetic susceptibility tensor. An expression similar to Eq. (8.3) can be formulated for the dipolar coupling in a magnetically ordered molecule [4].

D;~s = -(1~~~) 7;i:~ [~Xa(3COS20 -1) + ~~XTSin20COS24>] (8.9)

where T is temperature, k is the Boltzmann constant, and ~Xa and ~XT are the axial and rhombic components of the magnetic susceptibility tensor (~Xa = Xzz - (Xxx + Xyy)/2; ~XT = Xxx - Xyy)·

Usually, measurements are done at two different magnetic fields, and the residual dipolar couplings observed in each case are subtracted. This allows removal of the isotropic coupling contribution. The subtraction of two total couplings yields the difference of the residual dipolar couplings.

Since the interaction is proportional to the square of the magnetic field intensity, it is preferable to use the highest field strength possible to increase the range of observed residual dipolar couplings. Even then, the method generally yields very small orientational orders (one order of magnitude lower than in a dilute liquid crystal medium), and only intrinsically large couplings can be observed. For example, the 15N_1HN

one-bond residual dipolar coupling is generally less than 2 Hz at 750 MHz considering magnetic ordering alone [4]. To be useful, these small couplings must be measured with high levels of precision (~ 0.1 Hz) using time-consuming NMR pulse sequences (see section 3.3). In addition, the measurement of such small contributions is hindered by the presence of other small field-dependent effects such as the dynamic frequency shift from relaxation interference effects [14].

The use of paramagnetic ions overcomes these difficulties as they have larger susceptibility anisotropies than diamagnetic molecules and thus provide larger couplings. Moreover, pseudo-contact chemical shifts originating from the paramagnetic centre can be used as an additional type of structural constraint. Cyanometmyoglobin complexed with an iron-bound heme group has been shown to orient significantly in a 750 MHz NMR spectrometer, with 15N_1HN couplings ranging from -3 to +4.5 Hz [3]. The experimental data (couplings and shifts) satisfactorily fit the tertiary structure,


determined by an independent technique. The method has been extended to other paramagnetic proteins such as cytochrome b5 [15], lanthanide-binding proteins [16] and rubredoxin [17]. Very few applications of couplings measured in diamagnetically ordered molecules have been published, except in a DNA:protein complex [4].

A clear advantage of direct magnetic ordering is that no change to the sample is required, and measurements are done under the same conditions at which other structural constraints have been extracted. Also, as noted by Prestegard et al., because the nature of the orientational order is known and predictable to some extent, the dynamical and structural interpretation of the dipolar couplings is less ambiguous and requires less data for the determination of the order tensor [7].

3.2 Orienting media Another means of inducing partial ordering in biomolecules is to dissolve them into

a dilute liquid crystalline or partially oriented medium [6]. This method is more widely used than direct magnetic ordering, for it is of more general applicability and produces larger couplings with the magnitude of orientational order in the range of 10-3 . It also allows adjustment of the degree of molecular orientational order by variation of the liquid crystal concentration [5].

Liquid crystalline media used for this purpose have to form homogeneous suspensions or solutions in water, the natural solvent for most biological macromolecules. Moreover, they should stay oriented at low nematogen concentration, such that the orientational order imparted to the solute is weak and does not broaden NMR signals due to multiple unresolved dipolar couplings. This is usually accomplished using lyotropic liquid crystalline mesogenic particles of high aspect ratio. The ideal medium should be stable over the wide range of experimental conditions (pH, ionic strength and temperature) that may be required for stability of the biomolecule. The degree of orientational order is normally monitored using the quadrupolar splitting of partially ordered D20 molecules.

Two contributions are recognized as being of importance in determining the components of the macromolecular order tensor. The first contribution is described by a steric model, in which orientational order is induced by steric exclusion interactions of the macromolecule with the liquid crystalline aggregates. In this case, it is the shape anisotropy of the macromolecule that dictates the Saupe order tensor elements. The principal axes of the order tensor will be roughly coincidental with those of the rotational diffusion or moment of inertia tensors [18]. The exact contribution of this short-range effect to the order tensor can be simulated numerically [19] or analytically [20] by considering the problem of rigid particles in the presence of large obstacles. The latter work approximates the protein as an ellipsoid, which enables a fast determination of the predicted coupling from shape, and can thus be implemented in a structure calculation protocol. However, the accuracy of these simulations seems to be reduced in the case of high rhombicity.

The other contribution arises from electrostatic interactions of the macromolecule with its environment. These interactions are far more complex than the steric ones.


They depend on the electrostatic charge distribution in both the analyte and the liquid crystalline particles. They vary with the pH and ionic strength of the solution, the pKa's of the acid-base groups on the surface of the molecules involved, as well as the orientation of molecular fragments with strong localized electric dipoles. Electrostatic interactions are particularly strong between charged particles, or particles with localized charges. It is well known that if the two partners have opposite charges, they will attract each other and this favours interactions that increase the orientational order of the molecule to values that can be larger than desired. Even if the macromolecule is neutral, a heterogeneous charge distribution can modify the interaction strength and therefore the orientation.

Some biomolecules may be incompatible with a particular orienting medium. Therefore, it is highly beneficial to have access to a large variety of orienting systems. Many research groups have contributed to the development of new media for measurements of residual anisotropic interactions in biomolecules. Here, we list some of their basic properties, advantages and drawbacks, in the light of the general principles described above.

3.2.1 Thermotropic lipid mixtures. A mixture of dimyristoyl phosphatidylcholine (DMPC) and dihexanoyl phosphatidy1choline (DHPC) in a 2.9: 1 molar ratio was the first liquid crystalline medium to be used for measurement of residual dipolar couplings in proteins [5]. Initially designed as models for biological membranes, they were proposed to form disc-shaped aggregates called bicelles, where the long-chain DMPC molecules form a bilayer and the rim of the discoidal particles is made from the shorter DHPC molecules [21]. Recently, another model based on translational diffusion anisotropy has been proposed [22]. The large planar surfaces punctured with disc-shaped holes cooperatively order with their normals perpendicular to the magnetic field, due to their negative magnetic susceptibility anisotropy (Fig. 8.2A).

The phase diagram of DMPC I DHPC mixtures shows a strong concentration, temperature, and ionic strength dependence. Below some critical temperature, the sample is isotropic, clear, and non-viscous. The temperature at which the isotropic-to-liquid crystal transition occurs is determined primarily by the salt concentration, the DMPC I DHPC molar ratio, defined as q, and the total lipid concentration [23]. For example, a 5% w I v mixture with q = 3.0 forms a stable ordered phase at 33 - 45°C, but only at NaCI concentrations below 100 mM. Typically, lipid concentrations between 3-7% w I v are used, and a q of 2.7-3.5. Addition of unsaturated lipids lowers the transition temperature by a few degrees [23]. The use of ditridecanoyl phosphatidylcholine (DTPC) has been proposed as a substitute for DMPC to lower the critical temperature. However, the isotropic phase of the latter is very viscous. This leads to broad spectral lines that prevent measurement of scalar couplings which therefore have to be measured in the absence of lipids. Another lipid mixture is that of dilauroyl phosphatidy1choline I 3-( cholamidopropyl)dimethylammonio-2-hydroxyl-1-propane sulfonate (DLPC I CHAPSO) which is liquid crystalline above 7°C in a 4.2: 1 molar ratio [24]. An advantage of using CHAPSO instead of DHPC is that the short-chain lipid is not hydrolyzable at low or high pH.

170

A

c

o 0 o

E ?

E ? E ? e=:=:=-?

e=:=:=-? E ? E ?


B

or

D E

Figure 8.2. Schematic representation of orienting media used to generate residual dipolar couplings in biological macromolecules. (A) OMPC / OHPC lipids in aqueous solution which form either punctured membrane planes or disc-shaped bicelles with their normals perpendicular to the magnetic field. (B) Nematic filamentous bacteriophages that order parallel to the magnetic field. (C) Oriented purple membrane fragments with their membrane normals parallel to the magnetic field. (0) A cellulose crystallite chiral nematic phase. The particles are orthogonal to the magnetic field, but the chiral twist is along the magnetic field. (E) A vertically stretched polyacrylamide matrix in aqueous solution.

Doping the membrane with charged amphiphiles can extend the temperature range of stable liquid crystallinity [25,26]. In the latter case, electrostatic repulsion between the bicelles prevents their aggregation. Examples of charged lipids that have been used are cetyltrimethylammonium bromide (CTAB, positively charged) and sodium dodecyl sulfate (SDS, negatively charged). An additional benefit of the latter method is that it provides a way of adjusting the mechanisms of orientational order for charged proteins and nucleic acids [25]. Without charged lipids, the bicelles are nearly neutral and interact sterically with the biomolecule.

3.2.2 Surfactant / hexanol mixtures. Alternative liquid crystalline surfactant solutions for biomolecular NMR have also been developed. The first one consists of a mixture of cetylpyridinium chloride / bromide (CPCl / CPBr), the corresponding sodium salt, and n-hexanol [27,28]. Cetylpyridinium / hexanolliquid crystals are stable at NaCI concentrations of 200-500 mM, while only 10-50 mM NaBr is required for a stable CPBr phase. The former exhibits an a-lamellar phase, the so-called Helfrich phase [27]. The phase diagram and the ordering properties of the latter have been thoroughly studied [28,29]. This system is lyotropic with the ordered phase being stable at total mass concentration of 2-6% and over a wide range of pH (2-8) and buffer types. The optimal molar ratio of cetylpyridinium to hexanol is 1: 1.33 (w / w). The morphology of the liquid crystal is still under debate, but evidence from translational diffusion anisotropy measurements indicates that the structure is an a-lamellar phase with the


bilayer normal perpendicular to the magnetic field [22]. In contrast, a recent electron microscopy study has shown a worm-like micellar structure [29]. Nevertheless, the phase does order with the magnetic field. It has been shown to induce observable residual dipolar couplings in several proteins [27-29]. The cetylpyridinium polar head is positively charged at neutral pH, and thus the mechanism of orientational order should be mostly electrostatic in the case of transiently interacting proteins and nematogens.

Another promising medium is the analogous lamellar phase formed from a mixture of alkyl poly(ethyleneglycol) (PEG) and n-hexanol [30]. This system also forms a lamellar phase, but it is different in that it is uncharged. It is also very stable upon pH and temperature variations.

3.2.3 Rod-shaped viruses. Rod-shaped viruses are long, highly anisotropic particles, made of single-stranded DNA or RNA coated with helical proteins. Three viruses have been proposed: fd, Pfl bacteriophages, and the tobacco mosaic virus (TMV) [31,32]. Their average lengths and diameters (in nm) are 880 / 6, 2000 / 6, and 300 / 18, respectively [33,34]. Aqueous suspensions of these viruses form lyotropic chiral nematic phases (Fig. 8.2B). The critical mass concentration for the liquid crystalline phase is fairly low for fd and Pfl bacteriophages, and depends on temperature, ionic strength, and contour length of the particle [35]. The Pfl andfd phage liquid crystals are stable over a wide range of temperatures (5-60°C) and salt concentrations (0-600 mM NaCl) [34,35].

In contrast to neutral bicelles, bacteriophages have a high negative charge at physiological pH with an isoelectric point around 4 [33,34]. At pH 7, the charge density is approximately 10 e- / nm in water. It has been reported that filamentous phages aggregate at pH 4 [33], but thefd phage was shown to be stable below pH 4 where it is positively charged [34].

Electrostatic interactions are the main contributions to the orientational order of solute biopolymers in phage. Consequently, the orientation and magnitude of the order tensor is highly dependent on the ionic strength and pH [34,35]. High salt concentrations screen the charge of the filamentous phage and reduce the effective diameter of the particle, resulting in a more steric type of orientational ordering. If the pH of the solution is such that the phage and the biomolecule have opposite charges, strong transient bonding results, and the protein NMR signals may vanish due to a reduced tumbling rate. In the converse case where charges are the same and proteins are repulsed by phage particles, the alignment mode will be a simulacre of a purely steric alignment mechanism. This was observed in the archaeal protein EFl,6, as shown in Fig. 8.3 [36]. The concentration range for proper weak orientational order depends on the type of virus. Approximately 1-20 mg / mL are used for Pfl, and twice as muchfd phage is required for similar magnitudes of order tensor elements, as their length is half that of Pfl with a reSUlting order parameter reduced by a factor of two.

3.2.4 Purple membrane fragments. Introduced by Koenig et at. [37] and Sass etal. in 1999 [38], the purple membrane (PM) system has been extensively studied by biochemists for years. Isolated from extremophile Halobacterium salinarium cells,

172

A

~ 20 N

'-' 15

~ 10 .s a. 5 j e O Pm"~r---~ .. ~ ;; -5

8. -10

is -15


B

z

,J "

-20 L--~ __ ~ _____ ~ __ ...!.-__ ,--~_~ __ -'

o 10 20 30 40 50 60 70

Residue number o 90

Figure 8.3_ Fold recognition and ordering properties of the archaeal translation elongation factor l-/3 from Methanobacterium thermoautotrophicum in 20 mg / mL Pfl bacteriophage. (A) 15N_1 H residual dipolar couplings plotted against amino acid number. Grey bars are for residues in a-helices and black bars are for residues in loops and l3-sheets. (B) Cartoon representing the three-dimensional backbone structure of the protein. Secondary structure elements are shown as ribbons. The principal axes of the order tensor are shown on the right. These axes are approximately coincidental with the principal axes of the moment of inertia tensor. N-H vectors are roughly parallel to the helix axes, and thus parallel to the magnetic field. Conversely, N-H vectors are perpendicular to the orientation of l3-strands which are parallel to the major principal axis of the order tensor_ The topology of the protein (l3-a-I3-I3-a-l3) is easily recognized from residual dipolar couplings_

they contain the integral membrane protein bacteriorhodopsin which uses energy from light to keep a steep ionic concentration gradient across the cell membrane in high salt water (5 M). They do not form a lyotropic liquid crystalline phase, but order completely in a strong magnetic field (> 10 Tesla), with their normal aligned parallel to the magnetic field (Fig. 8.2C). This is attributed to the large size of the PM fragments (0.5 to 2 /-Lm) and to the magnetic susceptibility of bacteriorhodopsin helices which are orthogonal to the membrane surface. The latter property is quite useful, as it sets no lower limit to the concentration of the orienting medium. PM fragments are also stable over a wide range of temperatures. However, they are very sensitive to the ionic strength. Salt concentrations greater than 50 mM cause aggregation of the PM fragments.

The large PM fragments have a high negative charge, and electrostatic interactions are the main contribution to the orientational order of macromolecules [37, 38]. As a consequence of their high surface charges, a very low amount of PM (typically 1-4 mg / mL) is required to obtain orientational order in the range of 10-3 . This has been demonstrated with ubiquitin, p53 [38], the Va domain ofthe T-cell receptor [37], and protein L [39].

3.2.5 Cellulose crystallites. Aqueous suspensions of high aspect ratio cellulose particles form a lyotropic chiral nematic liquid crystalline phase [40,41]. Crystallites can be prepared by sulfuric acid hydrolysis of Whatman paper and dialysis against water [40]. They measure about 200 by 10 nm and they possess a negative


susceptibility anisotropy with respect to their long axis. Thus they align orthogonal to the magnetic field, with a chiral twist along the field orientation (Fig. S.2D). The mesogenic concentration range is from 4.5-15%, as higher concentrations are too viscous. Their stability against different experimental conditions is still unknown. However, they are chemically robust and should resist wide variations in pH.

The medium has been shown to be suitable for biomolecular residual anisotropic interaction measurements [41]. Crystallites are negatively charged due to the presence of sulfate ions, and the mechanism of orientational order should be partially electrostatic, but this remains to be characterized.

3.2.6 Strained polyacrylamide gel. Anisotropic deformation of a crosslinked polyacrylamide gel was shown to induce dipolar couplings in biopolymers [42, 43]. Initially, the aqueous gel has pores that can be considered spherical. Homogeneous compression or stretching of the gel distorts the pores and induces partial orientation of dissolved biomolecules by steric effects (Fig. S.2E). Vertical or radial compression results in oblate or prolate shaped pores, respectively. Techniques for the preparation of strained gels are described elsewhere [42-44]. Biomolecules are incorporated into the gels either by diffusion or by inclusion prior to the polymerization of the acrylamide.

The polyacrylamide gel matrix is uncharged, and thus imparts residual orientational order to biopolymers through steric effects. In addition, it is robust and resists a wide range of experimental conditions. The effects of varying the total acrylamide concentration, the degree of compression (stretching), and the amount of cross-linking N,N'methylenebisacrylamide on orientational order and relaxation properties of proteins have been studied extensively [43-45]. The degree of orientational order increases quadratically with the gel concentration, but higher gel concentrations may slow down rotational and translational diffusion.

3.2.7 Polymer-stabilized liquid crystals. Orientation of magnetically ordered particles can be achieved using a hydrogel matrix. This has been done for PM fragments and Pfl phage, both dissolved in polyacrylamide [36,43]. The addition of the unstrained polymer matrix seems to have very little influence on the order tensor of solute biomolecules. Samples are polymerized in the magnet in order to freeze the alignment of the liquid crystalline particles.

One advantage of using such media is that the orientation of the liquid crystal director with respect to the magnetic field can be varied [36]. A scaling of the dipolar couplings results, with the scaling factor being equal to ! (3 cos2 (3 - 1), where (3 is the angle between the magnetic field and the liquid crystal (LC) director. This enables measurement of scalar and anisotropic interactions in a single sample.

3.3 NMR pulse sequence methods for measurements of residual couplings

Once the proper ordering method is determined, anisotropic NMR interaction results are normally acquired in two steps:


1 Scalar couplings or isotropic shifts are measured in an isotropic medium.

2 The total couplings or chemical shifts are measured in the partially oriented phase, and the extrapolated isotropic contributions removed to obtain the residual dipolar coupling or chemical shift anisotropy (CSA).

The scalar J-coupling is nearly invariant under different experimental conditions and can be estimated quite accurately. However, the isotropic contribution to the shielding can change upon variation of experimental parameters such as temperature. Therefore, it is very important to measure the isotropic chemical shift (J'iso under conditions as close as possible to the liquid crystalline environment. In lipid bicelles, one can record (J'iso

in the isotropic phase at two or more different temperatures. The temperature is then increased such as to obtain a nematic phase, and the chemical shielding is measured. The isotropic contribution to the latter is extrapolated from the data obtained at lower temperatures, assuming linear or quadratic relationships between the isotropic shift and the temperature [46].

Numerous methods have been described in the literature for measurement of residual dipolar couplings in solution. Instead of describing each of those methods, we will briefly examine the building blocks that constitute the NMR pulse sequences that are commonly employed. For practical purposes, it is sometimes easier to modify an existing experiment than to implement a new one, and the choice of a suitable pulse sequence to measure a desired coupling will often be dictated by those readily available to the spectroscopist. We refer to a review by Bax et al. for a more detailed overview of experiments for dipolar coupling measurements [9].

3.3.1 Measurements of couplings in multidimensional experiments. Most pulse sequences used to measure heteronuclear residual dipolar couplings in proteins are modifications of two- or three-dimensional experiments designed for the assignment of l3C and l5N backbone resonances. In the unmodified experiments, full decoupling is accomplished in every dimension by using 1800 inversion pulses or broadband decoupling methods. Removal of a decoupling element introduces a multiplet pattern that enables direct measurement of a coupling in the frequency domain. This is the most common and simplest method of measuring spin-spin couplings. The choice of the dimension in which the coupling is measured is dictated by the linewidth of the resonances in the various dimensions. An ingenious modification is the couplingenhanced accordion pulse sequence element which consists of an evolution period of duration Atl where the chemical shift, but not the coupling, is refocused. The result after Fourier transformation is an apparent splitting equal to (1 + A)~v, where b..v is the real coupling. This technique is particularly useful to resolve multiplets in which the linewidth is equal to or greater than the splitting.

Another method for measuring couplings consists of a constant-time coupling evolution period, where the size of the coupling is encoded in the intensity of the resonances. This is the method of choice for very small couplings (such as those in magnetically ordered molecules) or intrinsically small interactions (such as one-bond


Table 8.1. Characteristics of various residual dipolar couplings in proteins

Type N a Teffb .r Dd References·

15N;_lHf' 1.041 90-96 1.00 [14,47-51]

13Cf-lHf J.l17 135-150 2.08 [51-55]

13Cf_13C'i 1.526 52-58 0.187 [49,51,55-58]

l3C'i_1-15Ni 1.329 13-17 0.111 [49,57-61]

l3Cf_15Ni 1.482 9-12 0.080 [57,58,62]

13Cf_l3C? 1.546 48-52 0.180 [58,63]

13C'i_l_1Hfl 2 2.065 4-5 0.329 [47,49,57,60,61]

13Cf_1Hf' 2 2.179 1-2 0.280 [57,58]

13Cf_l-1Hf' 3 2.438 0.2-0.8 0.200 [49,57]

IH_IH n.aJ n.a. n.a. n.a. [6,64-70]

aThe number of intervening bonds bEffective internuclear distance (A)[71j. cScaiar coupling range (Hz). dRelative maximum absolute value with DNH set to 1.00. ·References for coupled NMR pulse sequences. fNot applicable.

13C_15N couplings). However, the method is rather time consuming since a very good signal-to-noise (SIN) ratio and multiple data points are required.

3.3.2 Dipolar coupling types. A wide range of couplings can be measured for biomolecules dissolved in ordered media. A list of the most important dipolar couplings in proteins is given in Table 8.1, along with references for the pulse sequences used to measure them. The 15N_1 H coupling is the reference dipolar coupling in proteins, as it is large and easily measured from the well-dispersed 2D HSQC spectrum [47]. Moreover, it only requires synthesis of a 15N-Iabeled protein which is less expensive than 13C-Iabeled material. Other interactions that can be measured in 15N-Iabeled proteins are 1 H_l H couplings. Thorough residual dipolar coupling studies include the measurement of 13C_X couplings, either to characterize with precision the order tensor of a protein with known structure or to determine the structure of a macromolecule with a low NOB density. There are many of these, but the most commonly measured are 13C<l<_1 Hel<, 13Cel<_13C', 13C,_1 HN and 15N_13C'. As discussed in section 4.2, only 13C' and 15N CSA have been shown to be useful in structural analysis of proteins.

In nucleic acids, the most commonly measured couplings are 1 H_l H [32,65], 1 H-13C [72,73], IH_31p [74] and IH_15N [72,75,76]. The only nucleus with a uniform and structurally useful CSA tensor in nucleic acids is 31 P [77]. Oligosaccharides are naturally limited to IH_13C and 1H_IH couplings [78]. In many cases, it may be difficult to obtain labeled material for these biomolecules, and natural abundance 13C and 1 H may be the only nuclei available. There are many C-H pairs per fragment

176

A

So, 6

&0 Asn39

Nn 33

Asn 17

Cit

Asn 80 Leu 51

00 8.20 8.10 8.00

IH (ppm)

B Asn 17 • Ser 6 • oe Asn39

A.n 33 Asn 80 Leu 51

eO

CD o 8.20 8.10 8.00

IH (ppm)


C 112.0

CD

~.n M'~

i i 96,4 ; ~l.o 1 @. ,

113.0 90.'1 • 114.0

E c..

115.0 5

C z ~

116.0

G 117.0

4 8.20 8.10 8.00

IH (ppm)

Figure 8.4. Spectral editing using the 2D IPAP-HSQC pulse sequence for measuring 15 N_l H residual dipolar couplings [47]. The example shown here is from the C-terminal domain of the poly(A)-binding protein from Trypanosoma cruzi, dissolved in 12 mg I mL Pfl phage. (A,B) Contour plots of the subspectra containing the downfield (A) and upfield (B) components of the N-H coupling. (C) Superposition of the two subspectra; the upfield components are shown as dotted contour lines. Couplings are shown in Hz. The superiority of the IPAP method is the elimination of overlapping doublets that prevent the measurement of some couplings.

in these biopolymers and the IH_13C couplings are generally sufficient to refine their global structures [73].

3.3.3 Spin-state editing. When doing measurements directly in the frequency domain, the multiplet components of different moieties of the biomolecule can overlap, preventing an accurate measurement of the couplings involved. To minimize these spectral overlaps, pulse methods have been devised to edit the spectra and to decompose the multiplet components into separate data sets. One such method is the spin-state-selective excitation (S3E) which uses a refocusing INEPT evolution period with an appropriate combination of 90° pulse doublets with different phases to prepare magnetization such that the In) and 1,6) coupled-spin states can be distinguished [79]. One inconvenience of this method is that elimination of doublet components is sensitive to the size of the coupling. The IPAP methodology consists of obtaining in-phase and anti-phase doublets in HSQC spectra [47]. The anti-phase subspectrum is produced by inserting a refocused INEPT element before the 15N chemical shift and coupling evolution period. Each doublet component is obtained by either subtraction or addition of subspectra (Fig. 8.4). Both methods can easily be implemented in most two- or three-dimensional heteronuclear NMR experiments.

3.3.4 Sensitivity, accuracy and precision. The degree of structural information contained in a coupling depends on its size and relative precision with which it can be determined. For example, an N-H coupling with a precision of ca. 1 Hz defines more precisely the internuclear vector orientation than a C'-N coupling with a precision


of ca. 0.2 Hz. The smaller the coupling, the higher the precision of the measurement required. Several factors influence the precision of couplings determined in the frequency domain. Obviously, the digital resolution of a spectrum sets a lower bound to the achievable precision of an experiment. A higher SIN ratio will also invariably result in a higher precision. This can be achieved either by increasing the number of scans or, for large proteins, by using the transverse-relaxation optimized spectroscopy (TROSY) methodology to utilize advantageously the differential relaxation between two doublet components [80]. The latter technique has been implemented in most heteronuclear multidimensional experiments. Another straightforward way to increase the SIN ratio in an experiment is to decouple any unresolved couplings that broaden the apparent linewidth. Adiabatic homonuclear decoupling pulse sequences, used simultaneously with acquisition of directly detected amide protons, have been shown to enhance significantly the SIN ratio in a 2D IH-coupled IPAP-HSQC experiment [81]. This is particularly important for strongly ordered samples, where most of the 1 HN broadening comes from unresolved homonuclear couplings to the aliphatic protons. The CO< resonance can be broadened by the large Co<-eP scalar coupling. This homonuclear coupling can be removed by using an appropriate constant-time evolution period.

4. Interpretation and applications of dipolar couplings and CSA in structural biology

Once a set of experimental residual dipolar couplings and CSA measurements are obtained, the next step is to interpret them in terms of structure and dynamics of the macromolecule. First, we describe the steps involved in the analysis of residual dipolar couplings. Applications in structural genomics will be presented. Finally, an overview of the analysis of internal dynamics in macromolecules using residual anisotropic interactions will be given.

4.1 Determination of order parameters

In order to solve a macromolecular structure, a preliminary determination of the components of the order tensor is required. The different procedures that have been proposed in the literature depend mostly on the amount of other structural information known a priori about a macromolecule, as well as on the total number of dipolar couplings measured.

In the case where the structure is unknown and residual dipolar couplings are to be used as structural constraints, the orientation of the principal axes of the order tensor can obviously not be determined because a molecular frame has not yet been defined. Therefore, only two parameters can be obtained prior to structure determination: the magnitude and the rhombi city of the order tensor. Two methods are available to estimate these values. One method uses a histogram of all dipolar couplings (in Hz) measured in a single medium, as shown in Fig. 8.5 [11]. The couplings are scaled to the NH couplings, using the factors shown in Table 8.1, for direct comparison. These factors are calculated from effective internuclear distances and gyromagnetic ratios and


~::DQ 0 10

o

~::DQ 0 10

~ ~ 0 ~ ~~ ~ 0 ~ ~

30~ 'E 20 5 0 10

o -40 -20 0 20 40

o (Hz)

o (Hz) 0 (Hz)

Figure 8_5_ Histograms of dipolar couplings for the C-tenninal KH domain of the heterogeneous nuclear ribonucleoprotein K. The individual distributions shown in the histograms for N-H (A), C"-H" (B), C'N (C), and C"-C' (D) do not reflect a powder pattern distribution due to the high content of regular secondary structure in the domain_ The collective distribution obtained by scaling the dipolar couplings to that for NH provides a better representation of a powder pattern distribution, as shown in (E)_ The pattern allows an accurate estimate of Da = 19_6 Hz and R = 0.3. (Redrawn from [82] with pennission).

take differential bond dynamics into account. If the ensemble of all internuclear vectors covers uniformly the surface of a sphere, a perfect powder-pattern shape is expected. The dimensionless principal components Szz, Syy, and Sxx of the order tensor are determined from the largest positive, largest negative, and modal (highest frequency) values in the histogram, after multiplying them by the appropriate normalization factor. The sum of the components should be equal to zero, since the order tensor is traceless (i.e. Sxx + Syy + Szz = 0). The magnitude and rhombicity of the order tensor are then estimated using Eq. (8.5).

The isotropic distribution requirement is one limitation of the method, but in practice it fails if only a sparse amount of data is available. In this case, the probability of finding extreme values in the experimental data is very low and order parameters cannot be used with confidence. In general, the probability of finding a vector with polar angle e is proportional to sin e. Consequently, it is difficult to determine Szz with satisfactory accuracy, as it corresponds to the orientation e = O. An improved method makes use of all information contained in the histogram using a probability function, convolved with a Gaussian error function, to calculate the most probable principal components from all dipolar couplings [83,84]. However, if internuclear vectors are anisotropic ally distributed, a histogram-based determination of order parameters is biased, and different methods should be used.

In the worst-case scenario where there are insufficient data to enable an accurate determination of Da and R, Grzesiek et al. have developed a method in which a structure can be elucidated without explicit knowledge of the order tensor [85,86]. In general, it is possible to adjust the order and structural parameters simultaneously.


In the case where the structure of a rigid molecular fragment is known, the complete Saupe order matrix for a set of N dipolar couplings can be determined using standard linear algebra methods. The N x 5 direction cosines matrix ~, built from the atomic coordinates written in an arbitrary molecular frame, is used to calculate the dipolar couplings from the structure [S7]:

(S.lO)

The 5 x 1 vector 8 contains the independent elements of the order tensor, and the N x 1 vector w contains the calculated dipolar couplings. This system of linear equations can be solved for 8 using singular value decomposition. With at least five measurements from interaction vectors that are not colinear, an exact or least-square minimized solution can be found. Diagonalization of the order tensor, constructed from the vector 8, yields the orientation and principal components of the tensor in the arbitrary molecular frame.

4.2 Structural interpretation of residual dipolar couplings and chemical shielding anisotropy

In the absence of other structural information, a single dipolar coupling does not define a unique (B,1) pair in Eq. (S.3). If the magnitude and rhombicity of the order tensor are known, the internuclear vector is restricted to the surface of a distorted cone and its inverse, centred around the principal axis of the order tensor [S8]. The deviation from the axially-symmetric cone shape increases with rhombicity.

This non-discrete degeneracy in the vector orientation can be lifted by the addition of dipolar couplings obtained in another medium, where the order tensor is significantly different [88,89]. For each internuclear vector in an unknown structure, two intersecting distorted cones are defined from the two data sets, and the intersections of these surfaces are the possible orientations allowed (Fig. 8.6). The number of discrete vector orientations is four or eight, depending on the orientation and rhombicity of the two order tensors. Addition of a third data set may reduce the degeneracy further. It is possible to estimate the Euler angles between the two order tensors in the absence of structural information by graphical analysis of correlation plots between the two dipolar coupling data sets [91]. In theory, the latter method could provide a set of possible (B,1) pairs for a particular internuclear vector directly.

For a set of couplings arising from a rigid fragment with known structure, there are only four possible orientations for the set of vectors in the principal frame of the order tensor [89]. These fragment orientations are related by inversion and rotation symmetry operations. In combination with other angular information, it is generally possible to assign degenerate, but discrete orientations to internuclear vectors using a dipolar coupling set originating from a single orienting medium [92]. In full analogy with the case where the structure is unknown, the degeneracy in orientation can be lifted by the addition of couplings measured in a second medium. In this case, the dipolar coupJings are stil1 consistent with two different order tensors for the rigid


fragment. Increasing the number of dipolar coupling data sets to three will not reduce the degeneracy because an orientation is always indistinguishable from its inverse.

A study by Comilescu and Bax [46] have focused on the empirical calculation of shielding tensors using dipolar couplings to determine the order tensor of the protein ubiquitin in lipid bicelles. Uniform shielding values have been used at all sites and were varied to obtain the best fit with the experimental results which were measured for carbonyl 13C', 15N and amide 1HN. Slightly different shielding values have been found in a-helical and extended regions for carbonyl and nitrogen atoms. The degree of carbonyl protonation and hydrogen bonding is known to affect 13C' shielding tensor values [93]. However, the variations observed have only a small effect on the accuracy of the back-calculation of chemical shift tensor elements from a known protein structure [94]. The scatter observed between predicted and observed 13C' and 15N shifts is dominated by errors in the atomic coordinates of a protein. This implies the potential use of chemical shift tensors as structural constraints. The proton shielding tensor has relatively small elements that vary considerably from site to site and are strongly influenced by solvent effects. Therefore, the use of 1 HN chemical shifts is of little value in structural refinement [46]. In nucleic acids, the chemical shielding tensor for the 31 P nucleus is quite uniform at all sites and is useful in structure determination [77].

-y

-x

Figure 8.6. Orientation of the Gln40 N-H vector in the protein ubiquitin using two sets of dipolar couplings. The coordinate frame is from the X-ray crystal structure [90). Strips A and B are compatible with dipolar couplings in undoped lipid bicelles and positively charged bicelles, respectively. The solid dot marks the orientation of the N-H vector in the crystal structure which is also one of only four orientations consistent with the two dipolar coupling measurements. (Reproduced from [881 with permission).

Observation and interpretation of residual dipolar couplings in biomolecules

4.3 Residual dipolar couplings as restraints in biomolecular structure determination

181

The classical way to solve biomolecular structures using NMR consists of finding a set of local distance and dihedral restraints from NOEs and J-coupling data [1]. These constraints are fed into a structure calculation programme that uses a simulated annealing algorithm to find the optimal ensemble of structures that best satisfies the experimental data and the appropriate molecular force field. The target function for the minimization includes an energy penalty function calculated from the sum of the squared differences between structure-based predicted and experimental distances and torsion angles. The convergence rate generally increases with the number of local constraints.

The simplest way to incorporate N dipolar couplings into a structure calculation is to define an analogous target energy function [4]:

N "" iii 2 E dip = ~ kdip(Dcalc - Dexp) (8.11)

where k~ip is the minimization constant for coupling i, and D~alc and D~xp are the calculated and experimental dipolar couplings for the ith interaction. In this approach Sxx, Syy, and Szz as well as the internuclear distances are used as input values. An expression related to Eq. (8.11), but using the absolute value, is used to incorporate dipolar couplings for which the sign is unknown [95]. This is the case for IH_IH couplings, whose sign is not straightforward to determine, especially in the absence of scalar couplings. Moreover, the distance between remote protons is not necessarily well determined, and also needs to be refined during the calculation.

Typically, the Euler angles of the principal orienting system in the arbitrary molecular frame are allowed to float during the minimization, but the input Da and R values are fixed. A major drawback of this approach is that the energetic function space is punctured with many deep local minima, and the optimal solution corresponding to the solution structure can be difficult to find [96]. This is attributed to the competition between dipolar couplings, in opposition to the cooperative nature of local distance restraints. The convergence rate is improved by decreasing the dipolar minimization constants (k~ip) early in the calculation, and gradually increasing them to arbitrary final values [4]. In the initial steps, NOE restraints predominate, and a global fold is circumscribed. This fold is then further refined using the dipolar coupling data at the end of the calculation. Nonetheless, the energy landscape is still very complex and there is low convergence, especially in cases where there are multiple competing couplings or when the NOE density is low and no fold is previously assigned to the macromolecule.

Order parameters can also be optimized in the calculation, either explicitly [73] or implicitly [86]. In the first method, the order parameters are treated as additional degrees of freedom and optimized simultaneously with atomic coordinates. In the second method, the error term r2 is minimized with respect to the vector a of the


internal degrees of freedom derived from the structure:

r2(5) = Ilwobs - cI>(5)cI>+(5)wobsIl 2 (8.12)

The matrix cI> (5) is the interaction matrix and is related to the cosine matrix described in section 4.1, and Wobs is the vector of the N observed residual dipolar couplings. The inverse matrix cI>+ (5) is computed using singular value decomposition. Notice that the order terms, equivalent to the product cI>+ (5)Wobs, are optimized implicitly during the optimization procedure. The final order tensor obtained is more accurate and therefore improves the quality of the fitting procedure. However, the convergence does not increase when employing this procedure in comparison with using fixed input values.

Methods have been formulated to avoid slow convergence rates. If sufficient couplings and CSA values are available for each rigid unit of a biopolymer for which a structure exists, discrete orientations can be found for each of those units [97]. These oriented fragments are then strictly translated using distance restraints and covalent bond information. The final structure is obtained from a force- field minimization using all experimental data.

It is also possible to convert dipolar couplings into direct angle restraints between internuclear vectors. For N dipolar couplings, a total of N(N - 1)/2 constraints is calculated. The constraints are intrinsically very loose, but their high number compensates and their inclusion has been shown to increase the precision and convergence rate of NMR structure determination [83,96]. In this case, the energetic potential is composed of shallower minima, which favours convergence of the simulated annealing protocol.

4.4 Accuracy and precision of the optimization procedure In a final ensemble of structures, the quality of the fit for N dipolar couplings or

anisotropic shifts can be evaluated using a statistical Rdip factor [98]:

(8.13)

A Q quality factor is also found in the literature [99]:

Q= (8.14)

If the sampling of internuclear vector orientations is isotropic in the macromolecule, the two factors are related by

(8.15)

The Rdip factor varies between 0 and 1, and a value below 0.2 is considered satisfactory. Differences between calculated and experimental data are attributed to errors


in atomic coordinates and small-amplitude dynamics. The precision of the structure increases with decreasing Rdip factor and can be adjusted by variation of the k~ip dipolar minimization constants in the simulated annealing calculation. However, higher precision is not necessarily associated with a higher accuracy, especially if there are errors in the experimental data or if the precision of the experimental dipolar couplings is overestimated. The accuracy of a structure can be estimated using a cross-validated free Rdip-factor. A random set of couplings (~ 10 %) is removed and the simulated annealing calculation is carried out using the remaining experimental data, called the working data set. Eq. (8.13) is then applied to those couplings that were not used, and a Rdip(free) is calculated. This is repeated for several working data sets and an average Rdip(free) is thus obtained, which quantifies the validity of the experimental data. An improved precision should be accompanied by a reduction in the average Rdip(free).

The accuracy of a biomolecular structure determination can be seriously compromised if the orienting medium used to extract anisotropic interactions modifies the structure or the dynamic behaviour of a molecule with respect to the isotropic solution. The use of a dilute liquid crystal has been shown to influence the structure and dynamics of several proteins minimally [44]. The inclusion of dipolar couplings in structure calculations does increase the accuracy and precision of protein structure determination (Fig. 8.7). In addition to Rdip(free), an independent indicator of structural accuracy in proteins is the Ramachandran plot which is a two-dimensional representation of the ¢ and 1/J backbone torsion angles for every amino-acid residue. Specific areas of this conformational space reflect more favourable conformations. High-accuracy structures should have a vast majority of data points falling into these regions, with none into the so-called disallowed regions.

4.5 Applications in structural genomics The use of residual dipolar couplings has an enormous potential for structural ge

nomic projects which aim to determine the fold of all proteins. Knowledge of a protein structure assists scientists in assigning functions to a gene. Very often, complete structure determination, i.e. backbone and side-chain positioning with an extensive set of distance restraints, is not required to investigate the function of a biomolecule. Much work has been done on devising rapid and accurate methods to determine protein folds using dipolar couplings and CSAs, in combination with a small number of distance restraints [97,101-104], paramagnetic constraints [105], ab initio structure predictions [106], or motifrecognition from structural databases [107-109].

The number of protein folds is smaller than the number of protein sequences and structural redundancy is expected. Homology in the amino acid sequence generally leads to structural homology, but there are also cases where proteins are significantly different in their primary sequence but have similar structures. Dipolar couplings provide a signature of a protein fold, independent of the primary sequence (Fig. 8.4). Meiler et al. have developed the software DipoCoup which accomplishes fold recognition from a structure database using solely residual dipolar couplings and pseudo-

184 NMR OF ORDERED UQUIDS

A 3 B 180

1 -T

2,5

::: 2 ..,90 ~

0 ~

~ 1.5 i' 0

'" ~ 1 ';;

c. 0.5 ·90

0 ·180 , ....

Residue in PABC ·180 -90 0 90 180 Phi (degrees)

C 3 D 180

2.5 90

- 2 ~ -~ 1.5 go 0 ::; ~

'" 1 .. c. ·90

0,5

0 ·180 -·180 -90 0 90 180

Residue ,n PABC Ph, (degrees)

Figure 8.7. Residual dipolar couplings increase the precision and accuracy of the NMR structure of the PABC domain from the poly(A)-binding protein of Saccharomyces cerevisiae [100]. (A, C) Plots of the root mean square deviation (RMSD) of N (black), C" (dark grey), and CfI (light grey) atoms from an ensemble of 30 structures, after a best-fit superimposition of backbone atoms. (B, D) Ramachandran plots of backbone torsion angles of the five lowest energy structures. Dark areas are the most favourable regions, where torsion angles minimize side-chain I side-chain bad contacts. The first ensemble (A, B) was generated using only NOB distance and 4> torsion angle restraints. The average backbone RMSD was 0.6 A, and 75 % of the residues were in the most favourable regions of the Ramachandran plot. Addition of 48 16 N_l H RDCs improved the RMSD to 0.37 A, with 84 % of residues in the most favourable regions of the Ramachandran space.

contact paramagnetic shifts [110]. Once a fold has been identified for a protein, a homology model can be built from it using a force field and the experimental residual dipolar couplings [111].

Another interesting alternative to the classical structure determination procedure is to use dipolar couplings in the assignment of backbone resonances. The frequency changes that occur due to media-induced CSA or dipolar couplings in coupled threedimensional NMR experiments can help to resolve resonances that would otherwise overlap in isotropic solution spectra [112]. A recent approach to backbone structure determination uses residual dipolar couplings to characterize structural motifs in the absence of explicit assignments [113]. This information is then combined with chemical shift correlations to assign resonances in a sequence-specific manner. The method has been used to build the backbone structure of the protein rubredoxin which was in good agreement with previously determined NMR and crystal structures.


B D

30

g20

~ ]10

.i It. 0 " w

· 10

o 10 20 30 Calculaled ROC (Hz)

Figure 8.8. (A) Structure of PBX homeodomain bound to a double-stranded DNA fragment. The complex structure was elucidated using restrained molecular dynamics with sparse intermolecular distance restraints and residual dipolar couplings measured in both members of the heterodimer. (B) Correlation plot of measured versus calculated residual dipolar couplings following structural refinement with measured RDCs. Twenty eight aromatic and Hl'-Cl' 13C_1H couplings (empty squares) were measured in natural abundance in the DNA. and sixty 15N_1 H backbone couplings (black diamonds) were measured in the 15N-labeled protein. A solution of 5% DMPC I DHPC bicelles was used to order the heterodimer. (Redrawn from [114] with permission).

One important application is the assembly of mUlticomponent complexes from residual dipolar couplings (Fig. 8.8). Detection of intermolecular NOEs is a difficult process and very often the number of NOEs collected is insufficient to define the relative orientation of monomers. The combination of sparse distance constraints with residual dipolar couplings measured in each member of the complex can aid in unambiguously determining the orientation of each monomer [115.116]. Moreover, if the structures of all components of the complex are already known, they can be considered as rigid bodies in simulated annealing protocols, thereby facilitating the calculation considerably [117,118]. Because it is often feasible to isotopically label only one of the components in a multimer, chemical shift redundancy is not a problem and large complexes can be studied. There are several examples of RDC-assisted complex assembly in the literature, ranging from protein / nucleic acids to protein / protein (72, 119].

4.6 Internal dynamics In addition to the strong structural significance of the anisotropic interactions, they

also carry important information about internal dynamics of the macromolecules over timescales that are faster than those required for motional averaging. Motions found in biomolecules range from fast, small-amplitude vibrations and librations to slow, large-amplitude conformational changes. Normally, dynamical studies are carried out for macromolecules that have known three-dimensional structures. Approaches


to the treatment of intramolecular motions in biomolecules using weak anisotropic interactions can be divided into two classes.

The first category considers small-amplitude motions that occur inside a macromolecule or molecular fragment with a well-defined structure. The assumption underlying this approach is that the overall order tensor of the macromolecule is not affected by these small-amplitude internal motions [120-122]. Each dipolar coupling is multiplied by a scaling factor Ardc that decreases the coupling calculated from the static structure solved without this particular coupling (see Eqs. (8.3) and (8.5)). This factor can be obtained independently from model-free analysis of 15N / 13C relaxation data [123], but it only covers motions in the nanosecond / sub-nanosecond timescale range. The scaling factor Ardc that takes into account the effect of internal dynamics can also be estimated from residual dipolar couplings measured in at least five different media that yield five independent alignment tensors for the molecule [121, 122].

The second class of approaches consists in defining small fragments within a biomolecule (e.g. peptide planes, nucleotides, folded domains) and in determining the order tensor for each of them separately [124,125]. No explicit discrimination is made between overall tumbling and internal dynamical averaging of the NMR interactions. Conclusions about the internal dynamics are drawn by comparison of order tensor elements between each fragment and the overall alignment tensor. This method is appropriate for analysis of slow motions such as large domain reorientation or for any case where the order tensor is different for all the major conformers in the fast exchange NMR limit. It has been successfully applied to the analysis of the dynamics in the domains Band C of the barley lectin protein [124]. The two domains contain well folded core regions which are connected by a short flexible linker. A comparison of order tensors determined for each core region allowed the determination of their average relative orientations and provided an estimate of the restrictions imposed by the linker to relative domain-domain motions.

5. Summary Residual dipolar couplings and chemical shift anisotropies represent a new and

highly informative type of constraint for biomolecular structure determination. Weak orientational ordering of biological macromolecules is achieved either through direct magnetic ordering, or more frequently by using a partially ordered medium. The NMR interactions obtained are first-order and are interpreted simply in terms of structural and dynamical properties ofbiopolymers. The methodology has numerous applications in structural biology, ranging from the assembly of complex structures and de novo fold determination to structural genomics and homology modeling.

Acknowledgments

This work was supported by a grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada to Kalle Gehring and an NSERC studentship to Jean-Fran~ois Trempe.


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Chapter 9

THE SEARCH FOR HIGH-RESOLUTION NMR METHODS FOR MEMBRANE PEPTIDE STRUCTURE

Christophe Fares and James H. Davis Department of Physics. University of Guelph, Guelph, Ontario, Canada

1. Introduction: NMR of orientationally ordered systems Biological and model membrane are lyotropic liquid crystals which not only exhibit

the characteristic positional disorder and orientational order of this class of materials, but also typically have a very complex composition. As in other liquid crystals, the molecules found in membranes are in an orientationally ordered state. Because of this, the application of nuclear magnetic resonance (NMR) to the study of these systems requires an appreciation of the dominant role played by the orientationally dependent second rank tensor interactions of the nuclear spins with each other and their electronic environment. Of equal importance is the dynamic disorder characteristic of liquid crystalline systems. The complex, anisotropic molecular motions found in such systems result in an averaging of these tensor interactions. The interpretation of the measurements of the residual interaction strengths in an NMR experiment needs to be based on an understanding of the symmetries, amplitudes and time scales of the molecular motions.

Membrane proteins perform an amazing variety of crucial biological functions and a firm understanding of how these functions are performed can best be achieved through a knowledge of the protein molecular structure and its organization and interactions within the membrane. While nearly a third of the genes in the genome code for membrane proteins, fewer than 1 % of the structures in the protein data bank are of membrane proteins. Gradual progress is being made, however, as techniques for crystallizing these amphiphilic molecules are being developed. Over roughly a 25 year period, solution state NMR methods were developed to enable structure determination of soluble proteins and now approximately one third of all new protein structures are obtained by NMR. There is currently a world-wide effort underway to develop solid state NMR methods for the determination of membrane protein structure.

In solid state NMR there are several approaches that can be used to obtain structural information. The most fundamental is to use the residual anisotropies of the orientation dependent interactions to obtain orientational restraints. For example, the anisotropic

191 E.E. Burnell and CA. de Lange (eds.), NMR a/Ordered Liquids, 191-213. © 2003 Kluwer Academic Publishers.


chemical shifts of 15N and 13C can be readily measured in orientationally ordered samples and, based on the principal values of the shift tensors and their orientations within the molecular group containing the nucleus, these chemical shifts establish an orientational restraint for that part of the molecule. Through the use of double isotopic labeling one can introduce defined spin pairs at specific locations. The measurement of the residual dipolar coupling between the members of such pairs also provides an orientational restraint (sometimes together with a distance dependence) for the labeled sites. Uniform isotopic labeling (where all sites of a particular type of nucleus are labeled, though not necessarily at the 100% level) can in principle provide a vast number of orientational restraints in a single series of experiments. The achievable spectroscopic resolution of macroscopically oriented samples may ultimately limit the applicability of this method. The utility of this procedure also rests heavily on the ability to assign all of the resonances in the spectrum, so initial efforts must be directed towards developing experiments aimed at improving resolution and solving the assignment problem.

A second approach is to use magic angle spinning (MAS) to eliminate the orientation dependent interactions (which scale as (3 cos2 0 - 1)/2, where 0 is the angle between the magnetic field and the spinning axis). This approach provides significantly higher resolution than the static approach described above. However, since the orientational information has been removed by MAS, some other methods must be devised to obtain structural information. A number of techniques for selectively reintroducing orientation dependent interactions such as the hetero- or homonuclear dipolar coupling between spin pairs or the anisotropic chemical shift of a specific type of nucleus, have been developed. Here too the use of specific or uniform isotopic labeling will be essential and, for the uniform labeling case, experiments must be designed to solve the assignment problem.

A third approach which will apply only to a restricted class of molecules relies on the rapid axially symmetric reorientation characteristic of a fluid membrane bilayer phase. If the molecular axial rotational diffusion about the local bilayer normal is fast enough, the orientation dependent interactions are projected onto the diffusion axis. This has the subtle but important effect of transforming the otherwise homogeneous, like-nucleus, dipolar broadening among abundant nuclei such as 1 H, into an inhomogeneous broadening which can be effectively narrowed by MAS. In cases where this approach will work the resolution obtainable for 1 H (the most abundant and sensitive nucleus in biological systems) can approach that obtainable in solution. Systems of this type can be studied by developing experiments analogous to those used for structure determination of proteins in solution. As for solution NMR, the principal problems to be solved are the sequence specific assignment of the resonances and the measurement of structural restraints. In addition to distance restraints obtained through nuclear Overhauser effect (NOE) spectroscopy, it should also be possible to obtain orientational restraints by selective reintroduction of dipolar interactions between 1 Hand 13C or 15N labeled positions within the molecule. The necessity for rapid axial rotational diffusion will almost certainly restrict the application of this approach to small molecules or fragments of molecules.

The searchfor high-resolution NMR methods for membrane peptide structure 193

It is beyond the scope of the present article to present a comprehensive review of the work being done on all of these different approaches. Instead, we will focus on the use of separated local field (SLF) spectroscopy for determining distance and / or orientational restraints in macroscopically oriented samples and on the use of 1 H MAS on small molecules in bilayer membranes. For recent reviews of these and other methods see references [1-7].

2. Theoretical background The orientation dependent second rank tensor interactions dominate the NMR spec

tra and relaxation in (partially) ordered systems such as solids, liquid crystals and membranes. The approaches discussed above are either designed to measure the residual tensor interactions or to eliminate them. The interactions of primary interest here are the nuclear dipole-dipole interaction HD and the anisotropic chemical shift Hcs. The notation to be used is that of [8].

The dipolar Hamiltonian (in energy units) describing the interaction between two nuclei having magnetic moments Jll and Jl2 separated by a distance r12 is just

'U _ Jll· Jl2 _ 3(Jll . r12)(Jl2· r12) I ~D - 3 5·

r 12 r 12 (9.1)

For practical spectral simulations it is most convenient to introduce spherical tensor notation wherein the Hamiltonians are the products of spatial and spin dependent operators

+2

HD = L (-l)m72,mFt,_m (9.2) m=-2

where the spin dependent operators 72,m are

(9.3)

(9.4)

(9.5)

and the spatial dependence, in the laboratory referenced frame, L, is specified by the spherical tensors FL which are defined in terms of the Euler angles (apL, (3PL, 'YPL) defining the transformation from the dipolar tensor principal axis frame, P, to the laboratory reference frame, L [9]

+2

Ff,m = L Fr.m,'D~Jm(aPL' (3PL, 'YPL). (9.6) m'=-2


The principal values of the dipolar tensor (in the reference frame, P, where this tensor is diagonal) are

Ffo = - V6'Y1'Y21i,2 /d2 ,

Ft.±1 = 0 (9.7)

Ff±2 o. ,

The nuclear gyromagnetic ratios, 'Yl and 'Y2, are defined by theyroportionality between the nuclear magnetic moment and the nuclear spin, J1i = 'Yinh

It is frequently necessary to introduce an intermediate reference frame N, for example, one whose Z N-axis is along the director or bilayer normal. In this case the transformation from dipolar tensor principal axis frame to the laboratory frame occurs in two steps and involves two sets of Euler angles

+2

Ff,m = L Ff,m,1)~Jm(aNL' j3NL, 'YNd (9.8) m'=-2

and

+2

Ff,m' = L Ft.m,,1)~J'm,(aPN,j3pN''YPN). (9.9) m"=-2

If ZN is an axis of symmetry, then only one angle j3NL is necessary for the transformation from N to L (since there is axial symmetry around the magnetic field direction ZL of the lab frame). For the dipolar Hamiltonian (which is also axially symmetric) only one angle is required for the transformation from P to N, j3PN' In this simple case,

L V6( 2 j3 )( 2 j3 ) 'Y1'Y2 n2 F20 = - - 3 cos N L - 1 3 cos P N - 1 3 4 r12

(9.10)

For unlike nuclei, the dipolar Hamiltonian can be treated as a small perturbation on the Zeeman Hamiltonian. In this case the nuclear spin energy levels are shifted by an amount hDo which depends on the orientation of the internuclear vector r12 relative to Bo, the value of 1/r~2' and the product of the two nuclear gyromagnetic ratios. For the commonly used case of bilayer membranes deposited on glass plates oriented such that the normal to the plates (and the bilayers) is at an angle j3NL relative to Bo, the dipolar coupling constant, assuming axially symmetric reorientation about the bilayer normal, is (upon converting to SI units)

Do = - J-Lo 'Y1'Y2 n (~cos2 j3PN - ~) (~cOS2 j3NL - ~) 811'2 r~2 2 2 2 2

(9.11)

where as above the angle j3 P N defines the orientation of the internuclear vector relative to the bilayer normal. The first order shifts in the nuclear spin energy levels for unlike

The search for high-resolution NMR methods for membrane peptide structure 195

spins split the resonance of either spin into a doublet with a dipolar splitting equal to 12Do + JI (see also Chapter 1). Normally the contribution from the indirect coupling J can be neglected. The magnitudes of these dipolar splittings depend on the types of nuclei involved and the magnitude and orientation of their internuclear vector [8].

The case of like nuclei is somewhat more complicated. Whether the dipolar Hamiltonian can be treated as a small perturbation relative to the Zeeman Hamiltonian now depends on the relative magnitude of the dipolar coupling and the difference in chemical shift of the two nuclei. In general, in this "strong coupling" situation the spin state eigenfunctions and eigenvalues both need to be evaluated so that the intensities of the resonances and their splittings can be obtained. There is also a complication in the case of MAS NMR since, when using average Hamiltonian theory to describe NMR of a rotating sample, the Hamiltonian, evaluated at two different times t1 and t2, for homogeneously dipolar broadened lines does not commute with itself in general [10, 11]. These important considerations are beyond the scope of this article and will not be considered further here.

In order to simulate NMR spectra of two unlike spins coupled by the dipolar interaction, we need to evaluate the resonance frequencies of the two components of the doublet. This is easily obtained now that we have the dipolar shifts. The lines of either of the doublets occur at frequencies

J..lo 'Yn2n (3 2 1) (3 2 1) 1/2 ± = 1/20 =F --- - cos (3PN - - - cos (3NL - - . , , 87r2 r3 2 2 2 2 12 (9.12)

Here 1/2,0 is the resonance frequency of the observed nucleus, including its chemical shift. These expressions will be used later in simulations of various types of separated local field NMR spectra.

The chemical shift Hamiltonian, 'Has, is

'Has = [t. 7J ·R (9.13)

where 7J is the chemical shift tensor. This tensor has an isotropic part (scalar or tensor of rank 0), an anti symmetric part (axial vector or tensor of rank 1), and a symmetric part (tensor of rank 2)

(9.14)

where '1 is the unit operator (in matrix notation, the unit matrix). In spherical tensor notation the chemical shift Hamiltonian is

2 k

L L (-l)m1k,mFk,_m (9.15) k=Om=-k


and the spherical tensor elements Fk,m, in terms ofthe Cartesian tensor elements aij,

are

Fo,o 1 .....

- y'3'YjTr[a]

F1,0 iV2'Yja12

Fl,±l 'Yj(a13 ± i(23)

VI 1 ..... (9.16)

F2,0 2'Yj{a33 - "3Tr [a]}

F2,±1 = 0

F2,±2 1 2'YAall - (22)

where 'Yj is the gyromagnetic ratio of the nucleus. The scalar part, Fo,o is independent of orientation, the anti symmetric parts, F1,m, are generally negligible and will be neglected, while the symmetric second rank components, F2,m, are orientation dependent and give rise to the characteristic broad line shapes seen in powders. This part of the chemical shift is called the anisotropic part. Like any symmetric second rank tensor there exists a principal axis reference system, P, where the tensor is diagonal. The principal values of the tensor, its diagonal elements, are generally referred to as aiI., afz and a~. The conventional assignment of the principal values is such that

la~ - 801 ~ laiI. - 801 ~ lafz - 801, where 80 = (1/3)Tr[u] is defined as the isotropic chemical shift. The chemical shift anisotropy is often defined as 8 = a~ - 80

and, when the chemical shift tensor is not axially symmetric, i.e., aiI. =I a22' the asymmetry parameter is defined as 1] = I (af2 - aiI.)/af31

The spin-dependent part of the Hamiltonian is

1 70,0 = - y'31zBo

7i,0 0

1i,±1 1 ± - -I Bo 2 (9.17)

ji1zBo 72,0 =

72,±1 1 ±

T21 Bo

72,±2 O.

To calculate the NMR spectrum we need an expression for the Hamiltonian in the laboratory reference frame. Introducing the Euler angles as above, we have the chemical shift tensor

2

F~m = L Ff.m,v~l,m(O:PL' (3PL, 'YPL)' (9.18)

m'=-2


If we introduce an intermediate frame, N, fixed in the bilayer the expression for the resonance frequency in the presence of an anisotropic chemical shift interaction is

1/ (3 2 1) I/o + 2 cos {3NL - 2

[(3 2 1).2 ]'Y8Bo X 2 cos {3PN - 2 + 2TJ sm (3PN cOS(2O:PN) ~

(9.19)

where the two sets of Euler angles are as defined above and the isotropic resonance frequency is given by

'YBo 6 I/o = -(1 - 80 x 10- ). 27r

(9.20)

In order to have asystematic approach for simulating separated local field spectra (see below) of proteins with known structures, we begin with the protein data bank coordinates, then transform to a coordinate system with origin at the centre of mass. We then calculate the moment of inertia tensor elements in this coordinate system. After diagonalizing the moment of inertia tensor we then transform to its principal axis system and finally from there to the laboratory frame. The Euler angles for the sequence of transformations required for a dipolar / chemical shift separated local field spectrum then are: (O:CSD, (3csD, 0) to go from the chemical shift principal axis frame to the dipolar frame (notice that this definition is slightly different from that used in [12] and [8]; (O:DMI, (3DMI, 'YDMI) to go from the dipolar reference frame (with principal axis directed along the vector joining the two dipolar coupled nuclei) to the moment of inertia (MI) reference frame; and (0, T, p) (using the notation of [13]) to go from the MI frame to the laboratory frame. In some circumstances we will introduce axial reorientation about the principal axis of the moment of inertia tensor. In this case we can average over the angle 'Y D MI·

In the following section we use the expressions derived above to simulate one type of separated local field experiment starting from protein data bank coordinates in order to illustrate some of the influences of structural variations and molecular motion on this type of spectrum. This approach will prove valuable both in designing new experiments [14] and in interpreting experimental results.

3. Separated local field spectroscopy As discussed above, the most direct way to obtain structural restraints is by mea

suring the residual magnitudes of the second rank tensor interactions directly from the spectrum of a macroscopically oriented sample. In multi-dimensional spectroscopy, separated local field (SLF) experiments are designed to allow the spin system to evolve under HI during tl, under H2 during t2, etc. In static samples the range of anisotropic chemical shifts and dipolar couplings can be quite large, which can overcome to some degree the relatively low resolution typical of solids due to their inherently broad lines


Figure 9.1. Molecular model of an a-helical alanine with its two adjacent planar peptide bonds. To support our discussion, we display the dipole-dipole interaction between directly-bonded 1 H_15N (dashed arrows) and the relative size and direction of the chemical shift anisotropy principal values 0'11 ,0'22 and 0'33 for 15N (straight lines). For 15N, (0'11, 0'22, 0'33) = (40, 68, 202) ppm and (acsD, (3CSD, "(CSD) = (900 , 1040 ,0°) [8].

(even for well oriented samples). In some cases significant resolution enhancement is achievable using homonuclear 1 H decoupling techniques such as the frequency shifted Lee-Goldburg sequence [15].

The problem of peak assignment may limit the application of SLF experiments for large uniformly labeled protein systems. Some assignment strategies based on the repeating pattern of protein substructures (a-helices, ,B-strands) have been proposed but require a level of rigidity and regularity unusual for proteins. However, the flexibility of this general approach should ultimately lead to a series of sequence specific experiments (such as those commonly used for soluble proteins) to enable the complete assignment of uniformly labeled proteins [2,16]. It is likely that experiments which measure the anisotropic interactions of nuclei along or near the protein backbone will prove to be the most useful. The authors of [14] review the broad range of possible SLF experiments and discuss their potential utility. We will focus our discussion on only one of these experiments that we consider to be both feasible and useful. This experiment which we will refer to as HNIN measures 15N chemical shifts and 1 H_15N dipolar couplings from the same peptide plane as seen in Fig. 9.1.

The search/or high-resolution NMR methods/or membrane peptide structure 199

'H ~I-------lcp ~ Oecoupling

15_N--'-_C_p----'_+X_--'-t1_-X----'~--

Figure 9.2. Pulse sequences for the HNIN two-dimensional separated local field spectroscopy experiment (PISEMA, [15]).

Fig. 9.2 shows the three-step pulse scheme for the HNIN experiment. Cross polarization is used to transfer coherence from the abundant IH spin population to 15N. Then the 15N magnetization is allowed to evolve under its dipolar coupling with the directly bonded 1 H during tl' The chemical shift interaction can be eliminated with an inversion pulse or by letting 1 Hand 15N evolve under a common spin lock. The use of Lee-Goldburg schemes during the tl interval, to suppress the 1 H_l H homonuclear coupling, has a dramatic effect (up to four-fold improvement) on the resolution in the WI dimension [15]. The homonuclear decoupling is achieved by setting the RF field offset such that the 1 H effective field is at the magic angle in the rotating frame. In this specific form, the HNIN experiment is also known as PISEMA (Polarization Inversion Spin Exchange at the Magic Angle [15]). Finally, after t1> the free induction decay is recorded in the 15N channel while decoupling hydrogen to leave only the 15N chemical shift during t2' So far this has been the most common of the SLF spectroscopy experiments applied to polypeptides in oriented membrane samples.

Besides the resolution improvement in WI, this experiment potentially has very good resolution because of the strong 1 H_15N dipolar coupling (up to 23 kHz) and the large range of 15N chemical shifts (0 = 100 ppm). It should be noted that although the potential range of dispersion for both interactions is large, they are not independent. Indeed, because directions corresponding to a33 and the NH dipolar coupling are almost colinear, as seen in Fig. 9.1, there is generally a clustering of the peaks in a restricted region of the spectral plane (for 2D experiments) in proteins with secondary structures with repeating patterns. Signal-to-noise is relatively good because the pulse sequence does not require any long transfer delays which are prone to suffer from relaxation effects and orientation dependent transfer efficiency. Furthermore, this experiment only requires the uniform introduction of the isotopic nuclear spin label 15N. This can be achieved biosynthetically on 15N minimal media, with the result of some overlap with side chain signal contributions in the spectrum (for Trp, GIn, Asn, His and Arg).


Finally, this experiment is technically relatively simple, since it requires only two channels whereas most potential SLF experiments employ at least three.

3.1 SLF of ideal polypeptide helices The transmembrane domains of membrane proteins are often formed from a-helices,

either as single transmembrane helices (neu, glycophorin) or as transmembrane helix bundles (rhodopsin, photosynthetic reaction centre, K-channel). Alternatively, they may form a ,6-barrel channel configuration (bacterial outer membrane porins). These secondary structures are defined by their repeating peptide backbone dihedral angles, namely <Pi (COi-l_Ni-C~-Coi), 'lj;i (Ni-C~-COi_Ni+l) and w (C~-COiNi+l-C~+1). In the a-helix, the first two angles have typical values of -65° and -40°, respectively, whereas in ,6-strands, these angles are typically -139° and 130°. Because of the resonant nature of the peptide bond, w is generally very close to 180°. In order to span a typical membrane, the protein backbone will form these regular structures over a significant stretch of amino acid residues. Such extended secondary structural domains have characteristic signatures in SLF spectroscopy.

Following the description above, using the principal axis frame of the moment of inertia tensor as an intermediate reference frame for an ideal polyalanine a-helix (with all (<p, 'lj;, w) = (-65°, -40°,180°», we can predict the corresponding SLF spectra for any orientation (tilt or azimuthal angle r) of the helix long axis with respect to the membrane normal/magnetic field and for any rotation or polar angle (p) about the helix long axis. For a helix with no irregularities, all the resonance peaks will lie on a smooth closed curve as is shown in Fig. 9.3C for 7 different values of r. These figures, called "PIS A wheels" (Polarity Index Slant Angle [13, 17]) in the case of PISEMA spectra, possess two interesting characteristics. First, as is evident from Fig. 9.3C, the overall position along the diagonal and the shape of these figures are indicative of the helical tilt angle. Indeed, for the HNIN spectrum, a helix parallel to the magnetic field (r = 0°) shows both a large chemical shift (0'33) and a large dipolar coupling resulting in a pattern of peaks near the upper left comer of the figure. However, when such a helix is tilted perpendicular to the field (r = 90°), the chemical shift is much smaller and the dipolar coupling is negative. A second property of SLF spectra of regular helices is related to the rotation or polar angle p. Although changing the polar orientation of the polypeptide about a given azimuthal axis direction does not alter the shape or position of the smooth curves (PISA wheel patterns), it does shift the relative position of any individual peak along the closed outline of each curve. Consequently, once one peak in a pattern has been assigned, one could in principle trace out the amino acid sequence and assign all the peaks of an ideal, regular a-helix since two adjacent residues are separated by 6.p = 100°.

The results of similar simulations on an ideal, regular ,6-strand with dihedral angles (-139°, 130°, 180°) are shown in Fig. 9.4. Since ,6-strands are very elongated and have only a small degree of helicity (Fig. 9.4A-B), the orientations of consecutive peptide planes will alternate by close to 180° relative to the long axis. The predicted spectrum in Fig. 9.4C shows how this peptide geometry translates into a binary pattern


A

c

Figure 9.3. (A) and (B) Configuration of an ideal 22-residue polyalanine a-helix and the definition of the tilt (r) and rotation (p) angles. (C) Typical patterns expected for an ideal alpha-helix (-65,-40) in the HNIN spectra. In (C) the dotted lines connect signal peaks according to the protein primary sequence, and the dashed curves give the expected locus of peak positions (as p varies) for the tilt angles given in the figure.

in SLF spectroscopy. Here we have restricted the simulations to a stretch of 14 residues (it takes a minimum of 7 to span a typical membrane bilayer) keeping the tilt direction approximately parallel to the peptide planes as is observed in all members of the bacterial outer membrane porin family. For tilt angles near 0°, the NH bonds are nearly perpendicular to the field so that the chemical shifts are near their minimum value and the dipolar couplings are negative in the HNIN spectrum; for a tilt T near 90°, the opposite effect is observed, with chemical shifts near their maximum value and large positive dipolar couplings. We also observe that for intermediate tilt angles the alternation in peak position arising from consecutive residues varies mostly in chemical shift in an HNIN spectrum; this effect is due to the alternation in the orientation of the chemical shift tensor with respect to the MI tensor principal axis (the long axis of the molecule).

It is important to stress here that these ideal patterns are simply spectroscopic signatures of the most common secondary structural elements of proteins. As we shall

202

A

C 25000

20000

f 15000

.1:' 10000

~ 5000 ~ R o .j! ·5000

;t .,0000


B

~ -=O~" -" __

···~o. - 'V-~

O' IS' · '5000 +-....... ~-.-~.-,........,_--,-'--~..-,_-.:-

200 180 160 140 120 100 80 60 40

"N Chemical Shift (ppm )

Figure 9.4. (A) and (B) ConfigW"ation of an ideal hydrogen-bonding antiparallel ,a-strand pair and the definition of the tilt (7) and rotation (p) angles. (C) Typical patterns expected for a 14-residue ,a-strand (-139, 130), when it is tilted as shown, in the HNIN spectrW"n. Dotted lines show the locus of peak positions (as p varies) expected for the given tilt angles.

see in the next section, there are many structural variations and fluctuations which will greatly affect the appearance of these spectra in real systems.

3.2 Effect of helical irregularities

The diversity of folds found in proteins leads to significant structural variability even within the secondary structural elements. For example, there are large ranges in the dihedral angles <p (with typical rms standard deviations of cr = 10°) and 7jJ (with cr = 10°) and in other internal angles [18], which still permit the typical hydrogen bonding patterns found in a-helices (NW to COi+3) or in ,B-strands. Statistical analysis of known structures shows that the peptide "plane" may also deviate from planarity with the w torsion angles having standard deviations of up to about 7° [19]. At least as significant as these structural variations, the chemical shift tensor principal values and their orientation within the molecule may vary significantly with residue type and local structure (variations in principal values of several tens of parts per million are typical). How do these common structural irregularities in complex proteins influence


the appearance of these solid state SLF NMR spectra? Using computer simulations on known membrane protein structures, for example, from the protein data bank (PDB) coordinate files (http://www.rcsb.org/pdbl), we can obtain the expected SLF spectra for any combination of second-rank tensor interactions.

As a model structure, we used that of the light-driven proton pump bacteriorhodopsin (PDB identification code: 1C3W, [20]), a typical helical membrane protein having 7 transmembrane a-helical segments. The structure was obtained by X-ray crystallography in a matrix forming a cubic lipid phase and is one of the the highest resolution structures available to date for a membrane protein (1.55 A). Fig. 9.5 shows the expected position of the peaks from the 7 transmembrane helices in the HNIN (Fig. 9.5A) spectra of this molecule oriented with the crystal unit cell z-axis aligned parallel to the magnetic field direction. Unlike the ideal helix case, the peak positions for anyone of the 7 helices do not follow any simple closed curve, but rather are clustered throughout the accessible region of the spectral plane. The obvious departure of the peak positions from the ideal curves can be attributed to large variations in a number of structural elements. Statistics on the structure show that there is a broad distribution of dihedral angles even within the 7 helical transmembrane domains. For different residues within these helices, 4> varies from -1200 to -500 and 'ljJ varies from -520 to +50 • These distributions have standard deviations of 10.30 and 10.20 , respectively. The planarity ofthe peptide linkages shows a distribution with a standard deviation of 4.8 0 in W, but this narrow range may result to some degree from the refinement procedures used in deducing the structure. Variations also arise from residue specific differences in the anisotropic chemical shift tensor principal values and orientations [8]. An important effect not included here is the sensitivity of the chemical shift tensor elements to the details of local structure. The tensor's principal values and its orientation within the molecular subunit can vary dramatically with the peptide dihedral angles.

To see even more clearly the effect of structural imperfections, we can compare the 8 leucine residues (position 87, 92, 93, 94, 95, 97, 99 and 100) ofbacteriorhodopsin's transmembrane helix 3, since all of these leucines have different positions (different values of p) along the tilted helix. The dependence of the predicted peak positions, for each of these leucines, on the angle p at a constant tilt angle, T = 8.80 , corresponding to the orientation of this helix relative to the crystallographic z-axis, is shown in Fig. 9.5B for HNIN. In this figure, the points give the positions of the peaks that would arise from a given leucine as p is varied from 0 to 21T'. For an ideal helix, each of these sets of points (one set corresponding to each leucine) would have traced out exactly the same smooth curve. Again the static structural irregularities in helix 3 have a large effect on the shape of these patterns, although their positions still roughly indicate the overall helical tilt orientation.

Of course, it is uncertain how accurately the crystal structure represents the structure of bacteriorhodopsin in its native membrane environment. There is some concern that extensive structure refinement may bias the experimental measurements towards ideal bond lengths and angles. Perhaps "on average" bacteriorhodopsin does adopt this structure, but the solid state NMR measurements are strongly influenced not only by structural variability such as that discussed above, but also are sensitive to dynamical


25"0 r----------------------------, A

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.... ... ..

0'

•

~ + HeIIlt2 .t. Hebl e Helil:04 CHerilx S O HilUx 6

~

o

soo.~-_---~---~--~---~---_---~ .. 0 200 180 ~ao 140 120 100

U N Chem ica l Sh ift (ppm)

2.000 '-S:---------------------;:---=:=,=.,=:.:;:,:::;-] --+- L.eu 92 ......... I.eu 93 22000

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........ lllu 94 -o-lilu 95 ~ll!u 97 -..-leu 99 -o-leu loa

10000 -I----_-~--_--~--_--_--~-~-----l 210 200 10. lao 170 '60 1$0 140 130 120

UN Chem ical Shift (ppm)

Figure 9.5. The effect of natural helical irregularities on HNIN spectra. (A) The expected spectra from the 7 static helices of bacteriorhodopsin based on the structure reported by X -ray crystallography (1 C3W, [20]) The dotted curves give the locus of peak positions (as p varies) for several tilt angles. (B) Individual peak patterns for each of the 8 leucines from helix 3, tilted at 8.80 • These curves are traced out as p is varied. For an ideal helix, all 8 leucines would trace out the same curve.

fluctuations in orientation of the particular molecular group being examined. In the next section we discuss the impact of motion on the observable solid state NMR spectra.

3.3 Effect of molecular reorientation In addition to their effect on NMR relaxation, any fast molecular motion that causes

the second rank tensor of either the chemical shift or the dipolar coupling to reorient in


the magnetic field will effectively reduce the strength of these interactions and will have to be considered in the interpretation of the residual couplings in terms of orientational restraints. Molecular structures obtained by X-ray crystallography are presented as static, since motions only affect resolution. Even at very low temperature, it is wrong to assume that NMR can be interpreted in the same way and can provide the same type of static structure. Rather, solid state NMR measurements provide an average tensor strength, where that averaging process is simultaneously an ensemble average over the distribution of orientations present within the sample and a time average over the orientational fluctuations. Assuming that the motions are in the "fast limit" regime, we can easily examine the influence of some simple types of molecular motion on the predicted NMR spectra. For the simulation of SLF spectra, the static structures available from the protein data bank are incomplete and we need to introduce molecular reorientation into the process. NMR relaxation studies can be used to characterize the form, the amplitude, and the time scale of some of these motions, and to provide a valuable starting point for our consideration of the effects of motion on the spectra [21-24].

In this section we shall consider several types of motion and evaluate their effects on the SLF spectra. We will look first at whole molecule axial diffusion about the moment of inertia tensor principal axis, then at fluctuations in the orientation of this axis ("wobble"), and internal "librational" motion of the individual peptide planes. Finally, we will use the results of full atom molecular dynamics simulations to model the types of motions that occur during the NMR averaging timescale and to see their effects on the SLF spectra.

As for membrane lipids, axial diffusion can be quite appreciable for small transmembrane proteins. Gramicidin A, a 2 kDa peptide which forms a transmembrane head-to-head dimer channel for monovalent cations, has been reported to rapidly diffuse about its long axis with a correlation time of ~ 7 x 1O-9s [24]. Neglecting all other motions, molecules with their axes of reorientation parallel to the magnetic field will appear invariant under this motion. As a result, for this orientation, this kind of motion does not reduce the magnitude of the tensor interactions and the NMR spectra are unchanged by the motion. In order to calculate the effect of axial diffusion about an axis inclined at an angle T relative to the magnetic field, we need to introduce an intermediate reference frame, as described above. For a long a-helix, where the moment of inertia principal axis can be considered to be the diffusion axis, we simply need to average the angle rDMI (the rotation angle p) from 0 to 211".

As a test molecule, we used the 36-residue transmembrane domain (TMD) of the rat growth factor receptor Neu, a receptor tyrosine kinase. The TMD primary sequence is QRASPVTFIIATV-V664-GVLLFLILVV-VVGILIKRRRQK-amide. What makes this peptide so interesting is that a single point mutation (V664:E) results in constitutive activation of the protein and cell transformation. The structure of the TMD was obtained by standard solution-state NMR methods based on nuclear Overhauser effect distance restraints. In tri-fluoroethanol, the TMD forms an a-helix [25] with some unraveling of the helix at each end. Due to its length and hydrophobic sequence, it is likely that it would form a transmembrane helix in lipid bilayers. Because the

206

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Figure 9.6. (A) The effect of axial diffusion on HNIN spectrum of neu transmembrane domain based on the structure reported by [25]. (B) Effect of ±1O degree wobble of an ideal helix with no tilt on the HNIN spectrum. (C) Effect on the HNIN spectra, of a librational fluctuation in the orientation of the (j - 1,j) peptide plane about the C~-l--C~ direction for an ideal helix tilted at 30°. In all spectra, the closed curves give the expected locus of peak positions (as p varies) for a given tilt angle.

hydrophobic segment is somewhat thicker than the typical fluid phase phospholipid bilayer, the TMD is expected to tilt in the membrane. In fact, molecular dynamics simulations of this protein in a phospholipid bilayer demonstrated that at the end of a 10 ns simulation trajectory, the helix tilts at an angle of 20° relative to the membrane normal [26].

A series of calculated HNIN spectra of the transmembrane domain of this polypeptide axially diffusing about its predicted moment of inertia and tilted at increasing angles is shown in Fig. 9.6A along with the traces ofthe rotation patterns (vs p) of an ideal helix with similar tilt angles. Whereas the locus of peak positions for an ideal axially diffusing helix would lie along a straight diagonal line (at the centres of each of the static rotation patterns), the peak positions for this imperfect helix are spread out over a significant area. As the tilt angle approaches the magic angle, the spread in peak positions becomes smaller and smaller. In the presence of axial diffusion, the peak dispersion is limited and renders resolution of the signals difficult. SLF spectroscopy is clearly better suited for non-reorienting proteins. However, rapid axial diffusion


becomes an advantage when using magic angle spinning 1 H NMR as it reduces the effect of homogeneous 1 H_l H dipolar broadening [11] (see below).

Another important whole-molecule motion is the wobbling motion whereby the orientation of the director axis fluctuates with time. This can be due either to the reorientation of the tilt axis of the molecule within the membrane or to local fluctuations ofthe membrane normal (sometimes referred to as "undulations"). These motions are therefore not necessarily restricted to smaller polypeptides. As an example of this type of motion, relaxation measurements on Gramicidin A in oriented lipid bilayers have indicated that the peptide's long axis reorients with a correlation time of the order of 6 x 10-6 s and an rms amplitude of about 16° relative to the normal to the glass plates (which were fixed parallel to the magnetic field) [24].

In order to simulate the dynamic effect on SLF spectra, we have chosen to average the values of the interactions over an evenly distributed population of orientations within a cone centred at the average tilt angle. An ideal helix with no average tilt, but with a wobble range of ± 10° will have an SLF pattern resembling that of a static helix tilted at about 5°, as shown in Fig. 9.6B for the HNIN spectrum. It is obviously crucial to be aware of the effects of this type of motion even when only estimating the tilt angle based on the pattern of peaks observed in SLF spectroscopy [17].

A third type of motion is the peptide plane librations which occur locally within the protein and thus may have different rates, amplitudes and average orientations depending on the position of the residue within the molecule. To model this type of motion, we allow the jth peptide plane to rock about the C~-l-C~ direction by varying amounts around an average orientation. Within a range of ± 10° around the average position, this motion has a relatively small effect on the HNIN spectrum (Fig. 9.6C). However, it does result in different shifts in peak positions depending on the position of the peptide plane within the molecule. At an average tilt angle of 30°, the larger 15N chemical shifts or 1 H_15N dipolar couplings are reduced more than the smaller ones. In any case, the shifts in this spectrum are of the order of 10 ppm for chemical shift, and 500 Hz for dipolar coupling.

As a final illustration of the effect of molecular motion on SLF spectra, we use "snap-shots" taken from a molecular dynamics simulation of the neu TMD in a fully hydrated dimyristoyl-sn-glycero-3-phosphocholine (DMPC) bilayer [26]. The internal and whole body motions that occur during the course of this 10 ns simulation trajectory lead to fluctuations in the orientations of the chemical shift and dipolar tensors relative to the bilayer normal (presumed parallel to the magnetic field) and, therefore, to fluctuations in the magnitudes of the couplings. At each instant of time (represented by a "snap-shot" in the trajectory file) we can calculate an HNIN SLF spectrum. The spectrum that would be observed experimentally would be an average over these rapid motions.

The simulated spectra shown in Fig. 9.7 A-C are for instants of time during the IOns trajectory. The differences between these three spectra illustrate the sorts of variations being averaged over during an actual NMR experiment. In Fig. 9.7D we show the evolution of two individual peaks in the HNIN SLF spectrum over the course of the

208

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Figure 9.7. Simulated HNIN SLF spectra of the neu protein transmembrane domain (TMD) using coordinates extracted from the trajectory file of a IOns molecular dynamics simulation of the neu TMD in a fully hydrated phospholipid bilayer [26]. Snap-shots taken: (A) 3 ns into the simulation; (B) 6 ns into the simulation; (C) 9 ns into the simulation. (D) The peaks corresponding to two of the residues located well within the stable a-helical core of the TMD. The solid symbols are for valine-663 while the open symbols are for valine-673. In (A), (B) and (C) the dotted lines connect consecutive points according to the primary sequence starting at ARG-6S2. In (D), the dotted line connects the positions of the peaks chronologically based on the 10 snapshots from 1 ns (large symbol) to IOns. The solid curves in all figures show the expected locus of points as p is varied, for an ideal helix tilted at the angles given.

trajectory. The simulation covers only 10 ns, and the NMR averaging timescale is much longer than this (several milliseconds) so that the motional averaging that occurs during the experiment is undoubtedly even more severe than what can be illustrated by such a simulation.

We have seen through this brief summary that different types of rapid fluctuations influence the appearance of SLF spectra to various degrees and in different manners. Whole body motions tend to reduce the spread of peaks in the spectrum in a generalized manner, whereas internal motions may result in essentially no effect to quite substantial shifts in peak position simply based on the location of the residue involved.


lH ~ t1 CP decoupling

n IW~w~ __ -15N or 13C CP J-rafocuslng

Figure 9.B. Pulse sequence for the two-dimensional cross-polarization (CROPSY) experiment.

4. High-resolution 1 H MAS NMR of small membrane proteins

Because of the sensitivity, ubiquity and natural abundance of hydrogens, highresolution 1 H NMR is the most successful NMR method for obtaining structures of proteins in solution, where the rapid isotropic reorientation of the molecules averages out all orientation dependent interactions. In combination with the isotropic part of the chemical shifts of 15N and 13C spin labels, it is conceivable that multidimensional experiments based on 1 H spectroscopy may provide enough resolution to solve structures of soluble proteins of up to several hundreds of kilodaltons. The assignment task is eased by strategic combinations of experiments based on the J-coupling among covalently bonded spins, while structural restraints are obtained mainly from nuclear Overhauser effect data between neighboring hydrogens and from internuclear J-couplings and small residual dipolar couplings. In solid systems, the strong homogeneous dipole-dipole coupling between many neighboring hydrogen nuclei is a major obstacle to the use of 1 H NMR for structural studies of large molecules.

The small constituents in membrane systems present an intermediate situation. A rapid axially symmetric motion projects the intramolecular IH_IH dipolar interaction onto the axis of motional symmetry. Consequently, the dipolar broadening scales as (3 cos2 () -1}/2, where () is the angle between the motional axis and the magnetic field. Effectively, the dipolar broadening becomes inhomogeneous and can be eliminated by MAS [10,11]. However, residual line broadening due to slow fluctuations in the IH_ 1 H dipolar interaction will, as always, limit the resolution. In practice, a rapidly axially diffusing (10-9 s < Tc < 10-7 s) polypeptide, such as Gramicidin A, still requires MAS rates of at least 10kHz to reduce the 1 H linewidth to a few tens of Hz.

Although the resolution approaches that obtained in solution, a few special considerations must be taken into account when designing experiments for such systems. For one, the magnitude of the naturally abundant spin-l 14N quadrupolar moment is so large (having quadrupolar splittings of the order of 1 MHz in non-symmetric systems)


that the 14N spins act as a strong lattice sink to the coupled backbone amide hydrogen (1 HN) magnetization. Since this quadrupolar nucleus cannot be decoupled by RF pulses, substitution with the spin-112 isotope 15N yields greatly improved resolution for 1HN.

Secondly, 1 H relaxation processes are so strong that relaying the signal through J-couplings (typically requiring mixing times of several tens of milliseconds) is not feasible. Liquid-state experiments such as the heteronuclear single quantum coherence (HSQC) or the homonuclear total correlation spectroscopy (TOCSY) thus become impractical in MAS membrane systems. For 15N-Iabeled proteins in multilamellar dispersions, the strong dipolar coupling between 1 Hand 15N may be employed as the transfer mechanism for their magnetization; this must be accomplished with a short cross-polarization exchange step since the membrane protein 1 H T 1p is typically of the order of milliseconds. Because of the through-space dipolar nature of this transfer, it will not occur uniquely through directly bonded HN pairs and will also depend on the HN bond orientation with respect to the reorientation axis. The pulse sequence for this sort of CROss-Polarization SpectroscopY (or CROPSY) experiment is shown in Fig. 9.8 in its simplest form. The 1H magnetization evolving during t1 is transferred to 15N via cross polarization and then the 1H-modulated 15N signal is recorded in t2. Of course, this experiment can also easily be applied to 13C-Iabeled proteins.

The results of this experiment can be seen in Fig. 9.9 for 15N-Iabeled Gramicidin A dispersed in chain and head-group deuterated DMPC-d67, and hydrated with a 50 mM phosphate buffer at pH = 6.0 [27]. The sample was rotating at the magic angle at a rate of 12 kHz while the sample temperature was maintained at 55° C. Even with average resolutions of 41 Hz and 164 Hz (measured at half-height) in the 15N and 1 H dimensions, it is possible to decipher a majority of the 20 HN pairs with only two strongly overlapping peaks (Trp-15 I Trp-ll) originating from the backbone. The tentative assignment provided in the figure was based on the similarity of this spectrum with the HSQC spectrum of the same molecule isotropically reorienting in a micellar dispersion of sodium dodecyl-sulphate (see supporting information to [27]). It is clear that this high resolution approach will only be viable if methods are developed to obtain direct peak assignments and structural restraints. In the latter case, it is likely that both cross relaxation (NOE) and dipolar recoupling approaches will be able to provide internuclear distance restraints. A combination of these restraints with orientational restraints from SLF experiments may provide an accurate high resolution structure of these small membrane polypeptides.

5. Conclusions The high-resolution molecular structures obtainable by diffraction methods are

based on experimental data which precisely define atomic positions through the electron density. To supplement the experimental data, refinement procedures use, to varying degrees, the well characterized bond length and bond and torsion angle distributions found in high precision structures of small molecules. Some refinement protocols also introduce interatomic potential functions defined for bond lengths, bond angles, di-


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15N

",

G:Ii _ .• ~ _____________ __ . __ • Et~ _.1 ________ ,,15

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130

Figure 9.9. 1 H, lSN CROPSY spectrum of lsN-labeled Gramicidin A dispersed in DMPC-d67 and hydrated with 50 mM phosphate buffer at pH = 6.0. The sample was rotated at the magic angle at 12 kHz and the temperature was maintained at 55° C. The labels represent the tentative residue assignments of the peaks based on minimum nns difference search of the chemical shifts with those in the HSQC spectrum of lsN-Gramicidin A in SDS micelles [27].

hedral angles, etc. Protein structure determination by solution state high resolution NMR is based on measurements of internuclear J -couplings, internuclear nuclear Overhauser effect distance restraints, and most recently, small residual internuclear dipolar couplings (see Chapter 8). Refinement protocols again introduce known bond length and angle distributions and interatomic molecular potential functions. Although there is a qualitative difference in the type of information obtained from the two types of experiments, diffraction vs solution NMR, both can be effectively used to generate "high-resolution" molecular structures. The principal difference between the structures obtained is that the NMR structures show the influence of molecular motion more readily. For this reason, a family of structures consistent with the NMR restraints is usually generated.

The situation with solid state NMR is fundamentally different. Here one can measure orientational restraints as well as distance restraints. In order to convert orientational


restraints to atomic positions it is necessary to use standard peptide geometry. In so doing it will be important to include bond angle variability just as is done in diffraction and solution state NMR structure refinement. Since the experiments measure average or residual second rank tensor couplings, the effect of molecular motion is inherently contained in the raw data. This too must be included when modeling the molecular structure.

In this Chapter we have illustrated how structural variability and dynamics effects can influence the measured tensor couplings through the use of simulations of SLF NMR spectra. The limitations involved in using solid state NMR to study protein molecular structure are evident from the examples presented here. A more difficult problem is the design of strategies to circumvent such problems and ultimately to arrive at a high-resolution protein structure.

The use of rapid MAS to obtain high resolution 1 H NMR spectra of small axially reorienting molecules in membranes is also a potentially powerful method which will require the development of sequence specific assignment strategies comparable to those used in solution state NMR. The use of uniform isotopic labeling, these effective assignment strategies, and the combination of distance and orientational restraints can potentially result in a complete structure determination capability.

Acknowledgments

We would like to thank Frances J. Sharom of the Department of Chemistry and Biochemistry, University of Guelph, for her guidance through countless discussions of membrane biochemistry. We also would like to thank Dr. Erick Dufourc of the Institut Europeen de Chimie et Biologie, Bordeaux, France for the congenial atmosphere which we both enjoyed working with him and his group in Bordeaux; Dr. Jin Qian and Scott Houliston for their work on TIp, Scott Houliston for the coordinates of the neu TMD, and Bryan van der Ende for the trajectory file coordinates from his simulations on the neu TMD. This work was supported by grants from the Natural Sciences and Engineering Research Council of Canada and the Canadian Foundation for Innovation.

References [1] Schwalbe, H., and Bielecki, A. (2001), Angew. Chem., Int. Ed. Engl., 40:2045. [2] McDermott, A., Polenova, T., Bockmann, A., Zilm, K. w., Paulsen, E. K., Martin, R. w., and

Montelione, G. T. (2000),1. Biomol. NMR, 16:209. [3] Opella, S. J., Ma, c., and Marassi, F. M. (2001), Methods in Enzymol., 339:285. [4] Denny, J. K., Wang, J., Cross, T. A., and Quine, J. R. (2001),1. Magn. Reson., 152:217. [5] Marassi, F. M. (2001), Biophys. J., 80:994. [6] de Groot, H. J. M. (2000), Current Opinion in Structural Biology, 10:593. [7] Bechinger, B., Kinder, R., Helmle, M., Vogt, T. C. B., Harzer, U., and Schinzel, S. (1999), Biopoly

mers, 51: 174. [8] Davis, J.H., and Auger, M. (1999), Progress in Nuclear Magnetic Resonance Spectroscopy, 35:1. [9] Zare, Richard N. Angular Momentum. Wiley, New York, 1988.

[10] Maricq, M. M., and Waugh, J. S. (1979), J. Chem. Phys., 70:3300.

The search for high-resolution NMR methods for membrane peptide structure

[11] Davis, J. H., Auger, M., and Hodges, R. S. (1995), Biophys. J., 69:1917. [12] Teng, Q., and Cross, T. A. (1989), J. Magn. Reson., 85:439.

213

[13] Wang, J., Denny, J., Tian, c., Kim, S., Mo, Y., Kovacs, F., Song, Z., Nishimura, K., Gan, Z., Fu, R., Quine, J. R., , and Cross, T. A. (2000), J. Magn. Reson., 144: 162.

[14] Vosegaard, T., and Nielsen, N. C. (2002), J. Biomol. NMR, 22:225. [15] Wu, C. H., Ramamoorthy, A., and Opella, S.J. (1994), J. Magn. Reson. A, 109:270. [16] Pauli, 1., Baldus, M., van Rossum, B., de Groot, H., and Oshkinat, H. (2001), ChemBioChem,

2:272. [17] Marassi, F. M., and OpelIa, S. J. (2000), J. Magn. Reson., 144: 150.

[18] Engh, R., and Huber, R. (1991), Acta Crystallographica Section A, 47:392. [19] MacArthur, M. w., and Thornton, J. M. (1996), J. Mol. Bioi., 264:1180.

[20] Luecke, H., Schobert, B., Richter, H. T., CartailIer, J. P., and Lanyi, J. K. (1999), J. Mol. Bioi., 291:899.

[21] Lazo, N. D., Hu, w." and Cross, T. A. (1995), J. Magn. Reson. B,I07:43.

[22] North, C. L., and Cross, T. A. (1995), Biochemistry, 34:5883.

[23] Prosser, R. S., Davis, 1. H., Dahlquist, F. w., and Lindorfer, M. A. (1991), Biochemistry, 30:4687. [24] Prosser, R. S., and Davis, J. H. (1994), Biophys. J., 66:1429.

[25] Houliston, R. S., Sharom, FJ., Hodges, R. S., and Davis, J. H. (2003), FEBS Lett., 535:39.

[26] van der Ende, B. M., Sharom, FJ., and Davis, J. H. (2002), unpublished.

[27] Fares, C., Sharom, FJ., and Davis, J. H. (2002), J. Am. Chem. Soc., 124: 11232.

III

THEORY, MODELS, AND SIMULATIONS

Arguably, the most interesting observables obtained from NMR spectroscopy of orientationally ordered molecules are their orientational order parameters, grouped together in the order tensor S [1]. These order parameters are the key to understanding the anisotropic intermolecular forces that are responsible for this orientational order in the first place. In this regard, small "rigid" well characterized solutes, whose NMR spectrum is amenable to analysis, are often chosen to probe the intermolecular potential. The order tensors of such solutes can be obtained with great precision from the measured dipolar couplings, provided that at least one internuclear distance is known from other sources (see Chapter 7).

The application of NMR to molecules that make up the liquid crystalline phases, both thermotropic and lyotropic, for the determination and interpretation of their orientational order parameters is a more difficult proposition. Such liquid-crystal molecules normally possess a plethora of coupled spins, and their high-resolution 1 H NMR spectra are often composed of a huge number of mostly unresolved lines. Substitution of deuterons for protons allows the use of 2H NMR for the determination of, for example, C-D bond order parameters. However, these parameters may have errors associated with imprecisely known C-D quadrupole coupling constants, and the possible lack of axial symmetry of C-D bonds. More importantly, most such molecules interconvert among many different conformations, and the NMR spectra measured are motionally averaged over all conformers. Each conformer has its own set of five (or fewer) independent order parameters which cannot be separated experimentally from the conformer probability [2]. Nevertheless, much has been learned from the detailed investigation of flexible molecules, and two of the following Chapters deal with this important problem: Chapter 14 discusses relatively "simple" flexible molecules, while Chapter 13 focuses on more complicated ones, with special emphasis on how to deal with the conformational problem of hydrocarbon chains.

The main question addressed in Part III is how to deal with the statistical mechanical problem of calculating the average orientational order of many types of solutes, ranging from the well-defined simple ones to the more complicated and flexible liquid-crystal molecules themselves. Several approaches will be described. These range from the use of simple phenomenological models that replace the liquid crystal solvent by an anisotropic mean field to attempts to formulate a proper theory of the anisotropic intermolecular potential of mean torque. An alternative approach involves the use of computer simulations. Each of the approaches has its advantages and limitations.

Chapters 10, 11, 13, and 14 are all concerned with the study of solute orientational order in an anisotropic environment. They discuss different simple models for the liquid-crystal environment, but have in common that they do not include any details of the molecular nature of the liquid crystal solvent. Rather, the liquid-crystal environment is viewed as a continuum that is "experienced" to be identical by every solute. These simple models suffer from their lack of detail, and might not be expected to be able to provide more than a qualitative description of the liquid-crystal intermolecular potential. Despite these obvious shortcomings, several of these phenomenological models are quite successful in rationalizing a host of experimental results with a minimum number of adjustable parameters [3]. For example, the CI model reviewed in

217 E.E. Burnell and C.A. de Lange (eds.), NMR of Ordered Liquids, 215-220. © 2003 Kluwer Academic Publishers.


Chapter 10 is capable of producing excellent fits to a set of orientational order parameters measured for more than 45 solutes in a special class of liquid crystal, labeled "magic mixture", with only two adjustable parameters [4]. These fitting parameters are transferable and can be used to predict molecular order parameters to an accuracy of about 10% in these special liquid-crystal mixtures. However, the phenomenological models do not seem able to do such a good job in other liquid crystals. However, they still do predict correct "trends" in order parameter values, and these trends happen to be borne out to a large extent by recent computer simulations (see Chapter 15).

An alternative approach to the problem is to describe both the solute and the liquidcrystal molecules in terms of their molecular properties, and to formulate an appropriate "exact" theory for the intermolecular interactions, as is done in Chapter 12. Clearly, such an approach is conceptually very attractive. In practice, higher than two-body interactions are difficult to deal with, and are usually discarded. The effect of the neglect of such terms is not always clear. However, much progress has been made in the theoretical treatment of problems of orientational order. As is clear from the various Chapters that form part III of this book, the theoretical considerations underlying such "exact" models present a valuable picture of the complications associated with a basic description of orientational order. These theoretical models constitute a true challenge to the oversimplified phenomenological approaches, and vice versa.

A quite different approach, that can be viewed as intermediate between experiment and theory, is to utilize computer simulations [5], as discussed in Chapter 15. Such simulations can be designed to treat liquid-crystal and / or solute molecules to whatever level of sophistication is desired, with a possible price to be paid in computation time. Of course, a simulation is as good as the quality of the intermolecular potential that it employs. Computers are getting sufficiently powerful that atomistic simulations of a solute amidst an abundance of liquid-crystal molecules are now feasible. However, simulations that use simpler pictures should also be able to assess the important features of the intermolecular potential, and to support or reject some of the proposed models and theories.

Part III of this book gives a representative overview of the different methods used to deal with the complicated problem of orientational order. These Chapters also show how much progress has been made in addressing problems of this type that only two decades ago where clearly beyond reach. This is not to say that there is general consensus about the scientific soundness of the approximations made in the various approaches, or about the success or the predictive power of the various methods advanced by different groups. However, what has become abundantly clear over the years is that the field is very much alive, and has benefited enormously from the mutual interplay and the often very critical discussions between proponents of one model or another. In the view of the editors the material presented in the book underlines the complementary nature of the different models for describing orientational order, and it offers a good case study of how science progresses in practice.

References [1] Saupe, A. (1964), z. Naturforsch. A, 19:161. [2] Burnell, E.E., and de Lange, C.A. (1980), Chem. Phys. Letters, 76:268. [3] Burnell, E.E., and de Lange, c.A. (1998), Chemical Reviews, 98:2359. [4] Zimmennan, D. S., and Burnell, E.E. (1993), Mol. Phys., 78:687.

219

[5] Allen, M.P., and Tildesley, DJ. Computer Simulation of Liquids. Clarendon Press, Oxford, 1989.

Chapter 10

SOLUTES AS PROBES OF SIMPLIFIED MODELS OF ORIENTATIONAL ORDER

E.E. Burnell Department of Chemistry. University of British Columbia. Vancouver, B. C. Canada

C.A. de Lange Laboratory for Physical Chemistry. University of Amsterdam. Amsterdam. The Netherlands

1. Introduction When a solute molecule is dissolved in a liquid-crystal solvent, the solute experi

ences an anisotropic intermolecular potential which causes solute orientational order. Since the intermolecular potential is extremely complicated in principle, predictions or even reliable estimates of solute orientational order have remained an elusive goal for many years. In the present Chapter we shall summarize the progress that has been achieved over a few decades, and we shall claim that the aim of predicting solute orientational order is no longer farfetched, provided the proper experimental conditions are created [1].

In an elementary picture, the potential caused by charge distribution B at the position of charge distribution A can be expressed in terms of a Taylor expansion around the centre of A. This so-called multi pole expansion [2, 3] results in the following interaction between A and B:

U = qA<pB(A) - Lfl~E~(A) - ~ LQ~{3F!(AA) + .... a a,{3

(10.1)

with <pB (A), EB(A) and FB(A) the potential, electric field, and electric field gradient (efg) caused by B at the position of A, and qA, flA and QA the charge, the electric dipole moment, and the electric quadrupole moment of A. For large distances between A and B the expansion converges rapidly, and an adequate long-range description for the interaction is obtained. For intermediate distances the convergence of the expansion becomes notoriously slow and the multi pole expansion becomes a less useful tool. At short distances where the charge distributions begin to overlap, Pauli repulsion starts to dominate and the multi pole expansion breaks down completely. A fully quantum-

221 £.£. Burnell and CA. de Lange (eds.), NMR of Ordered Liquids, 221-240. © 2003 Kluwer Academic Publishers.


mechanical treatment of short-range interactions runs into great difficulties, and hence a more phenomenological treatment of short-range interactions is called for. As will become clear later, solute size and shape anisotropy will playa crucial role in modeling the short-range interactions.

In an anisotropic liquid, with many charge distributions interacting with each other at all kinds of distances, a very complicated intermolecular potential arises which would be expected to be composed of both long-range and short-range contributions. The long-range part of the interaction can be expanded into terms occurring in the multi pole expansion. A pragmatic description of the short-range interaction requires phenomenological modeling in which solute size and shape anisotropy will be seen to be the determining factor. To make any headway, simplifying assumptions cannot be avoided. One of the most successful assumptions to describe orientational ordering in nematics and which, despite its apparent crudeness, has provided a useful framework for further developments, is the "mean field" approach of Maier and Saupe [4,5]. The nature of the intermolecular potential is not specified. The key feature is the P2 (cos 0) dependence of the "mean field" on the degree of molecular orientation, leading to the characteristic liquid-crystal orientational order.

The study of orientational order parameters provides an excellent means of exploring the anisotropic intermolecular potential in anisotropic fluids. As shown in other Chapters of this book, NMR is an invaluable technique for the determination of these order parameters which can be obtained from the magnetic dipole and electric quadrupole splittings observed in orientationally ordered molecules. For example, deuteron NMR provides information about the order parameters of CD bonds in molecules that have been selectively or completely deuterated. For rigid molecules the interpretation of NMR observables in terms of molecular orientational order is relatively straightforward. In the case of liquid-crystal molecules, the flexibility and lack of molecular symmetry lead to problems in the precise interpretation of NMR couplings in terms of molecular orientational order parameters. In this case the NMR observables are averages over all the conformational motions, with conformer probabilities that are often unknown. The normal approach is to use some model for the anisotropic intermolecular potential, and the order parameters calculated from the assumed potential are compared to the experimental couplings measured in NMR experiments (see Chapters 13 and 14). The success or failure of such a procedure is indicative for the quality of the assumed intermolecular potential.

The use of small, well chosen solutes as probes of the anisotropic environment in liquid crystals is an attractive approach to investigating the anisotropic intermolecular potential. As indicated above, the study of rigid molecules offers distinct advantages. Solutes that are essentially rigid (barring normal vibrations), and which have any desired symmetry are easily chosen.

In the cIassicallimit, the af3 component of the traceless second-rank orientational order tensor of a "rigid" solute as measured by NMR depends on the anisotropic

Solutes as probes of simplified models of orientational order

potential as ~(~ J(~coseal}/3 - ~8a/3)e- B do'

Sa/3 = ~ Je-kBTdn

223

(10.2)

where U (0,) is the solute-liquid crystal intermolecular potential, e a is the angle between the a = x, y, or z molecular axis and the Z director axis, and the integral is over all possible orientations 0, of the solute molecule. For a solute with axial symmetry only one independent order parameter exists.

There are two main philosophies followed in using Eq. (10.2) to interpret orientational parameters obtained from NMR experiments. The philosophies differ in the assumptions made about the intermolecular potential, U(n).

In the first approach, it is assumed that all solutes "feel" precisely the same average liquid crystal "field". For simplicity we assume that this field is axially symmetric about the director taken to lie along the Z direction. This is the situation that applies for calometric nematic phases that align along the magnetic field in an NMR experiment. Then the anisotropic solute-solvent intermolecular potential is assumed to be written as the product of a solute part and a liquid-crystal part [6]:

u(n) = Asolute(n) X Fliquid-crystal. (10.3)

For example, the solute part could be the polarizability anisotropy, and the liquidcrystal part the Z Z component of the mean square electric field "felt" by the solute. If the solute property Asolute(n) is available from other experiments or from theoretical calculations, then the general idea is to see to what extent such an approach can rationalize the results of NMR experiments. The second approach recognizes the fact that the separation indicated in Eq. (10.3) above is not rigorous, and that the precise details of solute-liquid crystal intermolecular potentials must be treated with more care. In other words, the average liquid-crystal "field" experienced by a solute may, in general, be different for different solutes. For example, one solute could on average spend more time in the aromatic regions of the surrounding liquid-crystal molecules, while another solute may have a preference for the hydrocarbon chain region. In such a case it is quite likely that the two solutes would sense a different average "field" FJiquid-crystal. A complete theory would take these differences into account, as for example is discussed in Chapter 12. An alternative approach to this problem is via the use of computer simulations, in which ideas as expressed above can be tested. Treatments along these lines are discussed in Chapter 15 of this book. In this Chapter we shall explore the first, simpler approach, and investigate to what extent the approximation that all solutes experience precisely the same liquid-crystal "field" is upheld. First we shall discuss the general problem of comparing orientational order among a set of chosen solutes. Then we shall review some examples drawn from our own work which indicate that the approximations involved with this approach are often quite reasonable. We shall focus especially on a special class of liquid-crystal solvents, the so-called "magic mixtures", in which the solute molecular hydrogen and its isotopomers are found to experience an interaction with the liquid-crystal "field",


and hence a degree of orientational order which is essentially zero. The discovery of these "magic mixtures" has provided a veritable breakthrough in unraveling different contributions to the intermolecular potential, as will be shown later in this Chapter [1].

2. Obtaining a self-consistent set of solute orientational order parameters

A series of experiments performed using D2 gas to generate different sample pressures points out some of the difficulties in comparing orientational order parameters [7]. Varying the external pressure of the D2 gas from 8.9 to 168.3 bar leads to two opposing effects. On the one hand, increased external pressure applied to a nematic phase would be expected to lead to increased solvent, and hence solute, order parameters. On the other hand, increased external pressure would cause an increase in D2 concentration. This larger solute concentration lowers TNI and therefore decreases the liquid-crystal and solute orientational order at constant temperature. Nevertheless, such experiments, when performed in a controlled manner, show considerable promise for investigating the intermolecular potential.

These pressure experiments highlight a general problem with using solutes as probes of anisotropic intermolecular forces in liquid crystals in that the solutes perturb the liquid crystal. Ideally, one would like to arrange that the solute is at infinite dilution. However, this is impossible in practice, and NMR spectra are obtained from samples with solutes at finite concentrations. Worse, it is often desirable to compare results from several solutes. For such comparisons to be fruitful, care should be taken to ensure that all samples are run under precisely the same experimental conditions. But what does one mean by the same conditions? One effect of the solute is to alter the phase diagram of the liquid crystal, such as to lower the nematic-to-isotropic phase transition temperature TNI. Thus, one possibility would seem to be to arrange that all samples are run at the same reduced temperature, TR = TT . Another possibility

NI is to use a spectral parameter (from either the liquid crystal or from a solute that is added to all samples) as a method of scaling order parameters among samples. A third possibility is to perform measurements as a function of concentration for each solute of interest, and to extrapolate the results to infinite dilution. For relatively small solutes with simple NMR spectra, this process is feasible. For larger solutes with many spins, the NMR spectrum can be quite complex, and such an extrapolation procedure may not be practical. It would seem worthwhile to explore the possibility of comparing orientational order parameters from solutes at finite concentrations. The danger here is that, in addition to interactions between solute and liquid crystal, solutesolute interactions can playa role and may affect the resulting orientational order.

We have used the nematic liquid crystal N-(4-ethoxybenzylidene)-2,6-dideutero-4'n-butylaniline (EBBA-d2) to explore the several possible schemes mentioned above for the determination of a consistent set of solute orientational order parameters [8]. We find that, for this liquid crystal, the most consistent scheme is to use the solute 1 ,3,5-trichlorobenzene (TCB) as an "orientational reference" for the purpose of a direct scaling of orientational parameters obtained from the NMR spectra of different sample


... '" .., '" E CII ... CII 0-... '" -0 ... 0

.. .... .. ...... . constant T _____ constant TR ______ constant TeB splitting

0.4 _____ constant EBBA splitting

... ~ .. ~ .. :;: .. = .. ;; .. i .. ~ .. == .. = .. == .. = .. = .. =.;, .. == ... = .. '-" .. :,:: .. = .. ~ .. = .. = .. .:"; .. .,. ....... ,,. ...... ~------- p - dcb ..

0,2 . = .. = .. ~ .. ~ .. ~ .. ~ .. ~ .. ~ .. _.~ __ ~~ __ ~ _________________ m-dcbD =:..;,;.--;;; .. ~ ...... ;;,; .. "' ....... "'" .. .., ...... --~~--~---------------- cbat __ ~~ ________________________________ o-dcb

u

"' .. ':" .. O': .. ~ .. ~ .. .., .. "". " ..... ..-..... -._~~ ..... ____________ o-dcb ..

° ~~~~ .. ~ .. ~ .. ~ .. :. .. ; .. :.: .. :":":H:WH:;":":"~~::::===================m-dcb .. :-:_ .. ~~ ___ . t.JiT" cbU'

_________________________ _ _______ benzene

~~ ... ~ .. ~ .. ~ ... ~ .. ~ .. . ~ .. ~ .. ~ ... ~ .. ~ .. ~ ... ~.~~ ... ~ .. ~ .. ~ ... ~ .. ~."~.--______________ p - dcbD

-0,2 ____________________________ lcb

dilute p·dcb ben cb o-dcb tcb m-dcb sample labeled by solute in excess

225

Figure 10.1. Comparison of order parameters scaled to TCB. The scaling is accomplished by mUltiplying order parameters from sample i by Syy(TCB, sample 6)/ Syy (TCB, sample i). To demonstrate the variation in order parameter with sample, parameters for all four sets of experiments are shown in one graph. The order parameters are plotted versus sample (labeled by the solute in excess, and presented in order of decreasing TNI). Note that sample 6 (excess TCB) was used as reference for the scaling because the spectrum run at 300.9 K for this sample was used for all sets of experiments. Hence, sample 6 is a common point for a particular order parameter for all four samples. The marks on the left of the graph repeat the order parameters for the excess TCB sample; this aids detection of deviations of lines in the graph from the horizontal. To avoid too much clutter, the Syy points for chlorobenzene, a-dichlorobenzene, mdichlorobenzene and p-dichlorobenzene have been omitted. Molecular axes are defined in [8]. (Reprinted with permission from [8]).

tubes, all run at the same real temperature. The scaling of the order parameters for a solute in sample i is then

S (1) - S·( 1 t) Sstandard sample(TCB) i, scaled so ute - ~ so u e x Si(TCB) . (l0.4)

Fig. 10.1 demonstrates the result of scaling four different experiments (taken in different ways on the same set of7 samples) using Eq. (l0.4). As indicated in the figure,


the experiments were performed either at constant real T, at constant TR, or by adjusting the temperature for each sample such that the dipolar splitting of TCB (or the quadrupolar splitting in EBBA-d2) was kept constant. A quick glance at the figure points out the success of the TCB scaling for experiments run at constant temperature, and the variation that exists when the real temperature is varied during the experiment, as is the case with the experiments run at constant TR and at constant TCB or EBBA spectral splitting. The constant-temperature TCB scaled order parameters for each of the 6 aromatic solutes at varying concentrations in 7 different samples agree to about 1 %, regardless of absolute concentration. As the total solute concentration exceeded 15 mol% for some samples, the agreement is quite a severe test of possible comparisons. It was found that using the liquid-crystal deuteron spectral splittings for scaling was not as successful as using TCB splittings.

Direct comparisons of order parameters among samples run at equal reduced temperatures, at equal TCB splitting, or at equal liquid crystal deuteron spectral splittings did not produce results as consistent as those in Fig. 10.1 for scaling the constant temperature results to TCB.

It is obvious that if one wishes to compare solute orientational order parameters to an accuracy of 5% or better, one needs to pay careful attention to the comparison method. More extensive, careful work on solute concentration dependence would seem to be useful.

3. Factors affecting solute orientational order The NMR investigation of small solutes such as the isotopomers of molecular hy

drogen [9-12] (see Fig. 10.2 for the spectra of three ofthese isotopomers) and methane [6,13,14] in nematic liquid-crystal solvents yields unique information about the intermolecular potential. In the case of 02 (see Fig. 10.3) and HD which are well characterized molecules, the quadrupolar splitting observed in the deuteron NMR spectrum can be predicted from the known intramolecular efg and from the solute orientational order obtained from the dipolar coupling measured from the same deuteron NMR spectrum. The measured value for this quadrupolar coupling disagrees with that predicted, and the disagreement varies with liquid-crystal solvent. The rationalization of the observed discrepancy involves an additional, external contribution to the efg tensor. This additional intermolecular contribution is taken as arising from the liquid-crystal environment. As a consequence of this supposition, a molecule that possesses a non-zero molecular quadrupole moment tensor Q 0I{3 is predicted to exhibit orientational order

when in the presence of an anisotropic efg, F~~t , the interaction energy being

1 -UQ = - 3QOI{3F~~t (10.5)

where F~~t is the af3 tensor element of the external efg. For an axially symmetric solute (with symmetry axis z) in a nematic phase this reduces to

TT lQ S Fext U Q = - '2 zz zz Z Z . (10.6)

Solutes as probes of simplified models of orientational order 227

Since the electric quadrupole moment is known experimentally and theoretically for H2 and its isotopomers, the order parameter Szz can be obtained for any value of Fzz using Eqs. (10.2) and (10.6) [1,6,10,15,16]. With this approach it is found that the interaction of the external efg (estimated from the above discrepancy) with the molecular quadrupole moment represents the dominant contribution (:::::: 90%) to the experimental order parameter. Detailed ab initio quantum-chemical calculations in which the orientational order of H2 and its isotopomers is calculated with Fzz as the only adjustable parameter show very strong support for the notion that for these molecules the predominant contribution to the orientational order is the interaction between molecular quadrupole moment and solvent external efg [17]. These quantummechanical calculations also predict interesting isotope effects that are observed in the experimental order parameters of H2 and all its isotopomers resulting from 2H and 3H substitution. An interesting and counterintuitive result is that the experimental order parameters of D2 are positive in some liquid crystals (alignment preferentially along the director), and negative in others (alignment preferentially perpendicular to the director). This observation derives immediately from the fact that the theoretical

I I

5000

TT

HT

DT

I I J I

o Frequency (Hz)

1 I

- 5000

Figure 10.2. 640.12 MHz tritium NMR spectrum of a mixture of T 2, H2, and D2 in EBBA at 300 K after irradiation of the sample tube in order to achieve isotope scrambling. (Reprinted with permission from [12]).


60

2H NMR of D2 in 1132 3J

* 1------28 -------

6 lobo 2doo 3000 Hz

Figure 10.3. 61.4 MHz deuteron NMR spectrum of D2 in ZLI 1132. (Reproduced with permission from [1]. Copyright 1998 American Chemical Society).

sign of the indirect coupling J DD is positive. Fig. 10.3 illustrates the situation for positive Szz' The change in sign of Szz is explained in terms of a change in sign of the solvent efg anisotropy "felt" by D2.

Experiments on methane and its deuterated analogues used as solutes provide additional evidence for the presence of a solvent efg. Despite the fact that anisotropic NMR couplings are not expected for solutes with tetrahedral symmetry, small dipolar and quadrupolar couplings are observed for all isotopomers just the same. These couplings arise from a mechanism that involves an interaction between the overall molecular reorientation and the non-totally symmetric vibrational normal modes of the solute. The dipolar couplings for various 1 H, 2H, 12C and 13C isotopomers of methane are measured and analysed in detail in terms of this vibration-rotation coupling mechanism. While the dipolar couplings can be readily explained, the predicted quadrupolar couplings show large discrepancies with the measured ones. However, when the same solvent efg required to rationalize the hydrogen experiments is considered, there is excellent agreement between experimental and predicted values. The fact that both the HD and D2 as well as the deuterated methane discrepancies between predictions based on intramolecular interactions alone and experiment can be remedied by assuming that methanes and hydrogens experience essentially equal solvent efg's is again strong evidence for the notions expressed here [1,6].


Both hydrogen and methane are rather small molecules, and the external efg they experience is not necessarily the same solvent efg that is "felt" by larger solutes. A set of experiments designed to measure independently the solvent efg seen by larger solutes indicates that they do "feel" an efg, but it is smaller than that experienced by the hydrogens. These experiments are based on the idea that a chi oro- and a methylgroup are essentially the same size. Hence, molecules such as 1,4-dichlorobenzene, 4-chlorotoluene and 1,4-dimethylbenzene all possess essentially the same size and shape, but have quite different electric moments such as quadrupole tensors. Such solutes are examined in liquid-crystal solvents that have positive, negative and zero-efg as experienced by dideuterium [18]. A more detailed discussion on zero-efg liquid-crystal solvents will be postponed to the next section. The results are analysed in terms of a sum of long-range and short-range contributions. For the long-range contributions to the intermolecular potential terms are included for interactions involving the molecular dipole (with the mean liquid-crystal local electric field, or induced field), the molecular polarizability anisotropy (with the mean liquid-crystal electric field squared), and the molecular quadrupole anisotropy (with the mean liquid-crystal efg). For the short-range contribution all that is assumed is that molecules of equal size and shape undergo identical short-range interactions. In this picture the character of the shortrange contribution is not specified any further.

A reasonable fit between experimental and calculated order parameters requires both long-range and short-range contributions. Of the long-range terms, the interaction between solvent efg and solute molecular quadrupole moment is found to dominate. The other long-range contributions do not appear to play an important role for these solutes. Using quadrupole tensors calculated with the GAUSSIAN 98 programme suite, the solvent efg experienced by the solutes can be estimated, and the values obtained are significantly smaller than, but of the same sign as those experienced by dideuterium in the same liquid crystal (see Fig. 10.4).

In order to include the short-range contribution in the fitting procedure, a formulation is taken that makes no assumptions whatsoever about the details of the size and shape interaction. It simply assumes that there is some solute-solvent interaction that can be expressed as

Uss = Ka,B(solute)F!!(liquid crystal) (10.7)

where Ka,B (solute) is kept the same for solutes of equal size and shape, and F!!Oiquid crystal) represents the short-range size and shape "field", and is a function of the liquidcrystal solvent only. It should be emphasized at this point that for the long-range interactions the solute multipole moments do vary with solute (see Eq. (10.3».

In general, it is quite difficult to separate the different terms in the anisotropic potential. Quantities such as molecular quadrupole moments are difficult to obtain experimentally, and their accurate computation presents significant challenges for quantumchemical methods. The experimental evidence discussed so far shows that short-range size and shape anisotropy effects dominate the anisotropic intermolecular potential for all solutes except the very small (such as H2) and those of very high symmetry (such as CH4 and other almost "spherical" solutes that show very low orientational order).

230

20

16

12

8

4 N

~N 0

- 4

-8

-12

-16

-20

~ ............. ~ .................. ...... . ......... . .... ~~- .................... ......... ,

- -.:.:.:.:.;'":'::-; ::::::. ....... ~~ . .:.: .~ ..

_ ._ .- .- .- .. all ... . ...... ..... ----para · ...... ········ .. .... ·meta ------- ortho .-.-.-.-.-.-. mono ------linear ----D2


....... . ',,, ........... ...

..... ..... . ........ ~ ... ........ ...

'. . ............ ~~:::.:-~ .. ...... . ..... .... '":,.

'...... ..... "

., ., ., ., "

'0

o 20 40 60 wt% EBBA

80 100

Figure lOA. The individually and globally determined efg's for the molecules acetonitrile, propyne, sym-trichlorobenzene, chlorobenzene, toluene, 0-, m-, and p-chlorotoluene, 0-, m-, and pdichlorobenzene, and 0-, m-, and p-xylene in three different nematic phases (ZLI 1132, EBBA, and a 55 wt% ZLI 1132/45 wt% EBBA mixture) from least-squares fits using Eqs. (10.5) and (10.7). Regardless of the size, shape, or electrostatic properties of the solute, all calculated field gradients for the same liquid crystal are the same sign and roughly similar in magnitude. (Reprinted with permission from [ 18]).

For molecules with significant orientational order, terms such as those that involve the solute molecular quadrupole moment provide a relatively small contribution to the overall anisotropic intermolecular potential. In order to investigate these small but significant long-range interactions, it is desirable to be able to describe accurately the dominant short-range interactions. In the next section we discuss an experimental approach to this problem.

4. Orientational order of solutes in ''magic mixtures" In the previous section it was emphasized that the notion of solvent efg's was

supported by independent experiments, viz. on dihydrogen and its isotopomers, on


a DEUTERIUM N~ OF D2

~~~~~~. ""

, I ' 400

, I ' -400 H;r

231

Figure 10.5. 61.4 MHz deuteron NMR spectra of D2 dissolved in nematic mixtures of EBBA and ZLI 1132 at 310 K. (Reproduced with permission from [16]. Copyright 1994 Elsevier Science).

methane and its isotopomers, and on a series of substituted benzenes. An important extension of these ideas in the investigation of the intermolecular potential in anisotropic liquids is the use of mixtures of liquid crystals, called "magic mixtures", for which D2 and HD experience zero external efg. Since dideuterium experiences a positive solvent efg in ZLI 1132 (Merck ZLI 1132 is a eutectic mixture of l,4-(trans-4'-n-alkylcyclohexyl)-cyanobenzene (alkyl=propyl, pentyl, heptyl) and 1 ,4-(trans-4'pentylcyclohexyl)-cyanobiphenyl [19]) and in (4-n-pentyl)-4'-cyanobiphenyl (5CB), and a negative one in EBBA, it is not surprising that an appropriate mixture of EBBA, either with ZLI 1132 (55 wt% ZLI 1132/45 wt% EBBA at approximately 301.4 K) or with 5CB (70 wt% 5CB 130 wt% EBBA at approximately 316.0 K), yields nematic phases with zero external efg. The deuteron NMR spectrum of dideuterium, dissolved in various mixtures of EBBA and ZLI 1132, including the "magic mixture", shows that the orientational order goes through zero as a function of liquid-crystal mixing composition (see Fig. 10.5).

We have seen before that there is strong evidence that in "pure" liquid crystals the overall degree of orientational order has contributions from long-range interactions, probably dominated by the coupling between solute molecular quadrupole moment and the average solvent efg, and short-range interactions, dependent on solute shape anisotropy. For larger solutes the available evidence indicates that short-range interactions dominate. In "magic mixtures" the long-range interactions are essentially "switched off", with only the short-range contributions remaining. Assuming that


the short-range interaction can be reliably modeled in a phenomenological fashion, the observed solute orientational order in "magic mixtures" may be understood with relative ease. In the following we shall focus on several short-range models that have been developed. It should be emphasized that even conceptually very simple models which only require a single adjustable parameter to fit a large series of unrelated molecules do an excellent job in reproducing solute orientational order in "magic mixtures". Slightly more complicated models which account somewhat better for the basic physics of short-range interactions can improve the situation still more. An alternative way of taking account of short-range interactions is through the use of Monte Carlo calculations. We note that, as discussed in Chapter 15, Monte Carlo simulations which model only hard-body interactions predict solute order parameters in excellent agreement with those predicted with the most successful phenomenological models for these short-range interactions.

In addition, it is interesting to note that the quality of the fitting of the solutes 1,4-dichlorobenzene, 4-chlorotoluene and 1,4-dimethylbenzene in various nematic phases and mixtures thereof produces the best results in the "magic mixture" [18]. The fit quality for these solutes is of somewhat lesser quality in ZLI 1132, and even worse for the results in EBBA. The poorer fits are possibly associated with the assumption that all solutes should experience identical liquid-crystal mean "fields". It will be interesting to see to what extent theories that take the liquid-crystal structure into account (Chapter 12), or computer simulations (Chapter 15), are capable of rationalizing the experimental results in even greater detail.

Many examples of phenomenological short-range models are discussed and reviewed in more or less detail in various Chapters of this book. Here we only review the implementation of two basic ideas. The first simple approach deals with the nematic liquid crystal as an elastic continuum, while the second slightly more complicated model involves an anisotropic interaction between the solute surface and the liquidcrystal mean "field". In the elastic distortion model, the liquid-crystal environment is viewed as a continuum which must be distorted in directions perpendicular to the director in order to accommodate the solute. One can think of placing the solute inside an elastic tube or container of infinite length which must then expand laterally to accommodate the solute. The extent of the expansion depends on the size anisotropy and orientation of the solute, as depicted in Fig. 10.6. The liquid crystal is assumed to provide a Hooke's law restoring force, such that the energy required to accommodate the solute is given by [21]

(10.8)

where k is the Hooke's law constant, and C the circumference of the projection of the solute onto a plane perpendicular to the director. This model, called the Continuumor C-model, only contains a single adjustable parameter, viz. the Hooke's law proportionality constant k.

An alternative distortion model, called the Integral- or I-model, is to assume that the solute pushes aside the liquid-crystal continuum in a manner so as to keep the "walls" of the liquid crystal always close to the solute [20]. One way of doing this is


C(Q}

Figure 10.6. The potentials given by Eqs. (10.8) and (10.9) depend on the solid angles!1 which describe the orientation of the director (the Z axis) in a molecular fixed coordinate system (the z axis is shown here). The size and shape of the molecule are characterized by Zp(!1), C(!1) and Cz(!1). Zp(!1) is the length of the projection of the molecule onto the Z axis. C(!1) is the length of the "minimum" circumference (the solid line) around the projection of the molecule onto the plane perpendicular to the director. C z (!1) is the "minimum" circumference obtained by including only the projections of the atoms where they intersect a plane located at Z. The potential in Eq. (10.9) is obtained by integrating C z (n)dZ (represented by the "ribbon" around the molecule) along Zp(!1). (Reproduced with permission from [20]. Copyright 1993 Taylor & Francis).

to calculate the circumference as a function of distance along the director, or

(10.9)

where ks is the Hooke's law constant for this model, and C(Z) is the solute circumference at distance Z along the director. The integral in Eq. (10.9) is equivalent to the


integral over the solute surface

(10.10)

where On (allowed to vary between 0 and 71") is the angle between the normal to the solute surface and Z, and sin On scales the interaction between the solute surface and the liquid crystal "field" such as to be minimum for surface normals perpendicular to Z and maximum for those parallel. Ferrarini et al. proposed a similar surface integral model using a P2 (cos 0) angle dependence (see [22] and Chapter 11 of this book). The two surface integral models (Eqs. (10.9) and (11.3)) give quite similar predictions. However, the Ferrarini et al. model has been criticized as not providing a good descriptor of the overall molecular shape (see Chapter 12 of this book).

The best fit of model parameters to order parameters obtained from a collection of solutes in "magic mixture" involves a combination of Eqs. (10.8) and (10.9). This model is called the CI-model [20]. The excellent agreement may be accidental, but the following physical rationale can be offered. The first model leaves too much free space above and below the solute, while the second has the liquid crystal fit around the solute rather like a glove. A combination of the two models makes intuitive sense in terms of the way solute and liquid-crystal molecules might be expected to pack.

5. Comparison of experimental and calculated orientational order in "pure" liquid crystals and "magic mixtures"

The picture that is emerging for the orientational order of a solute is that in "pure" liquid crystals two different mechanisms playa role, a short-range one and a long-range one. The short-range contribution is responsible for most of the orientational order. It can be modeled adequately with simple models as discussed in the previous section. The long-range contribution is generally small compared to the short-range one and is dominated by the interaction between the solute molecular quadrupole moment and the average solvent efg. This contribution can be calculated when these quadrupole moments are known, either from experiment or from quantum-chemical calculations. For the special case of "magic mixtures" the long-range order contributions are unimportant and only the short-range contribution persists. Since the proof of the pudding is in the eating, we should now consider whether the above statements are upheld when a range of unrelated solutes dissolved in different nematic phases and mixtures thereof is considered.

A careful study on the orientational order of the molecules acetonitrile, propyne, sym-trichlorobenzene, chlorobenzene, toluene, 0-, m-, and p-chlorotoluene, 0-, m-, and p- dichlorobenzene, and 0-, m-, and p-xylene has been carried out in the three different nematic phases: ZLI 1132; EBBA; and a 55 wt% ZLI 1132/45 wt% EBBA "magic mixture" [18]. The experimental order parameters are now compared with the orientational order that can be calculated and modeled in various ways. It is important


ZLll 132 ZLll1321EBBA

0.1 Al A2

1 • 0

.. 1

, 8 rn i

-0.1 _0- .... 0: .. - .. .,. 1MB: D - o.ae: A- g.ll

0.1 Bl

1 1/ 0

1 0

rn

-0.1

-0.4 -0.2 0 0 .2 0 .4 ·0.4 ·0.2 0 0.2 0 .4 s-' S..,!

A3

Jj ,

-0.4 -0.2

EBBA

II

lie 1/

0 0 .2 S ....

235

o

0.4

Figure 10. 7. S~Xpl minus scale results are plotted against sexpl for the molecules and liquid crystals listed in the caption to Fig. 10.4 [18]. The principal (largest absolute diagonal value) order parameter element is represented. Note that the vertical scale is expanded by three compared to the horizontal scale. Series A is the results from a least squares fit to Eg. (10.7) (D) and Eqs. (10.8) and (10.9) (6); the fits to the two short-range interactions. Series B is the result from fits to Eqs. (10.5) and (10.7) (D) and to Egs. (10.5), (10.8) and (10.9) (6); the short-range and quadrupole potentials. RMS errors (x 10-2 )

between sexpl and scale are reported within the graphs. The quality of the fits is represented by the scatter of points from the horizontal straight line at zero. For an exact fit all the points would lie on this horizontal line. Note that the fits improve from A to B, and that the fits in column 2 are superior to those in column 1 which are superior to those in column 3. (Reprinted with permission from [18]).

to use appropriate solute order parameters sexp/, derived using the preferred scaling procedure based on TCB splittings as discussed before. First, an attempt is made to fit the experimental results to the orientational mechanism involving the coupling between solute molecular quadrupole moment and solvent efg, while completely neglecting any short-range contribution. The results of these fits in all three liquid crystals are disastrous (not shown). Next, we do two types of comparisons involving the shortrange contribution which is modeled either using Eq. (10.7) or the more specific CImodel. As expected and as is shown in Fig. 10.7, where the solute experimental order sexpl is plotted against sexpl - scale, we see that the fit is excellent for the results obtained in the "magic mixture" (frame A2), but rather poor in the component liquid crystals (frames Ai and A3). Secondly, we compare the experimental order parameters to the results obtained when both the short-range contribution (again treated with Eq. (10.7) and the CI-model) and the long-range contribution between solute molecular quadrupole moment and solvent efg are incorporated. The solute quadrupole moments are computed with the GAUSSIAN 98 quantum-chemical programme suite.


-------. 0.. X (])

'--" N

~r:. »: r:. r:. lX N

U) X x

I 0 r:. x -------. x x

() x r:.r:. r:. .......... x (t\ r:. x

x ~ () x

'--" N N

U)

-OJO.5 0.5

Figure 10.8. The difference between the calculated and experimental values of S .. versus the experimental values of S .. , where S .. is the diagonal element of the experimental Sct{3 with the largest absolute value. Note that the vertical scale is expanded by five compared to the horizontal scale. The 6., x, and o correspond to a group of molecules with C3 1) or higher symmetry, a group with C2v or D2h symmetry, and a group with Cs symmetry. The calculated values were obtained using using Eqs. (10.8) and (10.9) with k = 2.04 dynlcm and ks = 48.0 dynlcm. (Reproduced with permission from [20]. Copyright 1993 Taylor & Francis).

The average efg's in ZLI 1132 and EBBA are introduced as a fitting parameter in the least squares procedure. As before, the fits in the "magic mixture" (frame B2) show by far the best agreement, but it is evident that inclusion of the solute molecular quadrupole-solvent efg coupling in the case ofZLI 1132 (frame Bl) and EBBA (frame B3) improves the quality of the fits. Since the long-range term is found to be rather small compared to the short-range effect, in order to get meaningful results for the long-range contribution it is important to use the best short-range model available.

Another way to view whether the type of modeling employed is effective is the following. Using the "magic mixture" as a solvent, we can dissolve a large range of unrelated solutes in it and do a comparison between the experimental order parameters and those obtained from sensible short-range models such as CI. An example is given in Fig. 10.8, where experimental values of S zz are compared to values of S zz (calc) -Szz(exp). For details about the solutes employed here and about the definition of


20

16

12

8

4

~ 0 ~

- 4

- 8

- 12

-16

- 20

-.-.-.-.-.. all

----para """""" """"" meta - ------ ortho .-.-.- .-.-.-. mono

------ linear ----D2

o 20 40 60 wt% EBBA

237

80 100

Figure 10.9. The individually and globally detennined efg's for the molecules and liquid crystals listed in the caption to Fig. 10.4 from least-squares fits using Eqs. (10.8) and (10.9) (the CI-model) together with Eq. (10.5). Regardless of the size, shape, or electrostatic properties of the solute, all calculated field gradients for the same liquid crystal are of the same sign and roughly similar in magnitude. (Reprinted with pennission from [18]).

molecular axes the reader is referred to the original paper. It is apparent that the quality of this comparison is excellent and that the model predictions are typically accurate to approximately 10% or better [20].

Another example of the usefulness of the CI-model is provided by the series of methyl- and chlorine-substituted benzenes discussed and modeled before with the use of Eq. (10.7) in section 3. When the same experimental results are now subjected to the CI-model, Fig. 10.9 is obtained. It is interesting to see that all the lines for the different solutes approximately cross for the liquid-crystal composition of the "magic mixture". Moreover, the solvent efg experienced by D2 is about twice that "seen" by the other solutes [18].

In the experimental NMR of solutes in liquid crystals the spectra get very complicated as the number of spins goes up and the solute symmetry goes down. This situation is aggravated when we consider solutes with large-scale internal motion in-


Experiment

Prediction

- 6000 - 4000 - 2000 o 2000 4000

Frequency

Figure 10.10. Experimental NMR spectrum of m-chlorotoluene (dilute) dissolved in the 55 wt% "magic mixture" at 300 K. The CI model was used for the prediction. This prediction was sufficiently close to the experimental spectrum that a fit to all the lines was obtained within two hours. The order parameters predicted from the CI model, and those calculated from the fit to the spectrum, are as follows: Sxx = 0.0737 (prediction), 0.0795 (experiment); Su = -0.2150 (prediction), -0.2166 (experiment); Sxy = -0.0846 (prediction), -0.0796 (experiment). The z-axis is taken perpendicular to the benzene plane, the y-axis points from the benzene ring to the methyl carbon, and the x-axis is defined such that the Cl atom has a positive x-coordinate. (Reproduced with permission from [1]. Copyright 1998 American Chemical Society).

volving several conformations. An example is the molecule butane with 10 protons. The NMR spectrum in a "magic mixture" (see Figs. 5.8 and 5.9 of Chapter 5) looks enormously complicated and unraveling such an NMR spectrum seems an impossible task. However, it was found that by using the CI model to calculate the order parameters of the various conformers and by assuming a reasonable trans-gauche energy difference, multiple-quantum NMR spectra for butane in the "magic mixture" could be predicted that were not unlike the experimental ones. This provided a good starting point for assigning sufficient lines in the multiple-quantum spectra to allow for a successfulleast squares fit (Figs. 5.10 and 5.11 of Chapter 5), leading to a full analysis of the normal NMR spectrum [23] (Figs. 5.8 and 5.9 of Chapter 5).

As a second example we present the molecule m-chlorotoluene which has only a plane of symmetry and hence requires three independent parameters to describe its orientational order. The NMR spectrum in a magic mixture is given in the top trace


of Fig. 10.10. Modeling of the short-range order using the CI-model led to the predicted bottom trace of Fig. 10.10. It is apparent that many spectral features can be assigned, sufficient for a detailed least squares fit. A fit to all the observed lines was achieved within two hours. Analysis of such a spectrum without any prior knowledge is notoriously time consuming.

6. Conclusions The prediction of the orientational order of solutes in nematic liquid crystals is no

longer a hopeless undertaking. The use of small, well characterized solutes as delicate probes of the intermolecular potential in anisotropic liquids, in conjunction with the development of zero-efg "magic mixtures", has contributed vastly to a situation where order parameters of solutes dissolved in these "magic mixtures" can be predicted at approximately the 10% level. The methods described in this Chapter have already played a crucial role in enabling and facilitating the analysis of very congested NMR spectra arising from solutes with many nuclear spins and / or geometries with low symmetry. They have also been invaluable in the interpretation of orientational order of flexible molecules that are undergoing conformational change. For molecules interconverting among several conformations, appropriate short-range models such as CI can be used for a rather reliable prediction of the orientational order for every conformer. Since the orientational order derived from the observed NMR spectrum is an average over all conformational motion, the conformer probabilities can in principle be extracted from a least squares fitting procedure. This is an interesting way of obtaining information about these probabilities. When sufficient care is taken, both experimentally and in modeling the dominant short-range interaction, reasonable estimates of solute molecular quadrupole moments based on these methods would seem feasible.

Acknowledgments

EEB thanks the Natural Sciences and Engineering Council of Canada for financial support. We thank E. Merck, Darmstadt for the gift ofZLI 1132. We are very grateful to the collaborators, students and postdoctoral fellows who have over the years contributed to this research.

References [1] Burnell, E.E., and de Lange, C.A. (1998), Chemical Reviews, 98:2359.

[2] Buckingham, A.D. (1970), An Advanced Treatise in Physical Chemistry, 4:349.

[3] Stone, AJ. The Molecular Physics of Liquid Crystals; Luckhurst, G.R., Gray, G.w., Eds. Academic Press, 1979, Chapter 2.

[4] Maier, w., and Saupe, A. (1959), z. Naturforsch. A, 14:882.

[5] Maier, w., and Saupe, A. (1960), z. Naturforsch. A, 15:287.

[6] Snijders, J.G., de Lange, C.A., and Burnell, E.E. (1983), Israel J. Chern., 23:269.

[7] Burnell, E.E., de Lange, C.A., and Gaemers, S. (2001), Chern. Phys. Letters, 337:248.

[8] Syvitski, R.T., Pau, M.Y-M., and Burnell, E.E. (2002), J. Chern. Phys., 117:376.

[9] BurnelJ, E.E., de Lange, c.A., and Snijders, J.G. (1982), Phys. Rev., A25:2339.


[10] Burnell, E.E., van der Est, A.I., Patey, G.N., de Lange, C.A., and Snijders, I.G. (1987), Bull. Magn. Reson., 9:4.

[11] van der Est, A.I., Burnell, E.E., and Lounila, I. (1988), J. Chern. Soc., Faraday Trans. 2, 84:1095. [12] Burnell, E.E., de Lange, C.A., Segre, A.L., Capitani, D., Angelini, G., Lilla, G., and Bamhoom,

I.B.S. (1997), Phys. Rev., E 55:496. [13] Snijders, I.G., de Lange, C.A., and Burnell, E.E. (1982), J. Chern. Phys., 77:5386.

[14] Snijders, J.G., de Lange, C.A., and Burnell, E.E. (1983), J. Chern. Phys., 79:2964.

[15] Patey, G.N., Burnell, E.E., Snijders, J.G., and de Lange, C.A. (1983), Chern. Phys. Letters, 99:271.

[16] Barker, P.B., van der Est, AJ., Burnell, E.E., Patey, G.N., de Lange, C.A., and Snijders, J.G. (1984), Chern. Phys. Letters, 107:426.

[17] Barnhoom, I.B.S., and de Lange, C.A. (1994), Mol. Phys., 82:651.

[18] Syvitski, R.T., and Burnell, E.E. (2000), J. Chern. Phys., 113:3452.

[19] Lounila, J., and Jokisaari, I. (1982), Progress in Nuclear Magnetic Resonance Spectroscopy, 15:249. [20] Zimmerman, D. S., and Burnell, E.E. (1993), Mol. Phys., 78:687.

[21] van der Est, AJ., Kok, M.Y., and Burnell, E.E. (1987), Mol. Phys., 60:397.

[22] Ferrarini, A., Mora, G.J., Nordio, P.L., and Luckhurst, G.R. (1992), Mol. Phys., 77:1. [23] Polson, J.M., and Burnell, E.E. (1995), J. Chern. Phys., 103:6891.

Chapter 11

MOLECULAR MODELS OF ORIENTATIONAL ORDER

Alberta Ferrarini and Giorgio J. Moro Universita di Padova, Padova, Italy

Introduction

The relationship between molecular structure and intermolecular interactions on the one hand, and the organization and orientational order of liquid crystals on the other, is certainly a very important issue in both theoretical and experimental research with possible spin offs for technological applications such as the design of molecular constituents that lead to ordered materials with optimal macroscopic properties. Starting from the early theories of Onsager [1] that describe excluded volume effects, and of Maier and Saupe [2,3] that rely on an approximate representation of dispersion forces, there have been many developments aimed at getting a better understanding of the different aspects of such a general problem. Efforts have concentrated on the improvement of statistical methods that are able to rationalize the order phenomena originating from anisotropic interactions (see Chapter 12). The attainment of such an objective has often required a schematic representation of the molecular entities, for instance approximating molecules as ellipsoids, rods or piles of spheres.

On the other hand, phenomenological approaches are often invoked when examining order properties of specific systems to correlate them with the molecular structure of the constituents [4]. For instance, in the analysis of the orientational order of flexible hydrocarbon chains, different methods have been proposed for the decomposition of the mean field potential according to group contributions [5-9] (see Chapter 13 ofthis book). In this framework emphasis is placed on the capability of estimating ordering properties on the basis of the three-dimensional structure of the molecule. From this perspective, surface models [4,10] have been quite successful. The basic assumption is that the anisotropy of molecular interactions is controlled by molecular shape.

In this Chapter we review the surface tensor model [10], and its applications to the analysis of orientational order. By describing the shape of a molecule via its molecular surface, and by decomposing the mean field potential in terms of contributions of surface elements with a functional form that is identical to that of the anchoring free energy of macroscopic surfaces [11,12], the orientational potential is easily derived. The orienting strength of the medium is the only free parameter. We shall also report the developments concerning the ordering contribution due to solvent electrostatic



polarization which, because of its long-range character, is complementary to the shortrange forces accounted for by the surface model. The representation of a molecule via its surface also provides a direct interface with the continuum treatment of electrostatic effects. Thus, by solving the Poisson equation for molecular charges in the presence of an anisotropic dielectric medium, one can determine the polarization contribution to the orientational potential [13,14]. Another development concerns the description of intermolecular forces on a statistical basis. Density functional theory [15] provides a general framework for treating condensed phases at the molecular level, but its application requires a suitable parameterization of the free energy. We shall review the use of the Van der Waals approximation [16-18] which allows one to recover the surface tensor model for the orienting potential of a solute by considering an approximate form of the short-range attractive interactions.

The Chapter is organized as follows. In the next section the surface tensor model is introduced in the framework of the phenomenological modeling of the orientational distribution of solutes in nematics. Some examples of its application are reviewed and a comparison with available experimental order parameters is made. The next section summarizes a method for evaluation of electrostatic interactions according to the dielectric continuum model. The results of calculations for some typical situations are reported in order to compare the effects of short-range interactions with those of electrostatic interactions. In section 3 density functional theory is introduced together with the Van der Waals representation of the free energy. Such a general methodology is employed for the description of orientational order of solutes. In the final section 4, remaining issues that originate from these investigations of ordering phenomena in liquid crystals are outlined.

1. Phenomenological models for short-range interactions In uniaxial liquid crystals, such as calamitic, nematic, and smectic A phases, rod-like

molecules preferentially align their long axes parallel to the director, while disc-shaped molecules tend to orient their short axis perpendicular to the director. However, real molecules are neither rods nor discs, and their orientational behaviour has a less trivial dependence on the molecular structure, whose rationalisation requires the understanding of the role played by intermolecular interactions. Deep insight into the orientational behaviour of molecules has been achieved using the NMR technique which provides accurate values of second rank order parameters. These are defined as the average values of Wigner rotation matrices V6m (see Chapter 1, where the equivalent representation in terms of the Saupe ordering tensor is presented), with the average performed with respect to the orientational (single molecule) distribution function

f(o.) = exp[-U(o.)jkBT] . J do.exp[-U(O)jkBT]

(11.1)

In this equation 0. are the Euler angles that define the orientation of the molecule with respect to the director, and U (0.) is the orientational mean field experienced by the molecule.

Molecular Models of Orienta tiona I Order 243

In principle, an accurate treatment of intermolecular interactions is essential in providing a complete picture of the orientational behaviour. However, the use of atomistic potentials within statistical approaches on one hand, or simulation methods on the other, are extremely cumbersome. Therefore, simple phenomenological models have been developed to provide insight into the mechanisms that are at the root of orientational order in liquid crystal phases. In this way, the mean field U(O) is simply parameterized in terms of the anisotropy of some molecular property that is deemed to be responsible for the tendency of the molecule to align in the ordered phase. Different choices have been presented over the years. They range from the molecular pol arizability [19], assumed to be the key molecular feature that controls the anisotropy of short-range interactions, in agreement with the original idea of Maier and Saupe [2,3], to properties that reflect the anisometry of the molecular shape, such as the inertia tensor [20] or the dimensions of a parallelepiped containing the molecule, following a suggestion by Straley [21]. More recently, other models have been proposed, based on the parameterization of the anisotropy of the orienting mean field in terms of the anisometry of the molecular surface [10,22]. The latter approaches have been used successfully for the interpretation of experimental data. The origin of their success lies in the fact that short-range intermolecular interactions are essentially modulated by the molecular surface. In fact, the surface parameterization of interactions is widely used in the context of macromolecular systems, e.g., to quantify the ability of molecular recognition. The definition of the molecular surface is not devoid of ambiguity: even though other representations have been used, the most common definition is that of an assembly of Van der Waals spheres centred at atomic positions. A more appropriate representation of the boundary really experienced by the surrounding molecules is obtained by smoothing the intersection of atomic "beads", e.g., by considering the surface traced out by a sphere rolling over the Van der Waals assembly [23,24]. Irrespective of the particular definition chosen, the morphology of the surface depends on the chemical structure, and its use therefore accounts for the specific features of the systems under investigation.

The phenomenological model proposed by Burnell and coworkers [22] relies on the picture of an interaction between probe molecules and the nematic environment in terms of restoring forces that oppose the director deformations produced by reorientations of the particle, by viewing the medium as an elastic continuum. The deformation is parameterized according to the projections of the molecular surface parallel and perpendicular to the director. In contrast, in the surface tensor model [10] the form of the orienting potential is defined in analogy with the anchoring free energy experienced by macroscopic bodies embedded in nematic phases [11,12]. Both approaches have been shown to be able to provide reliable predictions of the dependence on molecular structure of the order parameters measured by NMR [4]. As a matter of fact, despite the different physical pictures, they can be proven to lead to analogous mathematical forms [25]. The restoring-force models are described in Chapter 10. Therefore, in the following we shall only summarize the main features of the surface tensor approach.

In analogy with the anchoring free energy on the surface of a body embedded in a nematic phase [12], the mean field experienced by a planar element dS of the molecular


surface is assumed to have the form

(11.2)

where P2 is the second Legendre polynomial, and n, s are unit vectors, parallel to the mesophase director and to the outward pointing normal to the surface element. The factor ~ is a parameter that expresses the orienting strength of the medium. The orienting mean field acting on the whole molecule is obtained by integrating the elementary contributions over the molecular surface S:

(11.3)

It is convenient, also for computational reasons, to decompose the rotation from the director n to the local surface normal s into the rotation from a laboratory frame, with the Z axis parallel to the director, to the molecular frame, followed by the rotation from the molecular frame to a local frame with the z axis along the surface normal. By using the addition property of Wigner rotation matrices [26], Eq. (11.3) can be rewritten as

2

Ust(!l) = -~ L T(2,m)*1)5m(O) (11.4) m=-2

where !l are the Euler angles for the rotation from the laboratory to the molecular frame. The quantities T(2,m) are irreducible spherical components of the so-called surface tensor T. They are defined as

(11.5)

where !ls are the Euler angles for the rotation from the molecular to the local frame on the surface. The surface tensor quantifies the anisometry of the molecular surface. When expressed in its principal axis frame, this tensor has only two non-vanishing components, i.e. T(2,O) and ~{T(2,2)}, related to the elongation of the molecular surface along the molecular z axis and to its biaxiality perpendicular to such a direction, respectively.

The mean field Eq. (11.3) is determined by the structure of the solute, via the surface tensor T, while the properties of the sol vent only enter through the orienting strength ~. This depends on temperature and on the nature of the nematic solvent. According to Maier-Saupe theory generalised to biaxial molecules [10,19, 27], ~ can be expressed in terms of the order parameters and the surface tensor of the solvent. However, in some cases, e.g., when comparing the orientational order of different solutes in the same solvent or when analysing the different tendency to alignment of the molecular axes of a molecule, ~ can simply be taken as a free parameter. Thus, if the order parameter for one axis in a given molecule is taken as the independent variable, the order parameters of any other axis and / or any other molecule in the same medium result in being independent of the solvent. This is at variance with experimental findings,

Molecular Models of Orienta tiona I Order

(J) I

(J)

0.3~-----------.

>- 0.2 >-

x x

0.1

ortho

0.0 +-~---r--"""'----r---"'-~ -0.3 -0.2 -0.1 0.0

s zz

245

Figure 11.1. Order parameter of ortho- and meta-dichlorobenzene: predictions of the surface tensor model (lines) [27] and values derived from NMR experiments in the nematic mixtures 55% ZLI 1132/ EBBA (filled symbols) and 70% 5eB / EBBA (open symbols) [31]. The molecular frame is chosen with the z axis perpendicular to the molecular plane and the x axis parallel to the C2 symmetry axis.

since the measured values display some solvent effect, whose magnitude depends on the solute-solvent pair [28-30]. The reason for this discrepancy may be related to electrostatic solute-solvent interactions, or to the influence of solvent structure on short-range interactions. Both these factors are neglected in the surface tensor model. While it is hard to identify a criterion for estimating the relevance of the latter, the former can be expected to be enhanced in solvents with large values of the dielectric anisotropy lJ.e = ell - e.l (from the microscopic point of view, solvent molecules carrying large dipoles). In fact, a correlation has been found between the change of order parameters with solvent and changes in the sign of the dielectric anisotropy lJ.e of the medium [28-30]. Such considerations suggest that solvents with vanishing dielectric anisotropy should be the most appropriate medium to use when comparing the predictions of the surface tensor model with experimental data. The same applies to any other analogous approach that neglects electrostatic interactions. The peculiar behaviour of the so-called "magic mixtures" [4] which are mixtures of nematics with opposite dielectric anisotropies can probably be explained in this way.

Fig. 11.1 shows the biaxiality of orientational order in the molecular plane as a function of the order parameter of the axis normal to the molecular plane for orthoand meta-dichlorobenzene. The values predicted by the surface tensor model [27] are


compared with those obtained from NMR measurements in two of the so-called "magic mixtures" [31]. The ability of the model to describe the dependence of orientational behaviour on molecular structure clearly follows from the figure.

In spite of its simplicity, the surface tensor model has been shown to be able to capture the main features that govern the ordering of molecules in nematic phases. The orientational order of solutes of various structures has been investigated, and good agreement with experimental data, mainly derived from NMR measurements, has been found [10,27]. Even though this Chapter focuses on molecular models for the orientational order of solutes, it is worth remembering that the surface tensor approach has a wider scope. For instance, it has been used to predict the dependence of order and thermodynamic properties of pure nematics on molecular structure [32]. This kind of application takes advantage of another feature of the model, i.e., the possibility of treating flexible molecules by taking into account the structure of different conformers. The surface method approach has also been extended to the correlation of molecular structure with other phenomena that occur in liquid crystals, e.g., the twisting ability of chiral molecules [33,34] and the flexoelectric effect [35]. In these cases a more general characterization of the morphology of the molecular surface is required, by defining, in addition to its anisometry, other features such as its chirality and polarity. In particular, one of the main achievements of the model has been its ability to predict the signs and magnitudes of the helical twisting power of dopants in nematic solvents [36], and the pitch of pure twisted nematics [37]. These predictions are based on the structure of the chiral molecules.

2. Modeling electrostatic interactions While there is a general consensus about the importance of short-range interactions

which are controlled by the molecular shape, the role of interactions involving the permanent charge distribution in the molecule is more controversial. Apart from the general considerations mentioned above, concerning the opposite effects expected for a given solute when dissolved in nematics with positive and negative A£ values, it is hard to figure out how the charge distribution of the solute can influence its orientational order, even at a qualitative level. The difficulty in disentangling electrostatic contributions from the whole of intermolecular interactions is associated with that of obtaining a consistent representation of the former, combining the long-range nature of these electrostatic contributions with the effects of short-range correlations. Essentially two approaches which emphasize either one or the other aspect, and therefore suffer complementary drawbacks, can be formulated. The first approach, based on the statistical averaging of interactions between pairs of molecules [38,39], is limited in its ability to take into account the long-range collective character of dielectric properties. The second approach, based on the reaction field approximation, does not explicitly contain the short-range pair correlations between molecules. The former treatments are considered in Chapter 12. In the following, the principles underlying reaction field methods will be briefly presented.


In the reaction field approximation the electrostatic interactions between molecules are modeled in terms of the interaction between the charge distribution contained in the molecular cavity and the polarization this induces in its surroundings, treated as a dielectric continuum. In the original Onsager formulation [40] a spherical cavity with a central dipole embedded in an isotropic dielectric is considered and, by using the formalism of classical electrostatics, a simple analytical expression for the reaction field is obtained. A more realistic description of the molecular features can be achieved at the price of resorting to a numerical treatment. Nowadays, the problem of an arbitrary charge distribution in a cavity of arbitrary shape can be handled by using available computational facilities. As a matter of fact, the reaction field approximation has been widely exploited in the framework of the analysis of solvation phenomena. It is used to describe the effect of the environment on the electronic structure of molecules [41,42]. For this reason it is included in standard packages for quantum-mechanical calculations [43,44]. Moreover, the reaction field method is adopted within more coarse-grained classical approaches for the prediction of electrostatic properties of macromolecules, e.g., the pKa of proteins [45,46].

In the case of isotropic and homogeneous media the polarization of the dielectric can simply be reduced to a charge density on the surface of the cavity, (jP = - P . s (see Appendix A). Efficient methods for calculating this quantity for arbitrary charge distributions and cavity shapes have been developed, e.g., by solving the electrostatic problem on the cavity surface with Boundary Element techniques [47]. In the presence of inhomogeneity or anisotropy the same methods can no longer be used, since in these cases the volume polarization should also be taken into account. However, it has been shown in the case of anisotropic materials, such as liquid crystals, that it is possible, essentially by using the Green formula, to define an effective surface charge density (j

that accounts for the polarization of the environment [48]. This is the solution of an integral equation of the form

(11.6)

where A and B are integral operators, defined on the molecular surface, which depend on the shape of the surface and on the dielectric permittivity of the surrounding dielectric, and ¢i is the electrostatic potential generated on the inner surface by the charges contained in it. The explicit expressions are rather cumbersome, therefore they are not reported here. The interested reader is referred to [13,14].

The electrostatic potential produced in the interior of the cavity C I by the polarization of the environment can then be calculated as

¢(r) = _1_ r dr' (j(r') 47rcQ ) S Ir - r'l (11.7)

In this equation dr' represents the integration element on the molecular surface S. A notation different from that of Eqs. (11.3 and 11.5) is used to highlight the position dependence of the integrand. The electric field and its derivatives at any point inside the cavity can be derived according to Eq. (11.7).


From the charge density a it is also possible to calculate the electrostatic free energy of the solute in its own reaction field [49]:

(11.8)

where the sum is extended to all the permanent charges qJ contained in the cavity. In the nematic phase the dielectric permittivity along the director, ell' in general is different from that perpendicular to it, e 1.. As a consequence the energy W R is a function of the molecular orientation n. This effect can easily be calculated for the case of a solute represented by a dipole in a spherical cavity. In nematics with dielectric anisotropy l:!.£ > 0, dipolar orientations parallel to the director would be energetically favoured over those perpendicular to it (the opposite holds for l:!.e < 0). The microscopic origin is obvious, since positive and negative dielectric anisotropies occur for solvents with longitudinal and transversal dipoles, respectively.

To rationalize the electrostatic effects on orientational order, it can be helpful to resort to the multipole expansion of molecular charges. The free energy Eq. (11.8) depends on the whole charge distribution. By introducing a power expansion with respect to the distance r it can be rewritten as:

WR = -81 {q r dr a(r) + IL' r dr a(r)V' (~) 'lrea is r is r

+~e·is dra(r) V' ® V' (~) + K}

(11.9)

where the dots and the symbols ® indicate tensor contraction and tensor product. The various multi pole moments are defined as:

(11.10)

(11.11)

e = ~ LqJ(3Q ®rJ -ir)) 2 J

(11.12)

with 1 representing the unit tensor. For neutral molecules the first non-vanishing contribution is the dipolar one. We note that the presence of this term is not in contradiction with the non-polar nature of nematic phases, since the electric field appearing in it, i. e. the electric field experienced by the molecule as a consequence of the polarization of its environment, depends on the orientation of the molecule which is characterized by inversion symmetry in non-polar phases. Convergence of the power expansion Eq. (11.9) can be slow. In general, the distance r J can be comparable with r, and truncation at the lowest order terms is not justified.


In our reaction field method we have developed a numerical procedure for the solution of Eq. (11.6) [13] whereby the molecular surface is represented by the RichardsConnolly model [23,24], while the molecular charge distribution is described in terms of partial charges qJ calculated with the Merz-Kollman-Singh procedure [50]. The molecular polarization has also been included [14] through a set of mutually interacting atomic polarizabilities [51]. This method has been used to estimate the effects of electrostatic-induction interactions on the orientational order of solutes [13,14]. The same kind of approach has been employed to predict the dielectric permittivity of nematics [52]. In our procedure the reaction field approximation for the electrostatic interactions is combined with the surface tensor model for short-range interactions. The whole orientational mean field is then expressed as

(11.13)

The latter contribution is conveniently expanded in a basis of Wigner functions [26]

L

WR(!1) = L L w(L,n)*Vtn(!1) (11.14) L(even) n=-L

with coefficients

w(L,n) = 2£ + 1 J d!1 W (0) VL*(O) 8rr2 R On . (11.15)

In view of the symmetry (D OOh ) of the nematic phase, the Wigner functions V~n with odd £ and non-zero n values are excluded from the summation in Eq. (11.14). If higher-rank terms are negligible, as generally occurs, the expansion Eq. (11.14) can be truncated at the second-rank contribution, and the mean field in Eq. (11.13) can be expressed in the usual second rank tensorial form

U(!1) = L [_~T(2,n)* + W(2,n)*] V5n(!1). (11.16) n

Fig. 11.2 shows the order parameters calculated for anthracene and anthraquinone dissolved in nematics of different dielectric anisotropies. For the case /:),£ > 0 values appropriate for 5CB have been used (eN! = 10.2, t::..eN! = 8), while for Ae < 0 values suitable for MBBA have been chosen (eN! = 5.2, t::..eN! = -0.5) [14]. The data for t::..e = 0 have been obtained by considering only the surface tensor contribution. The figure shows that electrostatic-induction interactions can have non-negligible effects in nematics with sufficiently high dielectric anisotropy. Opposite changes are predicted for anthracene and anthraquinone. The biaxiality of orientational order in the molecular plane is predicted to decrease for the former, and increase for the latter, when dissolved in solvents with positive dielectric anisotropy. Thus, if Ae is high enough, the order biaxiality of anthraquinone should become higher than that of anthracene, in contrast with what would be expected simply on the basis of the shape anisometry. The theoretical predictions obtained with the reaction field approximation and shown

250

~ (J)

I x x if)

~ CJ)


0.8.,.------------,

ceo 0.6

0.4

0.2

0.0 -'--~-____.--_.___-~---1 -0.4 -0 .3

S zz

-0.2

0.8 -.-------------,

0.6

I ~0.4 CJ) LlE> 0

LlE = 0 LlE < 0

0.2

0.0 -'--...,-----r----,-----.,....----t

-0.4 -0.3

S zz

-0.2

Figure 11.2. Order parameters of anthracene and anthraquinone calculated for nematics with different dielectric anisotropy [14]. The molecular frame is taken with the z axis perpendicular to the molecular plane and the x axis parallel to the long molecular axis.

in Fig. 11.2 are found to be in agreement with the experimental behaviour reported in [53-55] for anthracene and anthraquinone. For the sake of comparison, calculations have also been performed by considering, instead of the full electrostatic energy in


Eq. (11.8), only the quadrupole term in the expansion Eq. (11.9) which is the lowest order non-vanishing multi pole contribution for anthracene and anthraquinone. It is found that such an approximation, while acceptable in the former case, is completely inadequate in the latter.

In conclusion, the reaction field method allows one to estimate the contribution of electrostatic interactions to orientational order in liquid crystals. The order of magnitude of the predicted effects is in agreement with experimental data. A quantitative correspondence between calculated and measured order parameters cannot be expected, because the variations of these with solvent are relatively small and electrostatic interactions probably represent only one of their causes. Deeper insight into experimental behaviour can only be achieved by also taking into account, in addition to the structure of the solute, that of the solvent. When modeling such interactions, the use of more sophisticated statistical approaches is required.

3. Density functional theory A variety of statistical techniques, each of them emphasizing a particular feature

at the root of the molecular correlations, is available. However, because of their intrinsic formal complexity, they are often applied to simplified representations of molecular entities, such as rods, spherocylinders, and ellipsoids. We shall review some applications of the density functional theory [15] that allow a good compromise between a sufficiently accurate solution of the many-body statistical problem, on the one hand, and the capability of dealing with a realistic representation of the molecular structure and intermolecular interactions, on the other.

Let us consider an ordered phase in the canonical ensemble, i.e. at given temperature T, volume V and number of particles N. We shall first examine a pure phase with one only component that has no internal degrees of freedom, such that a particle configuration is specified by the set of coordinates q == (R, n) for its position R and its orientation n. Correspondingly we introduce the probability density p(q), normalized as the number of particles

J dqp(q) = N = Vjv (11.17)

with v being the volume per molecule, i.e., the inverse of the particle density. Let us introduce the free energy as a functional F[pJ of any possible distribution p( q) satisfying the normalization constraint of Eq. (11.17). If with suitable external constraints one could realize an arbitrary distribution p( q), F[pJ would be the corresponding free energy measured for the system. Then, in the absence of external constraints, the equilibrium state of the system is determined by the free energy minimum which can be derived from the stationary condition

8F[pJ = O. 8p(q)

(11.18)


In this way the issue becomes that of modeling the functional dependence of the free energy: once .r[p] is specified, the equilibrium distribution is derived as the solution of Eq. (11.18) by employing suitable numerical methods.

In this framework, different liquid crystalline phases can be treated separately, by constraining p( q) to the functional space with the suitable symmetry. For instance, nematic phases are described by probability densities that are independent of the position R, and that have an axial dependence on the orientation n with respect to the director. Of course, a constant probability density is assigned to the isotropic phase.

Different methods, with different degrees of accuracy, are available for modeling the functional dependence of the free energy [16,17,56--58]. The simplest one derives from the perturbational analysis of molecular correlations in the fluid [59] which generalizes the Van der Waals approach by separating the effects of excluded volume from the attractive part of the interactions [16-18]. The free energy is partitioned into three different contributions

(11.19)

where .rid is the free energy for the ideal system (i.e., in the absence of interactions)

(11.20)

with A a generalized form of the De Broglie thermal wavelength. The repulsive part .rrep is modeled according to a hard object representation of the molecule, by considering at each atomic position a hard sphere with the proper Van der Waals radius, which allows the computation of the following functional

kBT J . .rrep[P] = -2-b(v) dq1 dq2 p(Ql)p(Q2)8excl (q2,1). (11.21)

The integration kernel 8 excl( q2,I), dependent on the relative coordinates Q2,1, determines the excluded volume: 8 excl = 1 when there is superposition between the spheres, and 8 excl = 0 otherwise. Notice that, if b( v) = 1, Eq. (11.21) becomes the second virial contribution to the free energy of hard objects. Therefore, b( v) represents the correction factor due to higher order virial contributions at the fluid density 1/ v. Theoretical estimates of such a correction factor are not available, apart from the simple case of spherical particles, where the Carnahan-Starling equation of state [60] can be used. Therefore, b( v) has to be evaluated from the comparison of theoretical results and the experimental density dependence of order parameters [18].

The perturbational treatment of the attractive part Vattr of the interaction potential with respect to hard object representation of particle correlations [59] leads to the following relation for the attractive part of the free energy

(11.22)

where 9ho is the pair correlation function for the hard object representation of the molecule. It should be stressed that the pair correlation factor 9ho is an essential ingredient of the perturbation analysis of the fluid, and that it cannot simply be neglected.


However, there are no straightforward methods for evaluating its angular and distance dependence in the case of molecules that have a complex structure. On the other hand, one expects that its functional dependence plays a secondary role with respect to the anisotropy of the interaction potential and of the distribution p( q) in determining the attractive part of the free energy of ordered phases. Then, a reasonable choice is that of neglecting the functional dependence of the pair correlation

gho ~ constant, (11.23)

thus simplifying the procedure for the calculation of the free energy. In this way, the functional F[pJ is determined by the two integration kernels, 8 excl for the repulsive part and Vattr for the attractive part, which can be modeled directly on the basis of the three-dimensional structure of the molecule.

We mention that the minimization procedure according to Eq. (11.18) leads to the following eqUilibrium distribution [18]

p(q) ex: exp{ -U(q)}jkBT

with the mean-field potential U (q) given as

U(qd = J dq2P(q2)[ghoVattr(q2,d + bkBT8(Q2,dJ·

(11.24)

(11.25)

Of course, the mean-field potential for nematic phases depends only on molecular orientation. In principle, however, the same method can be applied to smectic phases, in which case the functional space for the distribution probability, and for the meanfield potential as well, should include the periodic dependence on the displacement normal to the layers.

Such a general procedure can have different implementations according to the choices made for the two integration kernels 8 excl and Vattr . The most detailed, and the most demanding one from the computational point of view, is generated by modeling excluded volume effects by placing hard spheres at atomic positions, and by representing Vattr according to the attractive part of an atomistic Lennard-Jones potential [61]. The computational bottleneck derives from the integration of the distance dependent attractive potentials. A simplified treatment which maintains a realistic dependence of Vattr on the molecular structure and shape has been developed by using the surface contact model for the attractive potential [18]. The basic idea is that a stabilization energy derives from two surface elements of the two molecules when they are in contact in parallel alignment. On the basis of this hypothesis, the Vattr kernel is easily generated by introducing some parameterization 'f/ of the molecular surface and, for the displacement ~R from the molecular centre and the unit vector s orthogonal to a surface element, the functions b.R(O, 'f/) and 8(0, 'f/) dependent on the surface element 'f/ and on the molecular orientation 0

Vattr (Q2,1) = - Eadh J dS1 J dS2 X

8(R1 + b.R(Ol, 'f/1) - R2 - b.R(02' 'f/2)) x

8(8(01, 'f/1) + 8(02,7]2)).

(11.26)


A double integration is performed on the surface elements dS 1 and dS2 of two molecules with parameterization T}l and T}2, respectively. The two Dirac delta functions enforce the conditions that a stabilization energy, scaled according to the adhesion energy parameter Eadh, is recovered only for surface elements in contact and with opposite orientations. Of course, this is a rather crude representation of the molecular interactions since it does not take into account the dependence of the stabilization energy on the nature of the atoms in contact. On the other hand it describes shape effects in simple terms, since in this way the anisotropy of attractive interactions is controlled by the three-dimensional structure of the molecules via their corresponding surfaces. Moreover, it leads to a free energy contribution whose computation is not overly expensive. In [18] we have described in detail the procedures for the numerical implementation of this model, and its application to the nematogen PAA. It has been shown that, with such a Van der Waals model for the interactions, one can rationalize the dependence of the order parameters on the density and on the temperature. In particular it has been verified that excluded volume interactions play an important role in determining the thermodynamic state (T, v) dependence of order parameters.

The density functional theory can be applied not only to pure nematogens, but also to solutions. The generalization is rather straightforward, once a probability distribution pCk) (q) is introduced for each component k. The repulsive and attractive parts of the free energy should include contributions from the interactions between any pairs of constituents weighted by their corresponding probability distributions. Here we shall consider the case of a very dilute solution of components k = 1,2, ... in the solvent labeled as k = O. In the limit of infinite dilution, the solutes have a negligible effect on the solvent properties and, therefore, no modifications are required in the method previously described for determining the probability distribution pCO) (q) of the solvent. The mean-field potential U(k) (q) acting on the kth solute can be derived in analogy to Eq. (11.25) by including only solute-solvent interactions (i.e., by neglecting solutesolute interactions)

UCk)(q) = ui:£r(q) + U~;J(q)

U~;J(ql) = b(k,O)kBT J dq2 p(O)(q2)e(k,O) (q2,1) (11.27)

(k) ) Uattr(ql = gh~O) J dq2 p(O)(q2)Va~~~O)(q2,1)'

In this relation, the repulsive and attractive contributions to the mean-field potential

have been separated according to the integration kernels e(k,O) and Va~~~O) specific for the (k, 0) solute-solvent pair. The methods previously described are easily generalized to the calculation of these kernels for two different molecules. Notice also that the factors gi:'O) and b(k,O) are dependent on the pair of components.

Let us now examine in detail the attractive part of the mean-field potential as derived from the surface contact model of interactions. Eq. (11.26) is generalized by integrating over the surface elements dS(k) and dS(O) of the solute and of the solvent molecular


surfaces ('TI(k) and 'TI(O) are the corresponding parameterizations of the surfaces)

Va~~~(q2,1) = - Eadh J dsik) J dS~O) x

O(RI + ~R(k)(Ol' 'TIik») - R2 - ~R(0)(02' 'TI~O»)) x (11.28)

o(s(k) (01, 'TIik») + s(O) (02, 'TI~O»)).

Here .6.R(m) and sCm) are the functions that determine the position and the orientation of a surface element for the mth compound. By substituting into Eq. (11.27) and by taking into account that p(O) depends only on the orientation, so that the integration of the Dirac delta function on the positions can be performed directly, the following relation for the attractive contribution is derived

(11.29)

where the function

(11.30)

can be interpreted as the density of solvent surface elements with orientation s. Because of the axial symmetry of the nematic phase, (1(0) (s) depends only on the angle between s and the director n. Then we can perform an expansion of (1 (0) (8) in Legendre polynomials of the cosine of such an angle

(1(0)(8) = L(1jO)Pj (8.n) j

(11.31)

with vanishing contributions from the odd parity terms, because of invariance with respect to the inversion of the director. Of course the zeroth rank contribution does not bring any orientational dependence to the mean field. Therefore, the second-rank term is the first relevant contribution for the angular dependence. If we assume that higher-rank terms have negligible effect, the attractive part of the mean-field potential becomes

U~:£r(O) = -Eadhgi:,O)(1~O) J dS(k) P2[n.s(k)(0, 'TI(k»)] (11.32)

which is equivalent to Eq. (11.3) that defines the surface tensor model, ~ being now specified through molecular parameters.

One can conclude that the surface tensor model which originally was proposed on a purely phenomenological basis has a statistical justification if we consider only the attractive part of the mean-field potential, and if we assume that the surface contact model is capable of capturing the effects of molecular shape on the anisotropy of interactions. In this way the major limitation of the surface tensor model becomes evident, since it cannot account for excluded volume effects which would require


an independent evaluation of U;:J. On the other hand, rather good predictions for solute ordering are obtained from the surface tensor model, even if it does not account for excluded volume interactions. An obvious justification would be that excluded volume interactions playa secondary role with respect to attractive interactions in determining the orientational potential of a solute. An alternative explanation is that the two types of interactions introduce similar orientational dependences into the meanfield potential U(k)(!l) of a solute. Then comparable values of the order parameters

would be recovered by considering u~:2r(!l) alone if it is properly scaled, as is the case in the calculation of the dependence of the biaxiality order parameter on the main order parameter (see Figs. 11.1 and 11.2). However, there remain open issues that call for detailed calculations of the solute ordering by taking explicitly into account the solvent distribution in the framework of the density functional theory.

4. Conclusion Molecular modeling of orientational order aims at understanding how the structure

of a particular molecule determines the measured order parameters. Surface models [10,22] have been very effective from this point of view, since they provide a simple interpretation in terms of molecular shape which can be applied to any sort of solute with substantial agreement with the experimental data. On the other hand, the approximate representation of intermolecular interactions is implicit in such a procedure, as long as the specific effects of different nematic solvents are not taken into account. Even if some progress has been made in this direction, for instance by treating self-consistently electrostatic effects within the dielectric continuum model [13, 14], some issues remain. In particular, the assessment of the role of excluded volume interactions, and of specific interactions as well, is essential in order to get a better understanding of ordering at the molecular level. In our opinion, these problems can be efficiently addressed by using the density functional theory with the Van der Waals approximation [16, 17] which allows for the analysis of molecular interactions by preserving a realistic representation ofthe molecular structure [18].

Acknowledgments The authors acknowledge the financial support from EU Commission through the

TMR Contract FMRX CT 97 0121, and from the Italian MIUR through PRIN ex 40%.

Appendix: Polarization induced by a charge distribution in a dielectric

Let us consider a cavity C with permittivity e = 1 in a dielectric with dielectric permittivity e, containing the charge distribution p. The electrostatic problem is defined by Maxwell equations [62]:

{ \7. D =-p \7·D=O

(ll.A.I)

Molecular Models of Orientational Order 257

and the boundary conditions on the surface S of the cavity:

{ v;=v. Ei . S = e:Ee . S

(I1.A.2)

In these equations V, E and D are the electrostatic potential, the electric field and the electric displacement, S is the outward pointing normal to the surface, C E and Clare space regions inside and outside the cavity, and the labels i and e denote points in the inner and outer surface. By recalling the relation between electric displacement and field, D = eoe:E, it is easy to show that in the isotropic phase the polarization charge density, pP = - V' . P, vanishes both in the cavity and in the dielectric, while the cavity surface supports the charge density a P = -p . s. By making use of Eq. (l1.A.2), the latter can simply be expressed as a P = eo(e - 1)e:- 1E i · s, where Ei is the electric field on the inner surface. This does not hold in anisotropic phases, since in this case there is also a non-vanishing charge density in the dielectric, given by pP = V' . (e:-1D).

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Chapter 12

MOLECULAR THEORY OF ORIENTATIONAL ORDER

Demetri J. Photinos Department of Materials Science, University of Patras, Patms, Greece

1. Introduction NMR experiments in ordered fluids can provide values of time averages of nuclear

spin interactions such as dipole-dipole, electric quadrupole and chemical shift. From these one can evaluate second rank orientational order parameters pertaining to the various nuclear sites (chemical shifts) or pairs of sites (dipolar couplings) or molecular segments (quadrupolar splittings) [1]. Such order parameters are useful mainly because:

(i) they provide quantitative measures of molecular orientational ordering from which information on the symmetry and structure of the ordered fluid as well as on certain physicochemical processes taking place therein, can be obtained;

(ii) they can be used for molecular structure determination, either of the molecules forming the ordered fluid phase itself or of molecules dissolved therein;

(iii) they can provide information on the underlying molecular interactions from which insights can be gained on molecular structure - macroscopic property relations in ordered fluids.

At present, (i) and (ii) essentially relate to the use of NMR as a standard tool for experimental studies and therefore the primary theoretical interest is focused on (iii). Accordingly, the main purpose of this Chapter is to explore the relevance of high-field NMR measurements to molecular interactions. The molecules forming the ordered fluid phases are normally too complex for their interactions to be analysed directly. It is therefore advantageous to use solutes of simpler molecular structure as probes of the orientational bias in the ordered phase. In addition, most of the experimental studies designed to elucidate the molecular interactions are carried out on the structurally less complex ordered fluids, mainly low molar mass uniaxial nematic and smectic phases. Naturally then, much of the attention in this Chapter will be given to the analysis of the ordering of relatively simple probe molecules in uniaxial liquid crystal "solvent" phases.



It is generally accepted that NMR measurements are a valuable source of information for molecular interactions. However, it should be stressed from the beginning that such information is in general rather limited and indirect: what is directly measured is just a small set of second-rank orientational order parameters, and these are nothing more than second moments of the orientational distribution of the molecules which in tum is an inextricably complex function of the molecular interactions in the bulk phase. In view of the extreme complexity of extracting inferences about interactions directly from the measured order parameters, theoretical efforts proceed essentially in the reverse direction: a form of the interactions is postulated and the order parameters calculated therewith are compared with the ones obtained from measurement. This procedure, however, cannot single out a unique form for the interaction, since the set of measurements to be reproduced is not sufficiently restrictive. For a rigid solute molecule in a uniaxial liquid crystal solvent the measurements give a maximum of five independent order parameters. As a result of this situation it is rather common for vastly different interaction models, particularly of the phenomenological type, to give very similar results for a given set of measurable order parameters. The usefulness of theoretical models is therefore judged on the basis of their ability to describe successfully not only the results of NMR experiments on systems with different molecular architecture, phase structure and symmetry, but also other properties which are measurable by other methods and are sensitive to the molecular interactions that underlie the ordering mechanisms. In this spirit, particular attention will be devoted to theoretical schemes of clear physical foundation and broad applicability to the molecular-scale description of ordered fluids in thermodynamic equilibrium. Ad hoc models, although numerous, will only be considered briefly and selectively.

In the next section, order parameters that are accessible by high-field NMR measurements are identified and their rigorous statistical mechanical relation to the molecular structure and interactions is given. Then, in section 3, approximation schemes to the rigorous relations are presented. Finally, in section 4 some of the models that have been used to analyse NMR measurements in relation to molecular interactions are discussed in the light of the material from the previous sections.

2. Order parameters, molecular structure and interactions

2.1 Time averages and order parameters Under the assumption that all the reorientational molecular motions are fast on the

NMR timescale [1], the time-averaged quantities obtained from the measurements are equivalent to orientational averages of molecular second rank tensors, to be referred to as segmental order parameters. The following Cartesian notation will be used for the segmental order parameters

0i/3 _ ( Oi/3(f'I. )) Sij = Sij Hs, rs , (12.1)

Molecular theory of orientational order 261

with the angular brackets denoting equilibrium ensemble averages and the averaged quantities denoting symmetric and traceless second rank Cartesian tensors of the generic form

(12.2)

Here the indices i, j refer to the axes of a frame that is fixed relative to the direction of the molecular segment s, the indices 0:, f3 refer to the axes of a macroscopic frame that is fixed with respect to the liquid crystal phase (the director frame), ns represents the Euler angles describing the relative orientation of the two frames, and ia denotes the projection of the unit vector of the i axis on the direction of the 0: axis. In the above equations, and others to follow, the subscript s, used to label the segments, should not be confused with the order parameter symbol s. The scalar function s( r s) is introduced for the sake of a generic treatment of the various nuclear spin interactions. For dipoledipole interactions s(rs) = r;3, where rs stands for the interdipole distance; for quadrupole and chemical shift interactions s(rs) = 1.

2.2 Global and internal molecular motions Knowledge of the segmental order parameters is useful input for the determination

of the molecular structure, as well as for the partial reconstruction of the statistical distribution of molecular orientations and conformations and could eventually provide, through such reconstruction, insights into the molecular interactions in liquid crystals. To illustrate how this can be done for low molar mass ordered fluids, it is convenient to decompose the reorientations of the molecular segments into internal motions, associated with changes of the molecular conformation, and overall motions, associated with rotations of the molecule as a whole. To this end, all the relevant segmental tensors are expressed in a common molecular frame. The choice of this frame is not unique and is usually dictated by molecular symmetry and structure considerations. With the indices I, J referring to the axes of the common molecular frame, the tensors in Eq. (12.2) can be written in the form

(12.3)

where the molecular frame tensors sr1 are analogous to the segmental tensor given in Eq. (12.2),

(12.4)

and summation over the full range of the repeated tensor indices I, J is implied in Eq. (12.3). This summation convention will be implied in all subsequent equations. The Euler angles describing the orientation of the common molecular frame relative to the phase fixed macroscopic frame are represented collectively by n, while Ws

represent the respective angles of the frame of the segment s relative to the common molecular frame. Since the orientations of the segments relative to the molecular frame depend only on the conformations and so does the distance r s, the segmental tensors sf! (Ws; rs) can be labeled by a single conformational index n and will hereafter be


written as sff (n) and referred to as conformation structure tensors (CST). With these definitions, the segmental order parameters of Eq. (12.1) can be put in the form

s~! = (2/3) (sff (n)SrJ(O)) = (2/3) ~ sf! (n) J SrJ(O)fn(O)dO. (12.5) n

Here fn(O) denotes the orientational distribution of the molecular frame relative to the macroscopic frame. The subscript n is to indicate that the molecular orientations are correlated with the conformations, or, equivalently, that each conformation is characterized by its own orientational distribution function. The probability for a molecule in the ordered phase to be found in conformation n is given by

Pn = J fn(O)dO, (12.6)

and the normalization is such that L: Pn = l. n

Defining the order parameter tensor (SrJ)n for each molecular conformation according to the equation

(12.7)

one can write an expression where the measurable segmental order parameters can be related explicitly to the molecular structure, through the CSTs sf! (n), and to the orientation / conformation statistics, through the order parameter tensor of the molecular frame (SrJ)n,

se:! = (2/3) ~Pnsff (n)(SrJ)n. (12.8) n

For rigid molecules or for molecules with a small number of conformations, it is

possible in principle to determine all the ordering tensors (SrJ)n to within a scale factor

proportional to the respective conformation probability, i.e. Pn (SrJ)n, and moreover to determine some of the CSTs in terms of others, if a sufficient number of segmental

order parameters se:! can be obtained from measurement (see also Chapters 13 and 14). In other words, a sufficiently detailed set of measurements and partial knowledge of the structure of each conformation makes it possible to reconstruct completely the structure of each conformation and the relative values of its orientational ordering tensor components. Herein lies the usefulness of high field liquid crystal NMR methods as tools for studying molecular structure and conformation statistics, for measuring orientational order parameters, and for probing mechanisms of molecular alignment. In addition to the numerous studies involving relatively simple solute molecules or neat liquid crystal phases, there are examples of rather detailed sets of measurements allowing critical analysis of systems with a considerable number of conformations [2].


2.3 Molecular symmetry and phase symmetry

For arbitrary choices of the segment-fixed frames and of the macroscopic (director) frame, and ignoring possible symmetries of the molecular structure or of the condensed phase, the symmetric and traceless tensors on the left hand side of Eq. (12.8) have in general 25 independent components, i.e. there are in general 25 independent order parameters for each segment. As shown below, however, in most cases of practical interest the number of independent order parameters is much smaller as a result of molecular symmetry or phase symmetry.

First, the tensors sJ can be put in diagonal form, with respect to the i, j indices for each pair of the a, f3 indices. This is done by properly choosing the segmentfixed frame of axes x, y, z to coincide with the principal axes system (PAS), for the particular segment and a, f3 indices. The diagonal form contains just two independent components, say s~~ and s~~ -s~e, but this alone does not really reduce the number of parameters because the PAS needs three Euler angles to be specified and thus the total number of parameters is still 5 for each pair of the a, f3 indices and for each segment. However, if the molecule has a symmetry element by which one of the principal axes is singled out, then only one angle is required to locate the other two principal axes. This reduces the total number of independent parameters to 3 for each pair of the a, f3 indices and for each segment. For example, all a-chiral molecules have at least one plane of symmetry [3]. Identifying the direction normal to the symmetry plane with, say, the x axis of any segment-fixed frame we have by symmetry s~e = 0 = s~~ and therefore x is a common principal axis for all the segments and all the pairs of a, f3 indices. If the molecular structure has an additional symmetry element to single out a second principal axis, then all the segments will have a completely specified common PAS and therefore the number of independent parameters will be 2. This is the case, for example, of an a-chiral molecule possessing an additional symmetry plane. Identifying the direction normal to that plane with the y axis of the (common) PAS of the molecular segments makes the third axis, z, a principal axis of (at least) twofold symmetry of the molecule. In the case of higher than twofold rotational symmetry about the Z axis one has s~e = s~e = -s~~ /2 and therefore a single parameter specifies the tensor sJ for each pair of the a, f3 indices and for each segment. Obviously, the consideration of segments separately is necessary only in the case of non-rigid molecules where the different segments can move relative to one another while sharing a common set of symmetry dictated molecular axes. For rigid molecules, all the segmental order parameters can be obtained from the order parameters associated with the common molecular frame in terms of the fixed directions of the segments relative to the molecular frame. This becomes apparent also by treating the rigid case as a single-conformation system, with fixed CSTs in the formulation of Eq. (12.8). The segmental order parameter SJ of any segment are thus

obtained from the single set of molecular order parameters (SfJ) given the respective fixed set of geometrical constants sf! for that segment.


Turning now to the implications of phase symmetry on the number of independent order parameters we note again that each s~f tensor, being symmetric and traceless with respect to the a, f3 indices, can be put in diagonal form by choosing the macroscopic axes X, Y, Z to which these indices refer to coincide with the PAS of the phase. This choice will in general be different for different segments and for different pairs of i, j indices and therefore, in the absence of any symmetry of the phase, there will be five independent parameters, say for example, the principal order parameters s£Z, si§X -

Sf? and the three Euler angles specifying the orientation of the respective principal axis frame relative to a macroscopic reference frame, for each segment and each i, j index pair. However, possible phase symmetries reduce the number of independent parameters. Thus an a-chiral phase possesses a plane of symmetry, the normal to which defines the direction of a principal axis, say the X axis, for all the s'tf tensors. In this case there are two principal order parameters and one angle defining the direction of the other two principal axes Y, Z. In the case of the smectic C phase this angle is the tilt angle when the direction of the principal Z axis is specified relative to the layer normal. For phases possessing a second plane of symmetry, perpendicular to the first, the principal axis frame is completely specified and therefore the only independent parameters are s£Z, si§ X - sf?' This is the case, for example, of orthogonal biaxial

smectics and biaxial nematics. Finally, in a uniaxial phase, si§X = Sf? = _s£z /2, and therefore there is just one independent order parameter per segment and i, j pair.

Considering for simplicity rigid molecules, we can now combine molecular symmetry with phase symmetry to obtain the following examples of independent order parameters, in descending degree of symmetry. Uniaxial molecules in uniaxial phases: s~zZ; in orthogonal biaxial phases' sZz sXx - sYY. . zz' zz zz , Molecules with two orthogonal planes of symmetry , "I h . zz zz ZZ. In umaxla p ases. szz 'Sxx - Syy , 'rth lb' 'al h . zz xx YY ZZ ZZ xx YY xx+ YY In 0 ogona laxI p ases. szz ,szz - szz 'Sxx - Syy ,sxx - sxx - Syy Syy. A detailed discussion of the order parameters in the case of molecules and phases where only one of the principal axes can be defined by symmetry, which includes the important category of tilted smectics, can be found in [4]. More detailed listings of order parameters according to phase and molecular symmetry, albeit in different notation from the one used here, can be found, for example, in [5,6]; see also Chapter 1 of this book.

It is apparent from the above examples that the lower the symmetry, of the molecules or of the phase, the larger is the number of independent order parameters that can, in principle, be accessed by NMR measurements, and therefore the larger the input for the study of the molecular interactions. It would thus seem desirable to pursue NMR studies on lower symmetry systems. However, this advantage is counterbalanced by the complexity, both of the experiments and of the analysis of the results on such systems, and the main trend thus far has been towards simpler systems, mainly uniaxial nematic fluids probed by solutes of relatively simple structure.


2.4 Relation to molecular interactions The measurable order parameters are related to the molecular interactions through

the orientational distribution function f n (0) of Eq. (12.5). It is possible, without loss of generality, to express fn(O) in terms of the so-called "potential of mean torque" Vn(O) according to the relation

(12.9)

where E~ is the intra-molecular energy of the molecule in its nth conformation, Gn =

(Ix1ylz)1/2 is the rotational kinetic energy factor, expressed in terms of the principal values of the inertia tensor of the nth conformation [7], and ( is the normalization factor

( = L Gne-E~/kT J e-Vn(O)/kT dO. n

(12.10)

A rigorous relation of the potential of mean torque to the potential describing the molecular interactions in the bulk phase can be established by comparing the right hand side of Eq. (12.9) to the exact statistical mechanical expression for the singlemolecule orientation / conformation distribution function fn(O), namely,

In(n) ~((~/Z) J dr ~ (fi ~) (12.11)

x J d{Oi} J d{ri}exp(-Un(r,O;{ri,Oi,ni})/kT).

Here, Un is the potential describing the full interaction in a system consisting of the "singled out" molecule, with position, orientation and conformation variables r, 0, n, respectively, and a collection of N identical molecules (but not necessarily identical to the "singled out" molecule), labeled by the index i = 1, 2 ... N, whose position, orientation, conformation variables are denoted collectively by {r i, Oi, ni}. The notational abbreviation (~ == Gne-E~/kT has been used and Z denotes the partition function of the system of N + 1 molecules

z ~ ~ (~ J dn J dr ~ (fi (~.) (12.12)

x J d{Oi} J d{ri} exp (-Un(r, 0; {ri' Oi, nil )/kT).

Comparison of Eq. (12.11) with Eq. (12.9) yields the following expression for the potential of mean torque

e-Vn(O)/kT =((/Z) J dr L (IT (~i) {ni} ~=1 (12.13)

x J d{Oi} J d{ri} exp( -Un(r, 0; {ri' Oi, nil )/kT).


Except for trivially simple cases of the full potential Un, Eq. (12.13) is clearly offormal significance only. In particular, it provides the statistical mechanical foundation for the development of general analytical schemes of approximation as well as for numerical evaluations of the potential of mean torque for specified interactions by means of computer simulations (see Chapter 15 in this book). Closed form relations between the potential of mean torque and the intermolecular potential can be obtained through approximations that allow some of the multi particle summations and integrations in Eqs. (12.11) - (12.13) to be executed. Such approximations will be described in the next section. Aside from these approximations, however, the concept of the potential of mean torque is useful even if attempts to relate it to the detailed intermolecular interactions are completely abandoned and a phenomenological approach is followed.

2.5 Phenomenological descriptions

The obvious advantage of a phenomenological description is simplicity, of course, at the expense of microscopic insight and predictive power. In view of the adjustable parameters usually involved in such descriptions, their successfulness is assessed not only from their quantitative accuracy in reproducing the results of measurements, but also from their conceptual clarity, the number and physical significance of adjustable parameters, as well as their transferability among different systems. Transferability is of key importance to the predictive power of a phenomenological description.

Phenomenological formulations of the potential of mean torque are based, first of all, on symmetry considerations. Quite generally, Vn(O) can be put in the form of a series expansion in Wigner rotation matrices D!nm'(O) with conformation dependent expansion coefficients Gn(l, m, m'):

Vn(O) = L Gn(l, m, m')D!nm'(O). (12.14) l,m,m'

Depending on the symmetries of the probe molecule and of the solvent phase discussed in section 2.3, only certain groups of terms will be present in the expansion, as the others can be eliminated by properly choosing the molecular and phase-fixed frames to yield vanishing values for the respective coefficients. For example, in the simplest most symmetric case of uniaxial rigid probe molecules in a uniaxial apolar phase, only the coefficients with m = m' = 0 survive and the series expansion in Eq. (12.14) reduces to an expansion in Legendre polynomials PI( cos (}) of even rank, I = 0,2,4 ... ,

V(COS(}) = L CZPI(COS(}), (12.15) l=even

where (} denotes the angle between the molecular symmetry axis and the phase symmetry axis (director).

The exact series expansions in Eqs. (12.14) or (12.15) are the starting point for phenomenological formulations of the so-called maximum entropy type [8], where one assumes that all the coefficients above a certain rank l* are negligible and then


proceeds to evaluate the coefficients of rank l ::; l* by directly fitting the experimental order parameter data, without any reference to molecular interactions. Since the focus of this Chapter is precisely on the connection between order parameters and molecular interactions, the maximum entropy and related approaches will not be considered further.

The assumption that only low-rank coefficients are important (the truncation rank is usually l* = 2) is justified in the low order regime by the results of molecular theories and computer simulations using a variety of intermolecular potentials [8-15]. The usefulness of the truncated expansion, when justified, lies in providing a parameterization of the potential of mean torque in terms of a small number of expansion coefficients. The expansion by itself, however, does not relate these coefficients to the molecular physics of the systems, aside from symmetry, and this makes it necessary to introduce into the phenomenological description some elements of molecular structure and interaction. Such elements are generally introduced as constraints that the ordered fluid environment imposes on the probe molecule. The solute molecule is described more or less with atomistic detail, but the molecular picture of the ordered medium is replaced by a number of effective constraints (hard or soft) that it presumably imposes on the positions, orientations and conformations of the probe molecules. Various types of models, mostly describing simple rigid solute molecules in nematic solvents, are based on the constraint picture. These include continuum models, where the solvent is modeled as an anisotropic continuum endowed with certain directional attributes to which the solute molecules couple by virtue of directional properties associated with their shape, inertia tensors, electrostatic moments, etc. [16-23]. Some of these models are discussed in detail in the other Chapters of Part III of this book and will also be considered briefly in section 4 of this Chapter. A different type of constraint model [24], combining orientational and stratification constraints, has been developed and used successfully to reproduce detailed NMR measurements of segmental orientational order parameters together with measurements of the spontaneous polarisation of chiral molecules in the smectic C* phase.

A concept that is often useful in supplementing the constraint formulation of the potential of mean torque is modularity. It refers to the construction of the potential of mean torque for more complex molecules from the constraints imposed on their segments [22]. The obvious prerequisite is of course a reliable description of the constraints experienced by the constituent segments. The segmental interactions are then combined to produce the overall interaction of the molecule with the ordered medium according to the general scheme:

S 8,S'

(12.16) + L v~3)(ns,nsl,nsll)+ ..... ,

s,s',s"


where v~l)(Os) denotes a single-segment interaction contribution associated with the sth segment, whose orientation is denoted by Os, when the molecule is in conformation

n. Similarly the terms v~2) and v~3) denote contributions associated with pairs of segments s, s' and triplets of segments s, s', s", and the summations cover all segments, and combinations of pairs, triplets, etc., thereof. Such formulations are particularly suitable for chain molecules consisting of segments whose directionality is simple to describe on the molecular scale. By combining then the segmental contributions in a way that conveys the connectivity of the segments in the chain molecule one obtains the potential of the mean torque for the latter. In most practical applications only terms up to segment-pair ones are considered, and of those, only the ones corresponding to nearest and next nearest neighbour segments are retained in the summation.

A successful example of the modular formulation is the chord model for the potential of mean torque of n-alkane molecules dissolved in liquid crystals [7]. One of the attractive features of this model which is presented in Chapter 13 of this book is the highly efficient way in which it handles large numbers of conformations of chain molecules. The modular formulation can be used both for the description of chain solutes in ordered solvents [2,7,25], and for the self-consistent description of ordering and phase transitions in neat phases of chain molecules [26]. The most primitive form of the modular approach is the additive one: contributions from the different segments are simply added to give the total potential of mean torque. An example of this approach is the additive generalisation of the Maier-Saupe description to chain molecules, originally proposed by Marcelja [27,28]. The simple additive formulation is usually inadequate because it does not account for segmental correlations originating from the specific connectivity of the segments in the chain molecule [7, 22].

3. Approximation schemes for the potential of mean torque

We now return to the problem of establishing a closed-form relation between the potential of mean torque and the intermolecular interaction, starting from the rigorous formal relation in Eq. (12.13). This naturally involves approximations, both on the interactions and on the statistical mechanics treatment. The purpose of this section is to specify these approximations, to discuss their effects on the final results, and to identify possible ways of compensating for these effects.

The obvious complication in the right hand side of Eq. (12.13) is the summation / integration over correlated variables of the ensemble of N + 1 particles. Accordingly, the objective of the approximations is to reduce, or to effectively remove, the correlations that do not involve directly the probe molecule. A systematic way of doing this, in stages of increasing accuracy (and complexity) is provided by the variational cluster method [12,29]. The lowest order approximation, corresponding to the two-molecule cluster contributions, will be outlined here. An equivalent derivation can be obtained in the context of density functional theory [30,31]. To simplify the illustration, the probe molecule (solute) and the molecules forming the ordered fluid phase (solvent) will be


treated as rigid and the solvent phase will be assumed to have positional disorder and uniaxial orientational order, i.e. to be a common nematic phase. Generalisations to flexible molecules and phases with partial positional order are straightforward and can be found in [12-14].

3.1 Approximations on the form of the interaction For a uniaxial nematic system composed of rigid molecules, Eqs. (12.9) and (12.11)

are written as

and

f(n) = e-V(fl)/kT / ! e-V(fl)/kT do'

0, _ J d{ni } J d{ri}e-U(r,fl;{ri,fl,})/kT

f( ) - J do' J d{ni} J d{ri}e-U/ kT

(12.17)

(12.18)

Assuming that the potential U of the interaction among the N solvent molecules and the solute molecule (whose position is taken to coincide with the origin of the phasefixed frame of axes) can be separated into a part U(N) describing the interactions among the solvent molecules and a sum of pair potentials u(n; ri, ni) describing the interaction of the solute molecule with each of the solvent molecules, one can write

N

U(n;{ri,ni}) = U(N)({ri,ni }) + LU(n;ri,ni). i=l

(12.19)

Strictly, this form already constitutes an approximation, as the pair-wise additive form of the solvent I solute potential does not allow a full description of molecular polarisability effects among the solute and the solvent molecules. Such effects could become important for solutes with large polarisability anisotropy. Deformations of the partial charge distribution caused by mUlti-particle interactions are not negligible for highly polarisable elongated molecules forming nematic phases. It should be noted, however, that at this stage no pair-wise additivity is assumed for the interactions among the solvent molecules and therefore the approximation In Eq. (12.19) would introduce inaccuracies only in the case of solutes with large polarisability .

Defining the probability distribution function p(N) for the ensemble of the N solvent molecules according to

p(N){ri,nd = e-U(N){ri,fli}/kT/ ! d{nd ! {dri}e-U(N)lkT, (12.20)

Eq. (12.18) can be written in terms of the ensemble average ofthe solvent I solute pair potential of Eq. (12.19) as follows

(12.21)


Comparison of this expression with Eq. (12.17) leads to the following relation for the potential of mean torque

e-V(f!)/kT = J d{Oi} J d{ri}p(N){ri, Oi}e - ~u(f!jrl>f!i)/kT. (12.22)

3.2 Approximations on the statistical mechanics The major statistical mechanical approximation is applied at this point and consists

in replacing the exact probability distribution p(N){ri, Od by a product of uncorrelated single-molecule effective distribution functions F( r i, Oi) = (1 IV) j (Oi), where V denotes the volume of the positionally disordered system of the N solvent molecules. The replacement

N

p(N){ri, Oi} ~ v-N II j(Oi), i=l

(12.23)

allows the expression for the potential of mean torque in Eq. (12.22) to be approximated by

e-V(f!)/kT ~ [(IIV) J dO'1(O') J dr'e-u(f!;r',f!')/kT]N, (12.24)

where the primed variables refer to a solvent molecule. Finally, noting that for large N it is possible to replace e-V(f!)/NkT by 1 - V(O)INkT, Eq. (12.24) yields the following approximate relation between the potential of mean torque and the solvent / solute pair potential

V(O) ~ kTp J dO'j(O') J dr' (1- e-u(f!;r',f!')/kT)lv. (12.25)

Here v denotes the molecular volume of the solvent molecules and p = N v IV denotes the solvent density, expressed as a packing fraction. The linear dependence of V(O) on the density p is due to the fact that the neglect of correlations among the solvent molecules leads to an expression that corresponds to the leading term in the virial (density) expansion and is therefore valid at low density. If this form is to be used at liquid phase densities then p is to be replaced in Eq. (12.25) by an effective density p that compensates for the neglect of correlations among the solute molecules [32]. The effective density p is a function of the actual density p. In general, the function p(p) grows very large as p approaches its maximal packing value and coincides with p at low densities. Detailed forms of p(p) can be obtained in the context of approximate summation schemes of the virial series expansion [33, 34].

It is often convenient to put Eq. (12.25) in aform where the sol vent / sol ute interacti on is decomposed into a hard body part uO and a soft, longer ranged part u L . The overlap function gO(O; r', 0') = e-uO(f!;r',f!')/kT can then be defined to vanish when the solute / solvent molecular cores overlap in space and to have unit value otherwise. Combining these considerations, Eq. (12.25) can be rewritten as follows


V(O) ~ kTp J dO' j(O') J dr'(l - gO(O; r', O')e-uL(rl;rf,rlf)/kT)/v. (12.26)

In this final expression, the potential of mean torque is related to the detailed form of the solvent / solute pair potential through the solvent orientational distribution function j (0') and the solvent effective density p. On a qualitative level, this expression is useful because it illustrates in a relatively simple way the building up of the potential of mean torque from the actual solvent I solute interactions. The illustration is of course not complete because of the removal of explicit correlations among the solvent molecules, but it does, for example, demonstrate explicitly that the contributions to the potential of mean torque coming from the soft, longer range part of the intermolecular potential u L

are modulated by the hard body part through the overlap function gO, and that therefore the end result of such contributions depends on the shape of both the solvent and the solute molecules. On a quantitative level, compensating for the removal of molecular correlations by means of the effective density p appears to improve substantially the accuracy of the approximation [9,32]. For example, comparisons with Monte Carlo simulations on hard ellipsoids [15] show that, with proper choice of the value of the effective density p, Eq. (12.26) gives an essentially exact reproduction of the solute orientational distribution function.

In practical applications of Eq. (12.26) to the analysis of solute ordering, it is necessary, in addition to postulating an explicit form of the solvent I solute potential, to provide the solvent distribution function j(O'). Usually, this function is only partly known through its leading moments, the orientational order parameters, which are more directly accessible to measurement. In such cases it is desirable to express the potential of mean torque in terms of the orientational order parameters of the solvent. These are generally defined to any rank l using the Wigner rotation matrices as follows:

(D!nmf ) solvent == J dO'D!nmf (0') j( 0'). (12.27)

To introduce these order parameters in Eq. (12.26) we note that the integration of the interaction term over the intermolecular positions will yield a function of the relative solvent / solute orientations w namely,

(12.28)

The positionally averaged interaction term q(w) may in turn be expanded in a series of rotation matrices with expansion coefficients q!nmf given by

(12.29)

Inserting this expansion into Eq. (12.26) and using the definition in Eq. (12.27) leads to the following expression for the potential of mean torque in terms of the solvent


order parameters

v (0) ~ kT P L q!nmll D!nm' (0)( D!n'mll ) solvent· (12.30) l,m,m',m"

It is worth noting that, on comparing this equation with Eq. (12.14) one obtains, for the case of rigid solutes considered here, a closed form expression for the coefficients C(l, m, m') of the general Wigner matrix expansion of the potential of mean torque, namely

C(l,m,m') ~ kTpLq!nmll(D!n'mll)solvent. (12.31) mil

The practical usefulness of the form of V(O) given in Eq. (12.30) is that for typical nematic solvents only a few, low rank order parameters contribute significantly to the series expansion.

The approximation of the potential of mean torque in Eq. (12.26) leads, on making further simplifying assumptions, to the forms obtained in well known approximate molecular theories of the nematic phase:

(i) On completely ignoring the soft part uL of the potential and using the lowdensity limit for p, Eq. (12.26) yields the potential of mean torque of Onsager's theory [35,36]

V(O)Onsager = (kTN/V) J dO' /(0') J dr' (1- gO(O; r', 0')). (12.32)

(ii) If uL is not ignored completely but is kept in Eq. (12.26) to leading order in the high temperature expansion e-uL /kT ~ 1 - (uL /kT) then Eq. (12.26) gives

V(O)vdW =(N/V) J dO' /(0')

x J dr' [kT{l - gO(O; r', 0')) + gO(O; r', O')uL(O; r', 0')],

(12.33)

which is the potential of mean torque for the generalised Van der Waals theory of nematics [37,38].

(iii) Finally, if the dependence of the overlap function on the orientations of the solvent and the solute molecules is ignored completely by setting gO (0; r', 0') ~ g(r') and, in addition, the high temperature approximation for e-uL /kTis used in Eq. (12.26), one obtains the form of the potential of mean torque used in the Maier-Saupe theory [39], namely

V(O)M-S = (N/V) J dO' /(0') J dr'g(r')uL(O; r', 0'). (12.34)


4. Molecular models Anisometry of the molecular shape, giving rise to molecular orientability via highly

directional short-range steric interactions, is generally accepted [38] as the molecular feature responsible for orientational ordering in low molar mass nematic, orthogonal smectic and columnar liquid crystals. More than fifty years ago Onsager showed that sufficiently long hard rods can form a nematic phase [35]. Early molecular simulations of hard-body models of rod-like and plate-like molecules [40-42] showed that shape anisometry alone is sufficient to produce not only orientational ordering but also partial positional ordering of the kind exhibited by layered and columnar phases. This of course does not mean that shape anisometry is the only molecular property of importance to orientational ordering in liquids, nor does it imply that all but short-range steric interactions are necessarily insignificant:

• Attractive forces cannot be ignored in a realistic description of the temperature and pressure dependence of orientational order.

• Amphiphilic and multiphilic molecules without particularly strong shape anisometry are known to give orientationally ordered fluids through the mechanism of phase micro segregation [43]. This mechanism is also important in stabilising layered and columnar ordering.

• Electrostatic interactions, particularly dipolar, are believed to affect the orientational ordering in simple nematic systems [12,44,45] or the structure oflayers in some orthogonal smectics [46,47], to give rise to dipolar association and phasereentrance phenomena [48-50], and to influence the relative thermodynamic stability of smectics and nematics [45,47,48,50]. Also, the interpretation of NMR experiments on noble atom solutes in nematics suggests that electrostatic interactions cause deformations of the electronic density of these atoms (see Chapter 6 and [51]). Finally, there is strong evidence that the orientational order of molecular hydrogen in nematic solvents can be described almost completely in terms of its electric quadrupole anisotropy (see Chapter 10 of this book and [9]).

• The anisotropy of the molecular polarisability, to which the ordering is entirely attributed according to the original Maier-Saupe theory [39], is considered to be an important factor for orientational order in some types of mesogens.

• Other site-specific interactions, such as hydrogen bonding, are known to induce orientation ally ordered fluid phases [52].

It is precisely the above interplay of different intermolecular forces and orientation mechanisms that one tries to elucidate with the help of NMR experiments. The use of probe solutes allows one to magnify certain molecular features, or to suppress others, by judiciously choosing the solute molecular structure. As most of the experiments to date have been carried out in nematic solvents, the majority of the studies concern


shape anisometry and electrostatic interactions which are thought to be of primary relevance to the ordering in the nematic phase [9]. In contrast, little is known from NMR studies, for example, on microsegregation which is present in smectics and colurnnars. Practically all the models developed to date in connection with NMR studies of probe solutes refer to the nematic phase. Some of them are briefly presented below. First we consider models based entirely on shape anisometry, then models combining shape anisometry with electrostatic interactions, and finally models based on complete atomistic interaction force fields.

4.1 Shape anisometry models. The simplest way to convey shape anisometry is through anisotropic hard-body

potentials. Hard spherocylinders, discs, cut-spheres, ellipsoids, etc., have been extensively used in theoretical and computer simulation models to explore the influence of purely repulsive short range forces on orientational ordering [36,40-42,53]. Other models employ distributed Lenard-Jones sites, Gay-Berne parameterisation, soft ellipsoids, etc. [to, 11, 15,47,54,55], to incorporate attractive forces and soft repulsions into the shape-anisometric description of intermolecular interactions. In all these cases the modeling of shape anisometry introduces a number of external parameters. In more detailed representations, the molecular shape is constructed by means of the Van der Waals hard spheres of the constituent atoms or united atom groups. Such representations are preferable in that they essentially involve no adjustable parameters if the molecular structure of the solvent is known. The number of adjustable model parameters is an important issue in the analysis of the orientational ordering, particularly of rigid probe solutes, since, as discussed in section 2, NMR measurements provide at most five independent orientational order parameters for a given solvent / solute pair. It is thus apparent that a quantitative reproduction of the measurements on any solvent / solute system is not difficult to achieve with models using several adjustable interaction parameters. Accordingly, for models to be meaningfully testable against experiment they should not involve many adjustable parameters and should provide a consistent qualitative and possibly quantitative description for a broad set of measurements on different molecular systems.

Shape anisometry models are intended for systems where other types of interactions are expected to be negligible. An example is provided by cycloalkane solutes which, being essentially free of electrostatic moments and virtually rigid, are suitable for testing shape dominated model interactions with a nematic solvent. Seven of these solutes were studied [32] by deuterium NMR in a nematic solvent (the mesogenic mixture ZLI 2452) over a very broad temperature range. These solvent / solute systems were modeled using the variational cluster approximation for the potential of mean torque in the formulation of Eqs. (12.26) - (12.31) with only hard body interactions. The shape of the solutes was constructed from hard Van der Waals spheres of carbonhydrogen united atoms (of standard radius RvdW = 0.177 nm) and the shape of the solvent molecules was described simply by a spherocylinder of diameter D = 0.52 nm and aspect ratio L / D = 3.31 (i.e. total length L + D = 2.24 nm). Very accurate


reproductions of the order parameters of all the solutes throughout the entire nematic temperature range of the solvent were obtained for all the solutes using just the second rank term in the expansion ofEq. (12.30). As a result of the assumed uniaxiaIity of the solvent molecular shape there is just one second-rank order parameter for the solute molecules, namely (P2)solvent. Accordingly, the potential of mean torque is scaled by the factor p(P2)solvent which in the absence of independent measurements of the solvent order parameter constitutes the "fitting parameter" in these calculations. In all cases of solutes the effective density p of the solvent was found to be larger than the actual one and to be slightly dependent on the solute molecule size.

Generally, the description of the systems in [32] by strictly hard body repulsions is considered successful, particularly in view of the, perhaps oversimplified, representation of the solvent shape by a hard spherocylinder. A more realistic representation of the solvent molecular shape in that study would complicate the calculation considerably since the solvent consists of a mixture of several flexible mesogens. In studies with simpler solvents, however, atomistic modeling of the solvent molecules can readily be employed as discussed below.

A step in the opposite direction, i.e. eliminating altogether any reference to solvent molecular structure, is taken in models treating the nematic solvent as a uniaxially anisotropic continuum in which the orientations of the solute molecules are biased according to some measure of their shape anisometry. Naturally, the elimination of solvent molecular structure limits the predictive power of these models. The expected gain is a simple and consistent, at least qualitatively, description of the ordering of different solutes according to their shape. A number of such models have been proposed, some using coarse features of solute molecular structure [17-20,56] and others using atomistic detail [23] in order to quantify the measure of solute shape anisometry that couples to the uniaxial continuum. Examples of the latter type are the elastic tube model (Chapter 10 and [16, 23]) and the surface tensor model (Chapters 10 and 11 and [21,23]). The solute potential of mean torque in the original elastic tube model [16] is given by

V(O) = (kj2)C2(O), (12.35)

where k is a solvent-characteristic parameter and C(O) is the length of the minimum circumference around the projection of the solute molecule onto the plane perpendicular to the director of the uniaxial medium. In the surface tensor model [21] the potential of mean torque is obtained by integrating the second Legendre polynomial P2(8.N) = [3(8· N)2 - 1]/2 over the solute surface B, i.e.

V(O) = kbT € J dBP2 (8 . N),

s (12.36)

where 8 is the unit vector normal to the solute surface element dB and N is the nematic director of the medium. The parameter € is a characteristic of the solvent medium.

Both of these continuum models have been used to describe a variety of nematic systems [23] for which the ordering is believed to result essentially from shape anisom-


(a) (b) (c)

Figure 12.1. a, b: A tangent-sphere elongated object directed parallel (a) and perpendicular (b) to the nematic director N. c: A different tangent-sphere object. The potential of mean torque obtained from the surface tensor model according to Eq. (12.36) vanishes in all three cases.

etry, either by specific choice of the solute, or by special tuning of solvent mixtures to produce a nematic medium that is thought to be free of electrostatic fields or gradients. We shall return to the latter point in the next subsection. The two models are similar in that the scale factor of the potential of mean torque depends only on the solvent medium while the functional form of the orientation dependence is determined exclusively by the shape of the solute. This factorisation into purely solvent-dependent and purely solute-dependent parts is not sustained by the rigorous expression of the potential of mean torque in Eq. (12.22), nor even by the approximate expressions in Eqs. (12.26) and (12.30). In fact it is clear from Eqs. (12.28) and (12.29) that the expansion coefficients q!n m' are determined from the molecular structures of both the solvent and the solute and that the same is true for the coefficients C(l, m, m') in Eq. (12.31). The effects of imposing the factorisation on the potential of mean torque in these models, and in several of their subsequent variants [23], are reflected in the solute dependence of the optimal values of the solvent-characteristic parameters k, c.

Another common feature of the two continuum models is that the respective ordering mechanisms are tailored by analogy to mechanisms that are applicable to macroscopic objects. The direct transfer of such macroscopic descriptions to molecular systems often leads to serious flaws in the underlying physics. For instance, the potential of mean torque of the surface tensor model in Eq. (12.36) vanishes, as it clearly should, when the surface S is spherical. However, being an additive potential, it also vanishes for any system of tangent spheres, irrespectively of their configuration. This implies, for example, that the medium does not discriminate orientationally between the objects (a), (b) and (c) shown in Fig. 12.1. While such lack of orientational discrimination would be acceptable if the spheres were of macroscopic dimensions, i.e. very large compared to the size of the molecules of the nematic solvent, it clearly shows that the orientability of molecular size objects in the nematic phase is poorly represented by the surface tensor. In fact, the integral in Eq. (12.36) acquires non vanishing values


Figure 12.2. a: A pair of overlapping spheres of equal diameter defining a "bond length" d and a ''bond direction" d. b: Overlapping sphere representation of a chain molecule with successive ''bond directions" shown.

as soon as the spheres in Fig. 12.1 begin to overlap, indicating that the orientability of these objects would not be due to anisometry of their overall shape but rather to some details of the shape representation. Thus, according to the surface tensor model, a rod-like molecule, for example, would have no orientational order if its shape were represented by an array of tangent spheres but would begin to show some orientability when represented by slightly overlapping spheres.

The above flaws refer to the physical foundation of the model. The molecular shapes in actual calculations with the surface tensor model [21] are, of course, represented by groups of overlapping spheres and this confers some orientability to the solute molecules. However, in that case the surface tensor model reduces to a simple second-rank bond-orientation model. This can be shown readily by considering the two overlapping spheres of equal diameter in Fig. 12.2(a). The surface integral ofEq. (12.36) forthis object is simply -S(djD)(l- d2jD2)P2(d. N) where Sis the surface of each sphere, D is the sphere diameter, d is the distance between the sphere centres and d is the unit vector along the inter-centre direction (the "bond" direction). According to this result, the potential of mean torque for the pair of overlapping spheres is just a second rank Lengendre polynomial of the projection of the bond unit vector along the director. This potential is scaled by a factor that vanishes when the two spheres are either tangent (d = D) or coincident (d = 0) and reaches its maximal value for d = D j v'3. Aside from this scaling factor which lacks a sound physical foundation and in any case is eventually rescaled by the model parameter c of Eq. (12.36), the functional form is simply the usual leading rank term in the standard tensor expansion of the potential of mean torque in Eq. (12.15). In the general case of several spheres, with different diameters etc., the evaluation of the surface integral in Eq. (12.36) results in a combination of second rank tensor contributions of the various "bond" vectors d i , with scaling factors determined by the sphere diameters and inter-centre distances. For example, in the case of a flexible chain molecule consisting


of an array of identical overlapping spheres spaced a constant distance d from their nearest neighbours, as shown in Fig. 12.2(b), the potential of mean torque evaluated according to Eq. (12.36) is, aside from an overall scaling factor, Li P2(di . N), where the summation extends over all the bonds in the chain. This is precisely the potential of mean torque for the bond additive model of Marcelja [27].

4.2 The inclusion of electrostatic interactions. A full description of the electrostatic interactions in liquid crystals requires the

partial charge distribution on each molecule to be determined as a function of its conformation and of its configuration relative to the other molecules surrounding it. In principle, this can be done given the partial charge distribution for the free molecule and given the polarisabilities of the molecular segments. Such a description, however, entails the use of potentials that are not pair-wise additive and, to date, has not been attempted in any theoretical or computer simulation studies of liquid crystals. Substantial simplification is obtained by treating the partial charge distribution on the molecular segments as fixed, i.e. by ignoring molecular polarisability altogether. The effects of this approximation on the orientational ordering are not expected to be severe for probe molecules of small polarisability anisotropy. A further step of simplification is to replace the partial charge distribution by the leading terms of its multipole expansion. The electrostatic interactions are in this case represented as originating from a set of permanent dipole or quadrupole moments fixed on the molecular frame. An obvious limitation of this representation is that the multipole expansion is valid only for distances that are large compared to the spatial extent of the distribution and therefore the leading-moment potential may deviate significantly from the actual partial charge potential at short intermolecular separations.

A molecular model combining shape anisometry with interactions among localized electrostatic moments was used in the context of the variational cluster approximation of Eqs. (12.26) and (12.30) for the interpretation of the ordering of a several rigid solutes of different sizes, ranging from molecular hydrogen to anthraquinone, as measured by NMR in two nematic solvents (EBBA and ZLI 1132) and mixtures thereof [16,57-59]. The molecular shape of the solvents was approximated by hard spherocylinders and the atomistic Van der Waals hard-sphere construction was used for the shape of the solutes. The electrostatic part of the solvent / solute intermolecular potential was built up from localized dipole and quadrupole moments, both permanent and induced, whose values and directions were determined from independent studies. In spite of the simplified representation of both shape anisometry and electrostatic interactions, these calculations account consistently and fairly accurately for the measured order parameters of all the solutes in both solvents and in their mixtures. The electrostatic interactions are found to have a substantial contribution to the ordering mechanism only for the small solute molecules, such as hydrogen and nitrogen which have low shape anisometry. This contribution is strongly solvent dependent, in contrast to the contribution of shape anisotropy, and in the case of hydrogen the two contributions generate orientational ordering in competing directions. The final outcome


of this competition can be adjusted in mixtures of the two solvents by varying their concentrations. In accordance with experimental observations, the competition leads to complete cancellation of the measured orientational order of the hydrogen molecule at a particular concentration (defining the so-called "magic mixture" [23]). An analogous cancellation is not obtained for nitrogen, neither is it observed experimentally, because shape anisometry and electrostatic interactions are not found to produce competing effects on the orientational ordering in either of the solvents. For the larger of the solutes, anthracene and anthraquinone, shape anisometry becomes the dominant ordering factor whereas for intermediate size solutes, benzene and naphthalene, the contributions of shape and electrostatics have comparable magnitude and are of the same sign, i.e. not in competing directions. In all cases, the electrostatic contributions are primarily due to interactions among permanent moments, dipole-dipole (when present), dipole-quadrupole and quadrupole-quadrupole. Induced dipoles have minor effects on the ordering. Finally, the effects of the electrostatic interactions are found to be sensitive to the shapes of the solvent and solute molecules and to the positions and directions of the permanent moments within the respective molecular frames.

A different picture of electrostatic interactions, and interpretation of experimental findings, is obtained in continuum models. In these models an electrostatic component of the potential of mean torque is added to the part associated with shape anisometry. The electrostatic component is assumed to represent the coupling of the molecular charge distribution of the solute to an electrostatic field produced by the nematic continuum. The apolarity of the nematic continuum does not allow for any dipolar interactions of the solute molecules with the solvent. In this naive picture [16,23,57, 60], the leading-rank electrostatic interactions felt by a solute molecule are associated with the coupling of the electric quadrupole moment of the solute molecule to an ad hoc property of the nematic medium bearing the physical dimensions of an electric field gradient. This picture was soon demonstrated to be inadequate [9,44,61] and it is now well established by several theoretical works and numerous computer simulations [12, 44,45,47,62-65] that residual dipolar interactions are present in apolar mesophases (nematic, smectic or columnar), where they can produce substantial effects on the thermodynamic stability of these phases [12,45], give rise to molecular dimerisation via dipolar association [48,49] and to phase re-entrance phenomena [50], and cause structural modifications in smectics [46,47] . On the other hand, it was demonstrated theoretically [9], and later found in computer simulations [66], that the ad hoc "electric field gradient of the solvent" is not strictly a solvent property but depends on the structure of the solute molecules as well. This interdependence is apparent from the exact form of the potential of mean torque in Eq. (12.22) as well as from the approximate one in Eqs. (12.26) - (12.30): the electrostatic contribution is built up by sampling the respective electrostatic part of the intermolecular solvent / solute potential over positions and orientations that are determined at short distances by the molecular shapes of both molecules. It is also apparent from these expressions that even if a strictly additive form of the intermolecular potential in terms of steric and electrostatic interactions is assumed, the resulting potential of mean torque cannot, except for trivial cases, be written as a sum of a purely shape-dependent term and a term that is determined


exclusively from the electrostatic interactions. More generally, additivity of the intermolecular potential does not lead to a corresponding additive form of the potential of mean torque. It is thus clear that continuum models convey an oversimplified picture of the interplay between steric and electrostatic interactions in liquid crystals. The principal electrostatic property of the nematic continuum according to this picture, the "electric field gradient" (efg) [23,57,60], varies in magnitude and sign from solvent to solvent and can be tuned to a null value by mixing of opposite sign efg's. Probe molecules dissolved in such "magic mixtures" are then supposed to experience no electrostatic ordering bias, irrespectively of their sizes or shapes. The vanishing of the order parameters of molecular hydrogen in such mixtures is often interpreted using this picture [16,58,60] which is of course very different from the interpretation obtained in the context of the molecular picture of the solvent discussed previously, where the vanishing of the order parameter is a result of balance between steric and electrostatic interactions that occurs only for hydrogen solute molecules in that solvent [9]. The possible utility of the continuum description of electrostatic interactions in rationalizing the ordering trends of various solutes through a simultaneous classification of solute molecules according to their quadrupolar anisotropy, and of nematic solvents according to their efg, is described in Chapter 10 of this book.

4.3 Atomistic force-field models As discussed in the previous section, the description of the electrostatic interac

tions in terms of the leading moments of the molecular charge distributions, rather than directly in terms of the distributions themselves, could be seriously inaccurate at short intermolecular distances. Moreover, the classification of interactions into shapedetermining short-range repulsions and longer ranged electrostatic, although very useful in forming a coarse picture of the ordering mechanisms in common nematic fluids, is incomplete, to some extent arbitrary, and therefore artificial. It could also become misleading if attention is not paid to the fact that electrostatic interactions, although long ranged, are very strong at short distances. On the other hand, it has been stressed in previous sections that the various additive components of the molecular interactions combine in a non-additive way to produce the potential of mean torque and therefore the omission or misrepresentation of a particular component could influence the effect of the other components on the construction of the potential of mean torque. It thus becomes evident that to go beyond coarse simplified descriptions and special systems where such descriptions might be reasonably applicable, it is necessary to consider the dominant interactions in a more realistic context rather than in simplified representations or in isolation from one another or in isolation from other interactions with which they are inevitably imbedded in real molecules. Such integrated treatments [67] are becoming readily possible with molecular modeling packages providing detailed atomistic force fields, partial charge distributions and molecular conformation energetics. Extensive energy maps of fully interacting pairs of molecules can be thus constructed and used as complete intermolecular potentials for the calculation of the potential of mean torque. A very useful feature of such calculations stems from the


possibility of modifying the strength of the various components, or selectively turning them off completely, which allows assessments to be made of the effects that particular interactions may have on the potential of mean torque.

Calculations based on the above atomistic force field modeling have been recently carried out [68] on the orientational order of two solutes l,4-dichlorobenzene and 1,2-dichlorobenzene, nominally "apolar" and "polar" molecules, as measured by deuterium NMR. The solutes were dissolved in two nematic solvents, hexyl- and pentyloxy- substituted diphenyl diacetylenes, DPDA-C6 and DPDA-OC5, consisting respectively of "apolar" and "polar" calamitic mesogens of very similar shape. Atomistic descriptions of the solvent and solute molecules were used to compute the pair interaction potential for all solute / solvent separations and relative orientations. The molecules were assumed to adopt a single, lowest energy conformation with the pendant chains of the solvent nematogens in their all-trans state. The intermolecular potential was used to evaluate the potential of mean torque according to Eq. (12.30) with only second rank terms retained. The principal order parameter of the solvent was measured independently in this study and therefore the only external parameter for the evaluation of solute ordering is the effective density of each solvent at a single reference temperature.

The theoretical calculations reproduce the experimental results with remarkable accuracy. In addition to quantitative agreement, the basic features of the atomistic solvent / solute potential provide a clear rationalization of the qualitative experimental trends. The influence of the electrostatic contribution to the solute ordering is found to be small for all four solute / solvent combinations. Furthermore, these calculations indicate that the important interactions are operative over short intermolecular distances for which the description of the electrostatic component in terms of the moments of the partial charge distribution of the molecules is not valid. They also show explicitly that the relevant features of the solvent / solute energy maps are constructed from localized interactions that are sensitive to the mutual proximity of specific segments of the interacting molecules.

On a general level, these calculations show that coarse representations of probe / solvent interactions, and far more so continuum models, are not providing sufficient resolution for the comparative analysis of the orientational order in systems with subtle qualitative differences in molecular structure. Furthermore, the fact that the implementation of such atomistic calculations is feasible with more or less standard computational resources, at least for simple probe solutes and structurally not too complicated solvent nematogens, renders the coarse representations of molecular interactions and continuum treatments of such systems obsolete.

5. Summary. The orientational order parameters that can be measured by NMR are introduced and

discussed in relation to phase symmetry as well as to molecular symmetry, molecular structure and flexibility. A general formulation of the rigorous statistical mechanical connection of these order parameters to molecular interactions is then presented and


is used to identify and to assess the assumptions, approximations and limitations of various theoretical approaches that have been proposed for the interpretation of NMR measurements of orientational order in liquid crystal phases.

The main focus is on what can be learned about intermolecular interactions from the analysis of the measurable order parameters of relatively simple rigid probe molecules dissolved in nematic solvents. The potential of mean torque experienced by the probe molecule is the central theoretical tool both in phenomenological approaches, where its formulation constitutes the starting objective, as well as in explicit molecular approaches, where it represents the end product of the statistical sampling of the interactions acting on the probe molecule.

Starting from the rigorous formulation of the potential of mean torque, an approximate closed form relation between the potential of mean torque and the intermolecular pair potential is obtained. In this approximation, the interactions of the probe molecule with the molecules of the ordered fluid are treated exactly, whereas the correlations among the latter molecules are neglected and compensated for by rescaling the effective density of the fluid. The resulting form of the potential of mean torque is shown to generate, on introducing various additional approximations, the potentials of mean torque that correspond to the Onsager theory, the generalised Van der Waals theory, and the Maier-Saupe theory of nematics. Applications to the analysis of the orientational order in various systems of solute probes in nematic solvents using molecular models that range from purely shape-dominated, hard-body interactions, to fully-atomistic, complete force fields, are presented.

Two general points that are worth stressing emerge from the theoretical connection between order parameters and molecular interactions:

(i) Even a complete and exact analysis of very detailed measurements of order parameters by high field NMR would give no more than the second rank moments of the single-molecule orientation-conformation distribution, and these alone cannot lead to unique inferences on the underlying interactions.

(ii) The measured orientational order is generally produced by a highly non-linear confluence of different types of molecular interactions such that the contribution of one type depends on the others. Orientational order in common nematics is basically generated by molecular shape anisometry, as manifested by steric interactions, but cannot in general be fully described exclusively in terms of these interactions.

Acknowledgments Many of the ideas and results presented in this Chapter are products of my longtime

collaboration with Ed Samulski, Andreas Terzis and Alexandros Vanakaras.

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[50] Cladis, P.E. (1988), Mol. Cryst. Liq. Cryst., 165:85.

[51] Ylihautala, M., Ingman, P., Jokisaari, J., and Diehl, P. (1996), Appl. Spectrosc., 50:1435.

[52] See for example Demus, D. (1998), Handbook of Liquid Crystals, Vol.I, edited by D. Demus, J.w. Goodby, G.w. Gray, H.W. Spiess, and V. ViII, (Wiley-Vch, NY): page 169 and references therein.

[53] Allen, M.P. (1994), Mol. Phys., 81:263.

[54] Emerson, A.P., Hashim, R., and Luckhurst, G.R. (1992), Mol. Phys., 76:241.

[55] Celebre, G. (2001), J. Chem. Phys., 115:9552.

[56] Abe, A., and Fuyura, H. (1988), Mol. Cryst. Liq. Cryst., 159:99.

[57] Emsley, J.W., Heeks, S.K., Home, T.J., Howells, M.H., Moon, A.,Palke, W.E., Patel, S.U., Shilstone, G.N., and Smith, A. (1991), Liquid Crystals, 9:649.

[58] van der Est, AJ., Kok, M.Y., and Burnell, E.E. (1987), Mol. Phys., 60:397.

[59] Yim, C.T., and Gilson, D.F.R. (1991), J. Phys. Chem., 95:980.

[60] Syvitski R.T., and Burnell, E.E. (2000), J. Chem. Phys., 113:3452.

[61] Photinos, DJ., and Samulski, E.T. (1993), J. Chem. Phys., 98:10009.

[62] van Leeuwen, M.E., and Smit, B. (1993), Phys. Rev. Lett., 69:913.

[63] Weis, J.J., Levesque, D., and Zarragoicoechea, GJ. (1993), Phys. Rev. Lett., 69:913.

[64] Sear, R.P. (1996), Phys. Rev. Lett., 76:2310.

[65] Berardi, R., Ricci, M., and Zannoni, C. (2001), Chem. Phys. Chem., 2:443.

[66] Burnell, E.E., Berardi, R., Syvitski, R.T., and Zannoni, C. (2000), Chem. Phys. Lett., 331:455.

[67] Wilson, M.R. (1994), Mol. Phys., 81:675.

[68] Dingemans, T., Photinos, D.J., Samulski, E.T., Terzis, A.F., and Wutz, C. (2003), J. Chem. Phys., (in press).

Chapter 13

VERY FLEXIBLE SOLUTES: ALKYL CHAINS AND DERIVATIVES

Edward T. Samulski Department of Chemistry, University of North Carolina, Chapel Hill, NC, U.S.A.

1. Introduction A detailed understanding of intermolecular interactions in liquid crystals at the

atomistic level should enable us to understand how this delicate state of orientationaIly ordered, fluid matter manifests itself in such a wide variety of molecular structures. However, as emphasized in the preceding Chapter (Chapter 12), the information that can be extracted with nuclear magnetic resonance (NMR) is extremely limited: we are only able to take the most rudimentary steps towards characterizing the various types of contributions that comprise the liquid crystal's mean field - the anisotropic part of the motionally averaged intermolecular interactions that persist in fluids comprised of orientationaIly ordered molecules. In this Chapter we review approaches to this challenging problem in the physics of liquids by focusing on the way in which flexible solutes (probes that can access distinct conformations) are ordered in nematic liquid crystal solvents. This, in tum, can help characterize the anisotropic part of the intermolecular potential acting on each solute conformer. This potential is derived from the conformer's interactions with the orientationally ordered solvent molecules and it is designated the potential of mean torque, V(O).

By way of background, recall that V (0) experienced in both thermotropic and lyotropic liquid crystals is capable of "orienting" monatomic solutes: quadrupolar atoms such as 131 Xe exhibit an incompletely averaged electric field gradient (efg), presumably by biasing the orientation of transiently nonspherical collision complexes wherein the electronic distribution about the atom's nucleus is distorted [1]. High-symmetry tetrahedral solute molecules, e.g., tetramethylsilane, neopentane, etc., also exhibit orientationally biased NMR interactions in nematic solvents [2]. The magnitude of these interactions increases in smectic solvents with more ordered (and stratified) mesogens and a correspondingly stronger potential of mean torque [3]. In this latter stratified phase, solutes experience uniaxial potentials within the array of quasiparallel mesogens as well as that potential associated with the inter-layer constraints. Both quadrupolar and direct dipolar interactions are apparent in the NMR spectra of the small, nominally

285 E.E. Burnell and CA. de Lange (eds.), NMR of Ordered Liquids. 285-304. © 2003 Kluwer Academic Publishers.


"spherical" molecule methane and its isotopomers dissolved in nematic solvents [4, 5]. Clearly, the mean field exerts considerable influence on solutes and in turn, such solutes can be exploited to characterize the nature of the anisotropic environment in this delicate state of matter.

In this Chapter we focus on the behaviour of very flexible solutes in liquid crystals: alkanes and their derivatives. Flexible solutes are also dealt with in Chapter 14. In principle, such solutes have unique capability as probes of the mean field, because the inherent conformational flexibility allows them to conform their (average) shape to the constraints of the potential of mean torque. However, obtaining insight from such malleable reporter molecules demands a proper description of flexible solutes in the mean field. In this Chapter we review observations and progress on describing (substituted) alkane solutes dissolved in nematic solvents. The nomenclature and symbols are the same as those introduced in Chapter 12 wherein the focus is on general theoretical formulation as applied to rigid probe molecules; references to equations in Chapter 12 are indicated by Eq. (12.xx).

1.1 Averaging over intra- and intermolecular motions

Is solute intramolecular isomerization fast or slow relative to the global molecular libration I reorientation? Are the relative timescales of intra- and extra-molecular motions even relevant to an interpretation of the NMR interactions exhibited by flexible alkane-based solutes in nematic solvents? The lifetime of an alkane's conformational state (enumerated by n), Tn, is estimated to be of order 10-10 s [6]. The reorientational dynamics of mesogens and, therefore, solutes dissolved in mesophases, are characterized by correlation times, Trot, on the order of 10-11 s [7]. Hence, the reorientational dynamics and the isomerization dynamics are convolved. In order to begin to answer the second question it is essential to first recognize that Tn ~ Trot are both much smaller than the so-called NMR timescale, 1"NMR. This timescale is inversely proportional to the interaction magnitude of the corresponding anisotropic NMR tensorial interaction, h'Yi"'fj/41fr;j ~1.2x105 S-1 for dipolar couplings between a proton pair separated by 0.1 nm, and 61fe2qQ /2h ~2.5 x105 s-1 for the quadrupolar interaction in aliphatic deuterons. The fact that both isomerization and solute reorientation are very fast on the NMR timescale implies that "motionally averaged" NMR interactions are measured, i.e., a "motionally narrowed NMR spectrum" is observed for the solute. A more complete answer, one that impacts how we could begin to frame models of flexible solutes in liquid crystalline solvents, requires an understanding of what is really being averaged. Of course the averaging applies only to "fast motion"; for "slow motion" one has the usual static superposition of spectra.

The relevant average is over the motion of the segment that carries the nuclear spin(s) or the electric field gradient. However, it is impossible for that segment (or for an observer) to tell whether a particular displacement I reorientation it experiences over time is the result of an "internal" (conformational) or "external" (global molecular) motion. Of course one can always arbitrarily define a molecular frame fixed in some part of the molecule, and define all the motions relative to that frame as conformational and all

Very Flexible Solutes: Alkyl Chains and Derivatives 287

the motions of that frame relative to a macroscopic frame as global molecular, but such distinctions in general depend on the choice of the molecular frame. Alternatively, for motions that are fast on the timescale of the measurement, we can exploit an ensemble average with the assumption that the system is ergodic. Extensive discussions of the applicability and equivalence of dynamical and statistical mechanical averages can be found in the literature [8,9]. However, it should be emphasized that when we assume ergodicity (to replace the time average over the motion of that particular segment with an equilibrium ensemble average), we must be certain that we average over all configurational phase space available to that segment. One way of counting all the possible configurations correctly is by integrating over all possible orientations of a (in principle, arbitrarily defined) molecular frame, and then integrating (or summing) over all the possible configurations of that segment relative to that frame. The latter integration or summation, when weighted using the respective "internal" energy, is said to be "over all conformations" of the flexible molecule. Of course, the fact that these two integrations (or summations) are carried out separately does not imply that the motions are independent, or uncorrelated. The separation is viable because the ensemble average does not deal with motions. It employs weighted averages, and the weighting function, Gne-E':../kT e-Vn(fl)/kT, is determined by the potential of mean torque Vn(O) which includes the orientation-conformation correlations (see Eq. (12.9) in Chapter 12).

In summary, despite the fast isomerization we cannot speak about an average flexible molecule. Rather, every conformer orientation relative to the director, Dn , is employed

to define each conformer's average orientation via (S~1)n' the nth conformer's order

tensor in a molecule-fixed frame, and, in tum, each of the segment order tensors, sf!. It is the latter that ultimately determine the motionally averaged NMR interactions (see section 2.2 below and Chapter 12).

1.2 Rotational isomeric state approximation In alkane-based molecules uncertainties in the dihedral angle energy, E(¢), where

¢ is the dihedral angle specifying relative rotations about C-C single bonds, in conjunction with a desire to expedite computations of average macromolecular properties, prompted consideration of discrete conformers of flexible molecules. Such a conformation n is specified by the set of j dihedral angles ¢ associated with each of the j C-C bonds in the chain. Additionally, the conformer's intramolecular energy is approximated by a sum, E~ = Lj E(¢). Furthermore, the intramolecular energy used in Boltzmann-weighted averages is derived from finite sums over s allowed discrete "states" of the variable dihedral angle as opposed to integrals over the continuous variable. That is, in the rotational isomeric state (RIS) approximation [10] the following redefinition applies:

(A(¢)) = J d¢A(¢)e-E1P ~ (A(¢)) = LA(¢)e-E~t"). s

(13.1)


Underlying this RIS approximation is the assumption that each bond's continuous dihedral angle energy function E( <p) can be replaced by a small, discrete sample, E(<ps) (8 is typically three), associated with selected dihedral angles <Ps. In alkanes, the sample typically includes the trans state with <Pt = 00 and two gauche states located near the ideal values <Pg± = ±21T /3. A subtle and frequently overlooked aspect of the RIS approximation is the criterion for choosing the location of the dihedral-angle states <Ps. As emphasized by Flory, the <Ps states need to be chosen in a way that insures that the average predicted by the sum over states is equivalent to that using the continuous integral representation, and moreover, the choice of location of the sample states may vary with the quantity being averaged [11]. Irrespective of this explicitly stated prerequisite for RIS modeling, in practice the RIS average is frequently carried out by merely using the minima observed for E( <p) derived from molecular mechanics calculations.

For simple alkanes such as hexane, <Pg± is approximately ±113°, somewhat smaller than the ideal values of ± 1200 , and the energy (relative to the trans RIS) associated with the gauche state E(<pg) is approximately 2.5 kJ / mol [12]. There are schemes of varying sophistication for incorporating contributions from nearest (and next nearest) neighbour non-bonded interactions into the dihedral angle energy. Finally, realistic atomistic calculations on mesogens ("substituted alkanes") should include remote steric interactions within the chain as well as such interactions between the chain and the covalently attached mesogenic core. In summary, for a given conformation n, the intramolecular energy of the chain with j C-C bonds may be expressed by

E~ = 2: [EfA(n) + EfB(n)], j

(13.2)

where the E(n) terms are sums of the individual C-C bond dihedral-angle energy contributions and the associated non-bonded steric energy penalties, respectively. At this level of approximation, the conformer probability is

o exp[-E~/kTl Pn = L exp[-E~/kTl' (13.3)

n

and for isolated chains, conformation-dependent quantities (X (n)), averaged over all conformations, are given by

(X(n)) = 2:p~X(n). (13.4) n

In general the conformer probability of an isolated alkane, p~, is not the same as Pn, the probability for finding the nth conformer in the liquid crystalline phase, as Pn is enhanced by the nematic's mean field for elongated conformers and attenuated for globular conformers. This enhanced probability is key to modeling NMR interactions in liquid crystals.


2. Chronology of NMR studies of flexible solutes Soon after Saupe and Englert's paper [13], the utility ofNMR for solutes dissolved in

liquid crystals in structure determination was recognized. However, it rapidly became clear that the number of rigid solutes amenable to this technique was limited. Rigid solutes with a large number of spins exhibited hopelessly complex, single-quantum, high-resolution proton NMR spectra. The implications for flexible solutes such as alkanes were even more problematic. A normal alkane solute, Cj+2H2(j+2)+2' with j dihedral angles has, in the usual RIS approximation, nmax = j3 conformers; (methyl rotamers are not counted as distinct conformers). In order to describe the proton spectrum, the orientational order of each conformer must be specified with an order tensor, leading to a total of 5nmax parameters! For example, let us consider the proton NMR spectrum of the simplest alkane with distinct conformers, butane (nmax =3), dissolved in a nematic phase (see Chapter 5). Noting the symmetry relations between two of butane's RIS gauche conformers (g+ and g-), there are only two inequivalent ones (the trans and either the g+ or g-). The direct proton-proton dipolar couplings ofbutane in the nematic will depend on the average orientational order of these two distinct conformers, i.e., 10 order tensor elements. This number of unknowns is decreased to 6 if an appropriate choice of symmetry-based axes is used for the description of the order tensors. However, the motionally averaged proton NMR spectrum only exhibits 7 independent dipolar couplings. As a result, minimal structural information can be derived from the proton NMR of this flexible solute. Obviously, the situation rapidly deteriorates for larger alkanes where the number of unknown order tensor elements far exceeds the number of independent dipolar couplings. Because of this intrinsic limitation, only flexible molecules with a high degree of symmetry and a limited number of conformations could be addressed in the decade following Saupe and Englert's discovery. Nevertheless, important advances in the understanding of conformational barriers in symmetric molecules such as cyclohexane, cyclooctatetraene, etc. [14] were achieved with NMR studies in liquid crystalline (LC) phases (see Chapter 19).

Although deuterium NMR had been employed early on in the study of mesogen orientational order [15], in the 1960's deuterium labeling was primarily employed to simplify proton spectra. The small magnetic moment of deuterium markedly lowers the magnitude of deuterium - proton dipolar couplings. Deuterium decoupling further simplifies proton spectra of partially labeled solutes. However, this passive use of deuterium gave way to direct applications of this ideal, nonperturbative label. In particular, the "quadrupolar splittings" in deuterium NMR spectra of orientationally ordered, labeled molecules were ideally suited to elicit information about order and mobility in model biophysical systems and mesophases [16]. Before this application of deuterium NMR, contemporaneous studies on model membranes were dominated by Electron Spin Resonance (ESR) investigations of spin-labeled model and biological membranes. Despite the perturbing size of the nitroxide spin label, insight into membrane fluidity was achieved via ESR employing labels incorporated in lipid-like molecules [17]. Sampling at various depths in the membrane led to the concept of a gradient of mobility across the membrane with increasing "motion" on moving from


the polar head group (at the water-lipid interface) to the terminal methyl-rich bilayer centre. This concept was reinforced with deuterium NMR studies of labeled lipids and amphiphilic molecules. In fact, the universal appearance of the quadrupolar splitting profile in the membrane was termed the fluid bilayer signature. However, it was shown that this splitting profile is inherent to all substituted alkanes dissolved in uniaxial solvents. This applies to the profiles exhibited by labeled lipid solutes in a thermotropic nematic phase which is a continuum organic phase that does not possess a water-lipid interface to constrain the orientation of the lipid polar head group; this constraint was thought to be essential to the fluid bilayer signature [18].

As soon as labeled mesogens were available, deuterium NMR was used to characterize the average orientation of C-D bonds in the aliphatic "tails" of calamitic liquid crystals [19]. Both prolate calamitics [20,21] and oblate discotic mesogens [22] could be probed in this way and qualitative insight into tail ordering gave way to a frenzy of theoretical efforts to rationalize the observations in both classes, calami tics [23-25] and discotics [26, 27]. Fuel was added to the fire when it became apparent that both the quadrupolar splitting patterns of labeled alkane solutes [28-30] and the proton dipolar couplings [31-33] were amenable to analysis. More subtle aspects of the solvent's potential of mean torque, i.e. the role of electrostatic contributions, could be detected when polar (substituted) flexible alkanes were used to probe the mean field [34,35]. Then, with the growing interest in thermotropic polymeric liquid crystals of the alternating "core-spacer-core ... " variety, deuterium NMR was exploited to study the flexibility and orientational ordering of labeled spacer chains. Exaggerated odd-even effects in dimers [36] and polymers [37] had associated quadrupolar-splitting "signatures" for the respective labeled spacer chains, and these attributes could be rationalized with appropriate modeling. In the remainder of this Chapter we shall highlight aspects of models used to describe the behaviour of flexible alkyl chains in nematics.

2.1 C-D bond order parameter profiles In the late 1970's and early 1980's deuterium NMR studies were conducted on

both calamitic nematics and columnar discotic phases containing a deuterium-labeled terminal alkyl chain(s) attached to the core of a mesogen (typically an aromatic ring) via direct-, ether-, or ester-linkages. These studies focused on the profiles and apparent odd-even alternation in the magnitude of the observed quadrupolar splittings, Llv s, at successive segments (the s methylene units) in the mesogen's tail(s). Representative plots of experimental C-D bond order parameter profiles are shown in Fig. 13.1. There are qualitative differences in the profiles for the two types of mesogens, with the calamitics (solid circles) exhibiting a concave profile, while that of the discotics (open squares) is convex. The absolute value of the S~_D profiles are shown, and the lines connecting data points merely emphasize the profile trends. In calamitics, S~_Dis negative because on average the C-D bond is normal to the director, while for discotics S~_D is positive.

Initial efforts to simulate !lvs profiles in calamitics were guided by the concept that there is a unique "symmetry axis" in the (conformationally averaged) mesogen, viz.

Very Flexible Solutes: Alkyl Chains and Derivatives

0.3

0.2

0.1

• 60.7

D THE6

0.0 +-___ - __ -_-_-_-.._ 2 3 4 5 CD3

methylene segment

291

Figure 13.1. The absolute value of the deuterium NMR C-D order parameter profile versus segment number for labeled hexyloxy chains in calarnitic nematic and discotic columnar phases. Solid circles: p-hexyloxybenzylidene-p-heptylaniline-d13 (no.m). Open squares: hexyloxytriphenylene-d78 (THEn); n = 6andm = 7.

the "long molecular axis" L. As a result, the relationship between the observed D..vs and the local C-D bond order parameter, S~_D' was simply expressed by a product of second-order Legendre polynomials,

(13.5)

where () represents the instantaneous orientation of L with respect to the nematic director n, as denotes the (conformation-dependent) angle that the C-D bond makes with respect to L, and n is the orientation of n relative the spectrometer magnetic field, the Z-axis of the laboratory frame. The angular brackets 0 denote an average over mesogen isomerization and global mesogen reorientation. On decoupling the extraand the intramolecular averaging, one gets

(13.6)

When applied to a mesogen with a labeled, flexible tail these equations assume: (i) a single order parameter for the mesogen, SL, where L is the so-called "long molecular axis"; (ii) a locally uniaxial electric field gradient (efg) tensor with principal value q along the C-D bond in the 8th methylene segment which in a given conformation has the bond located at an angle as relative to L; and (iii) positive diamagnetic anisotropy with n parallel to the field (n = 0°). This separation of intra- and extramolecular


averaging fails when applied to flexible alkyl-based molecules. The independent computation of an intramolecular conformational average (P2(cosas )) does not consider the differential biasing of each conformation's ordering in the mean field, i.e. the conformer probability of an isolated alkane, P~' is not the same as Pn, the probability for finding the nth conformer in the liquid crystalline phase. Hence, application of Eq. (13.6) using the free-chain conformational average (Eq. (13.4)) predicts [18] that the conformationally averaged electric field gradient, q (P2 (cosas )), or equivalently, S~_D' diminishes very rapidly with increasing distance of the methylene unit from the mesogen's core, in contrast to the experimental data shown in Fig. 13.1. What is missing from this uncoupled segment description is the influence of the mean field on the flexible part of the mesogen. In a primitive description, the mean field constraint can be mimicked by exaggerating the lowest-energy conformer probabilities by, for example, artificially inflating E~ (by adjusting the trans - gauche energy difference) or equivalently, by drastically lowering the temperature (to", 50 K) on application of Eq. (13.6). However, the necessity for such ad hoc enhancements of certain moreextended chain conformers, i.e. penalizing conformers having gauche states with an artificially low Boltzmann factor, is merely a graphic indicator of the deficiency of Eq. (13.5) as applied to conformationally flexible molecules.

2.2 Early mean-field models Comparisons between experimental and calculated S~_D values made it obvious

that additional extramolecular constraints needed to be incorporated into models in order to reproduce observed experimental C-D bond order profiles. In the earliest attempts to include such considerations, the starting point was the seminal paper by Mareelja [38]. In this paper he used the RIS approximation, and explicitly included the extramolecular energetic contributions associated with having the mesogen's alkyl C-C bonds in the nematic mean field to calculate conformer probabilities. The selfconsistent statistical findings of Mareelja included the replication of the well-known odd-even oscillations in thermodynamic properties exhibited by homologous series of nematogens. The intrinsic conformer probability P~ is modified by each conformer's extramolecular energy derived from intermolecular interactions between the conformer and the nematic solvent. In the "independent segment" approach of Mareelja that was used in the early models of SC-D profiles, an additive potential of mean torque acting on the mesogen's s segments was employed

(13.7) s

where l/~l)(!1s) denotes a single-segment interaction contribution associated with the sth segment having orientation !1s when the molecule is in conformation n. This corresponds to the first term in a more general expression of the potential of mean torque given in Chapter 12 byEq. (12.14). The directions !1s ofthemesogen tail's C-C bonds in each of its n conformations is very straightforward to derive. The distribution


function, in(O), and conformer probability, Pn, (Eqs. (12.9) and (12.6)), enable one to derive the segmental order parameters s,:/ (Eq. (12.5)). The associated bond-order parameters S~_Dcan be obtained as follows. Consider a local 1,2,3 frame fixed to a methylene segment (the 1-axis bisects the D-C-D valence angle 2(3, the 3-axis is normal to the D-C-D plane, and the 2-axis completes a right-handed Cartesian frame). This frame is independent of chain conformation and is a PAS frame for the segmental order tensor sjf. Defining the electric field gradient tensor for a methylene deuteron in its a, b, c PAS (on the C-D bond with the b-axis along the bond and the a-axis parallel to the 3-axis), the relationship between the bond order parameter, quadrupolar splitting, and segmental order tensor is given by

Sb-D 2t::. lIs /3qbb

SrlZ cos2 (3 + srl sin2 (3 (13.8)

+ 1]( sfl - Srlz sin2 (3 + Sr2Z cos2 (3) /3,

where qbb is the principal value of the quadrupolar interaction tensor, and 1] = (qaa -qcc)/qbb, is its asymmetry parameter. For methylene deuterons, 1] ~ 0 and qbb ~ 163 kHz [39].

A variety of approximate schemes for parameterizing Vn(O) was pursued, but a limiting aspect of all these early single-segment interaction schemes was a belatedly recognized flaw: the single-segment couplings depend only on the orientation 0/ each segment relative to the director; independent o/the orientations o/neighbouring segments. As a result, very different conformations of the chain with vastly different global shapes are not distinguished in the mean field, i.e. they have identical angular dependent contributions (Vj(O) = Vk(O)). This is graphically obvious by considering

the idealized mesogens where the relevant lI~l)(ns) coupling involves the tail bond segments. In Fig. 13.2 idealized models of the calamitic mesogen 8CB show that, for independent bonds, both of the conformers have the same Vn(O), and consequently Pn will be the same for both; (in both conformers, the bonds along the director are bold-faced). This flaw also plagues more elaborate tensorial interaction models if the tensor relies on a sum of bond contributions from each conformer, e.g., the electric polarizability tensor [40].

3. Models of flexible molecules 3.1 Independent bond model

A representative illustration of the so-called independent segment or bond models considers that a mesogen with an alkyl tail in conformation n will experience an additional energetic contribution from the disposition of its tail's C-C bonds in the nematic mean field (Fig. 13.2). In its simplest form (uniaxial mesogenic core and bonds), the potential of mean torque may be expanded in even-order Legendre polynomials


Figure /3.2. Schematic diagrams of the 8eB mesogen in two different conformations that exhibit the same angular-dependent energy in the mean field using the independent-bond model. The tail bonds parallel to the director (vertical axis) are bold-faced.

(truncated at rank 2) and represented by

(13.9) s

where Ocore and Os are the orientations of the core segment and the 8 th C-C bond segment relative to the nematic director. The Os will depend on conformation n via the conformer's internal geometry, and on the global orientation of the mesogen's molecule-fixed x, y, z frame. With zilL and an assumed uniaxial core, SL is the single, dominant element of the order tensor, denoted (S;J)n in Chapter 12. The parameters /lcore and /lbond describe the strength of the coupling between the mesogenic core and the bond segments, respectively, and the mean field. These coupling parameters are selected to optimize agreement between experimental and calculated (conformationally averaged) bond order parameters, S~_D. The latter are computed using the conformer probabilities in the liquid crystal, Pn, through the probability distribution, fn(rl) which in tum is expressed in terms of the parameterized potential, Vn(rl) (see Eqs. (12.6) - (12.8». In other words, S~_D is obtained from each of the

computed (conformationally averaged) segmental order parameters 8 ~; s~ describes the average orientational order of the local methylene-fixed 1,2,3 frame relative to the director (Eq. (12.8) with 0: = (3 = Z),

DI./3 2", IJ( ) (SDI./3) 8ij ="3 LJPn8ij n IJ n· (13.10)

n


Armed with this idealized model for Vn(O), researchers focused on accounting for the general shape of plots of observed S~_D versus segment number s for calami tics of the type shown in Fig. 13.1. Specific labeling [41] and spin-echo-refocusing decoupling of core protons [42] can be used to confirm the assignments of the alkyl tail's b.lIs . The challenge in the 1980's was to merely reproduce the S~_D versus s profile, including its odd-even alternation pattern. Coarse agreement between calculations and observations for both calamitic and discotic mesogens could be achieved with the independent bond model. However, these fits were not very sensitive to the details associated with the ad hoc formulations of Vn(O), including refinements such as including the biaxiality of the mesogenic core and the independent segments.

3.2 Inertial frame model The inertial frame (IF) model, introduced in 1980 [27], probably does not warrant

the designation "model." It was originally suggested in an effort to remove the arbitrariness associated with a single order parameter (SL) description of deuterium NMR data (~lIs) in labeled calamitics. Initially, practioners ignored the extreme sensitivity of calculated S~_D profiles to the location of the ill-defined "long molecular axis" L, in the (assumed uniaxial) mesogen's molecular framework. Typically, L was taken to be along the bond connecting the alkyl tail to the mesogen's core, usually along the core aromatic rings' para-axis. However, in the early 1980s practitioners understood that the modulation of the odd-even alternation in the calculated S~_D profiles was very sensitive to the exact definition of L. Insofar as molecular shape might justify the concept of a long molecular axis, one had to question the conformational independence of L in the early independent-segment models. The IF model emerged from efforts to develop an algorithm to define a conformation dependent L( n). An attempt to remove the arbitrariness from the specification of L was achieved by identifying L(n) with the minor principal moment of inertia of each conformation. While there were some reports in the literature suggesting that the PAS of the order tensor of rigid solutes coincided with the PAS of the solute's inertia tensors [43], the proposed algorithm for identifying a conformation-dependent L(n) was nothing more than a facile computational scheme for coarsely characterizing a molecular shape-related long molecular axis.

The ultimate value of the IF model resided in the inadvertent realization that such a scheme could be used to explore the behaviour of perhaps the most flexible probe solutes of all, the previously unexamined n-alkanes [28]. In fact, in the case of alkanes the results of the algorithm are independent of which tensor is employed to define L( n), because of the near-equivalence of the methylene and methyl masses, polarizabilities, and Van der Waals radii; essentially the same results are obtained irrespective of whether or not L( n) is identified with the PAS of the inertia, polarizability, or "steric" tensors. The latter tensor can be constructed by using Van der Waals radii instead of masses in the inertia tensor [44].

Identifying L(n), a "symmetry axis" of each conformer, in conjunction with the uniaxial mean field in the nematic solvent, enabled one to construct shape-dependent


order tensors that are diagonal in the PAS of the inertia tensor of each conformer. The principal value (Sll (ai))n was simply scaled by ai, the magnitude of the semi-axis of the inertia tensor [28]. Moreover, rather than evaluating Vn(O) explicitly, a simple scheme for affecting a mean-field type constraint was based on an entirely intramolecular criterion. Conformations containing atoms that exceed a critical distance r c from L, i.e., conformations that penetrate a cylinder of radius r c circumscribed about L, were penalized (rejected); (a related empirical scheme for biasing flexible solute conformer probabilities in nematic solvents considers the "elastic restoring force" associated with the projection of the solute's circumference onto a plane normal to the director [45]). This rejection mechanism was tantamount to parameterizing the conformer probabilities to affect the transformation from the set of ideal probabilities P~ of an isolated alkane, to a set Pn(rc) that coarsely approximated the influence of the solvent mean field. A preference for more elongated conformers was found. In the computations, r c was optimized iteratively to give the best agreement between experimental and calculated S~_D values using a modification ofEq. (13.10) or (12.8),

stZ = ~ L: s£L(n) (Sll(ai))n' Pn(rc}

(13.11)

Despite the crude approximations of the influence of the mean field on flexible molecules, the IF model could reproduce the observed S~_D profiles in calami tics [22], and it anticipated the qualitatively different S~_D profiles observed for labeled chains in columnar discotic mesophases (see Fig. 13.1) [25].

3.3 Modular models The IF model provided for the first time a prescription for calculating the NMR

spectra of completely flexible solutes such as n-alkanes dissolved in a nematic phase. The IF algorithm locates a solute fixed frame in any arbitrarily shaped conformation of a completely flexible solute, i.e., it "rigorously" defines L(n), and approximates the influence of the mean field on the conformer probabilities. This was an exciting prospect as flexible solute probes readily acquiesce to the constraints imposed on them by the anisotropic solvent, and help quantify the dominant intermolecular interactions operative in the potential of mean torque. However, before new insight could be realized, a more physically realistic model of a flexible solute was required, in particular a model that avoided a major shortcoming of all prior, segment-wise additive constructions of Vn(O). As Eq. (13.7) only depends on the orientation of each C-C bond relative to the director separately without regard to orientational correlations between nearest, next-nearest, etc. bonds, conformations with vastly different global shapes exhibit the same angular-dependent extramolecular energy in the mean field. The bond-additive potential (Eq. (13.7)) does not discriminate between most alkane conformers. A more general formulation that considers correlations between segments is required to describe accurately Vn(O), e.g., see Eq. (12.14).

In 1990 Photinos, Samulski and Toriumi recognized this flaw in all extant models and embarked on a rigorous, self-consistent description of flexible alkanes in a nematic


/,.: ~ .'. h: / ti "" /.: .. ""...... ...... bN

• • • D 2 N-I

.'. . . ..

~I ~s • : Cl •••• e M- l

~A,) ---+ 7 ..... . .... ~" • • • • o 2 N-l

Figure 13.3. Different representations of alkane segments: Bond representation (top); Atom-Block representation (middle); Chord representation (bottom).

mean field [29,30]. By regarding the flexible solute as a collection of rigid submolecular segments that assume different relative positions in each conformation n, the potential of mean torque Vn(O) could be formulated in a modular fashion by summing the tensor contributions that account for the orientation of such rigid subunits. As long as the local subunit is fixed for all n, this subunit may correspond to atoms, groups of atoms, bond vectors, etc. Moreover, they pointed out that a rigorous expansion of the potential must preserve the equivalence of the representation, atoms, groups of atoms, bond vectors, etc. Additionally, the subunit definitions should lend themselves to a physically sensible and computationally reasonable truncation of the expansion of Vn(O). In tum, the truncated expansion of the potential of mean torque will assume different forms depending on the representation [46]. Some forms enable one to readily infer the physical significance of the subunits and the interactions that cause alignment of solutes in mesophases. Others are more obscure. The dominant excluded volume interactions can be very successfully described when the subunits of alkanes are taken to be slightly biaxial atom-block segments (methylene units). The atom-block representation readily reduces to the chord segment representation, i.e. vectors connecting successive bonds in the chain, in the limit of vanishing atom-block biaxiality. We start with three mathematically equivalent representations of Vn(n).

3.3.1 Bond representation. Ignoring the small differences between methylene and methyl united atoms, consider an alkane with N + 1 atoms labeled by i = 0, 1, 2, 3 ... N. The configuration of such a solute in a nematic is completely specified by the set of N bond vectors b i that connect successive atoms in the chain (See


Fig. 13.3). Photinos et al. showed that the most general form of the potential of mean torque (accurate to tensor rank 2) is given by

N-IN-m

V(n)n = - L L wmP(bi,bi+m), (13.12) m=O i=1

where P(bi, bi+m) = (3cosOicosoi+m- b i ·bi+m)/2 and Oi is the angle between the ith

bond and the nematic director. Moreover, they showed that within the experimental uncertainties of NMR measurement of quadrupolar and dipolar interactions, a sufficiently accurate form of the bond representation of alkane solutes is obtained with only two terms weighted by the associated coupling constants Wo and WI :

N N-I

V(n)n = -Wo LP(bi, bi) - WI L P(bi, bi+1). (13.13) i=1 i=1

This truncated representation of the rigorous expression in Eq. (13.12) very adequately describes NMR data. Higher terms in the expansion give fits that exceed the uncertainties in experimental data. However, good agreement between calculation and experiment obtained with V(n)n given by Eq. (13.12) does not imply that the orientation and conformer probabilities of the alkyl chain in the nematic are driven by the alignment of individual bond segments. Equally good agreement between experimental and calculated S~_D can be achieved with mathematically equivalent subdivisions of the chain into a different set of rigid modules.

3.3.2 Atom block representation. An alkane chain may be conveniently divided into N - 1 blocks consisting of the methylene carbons, each with its local Cartesian frame (Xi ,yi ,Zi axes; see Fig. 13.3). By differentiating the frames on methyl atom blocks (0 and N) from the methylene ones, Eq. (13.13) can be transformed readily into the atom block representation [47]:

V(n)n = - ~wo[P(zO, ZO) + p(zN, zN)] N-I

- L {(A + B)p(Zi, Zi) + B[P(xi, Xi) - p(yi, yi)]} (13.14)

i=1

with A = sin2 (OB/2) (WO + WI)' and B = cos2 ((}B/2) (WI - wo)/2, where (}B is the C-C-C valence angle. This representation is mathematically equivalent to that in Eq. (13.13). The fact that a single summation is present in Eq. (13.14) means that the atom block representation subsumes nearest-neighbour bond-correlation terms and therefore is equivalent to a simple, segment-wise additive scheme. Agreement between experimental and calculated S~_D profiles using this representation suggests that the methylene atom block is only slightly biaxial (ry = B /(A + B) ~ -0.03) and is indicative of the dominance of steric considerations in chain ordering wherein the chain's global contour is aligned along the director to minimize excluded volume interactions with the solvent molecules.


0.3 r-------------------,

0.1

2 3 4 5 6 7 8

Figure 13.4. The application of the independent-bond (thin line) and the chord model (thick line) to the quadrupolar splitting profile of the octyloxycyanobiphenyl-d17 mesogen. (After [49]).

3.3.3 Chord representation. In the limit of vanishing biaxiality of the atom block (wo = WI)' a representation ofVn(O) is obtained in terms of only the Zi vectors, and vectors that span the mid-points of successive bond vectors (see Fig. 13.3). We refer to the latter vectors as "chords" and denote them by ci . The translation of Eq. (13.13) to the chord representation yields

This representation requires only a single coupling parameter, 2wo, expressing the strength of the (uniaxial) chord's interaction with the mean field. The order tensors and quadrupolar splitting profiles derived from alignment stemming from these independent chords are quite different from those resulting from alignment in the independent-bond model (the second term in Eq. (13.8». The formal equivalence of these two expressions does not correspond to similar physical interactions, but this important difference appears to have been lost in subsequent comparisons of the chord model with other models [48,49]. The chord model contains nearest neighbour bond correlations and helps to distinguish the energy of isomers such as those shown in Fig. 13.2. In cases where the orientational ordering of the chains is dominated by the ordering of the mesogenic core, the distinction between the chord model and the independent bond model is subtle, but nevertheless evident (Fig. 13.4).

The chord model is not an extension of the independent-bond additive potential model. It is a contraction of the exact (to rank 2) representation in Eq. (13.12). In an effort to emphasize this important distinction, consider rigid solutes in nematics. Although the orientational order of rigid molecules can be handled in many ways (see Chapter 12), perhaps the important difference between the independent-bond and

300

Bond

©r I I \. ,

Tolu'-'lnt

Oord _. ~ v. .C>

©© ()() Na""I"I""~


©©© ( :··X .. X)

Figure 13.5. Independent-bond and chord representations of aromatic solutes; on the right, the chords are illustrated by the bold lines connecting the midpoints of all of the C-C bonds.

chord models can be more readily appreciated by applying both models to a set of rigid aromatic solutes (Fig. 13.5). In the case of benzene, obviously there is no difference between the independent-bond representation and the chord representation. However, when the ring is substituted or fused to give more extended structures, the number and orientation in the two different representations change and the chord representation more faithfully represents the spatial dimensions of the more elongated solutes. This is reflected in the angular dependence of V (0) and in the simulated order tensors SZZ - J SZZ(O)e-VCf2 )/kT dO/ J e-VCf2 )/kT dO where Z is the orientation of the IJ - IJ ' nematic director in the laboratory frame (magnetic field direction), and IJ=x,y,x are the solute-fixed frames; (the y-axis is normal to the aromatic ring planes and, because of the simple solute symmetries, these are PAS frames) . There is an obvious qualitative improvement in the simulation of the experimentally determined order tensors in this class of rigid solutes when the modular, one-parameter chord model is used instead of the independent bond model (Fig. 13.6). Despite this superior representation of rigid solutes, it should be emphasized that the utility of the chord model is its ability to handle flexible alkyl-containing molecules with many internal conformations.

It may be of interest to note that the concept of a chord being the primary interaction entity in mesogenic alkyl chains was introduced in the early 1970's. In an attempt to keep the simplicity of his independent bond model [38] and simultaneously force lipid chains to extend normal to the water / lipid interface, Mareelja chose chords to construct Vn(O) without recognizing the full implications of his inadvertent introduction of correlations between nearest neighbour bonds with that representation [50].

The chord model has been tested against very restrictive experimental data [32] and found successful, not only in being quantitatively accurate, but also in doing so with


Figure 13.6. Order tensors detennined from deuterium NMR for the aromatic solutes shown in Fig. 13.5. The independent-bond model is unable to represent the shapes of the fused aromatic ring solutes (top), while the chord model shows qualitatively better agreement with experiment. The magnitudes of the bond coupling constants are 0 .295 and 0.350 for aromatic and aliphatic C-C bonds, respectively; the coupling constants for aromatic chords and the aromatic-aliphatic chords are 0.295 and 0.303. (Photinos, D. J., Sarnulski, E. T., Toriumi, H. (1992), unpublished work).

standard input for the conformer energetics [32,47]. Furthermore, the basic analytic structure and assumptions in the chord model's realization of the modular formulation are supported by calculations [51] using the cluster expansion method (see Chapter 12, section 3) for short n-alkanes (hexane to decane) interacting with hard-rod nematic solvents. When comparing the predictions of the modular formulation with those of molecular theories or computer simulations, it should be noted that the values of the coupling constants Wm depend on the assumed form of the interaction of the chain solute with the solvent molecules. This dependence does not refer only to the numerical values of the leading constants Wo, Wi, but also to how rapidly the coupling constants for more distant neighbours die out, the adequacy of only second rank terms, etc. While the modular formulations can accommodate forms of Vn(n) for very different interactions, it cannot be expected that a particular form, such as the simple chordmodel representation used to analyze experiments on alkanes in common nematics, can be used for the description of solute-solvent systems of any conceivable interaction or conformational structure. For example, comparisons of the predictions of the simple chord model with the results of simulations [48,49] wherein a hexane molecule is fixed at conformations corresponding to arbitrary values of the central torsion angle, while


interacting with a solvent consisting of a linear array of four Lennard-Jones centres, are not meaningful. Furthermore, the comparisons attempted with such systems in the smectic phase [48] are not meaningful, since the chord potential in Eq. (13.15) explicitly refers to positionally disordered fluids, such as nematics. An alkane molecule dissolved in a stratified medium, such as a smectic liquid crystal, would not necessarily sample all positions across the strata with equal probability, and therefore the potential of mean torque should convey the respective positional dependence [52].

3.4 Other models

3.4.1 End-to-end vector. Attempts to describe the averaging of flexible molecules in liquid crystals have also considered parameterizing the global attributes of each conformer in the construction of Vn(O), e.g., the molecular polarizability. One such global characteristic of a flexible alkane is a conformer's end-to-end vector, R= I: b i , a sum of the chain's bond vectors. This vector has been used to model quadrupolar interactions in flexible solutes [53]. This sum can be inserted into Eq. (13.7) with

l/~l)(Os)= -WBP2(COS()B) to give an expression for Vn(O). This results in equal magnitudes for all of the coupling constants in the analogue of Eq. (13.12), which is at variance with the experimental findings. Clearly, the global representation Rn is too crude to use in the parameterization of Vn(O) for the case of flexible molecules in nematics [46].

3.4.2 Maximum entropy. The maximum entropy analysis of incompletely averaged NMR interactions, proposed by Zannoni [54], has been applied to flexible alkanes [55]. In its minimally constrained format, this methodology uses NMR data to infer the nature of the intramolecular dihedral angle energetics. In the case of alkanes, it could shed light on the applicability and limitations of the RIS approximate enumeration of conformers for computing conformationally averaged quantities. With the inclusion of constraints associated with the RlS description, the maximum entropy analysis enables one to infer the magnitude of the mean field's influence on the conformer probabilities, i.e., the transformation P~ -+ Pn. However, it would appear that the utility of this methodology is strongly dependent on the size of the data set, and it will be most useful in conjunction with other techniques [56].

3.4.3 Surface tensor model. In this model Vn(O) is obtained from the surface enclosing the shape of the conformers [57]. It is outlined in Chapter 12 and described in more detail in Chapter 11. When applied to chain molecules it reduces, as is pointed out in Chapter 12, to a bond-additive model of the generic type presented in section 3.3. The coupling constants are determined by the details of the shape representation. Thus, it appears that whatever success or failure this model has met with [48,56] is traced back to its reduction to a special case of an independent bond additive functional form.


4. Conclusions Flexible solute probes will continue to play an important role in refining the nature

of the mean field in liquid crystals. Judiciously chosen flexible solutes, e.g., the a, wdihalogenated alkanes [34], can help in quantifying the relative importance of excluded volume and electrostatic contributions to Vn(fl). More substantive a, w-substitutions, e.g., aromatic rings and mesogenic cores, emphasize the importance of excluded volume interactions in equilibrating the disparity between the coupling strengths of the chain modules and the core modules [35]. In tum, proper segment representations of flexible chains enable one to extend these ideas to self-consistent theories of polymeric liquid crystals [37]. It goes without saying that the original motive for studying solutes in liquid crystalline solvents, viz. solute structure determination, remains a viable goal even in the case of completely flexible solutes. For example, Vanakaras and Photinos have shown that the internal electrostatic contributions in ethylene oxide chains are manifested in the direct dipolar and quadrupolar interactions exhibited by such solutes [51].

Finally, the chord model has been used to gain valuable insight into the origin ofthe spontaneous polarization in ferroelectric liquid crystals (FLCs). In conjunction with stratification constraints imposed on Vn(fl), it is possible with this model to determine which subunits in the mesogen contribute to the configurationally averaged transverse polarization in chiral FLCs [52].

Acknowledgments

My liquid crystal research program is supported by NSF (Grant DMR-9971143)

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[4] Loewenstein, A. (1976), Chem. Phys. Lett., 38:543. [5] Snijders, J.G., de Lange, C.A., and Burnell, E.E. (1983), Israel J. Chem., 23:269. [6] Levine, Y.K., Birdsall, NJ.M., Lee, A.G., Metcalfe,1.C., Partington, P., and Roberts, G.c.K. (1974),

J. Chem. Phys., 60:2890. [7] Janik, J.A, Krawczyk, J., Janik, 1.M., and Otnes, K. (1979), J. de Physique, Colloque C3, 4D:169.

[8] Emsley, J.W, Khoo, S.K., and Luckhurst, G.R. (1979), Mol. Phys., 37:959. [9] Burnell, E.E., de Lange, C.A., and Mouritsen, O.G. (1982), J. Magn. Res., 50:188.

[10] Flory, PJ. Statistical Mechanics o/Chain Molecules. Wiley Interscience, New York, 1969.

[11] Flory, PJ. (1974), Macromolecules, 7:381. [12] Photinos, DJ., Samulski, E.T., and Terzis, A.F. (1992), J. Phys. Chem., 96:6979. [13] Saupe, A., and Englert, G. (1963), Phys. Rev. Lett., 11:462. [14] Luz, Z. (1983), Israel J. Chem., 23:305. [15] Rowell, J.C., Phillips, WD., Melby, L.R., and Panar, M. (1965), J. Chem Phys., 43:3442. [16] Seelig, 1. (1977), Q. Rev. Biophys., 10:353. [17] Griffith, O.H., and Jost, P.c. Spin Labeling Theory and Applications, L. 1. Berliner, Ed., chapter 12.

Academic Press, New York, 1976.


[18] Samulski, E.T. (1983), Israel J. Chem., 23:329. [19] Hsi, S., Zimmermann, H., and Luz, Z. (1978), J. Chem. Phys., 69:4176. [20] Boden, N., Bushby, RJ., and Clark, L.D. (1979), Mol. Phys., 38:1683. [21] Bos, P.J., Shetty, A., Doane, J.w., and Neubert, M.E. (1980), J. Chem Phys., 73:733. [22] Goldfarb, D., Luz, Z., and Zimmermann, H. (1983), Israel J. Chem., 23:341. [23] Samulski, E.T., and Dong, RY. (1982), J. Chem. Phys., 77:5090. [24] Emsley, J.w., Luckhurst, G.R, and Stockley, C.P. (1982), Proc. Roy. Soc. London, SeT. A, 381:117. [25] Photinos, DJ., Samulski, E.T., and Toriumi, H. (1991), J. Chem. Phys., 94:2758. [26] Samulski, E.T., and Toriumi, H. (1983), J. Chem. Phys., 79:5194. [27] Photinos, D.J., Luz, Z., Zimmermann, H., and Samuiski, E.T. (1993),1. Am. Chem. Soc., 115: 10895. [28] Samulski, E.T. (1980), Ferroelectrics, 30:83. [29] Photinos, DJ., Samulski, E.T., and Toriumi, H. (1990), J. Phys. Chem., 94:4688. [30] Photinos, DJ., Samulski, E.T., and Toriumi, H. (1990), J. Phys. Chem., 94:4694. [31] Rosen, M.E., Rucker, S.P., Schmidt, C., and Pines, A. (1993), J. Phys. Chem., 97:3858. [32] Photinos, DJ., Janik, B., Samulski, E.T., Terzis, A.E, and Toriumi, H. (1991), Mol. Phys., 72:333. [33] Photinos, DJ., Samulski, E.T., and Terzis, A.E (1992), J. Phys. Chem., 96:6979. [34] Photinos, D.J., Poon, C.-D., Samulski, E.T., and Toriurni, H. (1992), J. Phys. Chem., 96:8176. [35] Photinos, DJ., and Samulski, E.T. (1993), J. Chem. Phys., 98: 10009. [36] Photinos, DJ., Samulski, E.T., and Toriurni, H. (1992), J. Chem. Soc. Faraday Trans., 88:1875 and

references cited therein. [37] Serpi, H.S., and Photinos, DJ. (1996), J. Chem. Phys., 105:1718. [38] Marcelja, S. (1974), J. Chem Phys., 60:3599. [39] Greenfield, M.S., Void, R.L., and Void, R.R. (1985), 1. Chem. Phys., 83:1440. [40] Janik, B., Samulski, E.T., and Toriumi, H. (1987),1. Phys. Chem., 91:1842. [41] Boden, N., Bushby, RJ., Clark, L.D., Emsley, J.W., Luckhurst, G.R, and Stockley, c.P. (1981), J.

Chem. Soc. Perkin Trans., 2:928. [42] Emsley, lW., and Turner, D.L. (1981), J. Chem. Soc. Faraday Trans. 2, 77:1493. [43] Anderson, I.M. (1971), J. Magn. Reson., 4:231. [44] Catalano, D., Forte, C., Veracini, C.A., and Zannoni, C. (1983), Israel J. Chem., 23:283. [45] Hoatson, G.L., Bailey, A.L., van der Est, AJ., Bates, G.S., and Burnell, E.E. (1988), Liq.Cryst.,

3:683. [46] Photinos, D.J., Samulski, E.T., and Toriurni, H. (1991), Mol. Cryst. Liq. Cryst., 204:161. [47] Luzar, M., Rosen, M.E., and Caldarelli, S. (1996), J. Phys. Chem., 100:5098. [48] Alejandre, J., Emsley, J.w., and TiIdesley, DJ. (1994), J.Chem. Phys.,101:7027. [49] La Penna, J., Foord, E.K., Emsley, J.w., and Tildesley, DJ. (1995), J. Chem. Phys.,I04:233. [50] Marcelja, S. (1974), Biochim. Biophys. Acta, 367:166. [51] Vanakaras, A.G., and Photinos, DJ. (1995), Mol. Cryst. Liq. Cryst., 262:463. [52] Terzis, A.E, Photinos, DJ., and Samulski, E.T. (1997), J. Chem. Phys., 107:4061. [53] Abe, A., Fuyura, H. (1988), Mol. Cryst. Liq. Cryst., 159:99. [54] Zanonni, C. Nuclear Magnetic Resonance of Liquid Crystals, J.W. Emsley, Ed., chapter 2. Reidel,

Dordrecht, 1985. [55] Berardi, R., Spinozzi, E, and Zanonni, C. (1998), Chem. Phys. Lett., 260:3742. [56] Berardi, R, Spinozzi, E, and Zanonni, C. (1996), J. Chem. Phys., 109:633. [57] Ferrarini, A., Moro, GJ, Nordio, P.L., and Luckhurst, G.R (1992), Mol. Phys., 77:1.

Chapter 14

NMR STUDIES OF SOLUTES IN LIQUID CRYSTALS: SMALL FLEXIBLE MOLECULES

Giorgio Celebre and Marcello Longeri Universitd della Calabria,

Dipartimento di Chimica, Rende, Italy

1. Introduction The study of the conformational distribution of "flexible" molecules (i.e., molecules

displaying large-amplitude, low-frequency torsional motions) in the liquid phase represents a fundamental subject, from the point of view of both theory (e.g., molecular modeling) and applications (e.g., therapeutic properties). In this context, NMR spectroscopy of molecules dissolved in liquid-crystalline solvents is unique, since no other technique can give such a large amount of experimental information.

Unfortunately, as is the case for all existing spectroscopic techniques applied to flexible molecules, the NMR technique is marred by the difficulties of obtaining unambiguous results from averaged experimental data. Hence, it is not surprising that conflicting results about the conformational equilibria of a given molecule abound in the literature [1]. Often, the extent of experimental data, i.e., very limited sets of observables that in many cases are only indirectly related to the desired molecular properties, is insufficient to extract reliable conformational information. In such cases "reasonable" assumptions (whose validity can sometimes be questioned) must be adopted.

Generally, an attitude of "wishful thinking" seems to prevail. The smaller the data set obtainable from a given experimental technique that would only allow for testing of the simplest of hypotheses, the higher the tendency to accept as reliable the simplest conformational behaviour that is consistent with the limited data. In other words, the simple picture obtained from limited data sets tends to obscure the complex nature of the problem, thus leading to a false sense of confidence. Many previous 1 H NMR studies, in which only very simple spin systems could be analyzed, suffer from these limitations. The same limitations affect recent 2H NMR spectroscopic studies of the order and conformational equilibria of meso genic molecules. In fact, 2H NMR spectra only provide a single quadrupolar splitting for each deuterated position. In a few cases (mainly limited to aromatic deuterons) 2H _2H dipolar couplings can also be measured.



Usually, due to the small size of these dipolar couplings (the deuteron gyromagnetic ratio is just 1/6 of the proton value) and to the large line widths (a consequence of the deuterium quadrupolar relaxation), the fine structure of the quadrupolar doublets is difficult to observe [2]. Most other techniques, for example isotropic phase NMR, give even more limited information. Moreover, proton NMR cannot be used in general to study mesogenic molecules themselves because of the complexity of the reSUlting spectra. The nature of the phase (solid, liquid or gaseous) and the bulk properties of the solvents (in the case of solutions) both influence internal rotation potentials. These considerations add further complexity and indeterminacy to an already very complicated problem.

2. Theoretical background Molecules are far from being rigid objects. They undergo a considerable number

of internal motions on a time scale ranging from "fast", small-amplitude vibrational motions (in the 10-12 s range) to large-amplitude conformational motions, typically internal rotations or ring puckering. The latter motions span time scales ranging from 10-9 s to 10-1 s for rotations about bonds with partial double bond character or for puckering in strained rings [1]. Since coupling between internal and reorientational

motions cannot be excluded, the second-rank tensor T:~o , representing a generic NMR interaction (see Chapter 1), can be written in spherical tensor notation as

T:~O = J T:~o ( { w }, { </J } ) PLC ( { W }, { </J} ) d{ w }d{ </J } (14.1)

where { </J} represents all the molecular internal degrees of freedom, {w } represents the Euler angles (a, (3, 'Y) (see Rose [3] for a definition) and PLC ({ w}, {</J}) is the singlet orientational distribution function in the liquid crystalline mesophase. The subscript d indicates a director frame X, Y' Z', where Z' is aligned along the director n. A convenient way to deal with the total singlet orientational energy Utot ( {w }, { </J}) is to assume that it can be written as a sum of an anisotropic, intermolecular potential, Uext({w}, {</J}) (which vanishes in the isotropic phase) and an internal rotational

potential, Uint ( { </J} )

(14.2)

With this assumption we obtain

exp {- [Uext ({w}, {</J}) + Uint ( {</J})] /kT} (14.3)

J exp { - [Uext ( { W }, { </J}) + Uint ( { </J} ) ] / kT} d{ w } d{ </J} .

It is worthwhile emphasizing that different conformers experience different orientational torques, leading to a dependence of the overall molecular orientation on the conformational distribution.

NMR Studies of Solutes in Liquid Crystals: Small Flexible Molecules 307

We can also define the quantities PLC ( { ¢}) and Piso ( { ¢} ), where

(14.4)

is the probability of finding a molecule in an anisotropic phase in a conformation {¢} independent of its orientation, and

exp {-[Uint({¢})]/kT} Pi • o({¢}) = J exp {-[Uint({¢})]/kT} d{¢}

(14.5)

is the analogous probability for the molecule in the same solvent and at the same temperature, but in a virtual isotropic phase. Note that Piso ( { ¢}) and PLC ( { ¢}) are in general different.

T:~o can be related to the corresponding tensor elements expressed in the moleculefixed frame according to

2

T:~O = 2: J T:~OI({¢})D~~r({w},{¢})PLC({W},{¢})d{w}d{¢} r=-2

2

= 2: J T:~ol ( { ¢ }) (D~~r ( { ¢ } ) ) d{ ¢ } r=-2

(14.6)

where the quantities

are the molecular order parameters for conformation { ¢}. DL is the m, n element of the Wigner rotation matrix of rank L. "".n

If the rate of conformer interconversion is in the slow exchange NMR limit, a separate spectrum would be obtained for each conformer. The analysis of these spectra would yield separate conformer geometries and order tensors. Conformer probabilities would be obtained from relative integrated intensities. However, most cases of interest are in the fast exchange NMR limit, and a single, averaged spectrum is measured. A rather formidable problem is then implicit in Eqs. (14.6 and 14.7): the experimental observables involve products of conformer probabilities and orientational factors that cannot be separated.

3. The conformational problem 3.1 General considerations

The aim of conformational equilibrium investigations is to obtain Pi so ( { ¢ }) which is completely determined by Uint • Uint can be expressed in terms of a limited number of adjustable parameters. We focus on the common, unfavourable situation in which there


is rapid interconversion among conformers such that a single, motionally averaged NMR spectrum is obtained (see Chapter 19 for discussion of the slow interconversion limit). The following considerations need to be taken into account.

(i) As many as possible observables that are sensitive functions of conformation should be available for multivariate analysis of Uint which should be modeled in terms of adjustable parameters that are related to these observables.

(ii) A "reasonable" model that describes the conformer dependence of the observabIes should be available. In the case of chemical shifts and indirect spin-spin couplings this would involve electronic structure calculations as a function of conformation. However, the situation is much more favourable for the dipolar couplings, since their Cartesian tensor elements in the molecular frame, Dij,afj,

are easily calculated from

(14.8)

where Tij is the internuclear distance and Xa is the a component of the internuclear distance vector in the molecular frame [4,5]. K is a constant involving the product of the nuclear magnetogyric ratios. One should be aware that the "experimental" dipolar couplings obtained from spectral analysis include the spin-spin coupling anisotropic contribution (see Chapter 1, Eq. (1.68».

(iii) It is clear from Eq. (14.7) that each conformer has its own independent order

tensor (D~~r ( { cjJ} ) ) . Because of the large number of independent conformer

order parameters, from which in general the conformer probabilities cannot be obtained, some strategy is required. This point is central to the applicability of NMR of solutes in liquid crystals for the investigation of conformational equilibria in solution. In the past, two quite different approaches have been attempted. The first is a "heuristic" approach [6-8] whose aim is to reduce Eqs. (14.6 and 14.7) to manageable expressions by using some physically justifiable approximations. Some models for this approach are discussed below. The second, alternative approach, is an "unbiased" method based on Information Theory, the Maximum Entropy (ME) method [9, 10] (see section 3.4.2).

It must be emphasized that the interpretation ofNMR observables in terms of conformational geometries, probabilities, and order tensors is complicated by many interlinked factors. First, the number of independent observables M versus the number of adjustable parameters N is often critical, and the MIN ratio should be kept as high as possible. Second, a key problem is how to describe the dependence of order parameters and conformational probabilities on { ¢}. Third, the sensitivity of the observables to the model parameters should be carefully checked in order to increase the confidence in the distribution obtained [11,12]. Fourth, a priori knowledge of conformer geometries is extremely helpful in the context of calculating dipolar interaction tensor elements.

NMR Studies of Solutes in Liquid Crystals: Small Flexible Molecules

3.2 Conformation and orientational order: decoupled description

309

To an excellent approximation, internal and reorientational molecular motions can be decoupled when the only internal degrees of freedom are small-amplitude vibrational motions. In this "rigid" molecule case the equation

T 2 ,O = (3cos2(3' - 1) ~ T 2 ,r / D2' ) >. 2 L...J >.rnol \ O,r

r=-2

(14.9)

holds, with (3' the angle between Z aligned along Eo and Z' along the director n. Accordingly, for rigid molecules, a single order matrix (see Chapter 1) is needed to describe the molecular orientational order.

In early papers that dealt with the investigation of molecules undergoing conformational change in liquid phases, the decoupled model Eq. (14.9) was applied to a few simple molecules [13-15, and references therein]. It should be noted that deceptively good results can be obtained from Eq. (14.9) if the dipolar couplings are insensitive to changes in {<p}, or if the solute orientational order is small. However, the approximations involved in the decoupled model when applied to large-amplitude conformational changes have been found to be too crude, and this model is no longer used.

3.3 Internal rotation potential as a discrete function

According to the Rotamer Isomeric State (RIS) approximation [16], the internal potential function can be written

N

- L Vn {j ( { <p} - {<Pn} ) (14.10) n

where N is the number of minima in the potential function and { <Pn} the set of internal coordinates corresponding to conformer n. Within the RIS approximation, Eq. (14.6) reduces to

2

T 2 ,o = "" n / D2') (T2,r ) >'d L...J L...J p \ O,r >.mol n

n r=-2 n

(14.11)

where the summation extends over the finite number N of rotamers, p n is the conformer

probability, (T::oJ n is the magnetic interaction tensor for rotamer n, and (D~~r) are the Wigner rotation matrix elements averaged over the conformer reorientation~ motions. Eq. (14.11) for the dipolar coupling can be written as

(14.12)

Note that


• the molecule is described as a "collection" of rigid conformations, each of which has orientational order Sr, and weight pn;

• the adjustable parameters in the fitting procedure of calculated versus experimental dipolar couplings are the products pn S'&' implying that all pn values that give physically meaningful Sr, values represent possible solutions;

• the number of parameters increases rapidly when equilibria among non symmetry related conformers are involved since each conformation is described by its own probability pn and its own ordering matrix Sr,.

In order to relax the restrictions set by the RIS approximation, some authors have modeled Uint { <p} as a continuous function of some internal molecular coordinate, while they kept the elements of the S matrix fixed to the value corresponding to the RIS minima of the internal rotation potential function [17,18]. A similar approach was to use an axis system, the so-called "Eckart Axes" (EA), that allows a well-defined separation of internal and reorientational motions. The order tensor related to these axes is then assumed to be invariant to conformational change [19-22].

3.4 Towards a complete orientational-conformational description

The RIS approximation is inappropriate when Uint { <p} is flat. In this case, a continuous dependence of PLC ' Piso , and order parameters on (for example) internal torsional angles that describe the relative positions of rigid subunits forming the molecule is expected. Since very little is known about the nature of the solute-solvent anisotropic interaction potential Uext ( {w }, { <p} ), an exact solution of Eq. (14.6) does not exist. In this case an ad hoc dependence of the elements of the ordering matrix on {<p} can be assumed [21]. Alternatively, Uext ({ w}, {<p}) can be modeled with the help of adjustable parameters.

One way of modeling Uext ( {w }, { <p} ) is based on the idea that the anisotropic intermolecular potential is dominated by short-range anisotropic interactions that depend on the conformer size and shape. For example:

• A phenomenological single-parameter "Restoring Force" model [23] has been reported [24] to have given a good fit for 46 different molecules dissolved in the so-called nematic "magic mixture" consisting of 55 wt% ZLI 1132:EBBA [25] (see Chapter 10);

• In the phenomenological "Surface Tensor" molecular field model [26], the potential Uext ( {w }, { <p}) is based on the interaction between the solute surface and the liquid crystal environment (see Chapter 11).

An alternative approach is to assume that the potential can be built from a bond additive Maier-Saupe approach. For example:

• Additive Potential (AP) methods are the ones that are most commonly used to describe the orientational potential [25,27]. For molecules containing one or


two rotor groups (see section 3.4.1) the so-called AP-Independent Fragment model [27] (henceforth referred to as AP) has been widely used. To deal with very flexible molecules such as alkyl chains, the AP-Correlated Fragment or "Chord" model which introduces correlations between non-bonded rigid subunits is preferable [28,29] (see Chapter 13).

Unfortunately any model chosen may introduce bias into the problem. Hence, conformational results may depend on the model used.

3.4.1 The additive potential-independent fragment model. As an example of how the conformational problem can be treated, we shall expand on the AP model. From Eq. (14.7), the principal elements of the Saupe order matrix defined in the molecule-fixed axis system x, y, z are related to Uext(O, cp, {¢}) according to:

Szz ({ ¢})

= Q(l¢}) J D:,o(O, cp) exp {-Uext(O, cp, {¢})/kT}sinOdOdcp (14.13)

and

SXX({¢}) - Syy({¢})

= Q~}) J ~D:,2(O, cp) exp {-Uext(O, cp, {¢} )/kT}sinOdOdcp (14.14)

with

Q( {¢}) = J exp {-Uext(O, cp, {¢} )/kT} sin OdOdcp. (14.15)

The symbols 0 and cp represent the polar and azimuthal angles defining the direction of ii in the Cartesian system. In this way the uniaxial nature of the nematic mesophase has been taken into account. ~D~'2 signifies the real part of the Wigner rotation matrix. If c2 ,Tn ( { ¢}) are elements of the solute-solvent interaction tensor expressed with respect to the principal axes for each conformation {¢}, according to the AP model [6] Uext (0, cp, {¢} ) can be written as

(14.16)

In tum the interaction tensors can be written as a sum of conformationally independent contributions from each rigid subunit j

C2,m ({ ¢}) = L L E~,pD;,m ({n:}) (14.17) p j

where the Wigner rotation matrix D;,m ( {n:}) relates the principal axes of Ej to the

molecular reference frame. In this way, with EJ2" as adjustable parameters, it is possible

,p

to express the continuous dependence of the order parameters on the set of internal coordinates.


3.4.2 Maximum entropy (ME) approach. The approaches just described follow from different ways of modeling the interaction potential. An alternative, fully unbiased, and completely general approach, based on information theory, has been developed. According to this so-called Maximum Entropy method [9], it is not possible to determine the "real" conformational distribution, or the molecular order parameters, but only the most likely PLC ({ w}, {¢}) distribution that is compatible with the set of observables. The potential is written

U({w},{¢}) = - LLAij[Dij({¢})h,mD~.m({w}) (14.18) i,j m

where [Dij ({ ¢}) h,m are the components of the irreducible spherical dipolar coupling tensor, and Aij are Lagrange multipliers to be determined by comparing observed with calculated dipolar couplings. Of course, other spectral parameters such as indirect couplings Jij or chemical shifts Vi could be considered, but their use is hindered by difficulties in finding the proper dependence of their tensor elements on {¢}.

In order to emphasize some of the peculiarities of the ME philosophy, the quantity p:CE ( { ¢} ) that is obtained from an ME analysis should be considered in some detail. A sharp distribution found with ME implies that the problem is well determined. In this case physically meaningful conformational distributions obtained using "biased" models (e.g. AP) should produce the same {¢max}. If this were not the case, the inferred biased distributions should be rejected as just a mathematical, not a physical solution. On the other hand, if the "real" distribution were very flat, different functions U ( {w }, { ¢ }) could be found giving good fits of calculated to experimental spectral parameters. In principle it is impossible to distinguish between physical and mathematical solutions. Paradoxically, in isotropic solution where a constant, completely flat distribution is obtained from an ME analysis, no dipolar couplings are observed and the set of Dij == 0 can be reproduced by any chosen internal potential, because there is no dependence on {¢}.

Solutes with low orientational order lead to flat distributions. Hence, only limited information can be extracted from spectral parameters. The reverse is not necessarily true, since if the dipolar couplings depend only weakly on { ¢}, very flat distributions can be obtained even when the solute is strongly oriented. As an example, when an internuclear direction makes an angle 54.7° (the so-called "magic" angle) with respect to Eo, a zero dipolar coupling results even when the principal molecular order parameter is as large as one. In general, extended data sets are likely to give more peaked distributions, and the use of strongly orienting mesophases is preferable.

4. Selected examples Some basic initial information must be available and some assumptions must be

made in order to study a conformational problem. First we need starting values of geometrical parameters such as bond distances and angles, the rigidity and linkage of subunits, and torsional features from which the molecular dipolar coupling tensor can be calculated. Secondly, we must choose a method of describing the solute-


solvent anisotropic orientational potential and the intramolecular potential from which PLC ( { <p} ), and possibly Pi80 ( { <p} ), can be obtained. In the following, a number of interesting cases will be presented, with particular emphasis on comparison between AP and ME methods. Also, the use of vibrational corrections for the case of flexible molecules, and the application of liquid crystal NMR techniques to the study of conformational equilibria of mesogenic molecules, or of smaller molecules that mimic part of them, will be discussed.

4.1 Biphenyl Conformational equilibrium in biphenyl has been investigated using different ap

proaches in dealing with Eq. (14.6). However, in this case differences among the methods employed in the analysis were found to be small. Biphenyl interconverts among four symmetry-related structures characterized by a dihedral angle between the two ring planes of: 32° for the RIS [17] model; 34° for the ME [12,30] model; 37° according to Chandrakumar et al. [21]; and 38° from the AP approach [30]. Although the results obtained are quite similar, the variation in values of the position of the minimum is worth noting. For more details, the reader is referred to the original papers.

4.2 Benzaldehyde

In general, the possibility of discriminating among the different methods of treating conformational analysis is hampered by the fact that accurate molecular structures are often not available. Unfortunately, with NMR methods information on the molecular skeleton is not easily available because of difficulties experienced when dealing with 13C anisotropic NMR spectra [31]. As a consequence, the interproton distances obtained from the analysis of 1 H NMR spectra can often be rationalized in terms of seemingly realistic, but unproven skeleton bond lengths and angles.

A noticeable exception is the benzaldehyde case. Diehl et al. [18] succeeded in analyzing all the five 13C satellite spectra of this molecule for a staggering total of 33 independent direct couplings from which only 13 independent structural coordinates had to be determined, assuming the rigid rotor approximation to be valid, and geometry relaxation to be absent. No additional geometrical assumptions are needed, and it was possible to reject the simple RIS outcome [32] on the basis of an unusually long CH distance of the formyl group. Allowing for a continuous description of Uint{ <p}, but keeping just one Saupe ordering matrix corresponding to the planar rotamer, a proper distance for the CH bond was found, albeit with a suspiciously high barrier.

4.3 Bifuryl, bithienyl and biselenophene

The use of 13C enriched samples, or of satellite spectra from "exotic" magnetically active isotopes, allows for more extended data sets, and can play a considerable role in studying conformational eqUilibria. Such is the case with the 2,2' -bi (VI group elements) penta-heterocycle homologues. The presence of 77Se in 2,2'-


() () X X

Figure 14.1. 2,2' -bi (VI group elements) penta-heterocycle homologues: X=O,S,Se. The dihedral angle r/J = 0° corresponds to the cis rotamer.

biselenophene [33] increases the data set by six independent dipolar couplings compared to bifuryl [34] and bithienyl [15,35]. Three of these dipolar couplings depend on the dihedral angle between the heteroaromatic rings. The RIS approach can be used assuming either a mixture of cis and trans planar rotamers, or a mixture of cisoid and transoid twisted conformers. Choosing the molecular frame with one axis directed along the inter-ring bond, and the second one in the plane bisecting the dihedral angle between the two rings, six independent order parameters are needed to describe the orientational order, three for each twisted conformer. Note that, within the RIS approximation, there are pairs of symmetry-related cisoids and transoids, with off-diagonal order parameters differing only in sign. Since, as discussed in section 3.3, only p C S~ and (1 - pC) S~ can be determined, a range of possible pC weights, associated with

values of S~ and S~ within their physically acceptable ranges, was obtained. The set of dipolar couplings cannot be fitted to two planar structures. No attempt has been made to relax the RIS approximation.

In a later study, an equilibrium between the two planar forms of 2,2'-bithienyl in the "magic mixture" was excluded [36]. The best fit was obtained for an equilibrium between the two ±24° cisoids and the planar trans form (ptrans ~ 60%). Finally, a Maximum Entropy Internal Order (MEIO) [11] study on 2,2'-bifuryl showed that: (i) there are only seven truly independent combinations of dipolar couplings; (ii) the "single rotamer" model does not reproduce the data for any ¢; (iii) the 78% O-trans 22% O-cis mixture, found in [34], gives a reasonably good fit although it does not reproduce the two relative maxima of PLC at ¢ ~ 45° and 105° produced by MEIO.

4.4 Butane Butane represents the simplest alkane that shows rotational isomerism. The anal

ysis of the one-quantum 1 H NMR spectrum of such a ten-spin 112 system is quite a demanding task, and was accomplished with the help of high-order multiple-quantum (MQ) NMR spectra in a "magic mixture" solution [37] (see Chapter 5). The set of dipolar couplings provided important orientational and conformational information. The orientational order was successfully described by "Chord" [28] and "Restoring Force" [23] models, while a description based on the molecular moment of inertia tensor [38] was found to be inadequate. The trans-gauche energy difference (Etg) was found to be in the range of 2.1 - 3.0 kJ mol-I, significantly lower than gas-phase


experimental values. This result, obtained by the RIS approach, seems to suggest that the gauche structure population is enhanced in the condensed anisotropic medium compared to the gas phase. This result is very similar to that found in the isotropic phase.

4.5 Anisole: cooperative rotations As previously mentioned, the AP model introduces some bias into the problem,

since a "reasonable" shape of the potential barrier must be assumed a priori. For single rotors, the choice is rather easy because a Fourier series consisting of a small number of terms with a limited number of adjustable parameters will be adequate. However, the reproduction of potential surfaces when coupled rotations are involved is not so simple. There are essentially two options. The first is to choose a very general function, which might entail excessive parameterization. The second is to fix a number of constraints, which could further bias the solutions. Anisole-methyl-13C is a very good test case [39] because the presence of the magnetically active isotope increases the number of dipolar couplings that depend on the conformational equilibrium. The simplest assumptions, such as a single non-planar structure, or jump motions about ¢1 and ¢2, led to poor-quality fits and were rejected. Next, the possibility of testing a coupled rotational potential V(¢t. ¢2) was explored. This potential can be modeled by imposing suitable constraints and dependences to reduce the number of adjustable parameters, while maintaining a "reasonable" physical picture. The barrier height for the rotation about ¢2 was assumed to vary from zero (free rotation) when the methoxy group is orthogonal to the phenyl, to V3,4 when 13C lies in the plane of the ring. Hence, the potential

has all the flexibility needed. The adjustable parameter V3 ,4 also modulates the fourfold term in the ¢1 expansion.

Figure 14.2. Anisole


The potential function that reproduces the experimental couplings has an absolute minimum at cPI = cP2 = 0°, and an absolute maximum at cPI = 0°, cP2 = 60° (27 ± 5 kJ mol-I). Plotting the minimum of V( cPt, cP2) as a function of cPI, the least-hindered motion of the methoxy moiety shows a maximum at cPI = 90° (13 kJ mol- l ), in perfect agreement with ab initio calculations. In addition to this solution, a description that assumes the continuous potential V ( cPI) = ~ V2 (1 - cos 2cPI) + ~ V4 (1 - cos 4cPI) for the rotation about cPI, and a "three-site jump" motion for methyl protons can also reproduce the dipolar couplings. Thus, it is not possible to discriminate between the two descriptions.

4.6 AP versus ME: I-nitro-4-(2,2,2)trifluoroethoxybenzene (NTFEB) and related compounds

With the right combination of a sufficient number of dipolar couplings and easily analyzable spectra, NTFEB dissolved in 152 and ZLI 1132 provided a suitable test for a comparison of AP and ME approaches. The presence of the nitro group seems to playa considerable role in giving a large degree of orientational order to the molecule. Therefore, the nitro group was preferred over other non-protonated groups, even though its presence might have a strong influence on the conformational equilibrium.

The advantages of using the ME approach emerge very clearly when investigating conformational equilibria of ethoxy groups. The cooperative potential function is now very complex, with three intramolecular rotations involved. Hence, it is not feasible to infer a suitable hypersurface that is both realistic and involves a small number of

adjustable parameters. An unbiased distribution p:CE ( { cP} ), assuming that the rotation about the C-CF3 (or C-CH3) bond can be treated as a decoupled "three-site jump"

Figure 14.3. l-nitro-4-(2,2,2)trifluoroethoxybenzene (NTFEB)


Figure 14.4. Distribution functions for NTFEB dissolved in 152 (left) and in ZLI 1132 (right). (Reproduced with permission from [40]. Copyright 1993 American Chemical Society.)

motion, is just what we need, and is presented in Fig. 14.4 for NTFEB [40]. From this figure we note the following:

(i) the curve shapes are very similar in both mesophases;

(ii) because the solute in the nematic liquid crystal ZLI 1132 has low orientational order, the only information obtainable indicates that the most probable conformer has rPl = rP2 = 0°;

(iii) there is a clear indication of a strong coupling between the rotations about rPl and rP2;

(iv) there are no indications for the existence of secondary PLO maxima for the gauche forms;

(v) there are strong similarities with the shape of PLO obtained for ethoxybenzenes using the ME approach [41].

By making use of the information obtained from ME, the AP approach can be used. The simplest function that describes cooperative rotations of the proper symmetry is

(14.20)

This gave an excellent fit for both sets of data, assuming that V{ = O. Moreover, the potential functions obtained for the two nematic solutions have essentially the same shapes. It is important to note that both ME and AP results point toward a distribution that is strongly peaked at rPl = 0°, rP2 = 0° with no secondary minima. These results seem to contradict those obtained from deuterium quadrupolar splittings measured


for mesogenic molecules, and from dipolar coupling data when interpreted with less sophisticated motional models, such as RIS or uncooperative rotations. In such cases information about the local minima that define the positions of gauche ± states is needed to fit the results.

The importance of having extended data sets when the ME approach is followed is demonstrated in a comparative study of 4-cyano-4' -ethoxy-biphenyl (20CB), deuterated on the cyanobenzene ring and in 2' positions, dissolved in ZLI 1132, and 4-cyano-4' -(2,2,2-trifluoroethoxy)-biphenyl (20CBF3), deuterated on the cyanobenzene ring, dissolved in the liquid crystal60CB [42]. While the probability distribution of 20CBF3 is very similar to that observed for NTFEB, an ME treatment of 20CB direct couplings [43] gave quite a different distribution, without correlation between the two rotations, and twin probability maxima at (PI = ±15°, (/>2 = ±30°. If an estimated value of the direct coupling between the 2 ' and methylene protons, obtained by scaling the corresponding couplings of 20CBF3 for the different orientations, is added to the data set, then the probability distributions of the protonated and fluorinated compounds become identical.

4.7 Vibrational corrections As stated in section 4.2, a lack of information on the molecular skeleton may af

fect conformational results. When 1 H _13C dipolar couplings (obtained from analysis of 13C selectively enriched samples or from proton satellite spectra) are available, further complications may be introduced. These complications have to do with the need to perform vibrational corrections. The neglect of vibrational corrections can produce an increase in rCH bond lengths as large as 5% [44] when obtained from uncorrected dipolar couplings. A procedure that takes account of vibrational corrections is essential when dealing with dipolar couplings between directly bonded nuclei, such as in CH bonds or between geminal protons in methyl and methylene protons [44]. Unfortunately, the extension of vibrational corrections to flexible molecules is not straightforward. There are two points to be considered: (i) the calculation of the required force field; and (ii) the application of vibrational corrections to situations that involve low barriers to intramolecular interconversion. As a general rule, vibrational corrections are relatively unimportant when only IH_IH dipolar couplings are observed. Other complications such as a proper choice of geometrical coordinates and the model chosen to deal with the internal motion may well have a larger influence on the conformational results. When, as in benzaldehyde, the complete set of 1 H _13C and IH_IH dipolar couplings is available, it should be possible to compare different approaches. For this reason, the dipolar couplings reported in [18] were used in conjunction with various approaches to the conformational problem. The AP (excluding and including torsional motion) and the RIS approaches were used in conjunction with vibrational corrections based on force fields calculated at different levels of approximation. The rCR's and aromatic CCH angles were monitored in order to probe the effects of the strategies followed [45]. The formyl CH bond length is quite sensitive to the approach used, and ranges from 1.06 A up to 1.13 A as a function of the model and


the force field. Since bond length values ranging from 1.112 A to 1.122 A obtained from theoretical calculations [46] and electron diffraction data [47] can be found in the literature, no firm conclusions can be drawn. As is discussed in [45], the AP approach excluding torsion is slightly better than with torsion, and is roughly equivalent to the RIS approach. Some dependence on the force field can also be recognized. All the conformational results indicate the presence of a very high rotational barrier with negligible contributions to the averaged dipolar couplings from conformations with torsional angle larger than 7°. Hence, the results are not surprising, but the question about the best strategy to be followed remains unanswered. Other suitable molecules should be investigated.

4.8 Conformational studies of mesogens

Mesogenic molecules are complex compounds with a variety of different functional groups, but their general formula can be summarized as X - Y - Z. Y, the so-called core, is made up from rigid moieties such as phenylcyclohexyls, biphenyl, benzaldehyde phenylimines, diphenyl diazene, while X and Z can be alkyl chains, alkoxy, carboxylic, alkoxycarbonyl, or acyl groups directly bonded to the rigid core [48]. Phase diagrams of mesogenic molecules can be very complex, displaying a large variety of phases over a wide range of temperature. Small structural differences such as the length of alkyl chains and the character of functional groups in the rigid core can influence thermodynamic behaviour. A reliable description of conformational equilibria in mesogenic molecules is of fundamental importance for a full understanding of these different factors. In order to achieve this, the most commonly used techniques are 2H NMR spectroscopy of perdeuterated mesogenic molecules and 1 H NMR of small molecules that mimic parts of the mesogen. In this section we shall present results obtained using 1 H, 13C, and 19F NMR.

4.8.1 Arylalkoxy linkages. Given their relative simplicity, the conformational problem in a few nOCB liquid crystals has been investigated using 2H NMR, in conjunction with the RIS approximation. This approximation represents the only feasible way of handling the many possible conformers. The AP or Chord models are used to calculate order parameters for each conformer [6,28,29]. Using the AP model to obtain the dependence of order parameters on conformation, it has been found that two secondary minima, corresponding to the two gauche forms, have to be included in order to fit the quadrupolar splittings [6,49,50]. A significant contribution of the gauche forms was invoked in a 1 H NMR study of l-ethoxy-4-chlorobenzene [51]. The weighted order matrices pst and! (1 - p )sg, calculated from the dipolar couplings, were compared with those predicted by mean field theory obtained from analysis of the quadrupolar splittings.

The IH NMR spectra of ethoxybenzene, and of its p-fluoro derivatives, have been analyzed and the conformational equilibria investigated by a similar model. In this case the MIN ratio is high [52]. Similar shapes for the potential barriers of these molecules were found, with two local minima at the gauche conformers.


It is also worthwhile to remember that an investigation of the conformational equilibria in lOCB, the simplest molecule that exhibits a nematic phase, and of lOCBF3, has been carried out by the AP approach [53]. The potentials for rotation about the inter-ring bonds gave very similar results for lOCB (minimum at (Pt = 37.8°) and for lOCBF3 (minimum at (Pt = 39.2°). However, significant differences between conformational equilibria in methoxy and triftuoro-methoxy groups were found.

4.8.2 Arylalkyllinkages. The 4-n-pentyl-4'-cyanobiphenyl (5CB) conformational equilibrium about the inter-ring bond has been investigated in the pioneering work of Pines et al. [54] by the use of MQ NMR spectroscopy. The dipolar couplings, fitted by the RIS and the AP approaches [55], produced a value of 38.4° for the minimum of the dihedral angle, very similar to that obtained by the AP approach for various substituted biphenyls. The rotation about the aryl-alkyl bond in 5CB has been investigated by 1 H _{2H} NMR spectroscopy [56], assuming a simple fourfold potential. A very deep minimum resulted when the carbon of the second CH2 of the chain lies in the plane perpendicular to the aromatic ring, with a barrier higher than 22 kJ mol-1 and the fourfold term nearly zero. It is interesting to remember that 1 H NMR studies of 4-chloroethylbenzene [57] and ethylbenzene [58] in four different nematic solvents gave a much lower barrier (3 kJ mol-1 and about 1.8 kJ mol-I, respectively).

Figure 14.5. Ethylbenzene

The conformational study of ethylbenzene proved to be very challenging, since no simple model for rotation of the ethyl group produced a satisfactory fit to the experimental dipolar couplings. A reasonable fit to the experimental dipolar couplings was only possible when a correlation between (Pt and the 0: and j3 angles of Fig. 14.5 was assumed. This case is, in our opinion, a perfect example of what is mentioned in section 1, i.e. even after having succeeded in analyzing the very complicated spectrum of fully protonated ethylbenzene, no firm conclusions about its conformational equilibrium can be reached.


4.8.3 Aryloxycarbonyllinkages. This case is very complex since, even in the simple case of phenyl acetate [59], there are three possible rotations to be accounted for, and there are no magnetically active nuclei involved in the rotation of the carboxyl moiety. However, even the use of 13C enrichment in the carbonyl group did not facilitate a complete analysis. The only conclusion that could be reached by NMR alone was that the minimum energy conformation is non-planar. In this case at least three very different rotational potential functions fit the data. One of them, with a minimum in the potential when the angle between the acetate and phenyl groups is 54.40 , agrees quite well with ab initio calculations. Therefore, this solution was assumed to be the most probable. In addition, this seems to be a case where the phase has little influence on the conformational equilibrium.

4.8.4 Rigid cores. In addition to the biphenyl group discussed above, other more complex groups constitute the rigid core of some mesogens. Three of these groups, viz. the tolan [60], trans-azobenzene [60], and phenyl benzoate [61] groups have been investigated by 1 H NMR spectroscopy. The AP model was used to describe the dependence of order parameters on conformations. Tolan and azobenzene were found to be planar, with a potential barrier that is shallower than that measured in the gas phase by electron diffraction. This suggests the existence of some solvent effect. It should be emphasized that these results could depend on the manner in which vibrational corrections have been included in the calculations. In the case of phenyl benzoate, the benzoyl moiety was found to be planar. The potential minimum for rotation about the (CO)-O bond occurred at 00 with the two aromatic rings in the trans position, and the phenyl plane rotated by 500 with respect to the benzoyl plane.

4.8.5 Fluorinated liquid crystals. The presence of the spin 112 19p nucleus can be exploited as an alternative to 2H NMR spectroscopy. The 13C_{lH} spectra can be recorded, and the DCF dipolar couplings can be used for investigating conformational equilibria. Although in principle the experiment is very simple, an efficient decoupling scheme for both aromatic and aliphatic carbons has not yet been found [62]. In order to circumvent these difficulties, the VASS NMR technique in combination with decoupling sequences such as WALTZ-16 or COMARO-2 can be used to reduce the values of dipolar couplings (see Chapter 3). Such an experiment has been performed for the 152 and 135 mesogenic molecules [63] in order to investigate their conformational equilibrium. The AP model was used to calculate the dependence of the ordering matrix on {<p}. Assuming a Flory RIS description for the molecular conformations, a quite satisfactory fit of calculated versus experimental couplings was found. A similar approach was followed for 2,2'-difluoro-4" -hexyloxy-4-pentyl-pterphenyl (HTP) [64]. A total of 34 DCF and 1 DFF dipolar couplings were obtained. In addition, the 13C -{ 1 H} and 1 H _{2H} spectra of 2,2'-difluoro-biphenyl which constitutes a fragment of the "rigid core" of HTP were recorded [65]. The DFF, DHH,

DHF and DCF dipolar couplings were used for investigating the inter-ring rotational potential in conjunction with the AP model.


5. Conclusions Liquid crystal NMR is a powerful technique for studying conformational equilibria

in condensed fluid phases. Often the technique is capable of providing invaluable information about stable structures and of discriminating between antagonistic solutions. However, the available experimental data are not always sufficient to obtain unambiguous results. When considering the examples discussed above, the message seems clear, viz. it is necessary to aim for the most extended data sets possible. Unfortunately, this could trigger a perverse, vicious circle, a process in which the "snake eats its tail". In order to discriminate better between different reasonable conformational distributions, more extended data sets are required. This, in turn, induces the formulation of more complex hypotheses that involve an increased number of adjustable parameters which, in order to be tested, require more experimental data, and so on. Of course, the complexity of the spectral analysis and the need for excessive computer resources [66] both increase dramatically with the number of interacting spins and the decrease in molecular symmetry. These considerations soon become the limiting factors in the process. In spite of this, the fast development of novel experimental techniques in conjunction with innovative theoretical approaches suggests a bright future for this research field.

Acknowledgments This work has been supported by MIUR PRIN ex 40%. The authors are grateful to

Dr. Pina De Luca for useful suggestions.

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Chapter 15

SIMULATIONS OF ORIENTATIONAL ORDER OF SOLUTES IN LIQUID CRYSTALS

James M. Polson Department of Physics. University of Prince Edward Island.

Charlottetown. PE. Canada

1. Introduction Orientational ordering of molecules in liquid crystalline phases arises from the

presence of anisotropic intermolecular forces. Examples of such forces are as follows: short-range repulsive forces which ultimately arise from the high energy associated with the overlap of molecular electronic wave functions, in connection with the Pauli exclusion principle; "electrostatic" forces, defined here as an interaction between permanent molecular electric multipole moments; dispersion forces which involve the interaction between induced electric multipole moments; and induction forces which involve the interaction between permanent and induced multipole moments. An important objective in the study of liquid crystals is to obtain a clear understanding of the effects of each of these component forces on the molecular organization within the phases. While such an understanding is not yet complete, substantial progress has been made using a combination of experimental, theoretical and computer simulation methods.

Of the various experimental methods used to study orientational ordering in liquid crystals, Nuclear Magnetic Resonance (NMR) spectroscopy is among the most effective. An analysis of the NMR spectra of orientationally ordered molecules yields various NMR coupling constants (notably dipole-dipole and electric quadrupole couplings) which in tum can provide second-rank orientational order parameters. However, the application of this method to the mesogens directly is complicated by the structural complexity and flexibility of these molecules. These factors tend to complicate the analysis of the NMR spectra to obtain the coupling constants, and to complicate the analysis of the coupling constants to extract orientational order parameters.

An alternative approach to studying the mesogens directly is to study the orientation of small probe molecules which are dissolved in the liquid crystal. The basic idea is that the solutes probe the same intermolecular forces as the mesogens themselves. As long as the solute symmetry is lower than Td, the solute will be partially aligned in

325 E.E. Burnell and C.A. de Lange (eds.). NMR of Ordered Liquids, 325-344. © 2003 Kluwer Academic Publishers.


the liquid crystal. Solutes can be chosen for specific advantages. For example, rigid molecules with a small number of nuclear spins and I or a high degree of symmetry yield NMR spectra which are easily analyzed to obtain NMR couplings and, thus, second-rank orientational order parameters. Alternatively, specific solutes can be chosen to highlight the effects of specific intermolecular forces on orientational ordering. In addition, the study of suitable small flexible solutes with tractable NMR spectra provides dipolar couplings which can be analyzed to investigate conformational behaviour of these molecules in liquid crystalline phases.

While the orientational order parameters of partially aligned solutes are governed by the intermolecular forces, information about these forces is not directly accessible from the order parameters. Instead, mean-field theories of solute ordering typically are used to predict orientational order parameters for specific combinations of intermolecular forces. In principle, a comparison between theoretical predictions and experimental measurements of order parameters can determine the contributions of various intermolecular forces to orientational ordering. However, in practice, this approach is complicated by the imperfections of the theory that arise from (i) the modeling of the molecules and pair potentials, and (ii) the mathematical approximations employed in the theory to derive the mean field potential of the solute. Clearly, some alternative means of testing the validity of the theories is essential.

Computer simulations of solutes in nematic liquid crystals can provide an effective bridge between experiment and theory. The molecular models employed in any theory can be incorporated into the simulations. A comparison between simulation results and theoretical predictions can thus provide valuable insight into the validity of the models for the pair potential and the mathematical approximations used in the theory. In addition, this approach provides a simple method for examining the importance of specific intermolecular forces to solute ordering. The freedom to design model solutes that interact via arbitrarily chosen potentials with the mesogens therefore provides a powerful means of disentangling the competing effects of various intermolecular forces.

Computer simulation methods have been widely employed to study liquid crystals, and, consequently, the number of such studies reported in the literature is enormous. On the other hand, there are only a small number of simulation studies focusing on orientational ordering of solutes in liquid crystals. The simulation methods employed in these studies are standard. In some cases, Monte Carlo simulations have been used [1-9], while other studies have employed molecular dynamics simulations [10-15] or a combination of the two [16].

A more important distinguishing feature for each study is the type of model used to describe the solvent and solute molecules. While early simulation studies employed lattice models for the solvent [1,16] to improve the efficiency of the simulations, and a few have used a mean-field description of the solvent [2,7,8], the majority have used an explicit, off-lattice model solvent. In one class of model, all atomic detail is omitted, and the shape anisotropy of short-range repulsive forces is considered to be the chief orienting mechanism of the mesogens. In studies of oriented solutes, model solvent

Simulations of Orientational Order of Solutes in Liquid Crystals 327

molecules of this sort include hard ellipsoids [3-5] and soft ellipsoids interacting with the Gay-Berne potential [6,10,11,13,14]. In a few such cases, electrostatic moments have been added to the particle [4-6,9]. At the opposite extreme, liquid crystal molecules have been modeled with atomistic potentials [12,15].

The principal feature of the simulation model affecting solute ordering is the model describing the solute itself, and its interaction with the solvent molecules (and possibly other solutes, in the case of finite solute concentration). Clearly, it is essential that the level of complexity of the solute model be commensurate with that of the solvent model. A variety of solutes have been studied. Examples include hard ellipsoids of a variety of sizes and shapes, with and without electrostatic moments, dissolved in a hard ellipsoid solvent [3-5], and a Gay-Berne solute in a Gay-Berne solvent, in which the particles also interact via electric quadrupole-quadrupole interactions [6,9]. Most studies, however, have considered more chemically specific solutes, including hydrogen [16], benzene [10,12,15], naphthalene [7], various halogenated benzenes [8], hexane [2, 11, 13] and biphenyl [14]. It should be noted that each ofthese studies was motivated by experimental measurements of NMR coupling constants, and thus orientational ordering, of these solutes in various nematic liquid crystals.

2. Orientational distribution functions and mean-field potentials

The orientational behaviour of a rigid solute in a uniaxial nematic liquid crystalline phase is fully described by the singlet orientational distribution function f ({3, 'Y), where {3 and 'Yare the polar angles specifying the orientation of the nematic director in the molecule-fixed coordinate system. The distribution function can be expanded in the following series:

even L

f({3, -y) = L L [(2L + 1)/47rj{'Dt,n *)Vt,nU3, -y), (15.1) L=On=-L

where vt n ({3, 'Y) is a Wigner function of rank L, and where (vt n *) are orientational order par~meters, defined as the averages of the Wigner functi~ns over f({3, 'Y). A special case of this expansion occurs for solutes with C3v or higher symmetry, in which case the distribution depends only on {3, the angle between the molecular symmetry axis and the nematic director, and can be written as a series expansion of Legendre polynomials PL (cos {3):

(15.2)

The expansions in Eqs. (15.1) and (15.2) converge with a rate that decreases with increasing orientational order in the nematic solvent. NMR measurements of coupling constants do not yield f({3, 'Y), but rather only elements of the Saupe ordering matrix:

(15.3)


where cos () a is the direction cosine between the molecule-fixed a-axis and the nematic director, and where the average ( ... ) is evaluated using f({3, ')'). By convention, the principle molecular axes are usually chosen so that Szz = (D60) == P2.

The potential of mean torque U({3, ')') is defined by ,

f({3, ')') = exp( -U({3, ')')/kT)

J d')'d{3 sin{3 exp( -U({3, ')')/kT) (15.4)

As in the case of the orientational distribution function, the potential of mean torque can be written as an expansion in terms of the Wigner rotation functions Df:t n ({3, ')'), ,

even L

U({3, ')')/kT = L L £L,nD~n(O, {3, ')') (15.5) L=On=-L

Typically, many theories employ a mean-field potential that is truncated at the second order term:

U({3, ')')/kT ~ £2,oD6,0(0, {3, 0) + £2,2D 5,2(0, {3, ')') + £2,-2D 6,_2(0, {3, ')')

£2,od6,0(f3) + 2£2,2d6,2({3) cos(2')') (15.6)

The coefficients £2,0 and £2,2 in Eq. (15.6) are related to the orientational order of both the solvent and the solute, and so are highly temperature sensitive. On the other hand, according to the molecular field theory of Emsley et al. [17] the ratio £2,0/£2,2

is expected to be temperature-independent, which contrasts with experimental results for several biaxial solutes. To understand the origin of this discrepancy, Hashim et al. carried out a Monte Carlo simulation study of the orientational order of biaxial solutes dissolved in a nematic phase composed of uniaxial mesogens [1]. All particles were confined to an FCC lattice and interacted with nearest neighbours only. Solvent-solvent interactions were given by UH = _£11 P2( cos {3ij) , and solventsolute interactions were given by

(15.7)

In the molecular field approximation, the coefficients £2,0 and £2,2 of Eq. (15.6) are derived from the interparticle interaction coefficients £~20 and £~22 in Eq. (15.7) by averaging the pair interaction over neighbouring particl~s under'the approximation that orientational correlations can be neglected. An analysis of the two independent elements of the solute order matrix, Sxx - Syy and Szz, yields the two expansion coefficients £2,0 and £2,2. The ratio £2,0/£2,2 was found to be temperature independent, in agreement with the predictions of molecular field theory, but in contrast with experimental measurements. Evidently, it is the highly simplified molecular model which is described by the interaction potential ofEq. (15.7) and which neglects many details such as mesogen flexibility, and not the molecular field approximation, that is responsible for the discrepancy between theory and experiment.


The mean field theory for solute ordering in a nematic solvent was also studied by Palke et al. [10]. In this study, molecular dynamics simulations were performed for a system composed of a single benzene molecule dissolved in a Gay-Berne fluid of 256 particles. The solute-solvent interaction was the sum of Lennard-Jones interactions between each of six sites on the solute (representing a united atom of carbon and hydrogen) and four equally spaced Lennard-Jones sites along the mesogen. In this case, no attempt was made to tune the parameters so as to produce a close match with the observed properties of benzene. The orientational order parameters Pfc and P2, i.e. the single independent element of the Saupe order matrix Szz for the Gay-Berne nematogen and for benzene, respectively, were measured as a function of temperature. As expected, the magnitude of each was found to increase with decreasing temperature.

Under the assumption that the intermolecular vectors are spherically distributed, the potentials of mean torque for the mesogen and benzene are given by

-ELCP 2( cos (3) - Es P2 ( cos (3)

where the energy parameters are given by

- -LC ELC = ULCP2

and

such that

(15.8)

(15.9)

(15.10)

(15.11)

(15.12)

Values of Eb and ELC obtained by the order parameters Pfc and P2 were found to vary linearly, in accordance with the prediction ofEq. (15.12), and in agreement with experimental results on various solutes.

The orientational behaviour of benzene dissolved in a nematic solvent was also the subject of two simulation studies by Sandstrom et al. [12,15]. In this case, molecular dynamics simulations employing a detailed, atomistic potential were used. The liquid crystal was chosen to be 4-n-pentyl-4'-cyanobiphenyl (5CB). In the first study, the simulation consisted of 10 benzene molecules dissol ved in a solvent of 110 5CB molecules, yielding a mol fraction of 8.3%. In this study, solvent and solute singlet orientational distribution functions, as well as orientational order parameters were measured. The orientational distribution functions were analyzed using the same mean field theory employed in the study of Palke et al. [10]. These functions were fit to Eqs. (15.8)(15.11) by adjusting the parameters ULe and us. The fits were excellent for both the solute and liquid crystal distribution functions, yielding values of ULC and Us that closely match the values obtained from experimental measurements, and predicting solute and solvent orientational order parameters P2, Pi, Pfc , and PtC , that closely match the observed values. They conclude that the second-rank truncated potential


of mean torque provides a reasonable description of true distribution functions. On the other hand, the simulations also demonstrated clearly that the distribution of intermolecular displacement vectors (SCB-benzene and SCB-SCB) is not spherically symmetric. Thus, a central assumption built into the derivation ofEqs. (1S.8) - (1S.11) is clearly invalid.

In a follow-up study, Sandstrom et al. carried out simulations on the same benzeneSCB system [IS]. While most of the calculations focused on benzene translational and rotational diffusion, as well as the conformational behaviour of SCB, they also return to an analysis of the singlet orientational distribution function. Just as the molecular symmetry of benzene results injust a single independent principal element ofthe order matrix, Szz = P2' it also leads to a singlet distribution function which depends only on f3, the angle between the molecular symmetry axis and the nematic director. Thus, f (f3) can be written as a series expansion of Legendre polynomials, as in Eq. (1S.2). The form of this expansion is also approximately true for a semi-rigid mesogen like SCB, even though this mesogen is neither rigid, nor possesses the appropriate molecular symmetry. In this case f3 is the angle between the para axis of the SCB phenyl rings and the director. Approximations to f (f3) can be made using truncation to different orders. Thus, measuring the values of lower-rank order parameters provides a means of recovering f(f3). Of course, NMR is capable only of measuring the second-rank order parameter P2. The simulations clearly show, however, that f(f3) reconstructed using only P2 produces a very poor match with the measured f(f3) for both benzene and SCB. The predictions were found to improve when P4 was used, especially for the less orientationally ordered benzene, in accord with expectations that the series expansion converges more rapidly for systems of lower orientational order. It was also found that a P2 ( cos (3) distribution will match the observed distributions if an optimized coefficient other than P2 is used.

An analysis of the orientational distribution functions of a partially ordered solute was also reported by La Penna et al., who carried out a molecular dynamics simulation study of hexane dissolved in a Gay-Berne solvent [13]. As in the case of the benzene study by Palke et al. [10], a six-site united-atom model is used to describe the hexane solute, with one site for each methyl or methylene group. Also, as in the earlier study, the solute-solvent interaction is described by a Lennard-Jones interaction between the solute sites and each of four sites on the GB particle that are equally spaced along its long axis. In contrast to another similar study of hexane in a Gay-Berne solvent [11] which is described in the next section, this study examined the orientational behaviour of specifically chosen rigid conformers of this molecule, rather than modeling it more realistically as a flexible molecule. The conformers were generated by a rotation from the fully all-trans structure through a torsional angle ¢ about the central C-C bond in six steps of 30°. Orientational distribution functions for each of the seven conformers were calculated and analyzed in the context of three different model potentials.

As each of the hexane conformers possesses a lower degree of symmetry than benzene, the distribution function must be expanded in the more general form ofEq. (15.1). The expansion coefficients [(2£ + 1)/41T](1)t n *) were observed to diminish rapidly in magnitude, although the two lowest order terms with £ = 2 and 4 were compa-


rable in magnitude. An alternative analysis of the series expansion of the associated mean-field potential in Eq. (15.4) was also performed, and a similar rate of convergence was observed. Despite the significant magnitude of the L = 4 terms, it was also noted that the NMR dipolar coupling constants which are second-rank quantities are far more sensitive to the second-rank, as opposed to fourth-rank, terms in Eq. (15.4). The principal second-rank orientational order parameters determined from an analysis of proton dipolar couplings that were generated for each conformer in the simulations were fit using the truncated mean field potential ofEq. (15.6) and treating E2,O and E2,2

as adjustable parameters. For each conformer, the parameter E2,O obtained from the fit was close to, though consistently less than, the value obtained directly from an analysis of f({3, ")'), while the agreement for E2,2 was reasonable, though slightly poorer.

The main goal of this study was to compare the coefficients E2,n (¢) with the predictions of three different mean field models. The first model is the additive potential model of Emsley et al. [18], in which the E2,n(¢) are assumed to be tensorial sums of the contributions from each fragment of the molecule, where in this case, a fragment is one of five C-C bonds which are assumed to be identical and spherically symmetric. Thus, only a single parameter, E2,O (CC) is needed to describe the interaction. Upon conducting a separate fit of each set of dipolar couplings for each conformer, it was determined that the fitted E2,O (CC) were roughly independent of the molecular shape, as assumed in this model, though the difference between the simulated and calculated couplings was large, 10%-25%, revealing a fundamental problem with the model. The second mean-field model employed in the analysis was the chord model of Photinos et al. [19]. This model is an extended version of the additive potential model which was introduced to address the fact that the latter model cannot distinguish between conformers that have identical numbers of C-C bonds with the same orientation but are in different positions in the molecule. To correct this defect, the chord model recognizes that the rigid subunits in the alkyl chains are the Ci-l-CI-Ci+l fragments. A detailed description of this model is presented elsewhere in this book (see Chapter 12). A three-parameter version of this model was used to fit the dipolar couplings generated in the simulations. For any single conformer, it was easy to fit, essentially exactly, the dipolar couplings. However, the best-fit parameter values obtained for the all-trans conformer did not yield a particularly good quantitative prediction of the order parameters for the other conformers. This result appears to suggest that the chord model is not a reliable predictor of orientational order of chain molecule solutes in nematic solvents. However, the apparent discrepancy between the simulation results and the predictions of the model could very well have been exaggerated by the procedure used in the comparison. An analysis conducted by fitting simultaneously the dipolar couplings from all of the conformers could have yielded a more consistent prediction of order for all the conformers. Finally, as La Penna et al. [13] point out, it is noteworthy that the model predictions most closely match the simulation results for the more probable conformations of a real flexible hexane molecule, and hence provide good approximations for calculating averaged dipole couplings for the real molecules.

The third mean-field model employed in the analysis of the dipolar couplings is that of Ferrarini et al. [20J. This model tries to correlate the molecular shape to


the orientational ordering in the nematic solvent. The model leads to a potential which is also of the form of Eq. (15.6), but with expansion coefficients which are proportional to the surface integral Is dB 'D5,m(0, (), X), where the surface is that formed by placing spheres with Van der Waals radii on each atom in the molecule, and where () and X are the polar angles made by the normals to the surface in the molecular frame. Again, the best-fit parameter values obtained from a fit of the couplings for the all-trans conformer were used to predict the principal orientational order parameters for the other conformers. The predictions of Bxx - Byy and Bzz were qualitatively correct, though quantitatively somewhat worse than those of the chord model. It was emphasized, however, that this is an expected outcome since the latter model has three adjustable parameters, while the shape model has only one. Again, a simultaneous fit of all couplings for all the conformers could have led to better agreement.

A mean-field model such as that of Ferrarini et al. (see Chapter 11) which incorporates the anisotropy of the molecular shape is in an indirect manner effectively accounting for the short-range repulsive forces which are now believed to be the principal ordering mechanism for mesogens, as well as most solutes in liquid crystalline phases. Burnell and coworkers have also developed a series of "size and shape" models to account for short-range repulsive solute-solvent interactions [21-24]. The details of these models are presented elsewhere in this book (see Chapter 10), and so we make no attempt to review them here. We note, however, two points regarding these potentials. First, they are intended for, and achieve the best results with, nematic solvent mixtures in which an associated electric field gradient (efg) vanishes. This is significant as there is compelling evidence that the interaction between the mean efg and the solute electric quadrupole moment is an important orienting mechanism for various solutes. We return to the importance of such electrostatic interactions in section 4. Secondly, these potentials are not of the form of a truncated expansion of Wigner functions, such as Eq. (15.6). Thus, such potentials will, in general, be described by an infinite expansion, as in Eq. (15.4). Nevertheless, their advantage lies in the fact that they are characterized by only one or two parameters.

Polson et al. carried out a series of Monte Carlo simulations to investigate the relationship between these size and shape models and short-range repulsive solute-solvent forces [3,5]. Unlike previous simulations of solutes in nematic solvents, they studied a system with hard-core repulsion and with no attractive forces. The solvent was modeled by a collection of hard ellipsoids with an axis ratio of 5: 1 at densities sufficiently high for a nematic phase to form. Rather than attempting to model chemically specific molecules, the solutes were also modeled as hard ellipsoids of a variety of sizes and shapes. In the first study, only prolate ellipsoidal solutes were studied [3], though oblate solutes were included later [5]. The orientational behaviour of each solute was studied in a separate simulation in the identical nematic solvent. Orientational order parameters Bzz were measured for each solute, as well as orientational distribution functions f(f3) in a few cases. As expected, Bzz increases with increasing solute shape anisotropy, and the f(f3) is peaked at () = 00 and is at a minimum at () = 900 •

The order parameters were analyzed using model potentials C and I which are both one-parameter models developed by Burnell and coworkers, as well as the two-


parameter CI model potential which is simply the sum of the other two. For a detailed description of these models, the reader is referred to Chapter 10 of this book, or to the review article by Burnell and De Lange [25]. The order parameters were also analyzed using the one-parameter shape model of Ferrarini et al. described above. For each model potential, the fit was performed for the order parameters of all the solutes simultaneously. It was found that all three one-parameter models performed comparably well, yielding roughly equal root-mean-square deviations. It is interesting to note that the defects of the model potentials C and I were the same as observed in fits to experimental order parameters: model C tends to underestimate S zz for shorter solutes and to overestimate Szz for longer solutes, while model I does the opposite. It is also noteworthy that the shape model of Ferrarini et al. behaves similarly to model I, to which it is closely related. Not surprisingly, the two-parameter CI model provided by far the best prediction of orientational order. The reason for its success is more than simply having more than one additional free parameter: addition of the C and I potentials causes the respective deficiencies to cancel out.

The most significant aspect of this study was the comparison with the analysis of experimental order parameters. The length scale was fixed by choosing the solvent ellipsoid length to be 20 A, roughly that of the nematogens employed in the NMR studies of Burnell and coworkers. As well, since the order parameter measurements in the simulations were carried out at a higher degree of solvent ordering than those obtained by NMR experiments, the best-fit parameter values were scaled. In this case, the scaling factor was a ratio of model parameters determined by fitting the orientational distribution functions for a specifically chosen solute in the more highly ordered solvent used in the measurements of the solute order parameters, and in a solvent characterized by the lower, experimental value of the solvent order parameter. The ratio of the bestfit parameter values for two very differently shaped solutes was used for scaling with each model potential. The scaled parameter values were all found to be very close to the values obtained from the fit to the NMR data. The conclusion is clear: the results of the analysis of the simulations of ordering of hard-particle solutes in a hardparticle nematic solvent firmly establishes the physical origin of these shape-dependent potentials as arising from the short-range repulsive intermolecular interactions in the nematic solvent.

3. Conformational behaviour of flexible solutes For a partially ordered rigid solute, the NMR dipole-dipole coupling constant

between spins i and j is given by

"V·"V·h (;i cos2 0 .. - 1) D .. - __ ,t_'_J_ 2 tJ 2 tJ - 47r2 r?

tJ

(15.13)

where rij is the distance between the spins, h is Planck's constant, 'Yi is the gyromagnetic ratio of nucleus i, Oij is the angle that the inter-spin vector makes with the director, and the average ( ... ) is over rotational and translational motions of the molecules, and any internal vibrational motion. Neglecting the effects of small amplitude vibrations,


this can be rewritten

Dij = j L Sa{3Dij,a{3 (15.14) a,{3

where Sa{3 is the Saupe order matrix defined in Eq. (15.3), and where the tensor Dij,a{3 is defined (in SI units) as

D J.Lo'Yi/j Ii (3 0 0 1 .I' ) ij,a{3 = - 87r2r~. "2 cos ij,a cos ij,{3 - "2ua{3

tJ

(15.15)

where o:i is the angle between the internuclear vector rij and the a-molecular axis. For a molecule that samples a distribution of internal conformations, this expression becomes

Dij = j L L p(n)Sa{3(n)Dij,a{3(n) (15.16) n a{3

where p(n) is the probability of the nth conformation, and where the tensors Sa{3(n) and Dij,a{3(n) are uniquely defined for each conformation. Note that the summation over n can be discrete or continuous. Also note that Dij,a{3(n) is determined solely by the molecular geometry of each conformation.

A key goal of many NMR studies of partially ordered flexible solutes in nematic solvents is the study of the solute equilibrium conformational behaviour, with particular emphasis on the effect of the condensed phase environment. Thus, one seeks to extract the conformational distribution function p(n) from the set of dipolar couplings, in addition to the associated Saupe order matrices, Sa{3(n). However, it is clear from Eq. (15.16) that a straightforward analysis of the measured dipolar couplings will only yield the products p(n)Sa{3(n). Consequently, most studies employ simple phenomenological models to approximate the potential of mean torque to calculate the Sa{3(n). Generally, these models are constructed to emphasize the molecular size and shape, and thus implicitly focus on the role of short-range repulsive solute-solvent forces. Examples are the chord model of Photinos et al., and the shape models of Ferrarini et al. and Burnell and coworkers described in section 2. Typically, the models are characterized by one or two adjustable parameters. In addition, simple models for describing the conformational probabilities p( n) characterized by one or two parameters can be employed in tandem with the model potentials. The parameters can be optimized to fit the calculated to the experimental dipolar couplings. In this manner, the conformational distribution p(n) and the order matrices Sa{3(n) can be separately calculated.

Of course, the model potentials required for the analysis described above are imperfect. Thus, the analysis will yield an estimate for the conformational distribution that is sensitive to these imperfections, and which will differ for each model potential. Computer simulations of the flexible solute in a liquid crystal can be used to calculate dipolar couplings which can be analyzed in a similar fashion to obtain estimates of


the conformational probabilities. These calculated probabilities can be compared directly with the true probabilities which are easily measured in the simulation. In this way, the simulation can be used to assess the validity of this method of estimating the conformational probabilities from experimental data, as well as to assess the quality of the model potential for predicting orientational order.

Alejandre et al. have carried out a molecular dynamics simulation of hexane dissolved in a Gay-Berne fluid in isotropic, nematic and smectic states [11]. The solutesolvent interaction was similar to that employed for benzene in a Gay-Berne solvent by Palke et al. [10]. The Ryckaert-Bellemans potential was used to describe internal torsional motion around the C-C bonds of hexane. The molecular conformations are characterized by the state of each of the three C-C bonds that connect methylene groups. Generally, the authors observe only a slight change in the conformational distribution upon changing from isotropic to nematic to smectic phases by increasing the solvent density at constant temperature. Specifically, there is a slight increase in the more elongated all-trans conformation when passing from the isotropic to the nematic state, in agreement with experimental results, while the transition to the smectic state has a negligible effect. On the other hand, moving from the nematic to the smectic phase by lowering the temperature generates a large increase in the number of all-trans conformers. By employing Monte Carlo simulations to study the conformations of a single hexane molecule in the absence of a solvent, it was shown that this enhancement was more significant than that predicted for an isolated hexane molecule at the same temperature. This is due to the high degree of orientational ordering in the smectic phase at this lower temperature, whose effect adds to that of the torsional potential which has a minimum in the trans state for each C-C bond.

Alejandre et al. also calculated proton dipole-dipole couplings for hexane in their simulation study, in order to facilitate a comparison with the experimentally measured couplings of hexane measured by Rosen et al. [26]. After scaling the simulation couplings so that the intramethyl coupling matched that of experimental results, the two sets of couplings were in reasonably good agreement. The unscaled couplings were analyzed using three model potentials: the model C of Van der Est et al. [21], a model based on an earlier model proposed by Straley [27] which uses the moment of inertia tensor of the molecule [28], and the chord model introduced by Photinos et al. [19]. (Note that it is now clear that Straley's model is inappropriate for describing a solute dissolved in a solvent composed of particles whose shape is different from that of the solute [25,29]; consequently, we omit those results from the present discussion.) In accord with the typical approach taken in the analysis of experimental couplings, the rotational isomeric state (RIS) model, in which only three discrete torsion angles about each C-C bond are considered, was used to describe the hexane conformations. In addition, they find the models of Van der Est et al. and Photinos et al. to be reasonably successful in predicting the conformational probabilities at higher temperatures in both the nematic and smectic phases. At lower temperatures, however, the models clearly underestimate the importance of the intermolecular contribution to the mean potential in that the hexane solute in the MD simulation is significantly more elongated than predicted by the fits for the two models. The study also found that the one-


parameter model of Van der Est et al. generally provided a more accurate estimate of the conformational probabilities than the two-parameter model of Photinos et al., the latter of which, for example, was found to underestimate the probability of the all-trans state. On the other hand, it was also noted that use of a discrete conformer distribution to approximate the real, continuous conformer distribution could diminish the significance of the comparison.

An indication of the effect of employing a more realistic description of chain molecule conformations in the analysis of NMR data is provided by the study by Luzar et al. [2], who carried out an alternative, though more limited, type of simulation study to analyze the dipolar couplings of partially oriented hexane. These authors carried out Monte Carlo simulations of a single hexane solute whose interaction with the nematic solvent is given by the model of Photinos et al. [19]. The calculated dipolar couplings were fit to the experimental couplings by adjusting the two mean field potential model parameters, as well as the effective trans-gauche energy difference E:/ f and the effective dihedral minimum gauche angle </>~f f. The simulation is a straightforward extension of the standard analysis of the dipolar couplings of partially oriented alkanes, except that in this case the Ryckaert-Bellemans torsional potential is employed in place of the RIS approximation of three discrete allowed torsion angles per C-C bond. Thus, the entire configurational space of the solute in a uniaxial environment is sampled with this approach. It was noted that the simulation and the RIS analyses give very similar predictions for the parameters E:/ f and </>~ff, as well as for the conformational probabilities. We note, however, that the simulation does predict a slight enhancement of the more elongated conformations, notably the all-trans conformation, relative to the RIS prediction. This result is clearly encouraging for the model of Photinos et al. which had generally underestimated these probabilities when the discrete-conformer RIS approach was used in the study of Alejandre et al. Thus, it suggests that the ability of the model to predict conformational probabilities of chain molecules could be improved with a more realistic modeling of the conformational distribution, as had been suggested by Alejandre et al.

Biphenyl is another significant flexible molecule that has been studied as a solute in a liquid crystal solvent by both NMR [30,31] and computer simulation [14]. Its importance in the field of liquid crystals lies in the fact that it forms the semi-rigid core of many mesogenic molecules. Besides the usual small amplitude internal vibrational motion, it undergoes internal rotational motion about the inter-ring C-C bond. An analysis of the proton dipolar NMR couplings appears to indicate that the average ring-ring dihedral angle, </>, is somewhat less than the 45° observed in the gas phase. Thus, as in the case of hexane discussed above (and, more generally, for alkanes), the conformational distribution of biphenyl is noticeably perturbed by a condensed phase environment.

Palke et al. have carried out an MD simulation study of partially oriented biphenyl dissolved in a Gay-Berne solvent [14]. The biphenyl solute was modeled in a similar fashion as benzene in the earlier study by Palke et al. [10], i.e. it was treated as twelve united CH atoms situated in two regular hexagons. The solvent-solvent parameters


in the Gay-Berne potential were the same as in that study, as was the solute-solvent interaction, in which each of the sites on the solute interacted with each of four equally spaced sites on the Gay-Berne particle with a Lennard-Jones potential. The rotational motion about the inter-ring bond was that of Tsuzuki and Tanabe [32], which potential has a broad minimum at ¢ = 45°. Thus, the torsional angle probability function P( ¢) for an isolated biphenyl molecule has a maximum at this angle. On the other hand, the simulations demonstrate that the maximum in P( ¢) is shifted to a smaller angle by roughly 3 ± 1 ° when dissolved in the solvent. This shift is approximately constant over a wide range in temperature and independent of whether the solvent is in the isotropic or nematic phase, though the distribution does broaden with increasing temperature, as expected.

In order to gain insight into the validity of the analysis of the experimental dipolar couplings for biphenyl, dipolar couplings were calculated for the model solute in the simulation. The simulated couplings were analyzed using the maximum entropy (ME) analysis [33]. This analysis yields a mean potential U({3, I, ¢), where (3 and I are the polar angles describing the orientation of the director in the molecule-fixed frame which is purely anisotropic. Thus, it predicts a distribution, PME(¢) which vanishes in the isotropic phase. The difference in the maximum in PME(¢) from the true value ranges from 9° smaller at the highest temperature in the nematic phase (1503 K) to 2° smaller at 301 K. In addition, the ME-calculated conformation-dependent order parameters Szz(¢) and Sxx(¢) - Syy(¢) show a much stronger dependence on ¢ than observed directly in the simulation, particularly at lower temperatures, and especially near ¢ = 0°. The couplings were also analyzed using the additive potential (AP) model of Emsley et al. [18], in which the same two free parameters characterize each phenyl ring. In addition, the conformationally dependent mean potential is divided into a sum of an external component which is essentially given by Eq. (15.6) with expansion coefficients dependent on ¢, and an internal component which does not vanish in the isotropic phase. The latter term is represented by a cosine series. The couplings were fit by varying the two coefficients in the anisotropic external potential term, and the coefficients in the cosine series expansion for the internal potential. Generally, the predicted distribution PAP ( ¢) is in good agreement with that calculated in the simulation, with a predicted maximum that is at an angle that is consistently larger than the measured value, but by no more than r. In addition, the AP-predicted ¢dependence of the order parameters Szz(¢) and Sxx(¢) - Syy(¢) is in somewhat better agreement with that measured in the simulations than that predicted by the ME analysis.

4. Electrostatic interactions While it is now clear that short-range repulsive forces constitute the principal order

ing mechanism of mesogens and most solute molecules dissolved in liquid crystalline phases, there is also direct experimental evidence that electrostatic interactions (i.e. interactions between permanent molecular electric multipole moments) contribute significantly to the orientational ordering of some solutes. The experimental NMR studies


of partially oriented deuterated and tritiated molecular hydrogen by Burnell, De Lange and coworkers indicate that a large part of the ordering of this solute can be described by the interaction of its electric quadrupole moment with an average electric field gradient (efg) caused by the surrounding solvent [34-38]. The form of this interaction for an axially symmetric solute in a uniaxial phase is given by

(15.17)

where Fzz == -(82V/8Z2) is the average of the main component of the efg tensor experienced by a solute in a nematic solvent, where V is the electrostatic potential at the site of the solute. In addition, the nematic director is chosen to be aligned with the Z axis, and Q zz is the single principal component of the electric quadrupole moment ofthe axially symmetric solute. Note that the value of Fzz at the sites of the nuclei can be measured in the experiment from an analysis of the NMR quadrupolar coupling constant. Employing this potential to calculate Szz, agreement between the calculated and measured coupling constants is obtained for a variety of liquid crystal solvents. This interaction also explains the anomalous orientational behaviour of solutes such as acetylene in some liquid crystals [21,39]. Studies of quadrupolar couplings of deuterated methanes [40,41] and methyl- and chi oro-substituted benzenes [5,42,43] yield a picture which is consistent with that of the hydrogen studies. In addition, the studies by Syvitski et al. show that the effects of permanent electric dipoles have a negligible influence on orientational order, while the effects of electric polarizability and short-range repulsive interactions could not be distinguished, though it is expected that the polarizability interactions are minor. In addition, the analysis of the order parameters in terms of the CI model for short-range repulsion plus the interaction of Eq. (15.17) produced the best fit, and yielded estimates of the efg that were consistent in sign with the measurements for deuterated molecular hydrogen, and roughly equal in magnitude. On the other hand, an earlier similar study of anthracene and anthraquinone suggested that the efg experienced by these similarly shaped solutes may differ significantly [44].

The question of the role that electrostatic interactions playas an orienting mechanism for molecular hydrogen has been investigated indirectly in a simulation study by Lounila and Rantala [16]. They conducted both Monte Carlo and molecular dynamics simulations of a hydrogen molecule in a rigid lattice consisting of parallel, infinitely long cylinders which interact pairwise additively with the H atoms of the solute. The interaction consisted only of a Lennard-Jones interaction that was integrated along the length of the solvent cylinders. Thus, no electrostatic interactions were included. The study found that the orientational order parameters of the solute, Szz were invariably positive at all densities and structures of the lattice, irrespective of the functional form of the interaction potential. The main role of the attractive forces is to pull the solute against the hard cores of the solvent cylinders, thus enhancing the effects of the repulsions (i.e. increasing Szz). The implication is that the negative order parameters measured for molecular hydrogen in several liquid crystals cannot be attributed to the effects of repulsion and dispersion forces. These results, coupled with the success


of the interaction of Eq. (15.17) in describing the ordering of molecular hydrogen, strongly imply that this interaction does provide the correct physical picture.

Another indirect investigation of the role of the quadrupole / efg interaction is provided by the recent Monte Carlo simulation studies by Celebre which examined the orientational order parameters of biaxial apolar, D2h symmetry solutes, including naphthalene [7], as well as l,4-diftuorobenzene, l,4-dichlorobenzene and dibromobenzene [8]. The simulation method is a cross between a standard empirical model potential of a solute interacting with a mean nematic field, and a standard simulation of a solute immersed in an explicit-particle nematic solvent. The simulation model treats the solute as a parallelepiped which is situated at the centre of a face-centred cubic lattice with a cylindrically symmetric solvent particle at each of the six neighbouring sites. The solute interacts with the solvent particles with a potential which is based on that of Straley [27], and that is designed to combine the tendency of each side of the parallelepiped to align with each neighbouring solvent particle and the propensity of each solvent particle to align with the nematic director. The average orientation of the solute relative to the nematic director is calculated by sampling over solute and neighbouring solvent particle orientational degrees of freedom. Clearly, this potential is designed to mimic the effects of anisotropic short-range repulsive solute-solvent forces. The single adjustable model parameter, €, is proportional to the inverse temperature. Thus, by increasing € in the simulation, one can effectively model the temperature-dependence of orientational order parameters. The principal order parameter Syy was plotted as a function of the anisotropy in the order matrix Szz - Sxx for various values of € and compared with results from variable-temperature NMR measurements of each of these solutes in three different nematic solvents, two with non-zero, though opposite signed, efg's, and another which was a mixture of the two and which is characterized by a vanishing efg. In each case, the simulated order parameters are in excellent agreement with the experimental order parameters corresponding to the zero-efg liquid crystal mixture, and deviate significantly from those corresponding to the non-zero efg nematic solvents. As the simulation model is designed to account solely for short-range repulsive forces, the simulation results imply that interactions of this type dominate the ordering of the solute in the zero-efg solvent, while the solute is subject to another important ordering mechanism in the other solvents. As the only clear difference between the three solvents is the magnitude and / or sign of the efg (as measured by dideuterium), these studies add credence to the postulate that the quadrupole / efg interaction described by Eq. (15.17) is the main ordering mechanism, besides anisotropic short-range repulsion, for some solutes in liquid crystal solvents.

Emsley and coworkers have considered the effects of an average efg on solute ordering in the context of a mean-field theory which is closely related to the MaierSaupe theory of nematics [44-46]. In this theory, the solute-solvent pair potential is averaged over the solvent positional and orientational degrees of freedom under the approximation that solute-solvent orientational correlations can be neglected. It was noted that the nematogen quadrupole moment is the lowest order electric multi pole moment that provides a non-vanishing contribution to the efg. The contribution to the


mean-field potential from an interaction between solute and solvent electric quadrupole moments has the form of Eq.(15.17) with an efg given by

F = -120 Q(v) p'(LC) 100 d PI (r)g(r) zz 7rp zz 2 r 3 ' o r

(15.18)

where p is the solvent density, Q~"2 is the nematogen quadrupole moment, pJLC) is the second-rank order parameter of the liquid crystal, g( r) is the solute-solvent radial distribution function, and PI (r) is a fourth-rank orientational order parameter describing the orientation of the solute-solvent intermolecular displacement at a distance r [4]. A notable feature of this expression is the prediction that Fz z vanishes in the the case where the distribution of the intermolecular vectors is spherically distributed.

Polson and Burnell carried out a Monte Carlo simulation study investigating the effects of quadrupolar interactions on orientational ordering of solutes in a nematic solvent [4]. The model system employed was similar to that used previously in their investigation of the relationship between model potentials and short-range repulsive forces [3]. The solvent particles were hard ellipsoids with an axis ratio of 5:1, while the single solute was a hard ellipsoid of a variety of sizes and shapes. In this case, however, solvent and solute particles also had a point quadrupole moment fixed at their centres. The orientational behaviour for a wide collection of solutes of different shapes and quadrupole moments was investigated. One key result was that the efg sampled by the solute was found to be highly sensitive to the details of the solute properties. In fact, for any non-spherical solute (prolate or oblate), the sampled efg actually changes sign, as well as magnitude, when the solute quadrupole moment changes sign. It should be noted that this observation is in sharp contradiction to certain key NMR measurements, notably those of Syvitski et al. in which the indirect measurement of the average efg experienced by several methyl and chloro-substituted benzenes of various quadrupole moments yield values reasonably consistent for each liquid crystal studied [5,42,43]. On the other hand, the value of the efg predicted by Eq. (15.18) was rather accurate. Despite this, the solute order parameters calculated using Eq. (15.17) for spherical solutes (for which the only anisotropic contribution to the solute-solvent interaction is the quadrupolar force) showed poor quantitative agreement with the simulation results, although they were qualitatively correct. Of course, a rigorously spherical solute possesses zero molecular quadrupole moment and therefore has all its order parameters equal to zero.

The qualitative discrepancy between simulation and experiment concerning the dependence / independence of the solute properties on the measured efg almost certainly lies in the drastic approximation of representing the electrostatic molecular properties by point quadrupoles, and thus neglecting all of the higher order electrostatic interactions which are likely very prominent at short solute-solvent distances. Thus, the molecular model employed is too simplistic to be used in mean-field theories. As well, the quantitatively poor agreement between simulation and the mean-field theory for the prediction of solute orientational ordering indicates that the mathematical approximations used in the theory are too severe. In light of these shortcomings, an accurate


theoretical description of the electrostatic contributions to orientational ordering of solutes in a nematic solvent is clearly not provided by the mean-field theory above.

The effects of quadrupolar interactions on the orientational behaviour of solutes in liquid crystal mixtures have also been studied by Burnell et al. [6]. As in the study of Polson et al. above, solute and solvent particles were both ellipsoidal in shape and had point quadrupoles embedded at their centres. In other respects, the model system was different. For example, the solvent aspect ratio was 3: 1 and interacted with other solvent and solute particles with a Gay-Berne potential. The same GayBerne parameters characterized both the solvent and solute which can thus be thought of as having the same size and shape as the solvent. In addition, the solvent was a binary mixture of particles with positive and negative quadrupoles of the same magnitude. Finally, there were 30 solutes out of a total of 1000 particles which were divided into equal subsets of those with positive and negative quadrupoles of the same magnitude, and with zero quadrupole moment.

It was observed that the quadrupolar interactions have essentially no effect on the orientational order parameter of the solute, consistent with the results of the study of the hard-ellipsoid system in the case where the solute has the same dimensions of the liquid crystal [4], though large quadrupoles were found to enhance orientational order. In an even mixture of the two solvent particles, the efg at solutes of vanishing quadrupole moment was found to be Fzz=O, as expected by symmetry. Thus, this mixture is analogous to the so-called "magic mixtures" of nematic liquid crystals for which the efg of D2 is measured directly from an analysis of the NMR coupling constants to be zero. As mentioned above, there is experimental evidence that various solutes experience similar efg's. By contrast, though consistent with the observation in the previous study by Polson and Burnell of hard ellipsoid systems, it was found here that the solutes of different quadrupole moments generally experience considerably different efg's. On the other hand, a special case was also seen where the efg is approximately constant for each of the three different solutes present for each of three different nematic mixtures studied in the limit of small solute and solvent quadrupole moments. Finally, it was also observed that the efg values change with liquid crystal composition in a manner that is independent of the sign and magnitude of the solute quadrupole, i.e. Fzz is proportional to the solvent quadrupole moment for lower values of this quantity. Such an effect is consistent with the observed change with the liquid crystal composition of experimental order parameters such as acetylene, benzene, hexafluorobenzene, methyl fluoride and D2, though it does not explain the negative order parameters observed for acetylene and D2 in a liquid crystal with a negative efg (as measured with D2)'

Again, the origin of the discrepancy between experiment and simulation is connected to the severe approximation of representing molecular electrostatic properties with a single point quadrupole. As pointed out in each of the two studies summarized above, the interaction between point quadrupoles leads to a strong molecular reorganization at short inter-particle distances that depends on the signs and magnitudes of the quadrupoles. These quadrupole-dependent changes in the interparticle displacement distribution in tum cause the variation in the efg measured at the site of one quadrupole (i.e. particle centre) produced by the surrounding quadrupoles. The fact


that the measured efg becomes independent of the solute quadrupole at low values of solute and solvent quadrupole moment is due to the fact that the perturbing effect of the quadrupolar interaction on the solvent structure around the solute vanishes in this limit.

In an attempt to circumvent the problems associated with point quadrupoles in simulation models of solutes in nematic solvents, Lee and Burnell have recently investigated the effect of "spreading out" the solute quadrupole over the the volume of the solvent [9]. In this study, five axially symmetric solutes (acetylene, allene, propyne, 2-butyne and 2,4-hexadiyne) were modeled as Gay-Berne ellipsoids with dimensions commensurate with those of the molecular structures. The liquid crystal solvent was modeled as a collection of Gay-Berne particles with an axis ratio of 3:1. As in previous studies, point quadrupoles were placed at the centres of the solvent particles. Three different solvents were considered: one involving positive quadrupoles, another with negative quadrupoles with the same magnitude as the first, and an even mixture of both types of mesogens. In contrast with the solvent model, multiple positive point quadrupoles were fixed on the solutes, close to or at the the sites of the chemical bonds, with the quadrupole moments varying proportional to the bond strength. The justification for this simple model follows from the fact that a positive quadrupole can be pictured as two negative charges on the centre and each of two positive charges along the same axis but in opposite directions. In this sense, it crudely resembles the distributions of excess negative charge in the region of the bond, and positive fractional charges associated with the neighbouring atoms.

The simulations reproduce the basic orientational behaviour of the solutes as determined from NMR measurements. Generally, the order parameter increases with increasing solute length in each solvent studied. This effect is due to the anisotropic short-range repulsive component of the solute-solvent force, and is consistent with earlier studies. More interestingly, variation of the solute ordering with liquid crystal composition was found to be consistent with experiment in every case. NMR experiments have demonstrated that the orientational ordering of each of these solutes (each of which has a positive quadrupole moment) increases when passing from a solvent in which the measured efg is negative, to the "magic mixture" where the efg vanishes, to a liquid crystal in which the efg is positive. In addition, the negative order parameter of acetylene, the solute in this study with the weakest shape anisotropy and whose orientational ordering therefore is most affected by long-range interactions, is reproduced.

The encouraging results of this preliminary study are consistent with the suggestion by Polson and Burnell that at least some of the inconsistency concerning the effect of molecular quadrupolar interactions between experimental studies and the previous simulation studies is due to the poor approximation that such interactions provide for the true electrostatic interactions at short intermolecular distances. It is expected that improving the modeling of electrostatic interactions by designing particles - solvent, as well as solute - with more realistic charge distributions will enhance the consistency between the simulation and experiment without the need to resort to atomistic detail

Simulations of Orienta tiona I Order of Solutes in Liquid Crystals 343

in the model, though the latter approach could also provide a useful complementary study.

5. Conclusions Computer simulation methods can provide an effective means to study the orien

tational ordering and the conformational behaviour of solutes in liquid crystals. The usefulness of this approach lies in its ability to aid in the analysis of order parameters obtained from NMR coupling constants, for example, by testing the validity of various assumptions inherent in any model employed in the analysis. In addition, it can be used to test the approximations employed in mean-field theories of solute ordering, and thus provide a guide for the improvement of the theories. In this way, it can elucidate the importance and effects of various anisotropic intermolecular forces on the orientational ordering.

The simulation studies described in this Chapter have demonstrated the effectiveness of this approach. For example, they have both confirmed the legitimacy of approximations [1] and predictions [10] of molecular field theories, as well as illustrated the shortcomings of such theories [4,12]. Other simulations have confirmed the conclusions of earlier experimental NMR studies of the importance of short-range repulsive interactions for solute ordering [3,7,8,13], as well as the importance of long-range electrostatic forces in general [7,8,16]. Other studies focusing on the conformational behaviour of flexible solutes have demonstrated the limitations of, and inaccuracies resulting from the use of model potentials and theories to extract the conformational distributions of flexible solutes from NMR couplings [11,14]. Finally, recent simulation studies by Burnell and coworkers investigating electrostatic interactions in nematic solvents in the context of a molecular quadrupole interaction with an average solvent efg have demonstrated the pitfalls of employing an overly simplified model for the interactions [4,6], and suggested a straightforward extension of the model to improve the agreement with experimental results [9].

It is important to note that the simulations in most of these studies employed relatively simple models, demonstrating the effectiveness of such an approach. With the rapid increase in accessible, powerful computational resources, it is expected that simulations employing more realistic, detailed molecular models will become more commonplace, and provide a useful complementary alternative to the simpler models, and thus provide even deeper insight into the orienting mechanisms of solutes in liquid crystalline phases.

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[8] Celebre, G. (2001), J. Chern. Phys., 115:9552. [9] Lee, J.-S. J. Undergraduate thesis, Department of Chemistry, University of British Columbia, 2001.

[10] Palke, W. E., Emsley, J. w., and Tildesley, D. J. (1994), Molec. Phys., 82:177. [11] Alejandre, J., Emsley, J. w., Tildesley, D. J., and Carlson, P. (1994), J. Chern. Phys., 101:7027.

[12] Sandstrom, D., Komolkin, A. V., and Maliniak, A. (1996), J. Chern. Phys., 104:9620.

[13] La Penna, G., Foord, E. K., Emsley, J. w., and Tildesley, D. J. (1996), J. Chern. Phys., 104:233.

[14] Palke, W. E., Catalano, D., Celebre, G., and Emsley, J. W. (1996), J. Chern. Phys., 105:7026.

[15] Sandstrom, D., Komolkin, A v., and Maliniak, A. (1997), J. Chern. Phys., 106:7438.

[16] Lounila, J., and Rantala, T. T. (1991), Phys. Rev. A, 44:6641.

[17] Emsley, J. w., Hashim, R, Luckhurst, G. R, Rumbles, G. N., and Viloria, F. R. (1983), Molec. Phys., 49:1321.

[18] Emsley, J. w., Luckhurst, G. R., and Stockley, C. P. (1982), Proc. Royal Soc. A, 381:117.


[20] Ferrarini, A., Moro, GJ., Nordio, P.L., and Luckhurst, G.E. (1992), Molec. Phys., 77:1. [21] van der Est, A. J., Kok, M. Y., and Burnell, E. E. (1987), Molec. Phys, 60:397.

[22] Zimmerman, D. S., and Burnell, E. E. (1990), Molec. Phys., 69:1059.

[23] Zimmerman, D. S., Li, Y., and Burnell, E. E. (1991), Molec. Cryst. Liq. Cryst., 203:61.

[24] Zimmerman, D. S., and Burnell, E. E. (1993), Molec. Phys., 78:687.

[25] Burnell, E. E., and de Lange, C. A. (1998), Chernical Reviews, 98:2359.

[26] Rosen, M. E., Rucker, S. P., Schmidt, C., and Pines, A. (1993), J. Phys. Chern., 97:3858.

[27] Straley, J. P. (1974), Phys. Rev. A, 10:1881.

[28] Janik, B., Samulski, E. T., and Toriumi, H. (1987), J. Phys. Chern., 91:1842.

[29] Rendell, J. C. T., Zimmerman, D. S., van der Est, A. J., and Burnell, E. E. (1997), Can. J. Chern., 75:1156.

[30] Celebre, G., De Luca, G., Longeri, M., Catalano, D., Veracini, C. A., and Emsley, J. W. (1991), J. Chern. Soc. Faraday Trans., 87:2623.

[31] Chandrakumar, T., Polson, J. M., and Burnell, E. E. (1996), J. Mag. Reson. Series A, 118:264.

[32] Tzusuki, S., and Tanabe, K. (1991),J. Phys. Chern., 95:139.

[33] Di Bari, L., Forte, C., Veracini, C. A., and Zannoni, C. (1987), Chern. Phys. Lett., 143:263.

[34] Patey, G. N., Burnell, E. E., Snijders, J. G., and de Lange, C. A (1983), Chern. Phys. Lett., 99:271.

[35] Barker, P. B., van der Est, A. J., Burnell, E. E., Patey, G. N., de Lange, C. A, and Snijders, J. G. (1984), Chern. Phys. Lett., 107:426.

[36] van der Est, A. J., Burnell, E. E., and Lounila, J. (1988), J. Chern. Soc., Faraday Trans. 2,84:1095. [37] Barnhoom, J. B. S., and de Lange, C. A. (1994), Molec. Phys., 82:651.

[38] Burnell, E. E., de Lange, C. A., Segre, A. L., Capitani, D., Angelini, G., Lilla, G., and Bamhoom, J.B.S. (1997), Phys. Rev. E,55:496.

[39] van der Est, A. 1., Burnell, E. E., Bamhoom, J. B. S., de Lange, C. A., and Snijders, J. G. (1988), J. Chern. Phys., 89:4657.

[40] Snijders, J. G., de Lange, C. A., and Burnell, E. E. (1983), J. Chern. Phys., 79:2964.

[41] Snijders,1. G., de Lange, C. A., and Burnell, E. E. (1983), Isr. J. Chern., 23:269.

[42] Syvitski, R. T., and Burnell, E. E. (1997), Chern. Phys. Lett., 281:199.

[43] Syvitski, R T., and Burnell, E. E. (2000), J. Chern. Phys., 113:3452.

[44] Emsley, J. W., Heeks, S. K., Home, T. J., Dowells, M. H., Moon, A., Palke, W. E., Patel, S. U., Shilstone, G. N., and Smith, A. (1991), Liq. Cryst., 9:649.

[45] Emsley, J. w., Luckhurst, G. R., and Sachdev, H. S. (1989), Molec. Phys., 67:151.

[46] Emsley, J. W., Palke, W. E., and Shilstone, G. N. (1991), Liq. Cryst., 9:643.

IV

DYNAMIC ASPECTS AND RELAXATION

Much of the material in the previous parts of this book has concentrated on the spectroscopy of NMR of orientationally ordered systems, and on the detailed information that can be obtained from the observable spectroscopic parameters. An additional important aspect ofNMR of orientationally ordered molecules lies in the time dependence, and hence in the dynamical aspects, of the physical processes that are involved. This part of the book focuses on the time-dependent phenomena that can be distinguished in this field of research. The contributions range from studies of time-dependent processes underlying the details of translational diffusion and of reorientational motion in anisotropic potentials, to the time-dependence of intramolecular motion such as interconversion among various conformations.

Relaxation processes in NMR have always been an important source of dynamical information in isotropic systems. It comes therefore as no surprise that NMR relaxation can also provide useful insight into the dynamics that underlie orientational order in the field of orientationally ordered species. Chapter 16 deals extensively with the theory of NMR relaxation in partially ordered systems. Although many different relaxation processes occur in principle, theory can provide reliable insight about which processes may dominate under different circumstances. The theory developed in this Chapter is supported by an abundance of experimental results.

An important problem with NMR relaxation processes is that they occur over a wide range of time scales, and that separation of the various contributions at one particular frequency in the power spectrum that describes the interactions with the lattice is never easy. NMR relaxation processes even take place at very low frequencies. This part of the power spectrum can only be addressed if measurements can be carried out as a function of external magnetic field. Chapter 17 addresses the experimental and interpretational problems that arise in the area of field-cycling experiments. Despite the technical sophistication required to perform such experiments, important and unique information has been obtained, particularly on extremely slow reorientational processes in spatially constrained liquid crystals.

Another time-dependent phenomenon in NMR experiments is diffusion of molecules. Of course it is well known that in the case of isotropic liquids the use of NMR for the study of diffusion processes has been invaluable. In Chapter 18 an overview is given of the scope and power of pulsed-field gradient experiments that are used to measure diffusion constants in orientationally ordered systems. Such anisotropic diffusion processes are studied in considerable detail in biological systems.

Slow intra- and intermolecular motions can affect the appearance of NMR spectra. In Chapter 19 an authoritative overview is given of the effects that can be expected and observed in orientationally ordered systems, where the time scale of the molecular motion is in special motional narrowing regimes. In such cases the NMR line shapes are affected by the type and the time scales of the motions. Dramatic effects are expected and observed when the correlation times for the motions are of the same order as the inverse of the NMR interactions that are modulated by these motions.


Chapter 16

SPIN RELAXATION IN ORIENTATIONALLY ORDERED MOLECULES

Ronald Y. Dong Department of Physics and Astronomy, University of Manitoba and Brandon University, Brandon, Man

itoba, Canada

1. Introduction Nuclear Magnetic Resonance (NMR) has been shown to be an extremely power

ful technique for investigating molecular orientational order and dynamics in partially ordered systems such as thermotropic and lyotropic liquid crystals (LC) [1]. In this Chapter, nuclear spin relaxation of orientationally ordered molecules is described. In addition, theoretical models explaining NMR observables are outlined for various dynamical processes in LCs. It is known that nuclear spin-lattice relaxation rates contain information on how a nuclear spin system exchanges energy with its surrounding "lattice", i.e., all degrees of freedom in the physical system of interest except those of the nuclear spins. Pulsed NMR provides a highly versatile tool for measuring various spin relaxation rates which can probe the entire spectrum of molecular motions in LCs. As in ordinary liquids, mesogenic molecules can reorient and translate, as well as execute internal motions if they are non-rigid. Furthermore, these molecules align preferentially in a certain direction labeled by the director no and possibly also arrange spatially to form various layered structures. When these molecules move collectively, the local director fluctuates both spatially and temporally. These unique motions are known as order director fluctuations (ODF). All the dynamical processes mentioned can contribute to the spin relaxation in LCs. In addition, cross relaxation due to possible couplings between different motions may also exist.

In relating the measured spin relaxation rates to molecular behaviour, several nuclear interactions may simultaneously contribute to the relaxation of a spin system. These may include the magnetic dipole-dipole interaction, the quadrupole interaction, the spin-rotation interaction, the scalar couplings of the first and second kind, and the chemical shift anisotropy interaction. Due to the need of estimating certain nuclear couplings and / or correlation times associated with different types of motions, considerable uncertainty exists in identifying and separating these contributions. The coupling between the nuclear Zeeman reservoir and the lattice is magnetic in all cases (e.g. dipole-dipole interaction) except one. The exception is an electrical



coupling between the nuclear quadrupole moment Q and the lattice via an electric field gradient (efg). When the quadrupolar coupling exists, it is generally more efficient than any magnetic coupling. Thus, a quadrupolar nucleus is preferred when other spin interactions exist, since its relaxation behaviour is more amenable to theoretical interpretations. Given that a deuteron has a small quadrupolar coupling constant e2qQ/h, typically 150-250 kHz, it is large enough so that relaxation is dominated by the quadrupole interaction, but small enough so that perturbation theory is applicable. The dipole-dipole coupling between a pair of deuterons is much weaker (:::; few kHz for methylene deuterons) than the quadrupole interaction, and one can normally consider deuterons in deuterated LCs as a collection of isolated spins. Although the deuteron spin eH, I = 1) has become a favoured probe in recent years, both proton and carbon-13 NMR relaxometry can provide complementary information on the motional behaviour of LCs.

The most difficult problem in any relaxation theory is the calculation of correlation functions or their Fourier transforms, the spectral densities of motion. Let Hq(t) be a component of the spin Hamiltonian which fluctuates in time owing to molecular motions. In addition to the mean square spin interaction < H~(t) >, the time autocorrelation function < Hq(t)Hql(t - T) > is needed for spin relaxation. The latter can often be approximated [2] by an exponential decay function of T, i.e.,

(16.1)

where the angle brackets denote an ensemble average, and Tc is a characteristic correlation time of the motion. Nuclear spin-lattice relaxation time (Td studies are insensitive to molecular motions that are either too fast (WOTe « 1) or too slow (WOTe » 1) with respect to the inverse of the Larmor frequency WOo To extend the spectral window, measurements of spin-spin relaxation times (T2) and / or spin-lattice relaxation times (TIp) in the rotating frame can prove useful.

2. Average Hamiltonian Due to anisotropic motions in partially ordered fluids, the spin Hamiltonian is not

motionally averaged to zero in time. The non-zero average spin interactions give rise to spectral splittings or a broad spectral pattern which can provide information on the molecular order and structure. Although the potential of mean torque is addressed elsewhere in this book (see Chapter 12), it is described here to show how deuteron spin relaxation rates and spectral splittings can be understood in a self consistent fashion, especially for the case of internal chain motions.

2.1 Quadrupolar Hamiltonian

The nuclear quadrupole interactions occur whenever the spin I 2: 1. In the principal axis (x', y', z') system of the negative of the electric field gradient tensor Va,B (eq = VZIZI and 'T7 = (VXIXI - Vylyl )/VZIZI is the asymmetry parameter), the quadrupolar

Spin relaxation in orientationally ordered molecules 351

Hamiltonian is written as

'HQ = IiYJQ { [I;, - ~1(1 + 1)] + ~(I;, - I;,) } (16.2)

where the frequency wQ = 3e2qQ/41(2I - 1)1i. The magnitude of the interaction is such that it completely determines the NMR spectrum for most nuclei centered at wo, even for a deuteron that possesses a small coupling constant (this magnitude can range up to several MHz for nuclei such as 14N). In the case of 2H, "axial" symmetry is usually a good approximation for the efg tensor, i.e. ry = 0 (ry rv 0.04 for aromatic deuterons) and the principal z' axis is along the direction of the C-2H bond. The principal axis frame does not necessarily coincide with the laboratory (XL, YL, ZL)

---+ frame defined by the external Bo field. 'HQ can be expressed in the laboratory frame using second rank irreducible spherical tensors T2,m and R2,m:

(16.3)

Table 1 lists T2,m and P2,m (which is R2,m given in the principal frame of the efg tensor). The R2,-m in the laboratory frame can be obtained in terms of P2,m using the

Wigner rotation matrices D~ m' (</J, (), 't/J) [3,4] ,

R2,-m = L D:''"m,m,(</J, (), 't/J)P2,m' (16.4) m'

TABLE 1. Irreducible spherical tenors P2,m and T2,m for various spin interactions

>. P2,±2 T2,o T2,±1 T2,±2

D 3 -3 0 1 - - =r~(Ii± liZ + Iizlf) lI±I± "2 r ii 7s[3Iizliz - Ii· Ii] 2 , J

Q ~ eq ~eq1J +S[3I; - 1(1 + 1)] =r~(I± Iz + IzI±) HI±?

CSa ~o ~01J FzBo =rF±Bo 0

au "'i3 are chemical shift (CS) tensor elements and 6 = U33 - ~ Tr&, Ull - U22 = 6T}.

Since 'HQ is much smaller than the Zeeman Hamiltonian 'Hz = liYJo1z in the high-field approximation, one can calculate the spectrum using first-order perturbation theory. Only the part of 'HQ (m = 0) that commutes with 'Hz is required:

Using the relation D:;: nCO) = (_l)m-n D~m -nCO), the above equation reduces to , ,


--+ where the polar angles (0, 'ljJ) give the orientation of Bo in the principal efg frame. For a deuteron (l = 1) to first order in 'HQ, a doublet is symmetrically displaced about Wo with a splitting

(16.7)

In a "powder" sample, the principal frames of the efg tensors are randomly distributed, and a powder spectral pattern results with 900 edges (0 = 900 , 'ljJ = 900 for yl axis) separated by a splitting of t5vQ = wQ(1 + "') /27r, and x' shoulders at ±wQ (1 - "') / 47r and z' shoulders at ±wQ /27r. The powder pattern for the case of", = 0 is the familiar Pake pattern [5]. To find the nonzero 'HQ due to anisotropic tumblings in LCs, one needs to introduce an intermediate molecular frame (e.g., a set of diffusion axes) which describes the molecular motion. Suppose for simplicity that the director no is along --+ Bo. The motional average of R2,0 in the (XL, YL, ZL) frame can be written in terms of P2,m using an extra transformation via the molecular frame

R2,0 = L L D5~mll(O, 'ljJ)D;;;II,m/({3, a)P2,ml (16.8) m'm"

where, = 0 is used for the case of the molecular z M axis possessing at least a three-fold symmetry with respect to the molecular motion. In R2,0, the time-averaged Wigner

rotation matrices D5 m(O, 'lj;) are the order parameters for molecules in a uniaxial , phase, and can be written as a Cartesian order tensor S [6] which is symmetric and traceless (see Chapter 1). The ordering of molecules in a mesophase may be described by a singlet probability function P((), 'ljJ). The molecular frame may often be chosen

such that P(O, 'ljJ) is an even function of the polar angles such that D5,±1 = 0 and

D52 = D5 -2' Consequently, one obtains from 'HQ a deuterium quadrupolar splitting , ,

(16.9)

where f(a,{3) = (3cos2 (3 - 1) + ",sin2 (3cos2a, g(a,{3) = [3sin2 (3 + ",(1 + cos2 (3) J cos 2a, and Sii are the principal elements of S. It is obvious that the two independent order parameters in Eq. (16.9) cannot be determined from a single quadrupolar splitting. When internal motions exist in a flexible chain, the geometric factors f( a, (3) and g(a, (3) need an additional conformational average (see Chapters 13 and 14).

2.2 Dipole-dipole Hamiltonian In terms of irreducible spherical tensor operators, the dipolar Hamiltonian (in SI

energy units) is

2

L (_1)m F2,-m(()ij, <Pij)T2,m (16.10) m=-2


where 'Yi is the gyromagnetic ratio of a nuclear spin, J.Lo is the magnetic vacuum permeability, rij is the internuclear distance between a pair of spins, and the functions F2,m describe the orientation (Oij, </Jij) of Tij in the laboratory frame:

F2 ,o(O, </J) F2,±1(O,</J)

F2,±2(O, </J)

= J3/2(3cos2 0 - 1)

=F 3 sin 0 cos 0 exp (±i</J)

(3/2)sin2 0exp (±2i</J).

(16.11)

Again 'HD may be treated as a first-order perturbation on 'Hz in the high-field approximation, and only the m = 0 part in Eq. (16.10) is retained to give a truncated 'H'r;. Using the coordinate transformation given in Eq. (16.8), the motional average of 'H'r; is

(16.12)

where (},'l/J are the polar angles of no in the molecular frame and 0:ij, {3ij the polar

angles of Tij (the principal z axis of the Dij tensor) in the same frame. As in the quadrupole case, the single Zeeman line now shows a dipolar splitting given by

(16.13)

2.3 Chemical shift (CS) Hamiltonian -->

When nollEo, the motional average of the chemical shift in the high-field approxi-mation is given by

(16.14)

where P2,m is given in Table 1 and the isotropic shift (in a liquid) O"iso = iTrc5-. In the principal (1,2,3) frame of fT, the difference in chemical shift 0" in the mesophase can be related to O"iso according to Eq. (16.14):

O"aniso - O"iso = ~833 [0"33 - ~(0"11 + 0"22)]

1 + "3(811 - 822 )(0"11 - 0"22)

(16.15)


where 8 11 , 8 22 , and 8 33 are not necessarily the principal components of the order matrix. In a liquid crystalline phase where the principal (x, y, z) frame of the molecular order matrix has been defined, a similar equation can be obtained:

O'aniso - O'iso = ~ [O'zz - ~(O'xx + O'YY)] 8zz

1 + 3(O'xX - O'yy)(8xx - 8yy ).

(16.16)

Since the principal elements of fT are not easily accessible in the mesophase, the calculated order parameters from the chemical shift anisotropy contain rather high uncertainties.

3. Spin relaxation theory In an orientationally ordered medium, the fluctuating dipolar Hamiltonian or quadru

polar Hamiltonian with an axially symmetric efg tensor is

'H.~(t) = 2: A2,mL {D~L,O[n(t)]- D~L'O} (16.17) mL

where m L is the projection index of the Hamiltonian onto the Z L axis,

(16.18)

for a deuteron, and for a dipolar Hamiltonian

A2,mL = -J318(fJ-o'Yi'Y/il1rr~j)T2,mL' (16.19)

The time dependence in Eq. (16.17) comes from the Euler angles n L,Q which denote the

orientation of the principal efg (or Dij) frame in the (XL, YL, ZL) frame. Assuming that the cross products between spin Hamiltonian matrix elements of different m L

values can be ignored, the spectral densities are

Jo.(3o.l (3' (Wo.(3) = 2: (aIA2,mL 1,6) (a'IA2,mL 1,6') * JmL (Wo.(3) (16.20) mL

where wo.(3 is the transition frequency between the eigenkets la) and 1,6) of the static Hamiltonian, and

JmL(W) = 100 GmL(r)e-iWT dr

with the time auto-correlation functions

(16.21)

GmL(r) = ({ D~L,O[n(t)]- D~L'O} {D;:L,O[n(t - r)]- D;;:L,O }). (16.22)

It should be noted that the JmL(w) are quantities that are obtained from experiments without reference to any dynamic model. By applying Redfield theory [7], the deuteron


T1z and TIQ relaxation times of the Zeeman and quadrupolar orders, neglecting contributions from the dipolar Hamiltonian, are given by [8,9]:

Tii = KQ [J1(wo) + 4J2(2wO)]

Tui = 3KQJ1(wo) (16.23)

where KQ = (31f2/2) (e2qQ/h)2. In aligned LC samples, both T1z and TIQ can be simultaneously measured using either the Jeener-Broekaert pulse sequence [10] or the broadband Wimperis pulse sequence [11-13]. When treating spin-spin relaxation of a deuteron spin, there are three independent observables (T2a, T2b, and T2D) [8, 9]:

T.- 1 2a = KQ [~Jo(O) + ~Jl(WO) + J2(2WO)]

T.- 1 = KQ [~Jo(O) + ~Jl(WO) + J2(2WO)] (16.24)

2b

T.- 1 2D = KQ [Jl (wo) + 2J2(2wO)]

The quadrupolar echo [14] experiment can be used to give T2a. The double quantum spin-spin relaxation time T2D can be obtained using the pulse sequence 90~ - T -

45~ 90~ - t - 45~ [9]. The first three pulses create the double-quantum coherence, and the last read pulse converts the double-quantum to single-quantum coherence for detection.

Treatment of spin-lattice relaxation of an isolated spin-~ pair by an intramolecular

dipole-dipole interaction is identical to that for a spin-l system. For two like spin-~ nuclei separated by a distance r, the Zeeman spin-lattice relaxation time T1z is given by Eq. (16.23), but with a different multiplicative constant KD = ~(/ko'Y21i/41fr3)2. Sim

ilarly, the spin-spin relaxation rate for a spin-~ pair is, according to T2~1 in Eq. (16.24), given by

Til = KD [~Jo(O) + ~Jl(WO) + J2(2WO)] (16.25)

The spin-lattice relaxation rate (T1-,,1) in the rotating frame is given by

(16.26)

where wI/'Y is the spin-locking Bl field. Finally, the Tl expressions for 13C relaxation

at the 13C Larmor frequency WI due to heteronuclear dipolar coupling and chemical shielding anisotropy are given by

Tid~p = Kb [~JO(WI - ws) + J1(WI) + 2J2(WI + ws)] (16.27)

-1 3 2 TICS = "2 (')'liBoa zz) Jl (WI) (16.28)

where K~ = ~(/ko'Ynsli/41fr3)2.


4. Motional models The evaluation of correlation functions is a daunting task for any spin relaxation

theory. To complicate matters further, several motional processes can occur simultaneously in the studied material. Thus, couplings between these processes must be properly treated. Besides the unique ODF [15], other dynamic processes that can cause spin relaxation in Les may include molecular reorientations, translational diffusion and internal rotations.

To an excellent approximation, deuterons are relaxed purely by intramolecular interactions. Translational diffusion which induces spin relaxation due to modulations of dipolar interactions between a pair of spins on two molecules does not directly affect deuteron spins. Intermolecular dipolar interactions must, however, be taken into account for proton spin relaxation. To evaluate GmL (T) in Eq. (16.22), the coordinate transformation from the principal efg frame to the laboratory (XL, YL, ZL) frame must be carried out through successive transformations to account for the fast motion of a molecule and the slow collective fluctuations of the director. Thus, n L,Q == (n L,N , nil, n' , n M,Q ), where: the Euler angles n M,Q transform from the efg frame to the principal molecular frame (x', y', z') defined, for example, via the rotational diffusion tensor of the molecule; n' is used to transform from the (x' , y' , z') frame to the instantaneous director (x" , y", z") frame; nil is used to transform from the (x", y", z") frame to the crystal (X, Y, Z) frame defined by the average director (no); and nL,N is used to transform from the (X, Y, Z) frame to the (XL, YL, ZL) frame.

---)

For simplicity, it is assumed that the sample has nollBo Uh,N = 0) and (for the mo-ment) that molecules do not possess intramolecular motions (i.e., f3M,Q is assumed to be time-independent). A major simplification in GmL (T) is achieved by separately averaging the director modes and the molecular reorientation. This is possible if the time scales for these motions are very different, and GmL (T) is given by [1]

GmL(T) = :L :L [d~M,0(f3M,Q)]2l:,i,m(T)g;;',mM(T) (16.29) mM m

where g~f,m(T) and g!$;,mM(T) are the reduced correlation functions for ODF and molecular reorientation, respectively. In general, there are three types of terms that contribute to Gm L ( T): order director fluctuations, molecular reorientation, and a crossterm [16] arising from both these motions.

4.1 Order director fluctuations ODF involve collective motions of a large number of molecules about no. These

motions are perceived as hydrodynamic phenomena and governed by molecular properties such as viscosities and elastic constants of the medium. The reduced correlation functions g~i,m in Eq. (16.29) are:

(16.30)


where the following conditions are assumed: (i) D;" m[n"(t)] = 0 is used as a result of random fluctuations of the local director; (ii) 'Y~l= 0 since there is cylindrical symmetry about the local director; and (iii) the azimuthal motion of the instantaneous director n around no is assumed to be slow or nearly frozen-in in comparison with its reorientation (/3" - motion). Thus, the director motion is solely described by the Euler angle /3". When a small angle is assumed (i.e., sin/3" ~ /3"), ODPs do not contribute to the spectral densities Jo(w) and J2(2w), but do contribute a frequency dependent term to J1 (w) that is different from the Lorentzian form given by BPP theory [2]. When higher orders of /3" are included, ODP do make small contributions [17, 18] to Jo(w) and J2(2w). Now the spectral density J1(w) due to ODP in the nematic (N) phase is, up to second order, given by

JPF(w) = A(I- 4a)S5 [d60(/3M,Q)] 2 U(wc/w)/ w1/ 2 (16.31)

where the cutoff function U (x)

U(x) = 1 I [x-ffx+ 1] 27r n x + ffx + 1

+ ~ [tan- 1( ffx - 1) + tan- 1( ffx + 1)]

(16.32)

is introduced to limit coherent modes in the ODP spectrum by a high cutoff frequency WC' The standard prefactor A is

(16.33)

with T the temperature, ry the average viscosity, and K the average Prank elastic constant. The parameter a = (kT /27r2) (rywc/ K3)1/2 is a measure of the magnitude of ODP, and So is the nematic order parameter of the molecule relative to the local director. So is related to the usual nematic order parameter < P2 > according to So =< P2 > /(1-3a). Typical a values are less than O.lforLCs. Purtherrefinements of Eq. (16.31) include the effects of finite sample size and chain flexibility [19]. In smectic A (SmA) phases, ODPs appear as layer undulations; the two-dimensional character of such phases leads to a linear frequency dependence in J p F (w ), i. e., JPF(w) oc w-1 [1].

4.2 Anisotropic rotational diffusion

The reduced correlation functions g;;;,mM in Eq. (16.29) which describe the overall reorientation of molecules are given by

g;;',mM(T) = (D~,mM[n/(t)]D~,mM[n/(t + T)]-ID;",mMI2). (16.34)

To evaluate these correlation functions for LCs, the small-step rotational diffusion model [20-22] may be used. The rotational diffusion equation must be solved in the presence of the potential of mean torque to get the conditional probability for a molecule in a certain orientation at time t given that it has a different orientation at t = O. This,


and the equilibrium probability for finding the molecule with a certain orientation, are required to evaluate g;;;,mM (T). In general, the orientational correlation functions for a rigid symmetric top can be written as a sum of decaying exponentials [23], and in the notation of the Tarroni and Zannoni (TZ) model:

g~n(t) = I:(A~n)K exp[(B!n)Ktj (16.35) K

where m and n represent the projection indices of a rank 2 tensor in the laboratory and molecular frames, respectively. (B!n)K / D ..L, the decay constants, are the eigenvalues of the rotational diffusion matrix. (A~n) K , the relative weights of the exponentials, are the corresponding eigenvectors. Now the decay constants contain the model parameters DII and D..L specifying rotational diffusion constants of the molecule (defined in a molecular frame) about its long axis and of the long axis, respectively. The spectral densities for a deuteron residing on a rigid uniaxial molecule are

(16.36)

where m = 0,1, or 2. Different approaches have been used to tackle the existence of a cross-term between the reorientational motion and ODE In particular, they produce different cross-terms, and with opposite signs. Debates still exist regarding how to properly obtain the correlation functions when the time scales of collective and individual molecular motions become comparable. It should be mentioned that other models, such as the anisotropic viscosity model, and the third-rate anisotropic viscosity model can also be used [1] to describe this motion.

4.3 Correlated internal rotations When deuterons are situated in the flexible chain(s) of a molecule, the C-2H bond

also experiences internal motions. To treat the quadrupolar splittings and spin relaxation rates of these deuterons, one needs to consider all possible conformations available to the chain. These conformations can be generated using the rotameric model of Flory [24]. The problem becomes rather complex due to a large number (N) of possible conformers available to the molecule. Furthermore, the ordering of each conformer in the potential of mean torque has to be specified. Of course, conformational transitions are fast on the deuterium NMR time scale and the observed quadrupolar splittings are time-averaged over all available configurations of the molecule. The quadrupolar

splitting can be related to the segmental order parameter 8gb of the C i -2H bond:

3 ( 2 (i) 6.Vi = '2 e qQ/h)i8CD (16.37)

Now 8gb can be calculated as a weighted average of the segmental order parameter

8~bi for the Ci-2H bond (along the b axis) of the molecule in conformation n, using the eqUilibrium probability Pn of conformer n:

Spin relaxation in orientationally ordered molecules

N

S (i) '" Sn,i CD = ~Pn bb'

n=l

359

(16.38)

Now for each conformer n, S~bi = E~'Y'z S:;a cos2 o~t where S:;a denote principal

components of the order matrix for the conformer nand 0:: denotes the angle between the Ci-2H bond in the conformer n and the Q axis of the principal axis system for the external potential of mean torque Uext(n,O). The potential energy U(n, 0) of a molecule in a particular conformer n having orientation 0 (= 0, 'Ij;) with respect to no is given by U(n,O) = Uint(n) + Uext(n, 0), where the internal energy can be approximated by Uint(n) = NgEtg with Ng being the number of gauche linkages in a chain. E tg denotes the energy difference between the gauche linkage and the trans.

Now Uext(n,O) is responsible for the alignment of conformer n and originates from the molecular field of its neighbors. In the additive potential (AP) method (see Chapters 13 and 14) [25,26], the molecule is divided into a small number of rigid segments. Each segment is associated with an interaction tensor that is independent of the conformation. The interaction tensor of the molecule is calculated by transforming the segmental interaction tensors from their local axis systems into a common molecular frame and then adding them together. Suppose that the aromatic core of the molecule has an interaction tensor E2 r and each C-C segment has an interaction tensor E~ r' If these local interaction tensbrs are assumed to have cylindrical symmetry, the u~ique components of E2 r and E~ rare Xa and Xc, respectively. The model parameters, Xa and Xc, are varied in ord~r to reproduce the observed quadrupolar splittings. In the principal frame of Uext(n, 0), S:;a can be evaluated in the same manner [27] as used for a rigid biaxial particle. Finally, the equilibrium probability of finding the conformer n in a mesophase is given by

1 Pn = Z exp [-Uint(n)/kBT]Qn. (16.39)

where Z = En exp [-Uint(n)/kBT]Qn is the conformation-orientational partition function, and Qn = J exp [-Uext(n, O)/kBT]dO is the orientational partition function of conformer n.

When internal rotations can be treated independent of the overall motion of the molecule, the so-called superimposed rotations model [1] or the decoupled model [28] can be used. Suppose the molecular N frame can be used to describe all possible conformations available to the flexible chain. The decoupled model gives for m L i= 0

GmL(t) = L L L D~MmN(OM,N)D;;;MmN(OM,N) g;:;LmM(t) mM mN mN (16.40)

x (D~NO[ON,Q(O)]D;;;NO[ON,Q(t)])

where D~NO = 0 has been assumed and the angle-bracketed factor denotes internal correlation functions. When mL = 0 there is an additional term in Eq. (16.40) due


to couplings between internal motions and the molecular reorientation [29]. To evaluate the reduced correlation functions, one needs Pn and p(i, tin, 0), the conditional probability that the molecule is in the conformer i at time t given that at t = 0 it is in conformer n. Now p( i, tin, 0) can be obtained by solving a master rate equation in which the transition rate matrix R contains phenomenological jump rates such as kl' k2 and k3 for transitions between different conformations via one-, two- and three-bond motions, respectively [28]. When the TZ model is used to describe the reorientation of an asymmetric rotor, the spectral densities for chain deuterons at the Cp site become

J!!:) (mw) ~ ~ ~ ~ ( t, d'mMo(!l,\';:~) exp [-imM"~~ 1 x}') xl') )

x (t. ~Mo(il~~) exp [-imM"~~l 4 ') xl,') )

x 2: (A~mMm})K [(B~mMm~)K + IAkl]

K (mw)2 + [(B~mMm~)K + IAkif

(16.41)

where Ak and £(k) are the eigenvalues and eigenvectors obtained from diagonalizing a symmetrized R matrix, and here the N frame is taken to coincide with the molecular M frame. There are N real and negative eigenvalues. One of these (k = 1) is zero,

and the corresponding eigenvector £(1) is given by xj1) = [pj]1/2. In summary, we have shown in this section how spectral densities of motion can

be obtained based on certain motional models. By fitting the experimental spectral densities with the predictions from a particular motional model, its model parameters can then be derived. However, the derived motional parameters are model dependent. Justification of NMR model parameters may be obtained by comparing them with those determined using other spectroscopic techniques.

5. Applications of spin relaxation In this section, we describe some recent relaxation studies of orientationally ordered

mesogens, most of which have been carried out in our laboratory using deuterium NMR spectroscopy. Relaxation studies by means of other nuclei such as 1 H, 13C, 14N will be briefly surveyed to reflect current interests in LCs.

The decoupled model used below was recently proposed by us [30] to overcome the difficulty of k3 rates that are too high (1017 -1 018 s -1) in many of the LCs studied. This motion involves interchanging two alternate bonds, i.e., {ijklm} -- {ilkjm}. According to Helfand [31,32], the three-bond rotation or crankshaft motion is labeled as type I motion, while both the one- and two-bond motions are grouped in his type III motion. His type II motion consists of two kinds, viz., tttt -- tg±tg~ is a gauche pair production or a kink formation, and ttg -- gtt is a gauche migration. This particular motion does not swing the chain but the chain does translate. Now the type I


motion should call for high activation energies because several bonds must be activated almost simultaneously. Thus, the type II motion is used in the new decoupled model to replace the three-bond rotation. The kg and k~ are for the gauche migration and gauche pair formation, respectively. In anticipation of using the model for longer end chains, the number of conformations has been drastically reduced by eliminating those conformers with low probabilities. For an octyloxy chain, those conformers with five or more gauche C-C bonds can be safely ignored. Even the conformations with two consecutive gauche C-C bonds (i.e. g±g± and g±g=f) can be ignored. For an alkyloxy chain, the O-Ca bond is taken to be fixed on the phenyl ring plane with LCarOCa = 126.4°. With these assumptions, N=171 for molecules with an octyloxy chain such as 4-n-octyloxy-4'-cyanobiphenyl (80CB). Now LCCC, LCCH and LHCH are assumed to be 113.5°, 107.5° and 113.6°, respectively. The O-C bond is treated the same as a C-C bond, and LOCC = LCCC. The dihedral angles (<p = 0, ±112°) are for rotation about each C-C bond or the O-C bond. The geometry of the molecule is needed in the AP method to construct Uext(n, 0).

5.1 Deuteron studies

Here some new analyses of relaxation data are presented for several LCs containing an octyloxy chain. In setting up the transition rate matrix R among 171 conformations, the following conditions are imposed: (i) no direct transition occurs between a g+ and g- state as this costs too much energy; (ii) transitions between gtgtt and tgtgt cannot proceed directly as these also cost too much energy. There are 584 conformational transitions for R when considering both forward and reverse transitions. Among the 292 forward transitions, there are 86 type III motions about the C6-C7 bond (k1 motions), 42 type III motions about the CS-C6 bond (k2 motions), and 164 type II motions. For the latter type, 82 transitions are for the gauche migration and 82 transitions are for the kink formation.

5.1.1 Mixture of80CB-d17 and 60CB. Fig. 16.1 reproduces the experimental

sgb for chain-deuterated 80CB-d17 as a function of temperature in the N, SmA and reentrant nematic (Nre ) phases of the 80CB /60CB mixture (72 / 28 in wt%) [33]. In fitting the segmental order parameter profile, an optimization routine (AMOEBA) [34]

was used to minimize the sum squared error f = Li (I sgb I-I sgbca1c I) 2 where the

sum included C1 to C7. The Etg(CCC) and EtgCOCC) values were set at 4000 J / mole and 5600 J I mol. The calculated segmental order parameters are indicated in Fig. 16.1 as curves. In spite of lowering the number of conformers, the fits are similar to those in [33]. The derived Xa and Xc are then used to find Pn for treating internal motions in the chain. The order parameters < P2 > and < Sxx - Syy > of an "average" conformer of 80CB are shown in Fig. 16.2.

The spectral density J1 (w) and h(2w) results at 15.1 and 46 MHz versus temperature for the chain deuterons are reproduced in Fig. 16.3. Note that all reported spectral

densities include the factor KQ. It is clear that the Jii ) (w) show substantial frequency

362

o u

C/)

0.3

- 0.2

0.1


300 310 320 330 340 350

T (K)

Figure 16.1. Plot of segmental order parameters of SOCB in SOCB 160CB vs temperature . • , ., x and T denote Cl, C3, Cs, and C7 sites, respectively. Open 6., 0, 0 and V denote C2, C4 , C6 , and Cs sites, respectively. The solid curves are the theoretical calculations for C1 to C7 starting from the top. Note that the experimental splittings of C3 and C4 are reversed from those predicted by the theory.

dependences at all carbon sites, while the JJi\2w) show less frequency dependences. Now the "slow" molecular reorientation can describe the frequency dependences of

Jii)(w) and JJi) (2w) in the SmA and Nre phases, while some ODF appear to be nec

essary in the high temperature N phase. The spectral densities Jii)(w) and J~i)(2w) for carbon 1 to carbon 7 are calculated using Eqs. (16.31) and (16.41), and fitted to their experimental values from all mesophases in a global target analysis. To get some ideas on the temperature behaviour of model parameters Dl., D II , kl' k2, kg and k~, individual target analyses (i.e. analyze spectral densities at each temperature) were first carried out. These analyses have indicated that the target model parameters varied smoothly with temperature, even across phase transitions. Furthermore, the rotational diffusion and jump constants obeyed simple Arrhenius relations:

(16.42)


0.04

A >-ch 0.03

0 o 0 Oro 0 0 0 0 0 o 0

>< >< C/) 0.02 V

0.01

0 0 o 0

0.6 0 0

0

A 0 C\I 0

G- 0.4 0

V 'b 0

0.2

0.0 300 310 320 330 340 350

T (K)

Figure 16.2. Plots of the order parameters < P2 > and < Sxx - Syy > of an "average" confonner of SOCB in SOCB I 60CB as a function of temperature.

ki = kfexp [-E!ijRT]

kg = k; exp [-E!g j RT]

o [ k' ] k~ = k~ exp -Eag / RT

(16.43)

(16.44)

(16.45)

where j =.1 or II, i = 1 or 2, D1, D~, kf, k; and k~o were the pre-exponentials and Ea with appropriate superscripts were the corresponding activation energies. Instead of Eqs. (16.42) - (16.45), these were rewritten in terms ofthe activation energies, and the jump and diffusion constants at Tref = 320 K which were first obtained by an individual target analysis. The ODP prefactor A was set equal to 6.7 X 10-6 s 1/2, which gave the "best" minimization of the mean-squared percent error (F), while a linear temperature dependence was imposed on wcl27f (= 90 MHz at 350 K) such that its value decreased to 3 MHz just below the N-SmA phase transition at 315 K. In this manner, the cutoff function U (x) --? 0 in the SmA phase. Since A and We are highly correlated, a different

364

.e ~ 40 'iii c: QI o ~ U QI a. (/)

(a) 40

T (K)

(b) 50


(e)

20 "1 '"

T (K)

" "

(d)

Figure 16.3. Plots of spectral densities vs the temperature in 80CB 160CB at 15.1 MHz (Figs. (a) and (c» and 46 MHz (Figs. (b) and (d». Closed symbols denote JJi) (w) and open symbols denote their

corresponding JJi) (2w). (a) 0, 6., ° and \l denote results ofC1 , C3, Cs and C7, respectively; (b) same as (a); (c) 0, 0, 6. and \l denote results of C2, C4, C6 and Ca, respectively; (d) same as (c). Some typical error bars are shown. Solid and dashed curves denote calculated spectral densities J 1 and J2 ,

respectively.

choice of We at 350 K is possible (see pure 80CB sample below). We found that at 15.1

MHz ODF contributed about 1.5-2.0% to Jii)(w) just below the N-SmA transition

(315 K), and made zero contributions to Jii ) (w) at lower temperatures. At the high end

of the N phase (350 K), ODF accounted for 30% to 35% of Jii)(w) at 15.1 MHz (about 14-16% at 46 MHz). Note that the temperature dependence of ODF came through We in the cutoff function. In the global analysis, 11 target parameters were used as kl was kept constant due to its insensitivity to temperature. The fitting quality factor Q is defined by

(16.46)

where the sum over i covers Cl to C7, the sum over w is for two Larmor frequencies, the sum over k is for fourteen temperatures covering N, SmA and N re phases, and m = 1 and 2. We have a total of 392 spectral densities to derive 11 global param-

eters. The calculated spectral densities are shown (Q =1.8%) as solid (Ji i ) (w) and

dashed (J~i) (2w)) curves in Fig. 16.3. Although there are some systematic deviations


..,... 'en 1013 --~-

• • • • 8 ••

o • 8 : o OQ.

VVVVVVV'ViJ9W.R 0 0 • .......

tiiii2i~ o 0

2.8 3.0 3.2 3.4

.-

107~~~~~~~

2.8 3.0 3.2 3.4

10001T (K)

365

Figure 16.4. Plots of jump rate constants kl (.6.), k2 (0), kg (V) and k~ (<», as well as rotational diffusion constants DII (0) and D.L (0) as a function of the reciprocal temperature. Solid symbols are for pure 80CB, while open symbols for the 80CB / 60CB mixture.

between the experimental and calculated spectral densities, the overall fits are quite satisfactory in view of the many simplifying assumptions used in the motional model. Fig. 16.4 summarizes the derived model parameters. The activation energies for DU and D.L are similar (about 41 kJ I mol), but the latter is not as well determined, a phenomenon often encountered in NMR studies of liquid crystals [1]. Since D.L is less than 4 x 107 S-1 just below the N-SmA transition, the tumbling motion produces some frequency dependences in both J1(w) and J2(2w) in the SmA and Nre phases as well as to a lesser extent in the N phase.

5.1.2 80CB. The fittings of experimental 8gb versus the temperature in the N and SmA phases of an 80CB-d17 sample, and the derived model parameters X a ,

X cc, (P2) and (8xx - 8yy) for the "average" conformer can be found in [30]. The experimental spectral densities for 80CB-d17 versus the temperature are reproduced in Fig. 16.5. To account for the spectral densities in the N phase, ODF were found


40 14

(a) 20 .' (b) (c) (d)

35 12

_- 30 ~

~ ~ 10

.?? 25 .2:-VJ 'iii c:: c:: 8 CD CD

Cl 20 Cl

~ '"' 'iU ... 6 U ~ "1..""" u

CD 15 Q) a. a.

C/') a \ ;;--...:.-..... ...

C/')

10 ~" 2 t .... "0 ...... . , .

' ''' ' -""~ .. ,,,,~-~ .... 5 2 "'1-

1 0 00 0 , '" . . 'o.~.~ 0.:::::

0 0 0 0 320 330 340 350 320 330 340 350 320 330 340 350 320 330 340 350

T (K) T (K)

Figure 16.5. Plots of spectral densities vs temperature in 80CB-d17. Closed and open symbols denote results at 15.1 and 46 MHz, respectively. (a) 0 and 0 denote Jii ) (w) of Cl and C2, while 6 and <> denote J~i) (2w) of Cl and C2; (b) 0 and 0 denote Ai) (w) of C3 and C4, while 6 and <> denote J~i) (2w) of C3 and C4 ; (c) 0 and 0 denote Jii) (w) of Cs and C6. while 6 and <> denote J~i) (2w) of Cs and C6; (d) 0 and 6 denote Jii)(w) and J~i)(2w) for C7. respectively. For Ca. 0 and <> denote Jii) (w) and J~i) (2w). respectively. Typical error bars are shown only in (a) for Cl and C2. Solid and dashed curves are calculated spectral densities for 15.1 and 46 MHz. respectively.

necessary. A was set at 9.9x10-6 Sl/2 andwc /271' (= 30 MHz at 350 K) was decreased linearly to 3 MHz just below the N-SmA phase transition at 335 K. At the high end of

the N phase (350 K). ODF contributed between 23% and 30% to Jii)(w) at 15.1 MHz (about 6-7% at 46 MHz). In using the Arrhenius relations. the motional parameters were found to vary smoothly across the N-SmA transition. By fitting the spectral densities of C1 to C7 deuterons at 15.1 and 46 MHz (we did not include C1 and C2 at 15.1 MHz in the minimization and chose to predict their values as before [30]) and seven temperatures covering the N and SmA phases. 11 target parameters were obtained from the minimization. A Q value of 2.2% was achieved. The rotational diffusion and jump constants are shown in Fig. 16.4 for direct comparison with those in the 80CB / 60CB mixture. while the calculated spectral densities for various sites are shown as curves in Fig. 16.5. The activation energies for k2• kg. k~. D 1., and DII are 130.7, 3.9, 89.4, 30.4 and 60.9 kJ I mol, respectively.

5.1.3 Octyloxy benzoyloxy cyanobipbenyl (SOBeB). 80BCB exhibits a N, a smectic Ad (SAd) and a Nre phase with a relatively high clearing temperature (Tc = 513

K). Fig. 16.6 reproduces the experimental s~b [35] as a function of temperature in the


0.3

o u

C/) 0.2

0.1

T (K)

367

Figure 16.6. Plot of segmental order parameters vs temperature. Solid., ., .. , and • denote Cl. C3 ,

CS,6, and C7 sites, respectively. Open 0, 0 and 6. denote C2, C4, and Ca sites, respectively. The solid curves are the theoretical calculations for Cl to C7, starting from the top.

Nre and SAd phases of80BCB-d17. Etg(CCC) = 3500J I mol and Etg(OCC) = 4900

J I mol were used to give the calculated 8gb shown as solid lines in Fig. 16.6. The derived interaction parameters Xa and Xc, < P2 > and < 8xx - 8yy > are plotted versus the temperature in Fig. 16.7. While < P2 > varies from 0.67 to 0.39 upon approaching the isotropic phase in 80CB, it changes only from 0.69 to 0.63 upon entering the SAd phase in 80BCB. Despite some obvious deviations between the

calculated and observed 8gb in Fig. 16.6, in particular 8~b and 8~b, the derived Pn and order parameters are quite satisfactory for treating the relaxation data.

The spectral density J1(w) and J2(2w) results at 15.1 and 46 MHz versus the temperature for all the methylene deuterons are reproduced [35] in Fig. 16.8. Again

Ji i) (w) show substantial frequency dependences at all carbon sites, while J~ i) (2w) show little or no frequency dependences. As in the N phase of 80CB, some ODF appear to be necessary in the Nre phase of 80BCB. The prefactor A was set at 4.7 x 10-6 Sl/2

with an appropriate high frequency cutoff such that the ODF contribution amounted to about 35% of the total rate at 15.1 MHz. Thus, wcl21'i was chosen equal to 35

368

8 -----o 7 E -... ~ l!? Q)

E ~ ctS a.. c: o

:0:; () ctS .... Q) -c:

6

5

4

3

2

.•................ . ..•.............. _-. __ ...

345 360 375 390 405


O.7~

0.6 (/) .... Q) -Q)

E 0.5 ctS .... ctS a.. .... 0.4 Q) "0 .... 0

0.3

0.2

0.1

345 360 375 390 405

T (K)

Figure 16.7. (a) Plot of interaction parameters Xa (solid line) and Xc (dashed line) vs temperature. (b) Plots of the order parameters < P2 > (solid line) and < Sxx - Syy > (dashed line) of an "average" conformer of 80BCB as a function of temperature.

MHz at 363 K and decreased to 25 MHz at the NrcSAd phase transition (393 K). This was consistent with the vanishing cutoff function U (x) in the S Ad phase at our Larmor frequencies. In Eq. (16.46), the sum over k is for seven temperatures. Since the signals from C5 and C6 deuterons overlapped and their calculated spectral densities were different, we have taken the average for these two sites in the minimization. A total of 196 spectral densities were used to derive 12 global parameters, giving Q = 1.5%. Although there exist some systematic deviations between the experimental and calculated spectral densities (shown in Fig. 16.8 as curves), the overall fits are quite satisfactory. All model parameters are summarized in Fig. 16.9. The Dill D J.. ratio varies between 30 and 90 in the Nre phase of80BCB. In comparison with the N re phase of the 80CB / 60CB mixture, the need of ODF in this sample is purely due to lower viscoelastic constants at this higher temperature range. The same motional model has recently been applied to the SmA phase of a smectogen with a decyloxy chain [36]. It was found that all jump constants are < 1014 s-1 for this chiral compound.

In summary, deuteron spin relaxation in LCs can be satisfactorily interpreted, at least in their uniaxial phases. In particular, the new decoupled model is able to give


24 10

• (a) (b) • (e) (d)

20 • • .-. 8 • 4

~ .-.

l:' 16 SAd

.~ SAd ~ N,. SAd N,. SAd 'iii l:'

c 'iii 6 Q) c 0 Q)

12 0 0

~ (ij 0 .... ti '~, .... d", d't!, Q) ti 4 • 2 a. Q) d'b (J) 8 •• a. 'c ••

0 " "'Ii?o (J) c •• 'l:!p '0 • ~",p ", 0 .. '~ ' .. 'b o ,~ 000 ' 0 ., .. 4 • A~ • 2 A~ 9:>0 0

4 A :

oeo ••• 00

ift" ~AA

0 0 360 380 400 420 360 380 400 420 360 380 400 420 360 380 400 420

T (K) T (K)

Figure 16.8. Plots of spectral densities vs temperature in 80BCB. Closed and open symbols denote

results at 15.1 and 46 MHz, respectively. (a) 0 and 0 denote JJi) (w) of C1 and C3, while <> and

t:" denote J~i)(2w) of Cl and C3; (b) 0 and 0 denote J?)(w) of C2 and C4, while <> and t:" denote

J~i ) (2w) ofC2 and C4; (c) 0 and t:" denote J?)(w) and J~i ) (2w) ofCs,6, respectively; (d) 0 and t:" are

JJi) (w) and J~i>C2w) of C7, respectively. Typical error bars are shown only for C3 and C4. Solid and dashed curves are calculated spectral densities at 15.1 and 46 MHz, respectively.

reasonable internal jump constants « 1014 S-l) in all LCs studied thus far. It is interesting to note that the tumbling diffusion constant D.l can be used to find the rotational viscosity coefficients based on statistical mechanical approaches. For 80CB, the derived rotational viscosity coefficients based on NMR results from the decoupled model agree well with the experimental values [37]. More work in biaxial phases and chiral materials [36] is still needed. The requirement of deuterated LC materials can, however, pose a problem because they are often costly and time-consuming to synthesize.

5.2 Proton studies Proton Zeeman spin-lattice (Tt) relaxation studies can be carried out in LCs over a

wide frequency range by using both conventional NMR spectrometers (from a few MHz to 800 MHz) and fast field-cycling spectrometers (down to a few hundred Hz), The field-cycling is discussed in detail in Chapter 17. Although proton spin systems (both within the molecule and with protons on neighbouring molecules) are tightly coupled by dipolar interactions, the Tl dispersion curve can be modeled based mainly on three kinds of dynamic processes: ODF; molecular self-diffusion (SD); and molecular

370

--'CI) -v

v

v 00

00

<><> ~oo <><> 0

Cc <><> c <><>

cc v c

c c

v

v

2.5 2.6 2.7 2.8

--'CI) -

NMR OF ORDERED UQUIDS

• • • • • • • •

•• ••• 8 • • __

10 -:

2.5 2.6 2.7 2.8

Figure 16.9. Plots of jump rate constants kl (D), k2 (0), kg (<» and k~ (V), as well as rotational diffusion constants Dn (D) and D..L (0) in the reentrant nematic phase of 80BCB.

rotations (ROT). Although the site specificity observed in 2H and 13C NMR is normally absent in the proton spin system, a proton Tl dispersion study remains to be the means for probing intermolecular interactions. It is usually assumed that internal rotations in the molecule are too fast to affect the proton relaxation. Thus, the proton spin-lattice relaxation rate is approximated by

T- 1 T-l + T-1 + T-1 1 = lDF lSD lROT (16.47)

in which possible cross-terms between different mechanisms are ignored. In the Tl dispersion curve, the frequency behaviour below 100 kHz is dominated by ODF, while above this frequency molecular rotations and SD are by far more predominant. It is noted that both ODF and SD are sensitive to the constraints imposed by the phase structure [38]. Hence, proton spin relaxation in the tilted smectic phases has recently attracted much attention. The Tl frequency dependence in the kHz region has been ascribed to nematic-like ODF in some studies, while in others it has been ascribed to undulations such as observed in the SmA phase. In a recent Tl relaxometry study [39], both mechanisms were detected close to the SmC-N phase transition temperature. This was attributed to the presence of pseudo-nematic domains in the SmC phase. In the chiral SmC* phase, it has been suggested that an additional term was needed to


account for the slow modes resulting from distortions of the helical structure [40]. Finally, both the undulation modes and rotations of the molecular director, without any distortion of the layers, are detected in the bilayer SmC2 phase of a ferroelectric LC (FLC). The latter motion has a nematic-like ODF behaviour [41].

In analogy to TIQ of deuteron systems, the proton spin-lattice relaxation time (TID) of dipolar order can be measured in LCs by the Jeener-Broekaert sequence. In the case of spin systems composed of equivalent, loosely coupled proton pairs, four quasiinvariants are expected: the Zeeman energy, the intra-pair dipolar energy, the inter-pair dipolar energy, and the population of the singlet state of the intra-pair interaction [42]. By studying TID-intra and TID-inter in 4-n-pentyl-4'-cyanobiphenyl (5CB), different dynamic processes (ODF, ROT, and / or alkyl chain rotations) may be differentiated from the spin relaxation [42] of the dipolar quasi-invariants.

5.3 Carbon-13 studies To achieve high resolution 13C spectra in LCs, one needs high-power proton de

coupling and perhaps also magic angle spinning (MAS). The interpretation of 13C spin-lattice relaxation is not as straightforward as 2H spin relaxation, simply due to the fact that both the dipolar interaction and chemical shift anisotropy must be taken into account (see Eqs. (16.27) and (16.28)). Thus, relatively few 13C relaxation studies of LCs have been reported in the past. However, recent interest in the dynamics of FLCs and anti-ferroelectric LCs (AFLC), as well as their chiral subphases, has made 13C TI relaxation [43] and linewidth measurements more attractive alternatives for these materials. For instance, the method has been used to study the SmA-SmC* transition [44] of a chiral LC. A recent I3C TI study [45] of an AFLC did not show dramatic changes in the dynamics at the SmA-SmC* transition.

5.4 Nitrogen-14 studies Thus far, I4N has not been fully utilized to investigate LCs, mainly due to its low

sensitivity and large nuclear quadrupole moment Q. Only under certain conditions can 14N be used successfully to investigate LCs [46]. The methodology developed for studying 2H spin relaxation can be applied to I4N if its large quadrupolar coupling has been motionally averaged to a reasonable value, such as in lyotropic LCs [47]. Recently, I4N was used to study the hexagonal (HI) phase of a binary mixture of 70 wt % dodecyltrimethylammonium chloride (CI2TACI) and heavy water. The sample shows a typical I4N Pake pattern with 6 being the quadrupolar splitting between the 90° edges of the powder pattern. The quadrupolar splitting and spin-lattice relaxation times T i z and TIQ were measured simultaneously [47] using signals at 90° edges to give separate spectral density parameters J I (w,e) and J2(2w,e), where e (= 90° in this study) is the angle between the external magnetic field and the phase symmetry axis (the phase director). The existing theory [48], based on a simple model of "classical" aggregates, for the lateral diffusion around the cylindrical aggregate surface was used to interpret the I4N quadrupolar splitting and spectral density results ofNMet cation in


C 12 TACl. The derived spectral densities of motion at seven temperatures, together with their corresponding quadrupolar splittings, were analyzed. Both fast local motion (J f in a white spectrum) and slower surface self-diffusion about the cylindrical aggregate axis with an azimuthal correlation time Taz were required to account for the 14N spin relaxation in this phase. The surface diffusion process in the HI phase contributed only to one of three spectral density functions [48]:

(16.48)

where

(16.49)

and XQ was the residual 14 N quadrupole coupling constant (given by 6. = 3XQ/8). Now Taz = b2/4Ds with Ds being the surface diffusion coefficient, and b the radius of the cylinder. In a global target analysis, T az and J! were assumed to obey simple Arrhenius temperature behaviours, i.e.

Taz = T~z exp [Ea/ RT]

J! = JJexp [E!/RT]

(16.50)

(16.51)

where Ea and E! were the activation energies for the azimuthal correlation time and fast local dynamics, respectively. The Taz was found to be tens of nanoseconds, while J! increased from 1.1 s-1 to 6.5 s-1 upon decreasing temperature in the temperature range 290 to 350K.

Acknowledgments

The financial support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.

References [1] Dong, R.Y. Nuclear Magnetic Resonance of Liquid Crystals. Springer-Verlag, 1997. [2] Bloembergen, N., Purcell, E.M., and Pound, R.V. (1948), Phys. Rev., 73:679. [3] Rose, M.E. Elementary Theory of Angular Momentum. Wiley, New York, 1957.

[4] Brink, D.M., and Satchler, G. R. Angular Momentum. Clarendon, Oxford, 1962.

[5] Abragam, A. The Principles of Nuclear Magnetism. Clarendon, Oxford, 1961.

[6] Saupe, A. (1964), z. Naturforsch. Teil A, 19:161. [7] Redfield, A.G. (1965), Advances in Magnetic Resonance, 1: 1. [8] Ahmad, S.B., Packer, KJ., and Ramsden, J.M. (1977), Mol. Phys., 33:857. [9] Jeffrey, K.R. (1981), Bull. Magn. Reson., 3:69.

[10] Jeener, J., and Broekaert, P. (1967), Phys. Rev., 157:232. [11] Wimperis, S. (1989),1. Magn. Reson., 83:509. [12] Wimperis, S., and Bodenhausen, G. (1986), Chem. Phys. Lett., 132: 194. [13] Dong, R.Y. (1992), Bull. Magn. Reson., 14:134.


[14] Davis, J.H., Jeffrey, K.R., Bloom, M., Valic, M.I., and Higgs, T.P. (1976), Chern. Phys. Lett., 42:390.

[15] Pincus, P. (1969), Solid State Cornrnun., 7:415.

[16] Freed, J.H. (1977), J. Chern. Phys., 66:4183 and references therein. [17] Void, R.L., Void, R.R., and Warner, M. (1988), J. Chern. Soc., Faraday Trans., 84:997. [18] van derZwan, G., and Plomp, L. (1989), Liq. Cryst., 4:133.

[19] Dong, R.Y., and Shen, X. (1997), l. Phys. Chern. A, 101:4673.

[20] Nordio, P.L., and Busolin, P. (1971), J. Chern. Phys., 55:5485.

[21] Nordio, P.L., Rigatti, G., and Segre, U. (1972), J. Chern. Phys., 56:2117.

[22] Nordio, P.L., Rigatti, G., and Segre, U. (1973), Mol. Phys., 25:129.

[23] Tarroni, R., and Zannoni, C. (1991), J. Chern. Phys., 95:4550.

[24] Flory, P.J. Statistical Mechanics a/Chain Molecules. Interscience, New York, 1969. [25] Emsley, J.W, Luckhurst, G.R., and Stockley, C.P. (1982), Proc. R. Soc. London Ser: A, 381: 117. [26] Marcelja, S. (1974), J. Chern. Phys., 60:3599.

[27] Luckhurst, G.R., Zannoni, C., Nordio, P.L., and Segre, U. (1975), Mol. Phys., 30:1345.

[28] Dong, R.Y. (1991), Phys. Rev. A, 43:4310.

[29] Caniparoli, J.P., Grassi, A., and Chachaty, C. (1988), Mol. Phys., 63:419.

[30] Dong, R.Y. (2001), J. Chern. Phys., 114:5897.

[31] Helfand, E. (1971),1. Chern. Phys., 54:4651.

[32] Skolnick, J., and Helfand, E. (1980), J. Chern. Phys., 72:5489.

[33] Dong, R.Y., and Cheng, M. (2000), J. Chern. Phys., 113:3466.

[34] Press, WH., Flannery, B.P., Teukolsky, S.A., and Vetterling, WT. Nurnerical Recipes. Cambridge University Press, Cambridge, England, 1986.

[35] Dong, R.Y., Carvalho, A., Sebastiao, PJ., and Nguyen, H.T. (2000), Phys. Rev. E,62:3679.

[36] Dong, R.Y., Chiezzi, L., and Veracini, C.A. (2002), Phys. Rev. E,65:41716.

[37] Zakharov, A.V., and Dong, R.Y. (2000), Phys. Rev. E,63:11704.

[38] Ribeiro, A.C., Sebastiao, P.J., and Cruz, C. (2001), Mol. Cryst. Liq. Cryst., 362:289.

[39] Acosta, R., and Pusiol, D. (1999), Phys. Rev. E,60:1808.

[40] Ferraz, A., Ribeiro, A.C., and Nguyen, H. (1999), Mol. Cryst. Liq. Cryst., 331:67.

[41] Carvalho, A., Sebastiao, PJ., Ribeiro, A.C., Nguyen, H.T., and Vilfan, M. (2001), J. Chern. Phys., 115:10484.

[42] Mensio, 0., Gonzalez, C.E., Pusiol, D.1., Zamar, R.C., and Dong, R.Y. (2002), Physica, 320:416. [43] Yoshizawa, A., Yokoyama, A., Kikuzaki, H., and Hirai, T. (1993), Liq. Cryst., 14:513.

[44] Yoshizawa, A., Kikuzaki, H., and Fukumasa, M. (1995), Liq. Cryst., 18:351.

[45] Tokumaru, K., Jin, B., Yoshida, S., Takanishi, Y., Ishikawa, K., Takezoe, H., Fukuda, A., Nakai, T., and Miyajima, S. (1999), lpn. J. Appl. Phys., 38:147.

[46] Tansho, M., Onoda, Y., Kato, R., Kutsumizu, S., and Yano, S. (1988), Liq. Cryst., 24:525.

[47] Dong, R.Y. (2001), Mol. Phys., 99:637 and references therein. [48] Quist, P-O., Halle, B., and Fur6, I. (1991), J. Chern. Phys., 95:6945.

Chapter 17

LOW-FREQUENCY NMR RELAXOMETRY OF SPATIALLY CONSTRAINED LIQUID CRYSTALS

F. Grinberg Sektion Kemresonanzspektroskopie, Universitiit Ulm, Ulm, Germany*

M. Vilfan J. Stefan Institute, Ljubljana, Slovenia

E. Anoardo Facultad de Matematica, Astronomfa y Ffsica, Universidad Nacional de C6rdoba, C6rdoba, Argentina

1. Introduction Liquid crystals (LCs) confined to structures of submicrometer characteristic lengths

exhibit many interesting phenomena related to their mesoscopic size and large surfaceto-volume ratio [1,2]. They represent an important field of current research both from the fundamental point of view and for applications in displays, electro-optical shutters, and telecommunications. Since the discovery of polymer dispersed liquid crystal materials (PDLCs) in 1986 [3], the interest in studying microconfined liquid crystals (first initiated by applications in electro-optical devices) has been steadily increasing. However, a rich variety of unexplained phenomena remains a driving force in this field of research which currently is undergoing rapid development.

Nuclear magnetic resonance (NMR) has proved to be a very successful method for the study of molecular ordering and dynamics in spatially constrained liquid crystals. Initially, the NMR research concentrated on structural studies of well-characterized confining geometries such as spherical, cylindrical, and planar cavities [4-8]. One of the most important goals was to determine the director field configurations, a problem that in submicrometer cavities is not amenable to conventional optical techniques. Almost simultaneously with theoretical and experimental NMR studies of static config-

• Current address: Max Planck Institute for Metals Research, Stuttgart, Germany

375 E.£. Burnell and c.A. de Lange (eds.), NMR of Ordered Liquids, 375-398. © 2003 Kluwer Academic Publishers.


urations [4-8], NMR relaxometry was applied to gain insight into dynamical processes in confined liquid crystals [9, 10]. Later, the focus of the research was extended towards more complex geometries, including randomly oriented surfaces in porous glasses and aerogels [11-16]. Special attention was devoted to liquid crystals with embedded thin polymer networks which stabilize specific configurations in cholesteric and ferroelectric LC displays. Relaxation studies gradually acquired equal importance with spectroscopic experiments, and provided clear evidence that cavity constraints and wall-LC interactions have a profound effect on molecular dynamics. An extremely broad frequency range associated with dynamical processes was established. These processes tend to cover the whole range of frequencies accessible by NMR, starting with fast reorientations of individual molecules or internal groups (characteristic times ~1O-8 s to ~ 10-11 s), down to ultraslow collective molecular motions on millisecond or even longer time scales. In order to cover such a broad time range, different NMR techniques have been applied to confined LCs. They include the measurements of conventional and field-cycling spin-lattice relaxation rates (T1- 1), longitudinal relaxation rates in the rotating frame (TI~I), the dipolar correlation effect, and transverse spin

relaxation rates (T2- 1) [17-19]. The process of spin relaxation is induced by the time-dependent part of spin interac

tions, and is therefore directly related to the intensities and frequencies of the motions of spin-bearing molecules. The relevant spin interactions are magnetic dipole-dipole interactions in the case of protons, "quadrupolar" interactions between the electric nuclear quadrupole moment and the electric field gradient for deuterons, and chemical shift anisotropies for carbon-13 nuclei. Whereas in isotropic liquids of low viscosity usually one type of motion, i.e. one relaxation mechanism, dominates on a given time scale, there is quite a number of non-negligible contributions from different mechanisms in bulk and confined liquid crystals. This is characteristic also ofliquid polymers in which molecular motions tend to range over many time decades [20,21].

Fast local molecular reorientations and conformational changes occur in much the same way in the spatially constrained state as in bulk LCs. Collective orientational fluctuations [22], usually denoted as order director fluctuations (ODF), are masked above a few MHz, but can be detected in the kHz frequency range. In cavities, the ODFs are hampered to a certain extent by the confinement. Their continuous wavevector spectrum, characteristic of the uniform bulk state, then transforms into a set of discrete fluctuation modes which depend on the size and interconnectivity of the cavities. In roughly the same frequency range, a non-collective but slow dynamic process is due to translational molecular diffusion. In bulk LCs with uniform director orientation, translational self diffusion modulates only intermolecular interactions. In confined LCs, it gives rise also to the time modulation of intramolecular spin interactions (on a slower time scale), because the molecules diffuse between non-identical local environments. Such different environments may exist, for instance, due to a varying local order parameter (in the presence of more and less ordered regions), or to a varying orientation of local directors imposed by solid interfaces. In particular, the relaxation mechanism produced by molecules that probe different surface orientations in the course of their self diffusion is known as "reorientation mediated by

Low-frequency NMR relaxometry of spatially constrained liquid crystals 377

translational displacement" (RMTD). This process has been most intensively studied by R. Kimmich et al. [23,24] for polar liquids confined in random pores. The RMTD mechanism was found responsible for a strong enhancement of the relaxation in the low frequency (kHz) range in confined LCs (see [25] and references therein). Additionally, in cases where the solid constraining matrix incorporates proton nuclei (e.g. in polymer-LC dispersions), a strong cross relaxation between protons belonging to the LC and those of the surrounding polymer exists [9]. This cross relaxation usually results in a much faster relaxation of the liquid crystal protons than expected. A more detailed classification and overview of different relaxation mechanisms can be found in earlier reviews [25,26].

The presence of multiple relaxation mechanisms in confined LCs gives rise to the problem of decomposing numerous overlapping stochastic processes along with the need to extend relaxation studies to the lowest possible frequencies (though ultimately limited by the spin-lattice relaxation rate). In turn, this has encouraged further developments of techniques and appropriate strategies for using available NMR tools. Some recent trends in this connection refer to progress in field-cycling (FC) relaxometry, an intensification of deuterium-based relaxation studies, and an adjustment of standard pulse techniques for monitoring ultra-slow correlated molecular motions. This Chapter is mainly devoted to these developments and has the objective of reviewing recent and current studies on low-frequency relaxation in spatially constrained thermotropic LCs. We focus particularly on the topics that have not been included in earlier reviews. It should be mentioned that relaxation mechanisms similar to those governing the relaxation in confined thermotropic liquid crystals appear in somewhat modified form in lyotropic liquid crystals with a spatially modulated structure, e.g. in the ripple phase [27], and in lipid vesicles which represent closed two-dimensional liquid-like layers in three dimensional space [28,29].

The review consists of four main sections. Section 2 deals with FC NMR relaxometry which is the most comprehensive and promising method in the relaxometry studies [17,30]. It allows one to "scan" stochastic processes over an extremely broad frequency range covering many decades. The frequencies addressed by modern FC NMR set-ups lie between several MHz and a few kHz in the case of protons. In experimental studies, the high-frequency end of this range can be extended further by combining FC results with those obtained using standard (high-field) NMR instruments operating at several different fixed frequencies. In section 2 we discuss general aspects of this technique. Special attention is given to the technical aspects of the so-called fast-field-cycling (FPC) relaxometry, including recent achievements in the development of related instrumentation.

Section 3 is devoted to the discussion of the most important results obtained by FPC in the investigation of LCs confined in porous materials. These include nematic LCs confined in polymer matrices, Anopore membranes and Bioran glasses.

A severe limitation of the FC technique is related to the rather strong (when compared to isotropic non-viscous liquids) residual local magnetic fields that are characteristic of anisotropic liquids. The local fields tend to impose a cut-off, typically at about a few kHz, on the low-frequency scale accessible to this method. This limit can be


(indirectly) extended by another three orders of magnitude if FPC relaxation studies are combined with the dipolar-correlation effect discussed in section 4. Such a combination is particularly promising in monitoring ultra-slow motions with characteristic frequencies that are less than the FC cut-off frequency.

Section 4 deals with the attenuation mechanisms of stimulated echoes that are specific for locally anisotropic liquids. The central topic is the dipolar-correlation effect (DCE) arising from ultra-slow fluctuations of residual dipolar interactions. The DCE permits one to monitor dipolar correlations over a time scale from roughly 10-4 s up to the order of spin-lattice relaxation times, and is therefore relevant for the longest time limit accessible to NMR. In general, motions such as ODFs in nematic LCs or reorientations of long chain backbones in polymers can be probed using the DCE. Section 4 will provide an overview of this rather recent method and its applications. Experimental examples include the DCE for a nematic 4'-n-pentyl-4-cyanobiphenyl (5CB) in bulk and confined in porous glasses.

Finally, section 5 briefly addresses deuteron relaxometry studies of cyano-biphenyls confined in cylindrical cavities of Anopore membranes, in polymer dispersed liquid crystals, and in porous glasses. Deuteron relaxometry gives information not only on molecular dynamics (complementary to proton studies), but also provides a unique way of determining the degree of surface-induced order above the nematic-isotropic transition.

2. Field-cycling relaxometry 2.1 The field-cycling method

The main difficulty encountered when using time-independent magnetic fields for Tl measurements is the well-known reduction of the free-induction-decay signal with decreasing strength of the magnetic field. FC NMR relaxometry is the method designed to extend Tl experiments to the lowest frequencies possible, while simultaneously avoiding any considerable loss of signal intensity. The alternative relaxation rates, such as T2- 1 or Tl~l, monitor roughly the same frequency (kHz) range of molecular motions, but are induced by different terms in the dipolar Hamiltonian. Hence, FC measurements of T1- 1 in the low-frequency regime cannot be replaced adequately by experiments performed at high field [31,32]. At the same time, in many cases it is helpful and instructive to compare information obtained by different techniques.

The two basic sequences used to measure Tl by cycling Zeeman magnetic fields are the so-called Pre-Polarised (PP) and Non-Polarised (NP) sequences (Fig. 17.1). The first is used to determine Tl at low fields, whereas the second is applied to fill the gap in the intermediate range below roughly 20 MHz, where standard static electromagnet field spectrometers are used.

In the PP sequence, a spin system is initially magnetized in a polarization magnetic field of strength Bp using a magnetic field polarization pulse of duration tp. The optimal polarization field must be the strongest one available with the given hardware, and must comply with the condition tp > NT! (Bp) with N usually between 2 and 5. Transients


!"If .... .... Bo ~l

I 11 J I! ! ) ii' , ,

'I t. ' I. i l td i 1, 1 I .1 r-! r

., i4i ~

Rr ill

B !111 : I II I ) I

)

tR IRF

Basic Pre-Polarized Sequence Basic Non-Polarized

Figure 17.1. Basic field-cycling pulse sequences for TI measurements.

that unavoidably arise from switching the field do not playa considerable role in the polarization step. In the next step, the magnetic field is switched to a lower value, B r ,

at which the corresponding relaxation time, TI (Br ), is to be measured. An important requirement concerning the switch-off time, totI, is that it must be short relative to relaxation times in order to avoid severe magnetization losses during the switching. However, this is at variance with the requirement that, in the course of switching, an evolution of the magnetic field in time should fulfill the adiabatic condition, Ida/ dtl < 'Y Bo, where 'Y is the gyromagnetic ratio and a denotes the angle between the directions of the applied and the effective magnetic fields. The effective field, as usual, refers to the vector sum of all possible contributions at local spin positions, including the Zeeman field itself, internal interactions, any externally induced but not properly compensated fields, etc. If the adiabatic condition is not fulfilled, this would eventually evoke an evolution of coherences during the relaxation period tn an effect sometimes referred to as non-adiabatic or zero-field conditions. After switching, the spin system relaxes in the field Br for a period tr towards a new eqUilibrium. After that, the magnetic field is switched to a high value Ba > Bp , where the remaining magnetization is measured. In the simplest case, the amplitude of the free induction decay following a 7r /2 pulse is measured. The latter is of course applied after transient processes have damped out. For a given Br , the whole cycle is repeated for a set of values of tr to obtain a series of signal amplitudes that decay at a rate TIl (Br), that is, as exp( -tr/TI (Br)).

As the relaxation field Br approaches the polarization field Bp , the difference in equilibrium magnetizations at both fields becomes so small that the evolution from one to the other cannot be measured effectively with the PP sequence. In these cases, one uses the NP sequence (Fig. 17.1). The evolution of the magnetization is sampled for a set of evolution times after the magnetic field had been switched on from a zero initial value up to the desired value of the relaxation field Br • Signal amplitudes measured at field Ba after a 7r /2 pulse now increase with an evolution time at the rate of interest, TI-I(Br ).


2.2 Instrumentation From a technical point of view, the FC method has limitations determined by the

homogeneity and stability of the magnetic field (specific for individual instruments), the maximum value of the operating magnetic field, the finite field switching times, the magnetic field pulse shapes, and the rather high complexity of the hardware in general. The (finite) stability and homogeneity of the magnetic field, for instance, ultimately limit the frequency interval that can be used between two successive measurements. A typical FC instrument usually achieves a frequency resolution (in proton Larmor frequency units) of the order of a few hundred Hz, depending also on the sample size.

The very first FC devices were based on mechanical or pneumatic systems designed to shoot the sample between two magnets with typical "flying times" of the order of a few hundreds of milliseconds. Needless to say, relaxation times considerably shorter than these "flying times" could not be measured in this way. During the sixties, Redfield, Fite and Bleich for the first time implemented transistors and current regulators at the IBM Watson laboratory [33]. Roughly at the same time, the first electronically field-cycled NMR instrument in Germany was constructed by Kimmich and Noack in Stuttgart.

The most essential parts of an FC apparatus are the electromagnet, its power supply and the cooling system for both the magnet and power parts. The rest of the instrumentation is basically the same as in any standard NMR set-up. Since the eighties, considerably improved air-core FFC electromagnets [34] were successfully used in academic laboratories, and further developed in an industrial environment [35]. Today, special electric networks combined with sophisticated air-core magnets allow the switching of magnetic fields between zero and 0.5 T in a few milliseconds (including both the transition and the duration of the transient). We refer to this relatively new generation of instruments as Fast Field Cycling devices.

3. Low-field proton relaxometry of confined liquid crystals

3.1 Field-cycling relaxometry of bulk liquid crystals

In the framework of classical continuum theories [22], the ODFs are expressed as overdamped normal fluctuation modes. The correlation function of these fluctuations represents a superposition of overdamped relaxation processes with a broad distribution of mode relaxation rates [18,22]. A conjugated intensity function relevant for spinlattice NMR relaxation rates shows (below an upper frequency cut-oft) an inverse square root frequency dependence as derived in [36]. In spite of numerous attempts, an experimental detection of the ODF relaxation by conventional high-field NMR relaxometry methods was hindered for a long time by the strong contributions from noncollective molecular dynamical processes which dominate in the MHz frequency range. The FC method finally revealed, in the kHz range, the predicted square root behaviour [37,38] associated with the ODFs. As an example, Fig. 17.2a demonstrates this type

Lowjrequency NMR relaxometry of spatially constrained liquid crystals

10'

10'

. . -.. Bulk aCB

'.

'_ ISOTROPIC 3231<

• NEMATIC 3O!IK

· 112 • -\,' . . ' , , .

, .' " '. .,

10' " Bulk 5CB • 5CB+B200

.. -" .

"

" "

10'

(b) \ '0 [Hzl

381

Figure 17.2. (a) Spin-lattice relaxation rate dispersions (a) in the nematic and isotropic phases of bulk SCB; (b) in bulk nematic 5CB at 303 K, and when confined in the Bioran glass with mean pore diameters of 200 nm at the same temperature.

of relaxation dispersion for bulk octy1cyanobiphenyl (8CB). Below 100 kHz, the phase transition from the isotropic to the nematic state is accompanied by a well-expressed change of the function T1- 1 (I/o) from frequency independent behaviour (typical for isotropic liquids at such low frequencies) to an inverse square-root dependence in the nematic state. The lowest frequency end of this dispersion is limited by local field effects [38,39] which produce (below a few kHz) a kind of a "false" plateau that is not related to real molecular dynamics. Another method of monitoring slow dynamical processes in the kHz range is relaxation in the rotating frame. In the case of ODFs, however, T~ 1 appears to be insensitive to modulations of dipolar interactions caused by this type of strongly anisotropic motions. The frequency dependence of T1-" 1 remains flat. A difference in the dispersion behaviour of the laboratory and the rotating-frame relaxation rates results from the fact that they are related to intensity functions of different order, and to different orders of spherical harmonics describing the angular part of the dipolar Hamiltonian. This point is not important in systems with spherical symmetry, but becomes crucial in the case of anisotropic, small-angle fluctuations such as ODFs (see [32] for more detail).

Recently, strong evidence for dipolar fluctuations due to ODFs was obtained in high magnetic field pulsed NMR studies using the effects produced on echo amplitudes [19, 40] by long-time scale dipolar correlations. This method permits one to extend the lowest frequency limit of the FC method by another two to three orders of a magnitUde, and therefore can be considered as complementary to FC.


Figure 17.3. Scanning electron micrograph (SEM) of a polymer dispersed liquid crystal, courtesy of A.K. Fontecchio and G.P. Crawford, Brown University, Rhode Island.

3.2 Field-cycling relaxometry of confined liquid crystals The first relaxometry studies of confined LCs were performed using nematic micro

droplets confined in an epoxi-polymer matrix, PDLC [9], Fig. 17.3. These materials are technologically most important for application in the production of switchable windows and in telecommunications [2,3]. In PDLCs, droplets of quite a uniform size are formed from a mixture of liquid crystal and a suitable oligomer in the process of polymerisation. The size of the droplets is controlled by the concentration and curing time, and ranges from 0.1 j.tm to 10 j.tm. The surface induced order and its effects on the molecular dynamics of the commercial liquid crystal E7 in epoxy polymer were studied using TI relaxometry in the MHz range (8-270 MHz) and measuring the TIp

dependence on the strength of the if-field in the kHz range. It was found that the spin relaxation in the PDLC system is mainly driven by the cross relaxation between the LC protons and those contained in the solid matrix. The exchange of magnetization occurs at the LC-polymer boundary. The prevailing impact of cross-relaxation over many frequency decades was later established by FC NMR relaxometry [41] of a very similar PDLC system. It was found that the measured TI- l in the MHz range is given by the weighted average of the proton LC relaxation rate, (TIl )LC, as it were in the absence of cross relaxation, and of the intrinsic polymer relaxation rate (Tl-l)P

TIl = PLC(TI-l)LC + pp(TIl)p. (17.1)

Here PLC and pp denote the relative fractions of protons in the LC and in the polymer, respectively. The agreement of the experiment with Eq. (17.1) indicates that the con-


dition for the fast cross-relaxation limit, i.e. ke » (TI- I )LC, (Tl1 )p, is fulfilled; here ke stands for the rate of magnetization transfer from the whole liquid crystal phase to the bulk of the polymer phase. The situation is different in the kHz range. The relaxation of the polymer protons becomes so fast here that (TI-I)p » ke, (TI-I)LC and [9]

(17.2)

The measurements in this frequency range thus allow for a direct determination of the effective cross-relaxation rate from relaxometry results. In PDLC materials, cross relaxation has been measured by other methods as well, i.e. by applying polarization transfer with off-resonance irradiation [42] and a selective magnetization inversion technique [43]. The results of various measurements showed that the cross-relaxation rate crucially depends on the compounds involved and on the temperature. Besides, it seems that in some cases ke is determined by the rate of transfer of spin energy across the phase boundary, and in others by the inverse time required for the magnetization to spread throughout the polymer [43]. The surface-dwell time of molecules seems to be less important in this context. However, it could be estimated from deuterium relaxation rates as shown in section 5 of this Chapter.

The PC relaxometry results, as well as TIp measurements of protons in PDLC materials, revealed that cross relaxation is not the only additional relaxation mechanism in the kHz frequency range. An additional contribution, roughly of the BPP-type (Bloembergen-Purcell-Pound), was found [9,41] and ascribed to a dynamical process (TR), which is now usually denoted as RMTD. This mechanism is expected for spatially non-uniform LCs with spatially dependent director orientation. It should be stressed that the molecular translational diffusion in non-uniform liquid crystals not only affects intermolecular spin interactions in the same way as translational diffusion in uniform bulk materials, but more important, also affects the stronger intramolecular interactions. The RMTD induced relaxation contribution for spherical nematic droplets was calculated in [44].

As far as the contribution of order director fluctuations in PDLC materials is concerned, no particular effect of confinement could be detected experimentally. It is clear that in droplets, the wavelengths of director fluctuations cannot be larger than the dimension of the cavity, assuming that the surface anchoring of molecules is strong. Therefore, a cut-off frequency (about 40 kHz) was introduced to account for this effect, assuming that ODP features will be significantly affected only below this value [9]. With the purpose of addressing this particular problem, Struppe et al. performed PC measurements of proton TI for 5CB in bulk and when confined in inorganic Anopore cavities where no cross relaxation [45] was expected. Moreover, with molecules parallel to the cylinder axis everywhere in the cavity, the RMTD mechanism is ineffective. The TI results were recorded for two orientations of the sample in the magnetic field. However, no difference between the bulk and the confined sample could be observed above 40 kHz, confirming that any effects of confinement on the ODP spectrum above this frequency are negligible.


The effects produced on the ODFs by interconnected pores of irregular shape were also considered. FFC relaxometry studies of 5CB in Bioran glasses [46,47] with different mean pore sizes manifested strong changes relative to the bulk. A pronounced increase and a much stronger (approaching vo2-like) frequency dependence of TIl compared to the bulk behaviour (a characteristic slow square root dependence) was observed below 100 kHz in confined samples, as shown in Fig. 17.2b for one representative (Bioran glass) sample. This indicates that, on the one hand, the ODF modes with wavelengths larger than the pore sizes are, very likely, quenched [46]. On the other hand, the low-frequency part of the mode spectrum seems to be transformed to a single mode or a narrow bunch of fluctuation modes as manifested by a strong Tl- l

dispersion in the confined sample. Such a scenario was supported also by the DCE studies [48]. At the same time, the problem of the origin of the low frequency modes still remains an open question. These modes may well point to the importance of ultra-long wave collective molecular fluctuations in strongly interconnected pores, or alternatively, they may be interpreted in terms of the RMTD mechanism as suggested in [47].

Before ending this section, it is worthwhile to draw the reader's attention to the problem of internal local fields relevant both for FC and rotating-frame experiments. One obvious consequence of rather strong residual dipolar fields in LCs is that they determine the lowest frequency limit of a "true" relaxation dispersion, and give rise to low-frequency plateaus eventually masking any slower stochastic processes. More severe problems might arise in systems with a non-uniform distribution of local fields, as is the case in confined LCs. In fact, it is still not clear how far reaching the relevance of local field effects might be for the above discussion.

4. The dipolar-correlation effect 4.1 Basic principles

4.1.1 Attenuation mechanisms for the stimulated echo. The dipolar-correlation effect denotes a specific attenuation mechanism for echo amplitudes due to ultraslow modulations of the dipolar interactions on the time scale probed by pulsed NMR techniques. This mechanism arises in macro- or microscopically anisotropic liquids like LCs [19,49], polymer melts [50] or networks [51]. In contrast to isotropic liquids of low viscosity, molecular motions in these systems tend to be locally restricted owing to an anisotropic molecular alignment or topological constraints. As a consequence, the secular part of the dipolar Hamiltonian is not completely averaged out by fast molecular motions in the evolution intervals of the echo pulse sequences. Depending on the strength, this intrinsic feature affects many measurable NMR variables in different ways. While an overview of the whole phenomenon is beyond the scope of the present discussion, we would like to point out some recent studies in this connection [52-55].

In the following, we consider the effects of residual dipolar couplings on the stimulated and primary echoes generated by the standard sequence of three 90° radio


frequency pulses [17, 56]

(17.3)

A residual dipolar coupling constant (in angular-frequency units)

- 2 2 nd = no < 1 - 3 cos Q;kl > ex 1 - 3 cos e (17.4)

(where no == 3/-Lo'",PIi/(87rr21)' rkl and Q;kl are the polar co-ordinates of the internuclear vector, and this definition of dipolar coupling constant differing by a factor 3 from that ofEq. (1.37)) remains after averaging over motions fast on the time scale t « 71, 72. The brackets in Eq. (17.4) denote the average with respect to the local symmetry axis defined, for instance, by an equilibrium director in Les or by an effective long chain axis in polymers. The residual constant, nd, has then a finite value determined by an angle e between an external magnetic field and that direction. In liquid crystals, residual dipolar couplings are further modulated on the time scale of the inter-pulse intervals 71 and 72 by much slower motions, such as ODFs, translational molecular diffusion, RMTD (see section 1). These ultraslow modulations cause an (additional) attenuation of echo amplitudes referred to as the DeE. The relevant time scale in the case of protons ranges typically from a few microseconds for the lower limit up to a few seconds for the upper limit. The latter is ultimately determined by spin-lattice relaxation and thus represents the longest time scale accessible to NMR techniques in general.

Experimentally, the DeE is elucidated via measuring the quotient of amplitudes of the primary (Apr) and the stimulated (Ast) echoes at the time points 271 and 271 +72

of their maximal refocusing, respectively. Magnetic field inhomogeneities in this experiment must be sufficiently large to ensure distinct echo signals, but be small enough to neglect attenuation caused by translational molecular or spin diffusion [57-59]. (This will be assumed throughout the whole of this section). The quotient of both amplitudes

Ast(271 + 72) '" Ade(271 + 72) _ Q Apr (271) '" Adc(271) = de

(17.5)

is usually measured as a function of 71 for a constant value of 72. Varying 72 as a parameter then produces a set of curves, Qde(7b 72). The subscript "de" refers to the attenuation produced by the DeE. Eq. (17.5) implies that the ultraslow dipolar fluctuations of interest are statistically (quasi)-independent of those due to fast individual or local intra-molecular motions with correlation times « 71> 72. In this case, the parts of the transverse relaxation due to slow and fast types of fluctuations are factorized. The attenuation factors associated with the fast fluctuations consequently cancel in the quotient since they are equal for both the primary and the stimulated echoes. Eq. (17.5) also implies the validity of quasi-static conditions for the coherence evolution due to single particle local resonance offsets, Aoff. This means that spins are assumed to experience the same local resonance offsets during the initial and final evolution intervals providing a complete refocusing of echoes at the times of their maximal amplitudes,


Aoff(2T1) = Aoff(2T1 + T2) = 1. Eq. (17.5) assumes that binary dipolar interactions dominate over other internal interactions in the system, so that, for instance, any cross-relaxation (spin exchange) [60,61] processes due to the presence of chemically inequivalent local environments are negligible (or compensated otherwise). A detailed density-operator treatment referring to a dipolar coupled spin pair and the evolution of spin coherences relevant for the formation of the stimulated and the primary echoes can be found in [17].

In general, the effects of slow dipolar correlations and local anisotropies result in the attenuation of the dipolar correlation quotient Qde(T1l T2) in the following form [19]:

Qde = ~exp {-l < on~ > c1}

X (1 + cos( < nd > T1) exp { -~ < on~ > c2}) 1

(17.6)

where Ond(t) == nd(t)- < nd > is the part of the residual dipolar coupling constant fluctuating on the time scale of the experiment about its mean value < nd >; (here and in the following the bar over the residual dipolar coupling constant is omitted for simplicity). The coefficients C 1 (T1 1 T2 1 Te) and C2 (T1, T2 1 Te) are functions of the inter-pulse spacings and the characteristic times of dipolar correlation losses, Te. The analytical expressions for these coefficients in the case of different types of correlation functions, < Ond(O)Ond(t) >, are evaluated in the literature [19,50].

The bar over the cosine term in Eq. (17.6) denotes the average over all orientations of local directors. It may be omitted for macroscopically uniform systems in which the mean coupling constant, < nd >, does not vary over space. The latter attains a value proportional to the order parameter of the system, S, and is finite in ordered systems. The cosine term in Eq. (17.6) produces an oscillatory behaviour of the echo amplitudes with a frequency equal to < nd >CX S. Any considerable oscillations vanish, however, in fully disordered systems with the isotropic distribution of local directors, that is, cos( < nd > T1) ~ O. The attenuation rate of Qde(T1, T2) as a function of T1 is determined by the coefficients C1 and C2 and the value of mean dipolar squared fluctuation, < on~ >. In non-viscous liquids, where dipolar interactions are completely averaged out by fast stochastic motions, no (extra) attenuation in addition to that caused by the fast motions is expected. In this case, the quotient, Eq. (17.5), is simply a constant as a function of T1.

4.1.2 Spin exchange. Another low-frequency echo modulation mechanism relevant for this discussion is due to cross-relaxation processes known as Nuclear Overhauser Effects [61,62]. They result from simultaneous flip-flops in dipolar coupled spin pairs and are best known in connection with two-dimensional NOESY NMR experiments [61,62] and the characteristic cross-peaks showing up in two-dimensional spectra.

In the context of low frequency mechanisms modulating echo amplitudes in the pulse sequence, Eq. (17.3), the relevant part of the cross-relaxation processes is that


due to transitions at the frequency determined by the difference in the chemical shifts. The rate of this process, T;; 1, is proportional to the intensity of the local field component fluctuating at this frequency and is thus indicative of the stochastic molecular motions that occur at low frequencies between zero and a few kilohertz. (In order to avoid any confusion with the established terminology of two-dimensional NMR, we shall refer to this mechanism as "spin exchange" throughout this section). Given that "spin exchange" is efficient on the time scale of the T2-intervals, the amplitudes of the stimulated echo are modulated as a function of T1 [63-65]. Note that "spin exchange" modulations should not be confused with the oscillatory behaviour of echo amplitudes owing to a finite value ofthe mean dipolar-coupling constant described by Eq. (17.6). The modulation frequency due to "spin exchange" is determined merely by chemical shift differences.

The magnitude of the remaining dipolar interactions is a crucial factor when considering the relative contributions of spin exchange effects and the DCE. If residual spin-pair dipolar interactions are much stronger than the chemical shift differences, the dominating attenuation mechanism for echo amplitudes is due to the DCE. In the case of somewhat weaker residual couplings (but not much weaker than the differences in chemical shifts), the situation becomes generally favourable for "spin exchange". Both mechanisms then may interfere with each other, so that an evaluation of the experimental results becomes complicated. In such cases, the modulations due to spin-exchange can be compensated using 71'-pulses inserted in the middle of each Tl

interval of the pulse sequence, Eq. (17.3). Provided spin exchange in the short Tl

intervals is negligible, these additional 71'-pulses refocus all phase shifts due to spin interactions (except those that have a bilinear form). The formation of the stimulated echo is then not affected by this kind of a modulation. Spin-exchange rates, on the other hand, can still be estimated by applying a standard, three 90° pulse sequence. Hence, the two mechanisms can be separated [51] when performing the experiment twice: once with the standard and once with the chemical-shift-compensating pulse sequence.

4.2 The DeE in liquid crystals 4.2.1 Bulk liquid crystals . The presence of long-time scale dipolar cor-relations attenuates the quotient Q de ( Tl, T2) as a function of Tl. A most illustrative example is the DCE in a nematic LC, 5CB [19,40]. Above the nematic to isotropic transition temperature, TNI, the experimentally measured quotient was found to be constant, as expected for non-viscous liquids. At the same time, a strong decrease of Qde(71, 72) with 71 superimposed on the oscillatory behaviour of the curves was observed for temperatures below the clearing point (bulk sample in Fig. 17.4a). The frequency of the oscillations is consistent with the dipolar splitting observed in the spectrum. The attenuation mechanism by the DCE was identified as the contributions of ultraslow collective molecular motions (the ODFs). This result is in good agreement with the findings from the FC relaxometry data which also revealed ODF relaxation in the low-frequency range, as discussed in section 3. The values of < 003 >, < 0 >,

388

313K

1 """",~!~o. td l l.r%Flt:r.~l:.:~.nJ1J .:: (lSdI')

As. '. ........ 15;::0- ,

Apr . ' ".~ I

:~ ... " " . 35 nm :m K

0.1 'I ' . (nom)

0.01

o

bulk

, 100 nm

0.2 0.4

(a) 0 .6

'1 ' ms

I 10· I

< IlO 2>

rad 2s·2 1

10' j

10 '

o


~

0.02 0.04

(b)

l I I

0.06 0 .08

R,1,nm ,1

Figure 17.4. a) Quotients of the stimulated and the primary echo amplitudes as a function of 71 in bulk 5CB and when confined in Bioran glasses with mean pore radii equal to 100,35 and 15 nm. Closed and open symbols refer to the temperatures below and above TN!, respectively. b) Mean squared fluctuation as a function of the reciprocal radius below TNI. (Reproduced with permission from [48]).

and the shortest mode correlation time, contributing to the DCE, could be determined along with their temperature dependences on the basis of fitting analytical functions to experimental results.

4.2.2 Confined liquid crystals below TNI. When confined to pores, the attenuation mechanisms for the quotient Ast/ Apr in LCs must be distinguished, not only relative to TN!, but also with respect to the characteristic size of the confinement. As shown for confined 5CB at temperatures below TN!, the main attenuation mechanism in relatively big pores (mean pore radii R 2: 15 nm) is still the DCE [48]. However, in contrast to the situation in bulk 5CB, no oscillations owing to a finite value of the macroscopic order parameter (considerthe cosine term in Eq. (17.6)) could be observed as shown in Fig. 17.4a. This suggests that local director orientations in the confined samples are governed by surface interactions, rather than by the external magnetic field. Any macroscopic preferential director orientation appears then to be destroyed by randomly oriented surfaces, so that cos( < nd > 71) in Eq. (17.6) is roughly zero. The attenuation rate of the quotient was found to exhibit a strong dependence on the pore size, as demonstrated in Fig. 17.4a. Its quantitative characteristics, < on~ >, in the limit of small angular amplitudes were shown to be proportional to the mean squared fluctuation of the transverse components of the instantaneous director relative to its eqUilibrium direction. The conclusion was that the decrease of < on~ > in smaller pores (Fig. 17.4b) reflects a damping of spontaneous ODFs by geometrical restrictions and surface interactions. The onset of bulk-like behaviour was estimated to take place in cavities 2: 60 nm. The correlation times of the fluctuation modes exceeded 1 ms, giving an estimate of ~ 103 s-1 for the mode relaxation rates. This


indicates the existence of extremely long wave modes that agrees with results ofFC relaxometry [46], as discussed in section 3. In the FC studies, such modes were detected well below the frequencies corresponding to finite pore size cut-off values, but could not be characterized quantitatively owing to the limits of the FC method. The findings of the DCE experiments thus provide a strong argument in favour of considering the observed increase of Til in the confined sample, Fig. 17.2b, as the "true" dispersion characteristic of molecular stochastic processes.

The above estimated mode relaxation rates can be used to evaluate formally the corresponding wave lengths, A = 27r / q, according to the continuum theory relation T-l(q) = (K/17 + D)q2, where K, 17 and D are the elastic constant, the viscosity and the diffusivity, respectively, and q is the wave number. (Typical values for 5CB can be found in [25]). The wavelengths evaluated in this way then appear to be extremely long (~ 2000 nm) exceeding the mean pore diameters of the investigated samples by to to tOO times. On the one hand, such a finding would fit in well with the results obtained by quasi-elastic light scattering [66] for a similar system (5CB confined in silica aerogel with a mean pore size of ~ 43 nm) where a slow fluctuation mode with a wavelength of order tOO mean pore sizes was also detected. In [66], these modes were naturally attributed to correlated interpore fluctuations due to the interconnection of the pores. On the other hand, the question arises whether such ultra-long wavelength fluctuations, detected now by at least three different methods, could easily survive unavoidable shielding effects due to pore walls. The question is of course of quite general importance and goes beyond the origin of the relaxation modes in the particular system under discussion. However, at this stage it is still too early to reach a clear conclusion on this point.

4.2.3 Confined liquid crystals above TN!. For temperatures above TN!, the question of which attenuation mechanism dominates again depends on pore size. As shown above, 5CB in the isotropic phase behaves as an ordinary liquid, with Ast/ Apr independent of Ti. The same remains true for all samples confined to relatively big pores (R2: 15 nm), see Fig. 17.4a (open symbols), but does not hold for smaller pores (R::;5 nm) [63,64]. The quotient Ast/ Apr in the case of such small pores appears to be strongly modulated, as shown for 5CB in 4 nm controlled porous glass (Fig. 17.5a). The modulation mechanism in small pores is due to "spin exchange", as discussed in section 4.1.2. It persists in a broad temperature range, both above and below the bulk clearing point. Incidentally, this is in contrast to the DCE which in low molecular weight LCs appears as the temperature goes below TN! and vanishes abruptly (')umplike") as the temperature goes above TN!.

The origin of the spin-exchange modulation mechanism can be understood in the context of an orientational anisotropy of 5CB molecules in the vicinity of surface interfaces. (This type of surface ordering is frequently referred to as "para-nematic" order and is discussed in more detail in section 5). Owing to the anisotropic alignment, averaging of dipolar interactions of near-to-surface molecules remains incomplete well above TN!. However, the magnitude of the residual couplings is much smaller than in the case of a nematic type of ordering. In fact, residual couplings are scaled down

390

o 2 4

(a) 6 'I ms

K 1 TR,

KEX

0.1


~ .......•. ~

2 3 4 5

(b) R, nm

Figure 17.5. a) Quotients A st / Apr as a function of 71 in seB confined to 4 nm controlled pore glass above TNI. The curve parameter 72 varies in the range between 10 IDS and 320 ms corresponding to the curves with the smallest and the biggest amplitudes of oscillation, respectively. b) Quantities K EX and KTR are the (normalized) square roots of 7;;',1 and the transverse relaxation rates, T2- 1 , measured above TN! as a function of the mean pore radius R. All values are normalized at the value of the smallest pore size, R = 1.5 nm. The solid and the dashed lines represent the fits of the exponential function to the experimental results for R < 5 nm with the characteristic length constant fitted to about 3 nm.

to the order of the chemical shift differences of cyano-biphenyl chemical groups as manifested by broadened and overlapped, but still partially resolved, spectral lines [63]. (In nematic 5CB, on the contrary, the spectrum is dominated by a dipolar splitting [19] and the chemical shifts of different groups remain completely unresolved). Such conditions are rather favourable for spin exchange processes with rates depending on temperature and pore size [64]. The absence of spin exchange modulations in relatively big pores above TN! must be the result of averaging due to molecular diffusion across the pore where the majority of molecules find themselves far from orienting walls.

Apart from the observed harmonic modulation (i.e., when considering only the "envelope"), the quotient remains nearly constant over a much longer time (a few milliseconds) compared to that of 5CB in bulk and in bigger pores below TN!. This indicates the lack of any irreversible attenuation of the quotient and leads to the conclusion that the DCE produces a minor, if any, contribution to transverse relaxation above TN!. On the other hand, a surface induced ordering was shown to be responsible for a strong enhancement of the relaxation rates in the kHz range [25,26]. The mechanisms discussed in this connection (RMTD, exchange losses, cross relaxation, etc.) arise mainly from modulations of residual dipolar or quadrupolar interactions reSUlting from translational diffusion between regions with different surface orientations and different levels of orientational anisotropy. The measurement of low-frequency relaxation rates thus provides information of major importance. In particular, it enables one to estimate a surface induced order [25,26], making use of the proportionality between the square roots of the relaxation rates and the orientational local order parameter. This is demonstrated, for instance, in Fig. 17.5b where the square roots of

Lowjrequency NMR relaxometry of spatially constrained liquid crystals 391

both spin exchange and transverse relaxation rates are plotted versus pore size. The decrease of the rates with increasing pore size is attributed to the averaging effect resulting from translational molecular diffusion between the regions of different degrees of local anisotropies. According to theoretical approaches [67], the local order parameter is expected to decrease exponentially with increasing distance from the wall as long as the radius of the cavity considerably exceeds the thickness of the ordered layer. The observed exponential decay of VT;;;l and VT2- 1 with increasing R (first three data points) in Fig. 17.6b thus reflects the change of the average (across the pore) S with increasing R. The characteristic length constant was found to be about 3 nm. Nearly identical behaviour of the spin exchange and the transverse relaxation rates as a function of R is quite remarkable, and shows that both relaxation mechanisms are governed by the same underlying dynamics.

5. Deuteron NMR relaxometry of confined liquid crystals Deuteron relaxometry studies of confined LCs started about ten years ago when it

became clear that proton relaxometry alone is not able to give conclusive evidence on molecular orientational order and dynamics in such systems. The relaxation of deuterons with spin I = 1 results predominantly from the interaction of their electric quadrupole moments with the time-varying electric field gradient tensor at the site of the nucleus. Their relaxation is therefore essentially intramolecular and depends only on the rotational and translational motion of the spin-bearing molecule. The advantage of studying deuteron relaxation over that of protons is certainly the smaller number of parameters that determine the relaxation rate. This makes the analysis of experimental results more conclusive. However, the drawback is the demanding synthesis of deuterated liquid crystalline compounds, and a more tedious application of experimental NMR techniques. The application of more sophisticated NMR techniques to deuterons was rare even with bulk LCs. To our knowledge, the frequency dependence of deuteron spin-lattice relaxation times over a range of many decades was measured only for one bulk LC (5CB) [68]. In the following, deuteron relaxometry studies of liquid crystals confined into cylindrical cavities of inorganic Anopore membranes and in PDLC will be reviewed.

Most of the work performed until to now focuses on the high-temperature region. This might appear controversial at first sight, as bulk LCs loose their essential property, i.e., orientational order, above TNI. However, in microconfined liquid crystals the hightemperature phase is not isotropic throughout the volume of the cavity [67,69,70]. In most cases the internal surfaces impose a certain degree of orientational order to LC molecules in the interface layer. Nevertheless, the term "isotropic" is still used to designate the high-temperature phase, as long as the nematic-isotropic transition is discontinuous and the surface-induced order smaller than in the nematic phase. The surface-induced order in the isotropic phase is limited to a thin interface layer, and therefore directly related to the interactions of liquid crystal molecules with the solid boundary. Therefore, the isotropic phase is more suitable for the study of surface interactions than is the nematic phase.


5.1 Liquid crystals in cylindrical cavities of Anopore membranes

Cylindrical cavities in commercial Anopore membranes with diameters ~ 200 nm have proved to be a particularly suitable matrix for the study of confined liquid crystals. The Al203 walls of Anopore rnicrochannels can be treated with various surfactants before filling the channels with liquid crystal. In this way, the liquid-crystal - solid interactions can be systematically controlled using different surfactant molecules for the coating of cavity walls. However, as far as the NMR experiments are concerned the most important feature of Anopore confined liquid crystals is their cylindrical symmetry. Cylindrical symmetry simplifies the interpretation of deuterium NMR spectra, and information on the orientational order can be obtained. Using 2H NMR spectroscopy, the director field was found to be parallel everywhere to the cylinder axis for 5CB and 8CB in non-treated Anopore cavities. In contrast, the director field in cavities treated with surfactants like lecithin or aliphatic acids is perpendicular to the axis of the cylinder, and forms a planar bipolar structure. This is because the surfactants force liquid crystal molecules to orient perpendicular to the surface. The cylindrical symmetry becomes particularly important in the isotropic phase. Here, the splittings observed in the NMR spectrum are given by [25,26,71]

3 e2qQ /1 2 ) 6.v = "2-h- \"2(3 cos ()B - 1) (S(fj) (17.7)

where e2qQ / h is the deuteron quadrupole coupling constant averaged over molecular conformational changes and fast reorientations around the long axis, () B denotes the angle between the nematic director in the surface layer and the external magnetic field, S(fj is the local orientational order parameter and <> denotes averaging over the cavity volume. The averaging is brought about by fast molecular translational diffusion, whereas small effects of director fluctuations and molecular biaxiality have been neglected in Eq. (17.7). When the sample is oriented with cylinder axes parallel to the magnetic field, () B assumes the same value everywhere at the wall, and the factor < (3cos2 ()B - 1) > is finite and relatively large. The spectrum is a well-resolved doublet with a splitting proportional to < S( fj > ~ 2dSo/ R, if the thickness d of the ordered surface layer is much smaller than the radius R of the cavity. So denotes the order parameter at the surface. The measurements of the NMR splitting showed that the surface order parameter So is very small and independent of temperature in non-treated cavities with molecules oriented parallel to the wall, e.g. for 5CB, ISol ~ 0.02 [71]. On the other hand, for surfactant-treated walls that induce perpendicular molecular orientation at the surface, So exhibits a strong pretransitional increase on approaching TN!. However, it does not reach the bulk nematic value at TN! (:=:::: 0.35).

Relaxometry studies of LCs in cylindrical cavities give information both on surface induced order and molecular dynamics. First, it was found that the spin-lattice relaxation time, measured either at 13 MHz or at 58 MHz, does not show (within experimental error) any difference between the bulk and the constrained LC in the isotropic phase [25,72]. This leads to the conclusion that the rate of fast molecular


reorientations that govern the spin-lattice relaxation in the MHz range is not affected by the confinement. The same conclusion extends to practically all other systems of spatially constrained LCs. The absence of any significant change in T1- 1 upon confinement was found also in POLC material and in porous glasses. Even the smallest cavities under study (with diameters of only ~ 7 nm) did not affect considerably the rate of deuteron spin-lattice relaxation in the MHz frequency range [16].

The effects of confinement become clearly visible if the relaxation is observed at lower frequencies. One of the methods for observing slow-motion contributions to the spin relaxation in liquid crystals is measuring the transverse spin relaxation rate T2- 1. A big difference between the T2- 1 of confined liquid crystals and that of the bulk is observed. In systems with uniform orientation of the surface director with respect to the magnetic field, the RMTD mechanism is not effective as all surface sites possess the same preferred molecular orientation. Therefore the only effective additional relaxation mechanism in the kHz range is the exchange of molecules from the ordered surface layer with the isotropic region farther away from the wall. The simplest expression describing such relaxation for BB = 0° is [73-75]

-1 9 2 e2qQ 2 (_)2 (T2 )exch = 47r 17(1 - 17) -h- So Texch (17.8)

where 17 ~ 2dl R denotes the fraction of ordered molecules in the surface layer and Texch their approximate average lifetime within it. The temperature dependence of (T2- 1 )exch depends essentially on the temperature dependence of the surface order parameter. In non-treated cavities, with temperature independent So, the exchangeinduced relaxation rate shows only a mild pretransitional increase with decreasing temperature, due to changes in the fraction of ordered molecules and the length of Texch which is estimated to be of the order ~ 10-5 s for 5CB and 8CB in Anopore membranes [25,72]. The pretransitional increase of (T2- 1 )exch is much more pronounced in cavities treated previously with aliphatic acid. It is particularly interesting for confined 8CB which possesses, in addition to the 7 K broad nematic phase, also the smectic A phase at lower temperatures [76]. In Fig. 17.6 it is clearly seen that there is a complete matching of T2- 1 curves vs temperature for the bulk and confined 8CB above T - TN! ~ 6 K. Obviously, the impact of the surface here is too small to induce an ordered interface layer. Then an abrupt change in the slope of confined 8CB occurs, indicating the onset of the first molecular layer at the surface which has a certain degree of both orientational and positional (smectic-like) order. About 3 K above TN!, the slope changes discontinuously once more as the second smectic layer next to the surface starts to form. The NMR results provide the first indication that a step-like onset of smectic ordering occurs already in the isotropic phase, even for a compound that has an intermediate nematic phase between the isotropic and smectic phases. The same phenomenon had been suggested earlier on the basis of optical measurements [77], and later by structural studies performed by atomic force microscopy [78]. It should be mentioned that order fluctuations might also influence deuteron slow-

394

'" '" 1000 - '" '"


• 3CB - ad2 bulk

'" 8C B- ad2 in treated Anopore> d-200nm

'" isotropic I paranematic phase

'" '" ... , ···.l..., 100 I;/TNI

o 5

.... ",.",., ".,., ... .......

I J "'-,

10 15 T-TN1[K]

Figure 17.6. Temperature dependence of deuteron T2- 1 for 5CB in cylindrical cavities of Anopore membranes and in the bulk (from N. Vrbancic-Kopac, Ph. D. Thesis, University of Ljubljana, 1997).

motion relaxation in the isotropic phase with an ordered surface layer. The spectrum of such fluctuations is characterized by the appearance of two slow modes that are not present in the bulk isotropic phase [79]. One of them corresponds to fluctuations in the thickness of the ordered boundary layer, and the other to nematic director fluctuations within the layer. Their characteristic decay times are of order 10-7 s and differ from the fluctuations appearing in the bulk in a narrow temperature range up to about 1 K above the transition temperature. However, the measurements of T2- 1 show deviations from bulk behaviour over several degrees K, and in most cases over several decades of degrees K above the nematic-isotropic transition. Therefore, one may conclude that order fluctuations cannot be identified with the additional kHz-range relaxation mechanism in confined liquid crystals.

5.2 Polymer dispersed liquid crystals (PDLCs)

Deuteron NMR relaxometry is of prime importance in the study of surface induced order and liquid-crystal- polymer interaction in PDLC materials [80,81]. Whereas the spectrum of a liquid crystal in Anopore cavities consists of a doublet even above TN!, only a single central line is observed in PDLC. Due to the almost spherical symmetry of the droplets, the distribution of surface directors in the cavity is isotropic and translational diffusion of molecules "blurs" the splitting. However, the full width of the line at half maximum (fwhm) exceeds that of the fwhm in the bulk isotropic phase. The broadening would be inhomogeneous and caused by the residual static quadrupole interaction of deuterons in the surface layer if motional narrowing were


not complete. In this case the linewidth is proportional to the surface order parameter. On the other hand, the broadening might result from dynamics, i.e. time-modulation of the quadrupole interaction on the slow time scale. This homogeneous contribution to the linewidth equals T2- l /1r. Comparison of the fwhm and the transverse spin relaxation rate measured in PDLC shows that the broadening of the line is almost completely homogeneous, and that it should be analysed in terms of the transverse spin relaxation rate [81].

The temperature dependence of the deuteron T2- l in the isotropic phase of selectivel y deuterated 5CB in PDLC droplets and in the bulk is presented in Fig. 17.7. The droplet diameter is about 500 nm. There is a significant difference between the relaxation rates of the two samples. The T2- l of the confined liquid crystal is larger by at least a factor of 2 compared to the bulk, and shows a well expressed pretransitional increase. It is reasonable to assume that the slow dynamical process which gives rise to an additional contribution to the relaxation rate is translational diffusion of molecules among surface sites with different orientations of the local director (RMTD mechanism). Its contribution to Til has been calculated by Vilfan et al. [81] for a spherical enclosure with uniform translational diffusion coefficient throughout the cavity. The resulting (T2- l )RMTD is proportional to the square of the thickness of the surface layer, i.e. approximately to (T - T*)-l. The comparison of experimental data with theory shows that an adequate interpretation requires an additional, temperature independent contribution. The bold solid line in Fig. 17.7 represents the fit of the expression

(17.9)

to the experimental results. A good fit is obtained by varying only two adjustable parameters: the surface order parameter So and the temperature independent contribution C. It turns out that this good fit implies a temperature independent So ~ 0.08. Over the whole temperature range this value is much smaller than the bulk order parameter at TN! and is, therefore, characteristic of partial orientational wetting of the surface with the nematic phase. This means that the polymer surface has an ordering effect at temperatures where the bulk is isotropic, and a disordering effect in the nematic phase. The temperature independent contribution C in Eq. (17.9), required to explain the experimental results, arises most probably from a slowing down of molecular translational diffusion in the first molecular layer next to the boundary.

Recently, a new kind of PDLC material attracted great attention. It is formed by illuminating a suitable monomer - LC - surfactant mixture by two coherent laser beams that form an interference pattern. The polymerization in the bright regions takes place more rapidly than in the dark regions, resulting in a stratified structure of the sample where liquid-crystal-rich layers alternate with pure polymer layers. Such materials are known as H-PDLCs (holographic polymer dispersed LCs) and are used as electrically switchable diffraction gratings [82,83]. The liquid-crystal-rich layers usually exhibit a partly-interconnected droplet morphology. The deuteron NMR measurements were performed with an H-PDLC sample of composition as close to that of standard switchable optical elements as possible. On one hand, a huge decrease of the nematic-isotropic transition temperature was found and ascribed to the presence

396

~

f.-'"


800 • , • I . ,-700

2 H \ 'L = 58 .34 MHz

600

500

400 POLe

300 -',

200 l~ bulk

100 .... · ~"::~~·-..... ·-... i .. -.. ...... _· .... ·_ ...... _·_ ... ·_. - ----=I

o -f-~r--~r--'---~'T - ..... • - ~ - -.:. .. . - - .... - - - - -

o 2 4 6 8 10 12 14 16

T-TNt (K)

Figure 17.7. Temperature dependences of the deuteron T 2- 1 of selectively deuterated 5CB in PDLC droplets and in bulk 5CB. (Reproduced with permission from [81]).

of foreign ingredients in the liquid crystal that act as impurities. On the other hand, a strong ordering effect of the rough polymer surface on the liquid crystal and a reduction of translational molecular mobility at the boundary for at least two orders of magnitude take place [84]. Furthermore, the NMR spectra and transverse spin relaxation rate show that some droplets are quasi-isotropic, even at room temperature, which might decrease the switching efficiency of the diffraction.

Acknowledgments

The authors would like to thank Prof. Dr. Rainer Kimmich for many helpful and stimulating discussions. Besides, EG. is cordially grateful to him for the excellent research atmosphere and conditions during her enjoyable multiyear stay in the "Kernresonanzspektroskopie" laboratory. E.A. thanks the Alexander von Humboldt Foundation for a research fellowship partially devoted to the study of confined liquid crystals and related problems.

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(2002), Phys. Rev., E 66:021710.

Chapter 18

NMR ON MACROSCOPICALLY ORIENTED LYOTROPIC SYSTEMS

G. Oriidd and G. Lindblom Department of Biophysical Chemistry, Umea University, Umea, Sweden

1. Introduction Solid state NMR spectroscopy is extremely useful for investigations of many dif

ferent aspects of lyotropic liquid crystalline phases, as can be inferred from the very large number of papers published on this matter over the last twenty years (see Web of SCI). Just to mention a few important examples, NMR methods have been shown to be informative and rapid for convenient determinations of phase equilibria as well as of phase structures [1], in particular the structure ofthe so called cubic phases. Information about solubilization, extent of hydration, orientation of various molecules and the dynamics of the lyotropic phase can be easily obtained. Here, we confine ourselves to NMR investigations of translational diffusion of water and lipids in macroscopically aligned lyotropic liquid crystals. We also put some emphasis on how to get a good orientation of a lamellar phase between glass plates. This is crucial for the use of pulsed-field gradient (pfg) NMR and it is also of importance for other solid-state NMR techniques as well as for various other methods, such as X-ray and neutron diffraction, fluorescence recovery after photo-bleaching (FRAP), linear and circular dichroism and electron spin resonance. Finally, we present some of our recent results on macroscopically oriented systems.

2. Orientation dependent NMR interactions Any lipid or surfactant molecule contains at least one nucleus that can be studied

by NMR, but some nuclei are more efficient and convenient to use than others. The nucleus used also depends on what kind of information one would like to get. If the object is only to get a phase transition temperature, the choice of nucleus is more or less arbitrary. However, if more detailed information about the lyotropic liquid crystal is required, the static interactions in NMR are often exploited. On the other hand, for determinations of lipid lateral diffusion, static interactions should not be present or methods to cancel them must be available. For a discussion of diffusion measurements by NMR, it therefore seems useful to touch briefly upon the various static interactions



that may appear in such experiments. For 1 H NMR the static dipole-dipole couplings, observed through broadening of the linewidths of the NMR peaks, are important, while for 31 P, 19F and 13C NMR the chemical shift anisotropy (CSA), observed in the lineshape of the NMR signal, is the appropriate NMR parameter to consider. For nuclei like 2H, 14N and alkali and halide nuclei quadrupole splittings are generally observed in anisotropic phases. The NMR spectrum for a general spin system is determined by the spin Hamiltonian, 'H which consists of a number of interaction terms, but only the following four terms are of interest (see also Chapter 1):

(18.1)

where 'H z represents the Zeeman term, 'He SA represents the effect of induced magnetic fields due to orbital electronic motions, i.e. the chemical shift, and 'HQ and 'HD are the quadrupolar and the dipolar Hamiltonians, respectively.

The great advantage of macroscopically oriented systems from an NMR point of view can be rationalized from the orientation dependence of the static interactions in a semisolid sample, such as a lyotropic liquid crystalline phase. As discussed below all of these interactions have a common scaling factor, namely! (3 cos2 OLD - 1) which is the second Legendre polynomial, P2( cos OLD), where OLD is the angle between the bilayer normal and the main magnetic field (Bo) [1].

The above scaling implies that the NMR signal will be spread out over a range of frequencies if the sample consists of randomly oriented microcrystallites. The reSUlting NMR spectrum, the "powder pattern", has often been used in the investigation of lyotropic phase behaviour and the determination of molecular ordering [1]. In such investigations calculation routines are often utilized in order to transform the powder pattern spectrum into the spectrum that would be obtained from an oriented system, since the intensity is then concentrated to a single frequency, thereby improving the signal-to-noise [2-5]. The ability to prepare oriented samples not only circumvents this transformation but also, and more important, it gives the opportunity to study angular-dependent NMR properties.

2.1 The quadrupole interaction Deuterium and all the alkali and halogen nuclei except fluorine have spin quantum

numbers 1 ~ 1 and consequently possess quadrupole moments. In an anisotropic, uniaxial liquid crystalline phase the quadrupole interaction does not average to zero and we get a static term which is small compared to the Zeeman term. Only the secular part of the Hamiltonian needs to be considered, and we have (in frequency units) [1]

where 3 eQVoM

IIQ = 2" 21(21 - l)h

(18.2)

(18.3)

NMR on macroscopically oriented lyotropic systems 401

and

(18.4)

YOM is the principal component of the electric field gradient tensor at the nucleus. The principal axis of this tensor is normally taken to lie along the C-2H or 0-2H bond, and is here designated by M, eQ is the electric quadrupole moment of the nucleus, S is an order parameter characterizing the orientational order of the electric field gradient tensor, e.g. a measure of the ordering of a C-2H or 0-2H bond with respect to the normal of the lipid bilayer, ()DM is the angle between the C-2H or 0-2H bond and the director, and () LD is the angle between the director and the applied magnetic field, Bo.

For a macroscopically aligned sample the 2H NMR spectrum exhibits two peaks of equal intensity separated by the quadrupole splitting, D.vQ, (c.f Fig. 18.7)

(18.5)

Since ions and water molecules in a liquid crystalline phase may reside in different sites, in the fast exchange limit Eq. (18.5) has to be modified according to

(18.6)

where the Pi represent the fractions of ions or water molecules in site i. For a powder sample (with a random distribution of the director axes) where all

values of cos ()LD are equally probable, the quadrupole splitting in the NMR spectrum corresponds to that for ()LD = 90· in Eqs. (18.5) and (18.6) and gives

D.VQ = LPivbSi = l(vQS)I. (18.7) i

From Eqs. (18.5) and (18.6), it is obvious that the (hD = 90· orientation of the director axis with respect to Bo would have a splitting of one-half that obtained at the () LD = O· orientation. Furthermore, the intensity of the () LD= 90· peak is largest and it declines steadily to the smallest value when () LD = 0·.

According to Eq. (18.7) it can be expected that the measured NMR quadrupole splitting depends on the phase structure, since at least one of the parameters Si, Pi or vb should be sensitive to a change in the mesophase structure. Thus, it is possible to investigate phase equilibria through measurements of quadrupole splittings.

2.2 The chemical shift The presence of electrons in the molecule containing the nucleus in question gives

rise to chemical shielding. The strong, external magnetic field, Bo, induces electronic currents that produce, in tum, an induced magnetic field B' that modifies the local magnetic field Bloc at the site of the nucleus in question. B' will be proportional to Bo, and will generally be diamagnetic; for molecules executing rapid and random rotation


in liquids B' = -O'oBo, where 0'0 is the isotropic chemical shielding parameter. In solids and liquid crystals it may be necessary to write B' = -0' Bo, where 0' is a second rank tensor. Thus, the Zeeman interaction between the magnetic field and the nuclear spins is influenced by the second-rank chemical shift tensor 0'(2). This is orientation dependent and carries the information about the molecular arrangement. In analogy with the treatment of the quadrupolar interaction, the anisotropic part of the chemical shielding of the Hamiltonian can be treated as a perturbation to the scalar part of the Zeeman interaction. In first order it is only the secular terms that give any contributions [6]

(18.8)

which after time averaging yields a relatively complicated equation showing that also the anisotropic part of the chemical shift is multiplied by (3 cos2 (hD -1), and contains the order parameters. B z is the Z -component of the magnetic field, and the other terms have their general meaning. Note that, since CSA measurements are obtained with their sign in the NMR spectrum, information about phase structure is quite easy to get. 31 P NMR is most frequently used for studies of lipids, and it is very useful for testing the macroscopic alignment of an oriented bilayer between glass plates. The effective chemical shift tensor is cylindrically symmetric, reflecting the cylindrical symmetry about the normal to the lipid aggregate in the liquid crystalline phase (i.e. in the director coordinate system). Therefore, the observed peaks in the NMR spectrum will depend on the orientation of the cylindrically symmetric tensor with respect to the magnetic field. An NMR signal will appear in a limited region of frequencies determined by (hD which can vary between O· and 90· (Fig. 18.1). The distance between the peaks at these two extreme values of (J LD can be measured and is usually called the CSA which is defined as t:::.O' = (0'11 - 0' ..l). A powder sample consists of microcrystallites exhibiting a random orientation of the directors and the NMR spectrum will consist of a superposition of signals from all the different values of () LD. The characteristic NMR line shape resulting from a CSA with axial symmetry can be seen in Fig. 18.1 along with the signal from the oriented part of the same sample.

2.3 The dipolar interaction Due to strong proton dipolar couplings in anisotropic liquid crystalline phases, 1 H

NMR exhibits broad featureless spectra. The interpretation of such NMR spectra poses problems, but at the same time there is a lot of information to be gained. In diffusion studies the use of protons with their large gyromagnetic ratios and high sensitivity would be advantageous were it not for the line broadening reSUlting from strong dipolar couplings. Fortunately, this problem can be circumvented in studies of lipid diffusion oflamellar liquid crystalline (Lo) phases. In the case that all the protons in the hydrocarbon chain in a lipid bilayer undergo fast rotational and translational diffusion, so that the intermolecular dipolar couplings average to zero, the dipolar part

NMR on macroscopically oriented lyotropic systems

30 ,---------,

20 f\v [ 10

CI. 0

-10

40

-50 0 50 100

8LD

20 o ppm

-20

403

Figure IB.l. Illustration of the dependence of the 31p chemical shift on the angle between the bilayer normal and Eo for a macroscopically oriented sample of dirnyristoylphosphatidyIcholine (DMPC)/gramicidin D/2H20. The first spectrum is taken at () LD = -54.7° and () LD is then incremented by approximately 12° for each spectrum. The sample chosen was only partially oriented in order to be able to observe also the powder pattern in the same spectrum, and the first spectrum shown was obtained with 16 times more acquisitions in order to increase the signal/noise. The signal from the oriented part of the sample can be seen to move between the edges of the powder pattern as () LD is changed. The chemical shift is scaled by P2 (cos () LD) as shown in the insert.

of the Hamiltonian is given by [7]

'HD = ~(3cos20LD -1) LFJ2\ij)A~2)(ij) (18.9) i>j

where the sum i > j is only over intramolecular pairs, and FJ2) (ij) and A~2) (ij) are irreducible tensor components representing the space and the spin parts, respectively. Again we see that the single factor !(3 cos2 OLD - 1) multiplies all the anisotropic terms. This fact results in the observation of a remarkable and characteristic bandshape, called super-Lorentzian, for powder samples of liquid crystalline phases in IH NMR. This spectrum is a superposition of spectra from microcrystallites with random orientations of their directors. Wennerstrom quantitatively reproduced such a spectrum for a surfactant - water system [7]. There is a dramatic change in the proton NMR spectrum at the gel to La phase transition, where the former phase shows a band shape typical of an organic solid. The basic reason for the difference between the spectra in


the gel and liquid crystalline phases is that the intermolecular dipolar couplings do not average to zero in the gel phase. In that phase the lipid lateral diffusion is slow and there is no cylindrical symmetry around the normal to the bilayers. Thus, a gel-type spectrum can be taken as an indication of slow lateral diffusion, D L ::; 10-13 m2s- 1,

while the observation of a super-Lorentzian bandshape for a liquid crystal implies a faster translational diffusion, DL ~ 10-13 m2s-1.

2.4 The magic angle

From the above expressions for the orientation dependent interactions it is clear that the factor ! (3 cos2 e LD - 1) scales all of these interactions. For the important case when cos e = ~, i.e. e = 54.70 , this factor becomes zero and the static interactions have "magically" disappeared. This is the reason that this angle goes under the name "magic angle". In the commonly used solid state NMR technique magic angle spinning (MAS), the interactions are first projected onto the sample spinning axis by the rapid spinning, and this axis is then adjusted to the magic angle with respect to the lab frame, thereby removing the static interactions. For ordered liquids, such as the La phase, the fast translational and rotational motion of the molecules project the interactions along the bilayer normal and, in order to remove those unwanted interactions, one needs to first orient the sample so that all microcrystallite directors are parallel, and then to place the sample with the director at the magic angle. These procedures results in NMR spectra with linewidths reduced from several kHz to the order of hundreds of Hz or less.

3. Lipid translational diffusion The NMR methods with pulsed magnetic field gradients provide attractive tech

niques for studies of molecular transport in lipid systems [8,9]. One of the most successful applications of pfg NMR is its use in extracting structural information about heterogeneous systems such as complex liquids and liquid crystals. Pfg NMR presents a method with which lipid lateral diffusion coefficients in an La phase can be directly measured [8, 10]. In recent years the applicability of NMR diffusion techniques has been growing fast due to the great improvement of the NMR equipment used for diffusion and NMR microscopy [11].

Two basic spin-echo experiments for diffusion measurements are illustrated in Fig. 18.2. The spin-echo sequence [12] (SE, Fig. 18.2 top) depends on the creation of transverse magnetization in a time interval, T, in which the nuclear spins with different precession rates are allowed to de-phase in the XY -plane. At time T the de-phasing process is reversed by the application of a 180· rf pulse and the nuclear spins begin to re-phase and eventually they meet again to form a spin-echo.

The monitoring of self-diffusion of molecules in a sample is accomplished by the application of magnetic field gradient pulses of strength 9 and duration <5 during the de-phasing and re-phasing periods. These gradients cause the nuclear spins in different local positions in the sample to precess at different Larmor frequencies, thereby


be extended to times comparable with the longitudinal relaxation time Tl which often is much longer than T2. In the STE experiment R is given by R = ~ + ir + In 2.

Several of the parameters in Eq. (18.10) may be varied in a diffusion experiment, but generally, all parameters except 8 or g are kept constant in order to eliminate effects arising from different relaxation times. The diffusion coefficient, D, is then extracted from a non-linear fit of data to Eq. (18.10) (Fig. 18.3).

In an anisotropic liquid crystalline phase the dipole couplings between protons will cause a very rapid relaxation of the spins, preventing any spin-echo from forming. The translational motion of lipids can still be measured with the conventional diffusion pfg NMR method if all static proton dipole-dipole interactions are removed. For measurements of the lateral diffusion of lipids in an La phase this can be accomplished by a macroscopic alignment of the bilayers between e.g. glass plates [10,14]. From Eq. (18.9) it is obvious that if the bilayer is macroscopically aligned and oriented at the magic angle fhD = 54.7", 1-lD becomes zero and the static dipolar couplings are

l.S ppm

o

• 1.1 ppIIl A 3.1 ppm • 4-.ppm

1 4 6 g(T/m)

Figure 18.3. Left: Stacked plot of NMR spectra obtained from a 1 H pfg diffusion experiment on an oriented sample of DMPCPH20. The three dominant peaks seen in the spectrum originate from water (4.6 ppm), choline methyl (3.1 ppm), and hydrocarbon chain methylenes (1.1 ppm). Right: The fits of the peak intensities of the three dominant signals to Eq. (18.10). In this experiment, the gradient strength has been varied non-linearly in order to capture both the fast decay of the water and the slower decay of the lipids. The decays are biexponential due to overlap of the signal intensities. The lateral diffusion coefficients obtained are 6.63 x 10-10 m2 /s (water) and 4.95 x 10-12 m2 /s (lipids).


900 1800

't 't .. t .. t

I~tg I. ~ 900 900 900

't T 't • t • t • ~

Figure 18.2. Basic pulse sequences used in the pfg diffusion experiments. Top: Spin-echo (SE) experiment. Bottom: Stimulated spin-echo (STE) experiment. The magnetic field gradient pulses are shown as hatched rectangles, the rf-pulses as black rectangles and the refocused spin-echo as a black triangle.

enhancing the de-phasing process. If the spins maintain their positions throughout the experiment, they will still refocus completely into a spin-echo by the SE pulse sequence. On the other hand, if they change their positions during the experiment, their precession rates will also change, and the refocusing will be incomplete, resulting in a decrease in the intensity of the spin-echo. The height or intensity of the recorded echo signal, S, will thus be sensitive to spin motions during the diffusion time, ~ - 8/3, and is described by the equation

In ~ = _(-yg8)2 D(~ - 8/3) - R So

(18.10)

where ~ is the time from the start of the first gradient pulse to the start of the second, So is a quantity proportional to the number of nuclei present in the sample, and R is the attenuation of the echo due to relaxation. For the SE experiment R is given by R = ~, where T2 is the transverse relaxation time.

In order to extend the diffusion time in systems where fast transverse relaxation prohibits longer T values, the magnetization can be stored along the Z-axis for a time period, T, between the de-phasing and re-phasing durations. In this so called stimulated spin-echo experiment [13] (STE, Fig. 18.2 bottom) the diffusion time can


reduced to zero. At this arrangement the sample becomes apparently isotropic and regains some of the properties of an isotropic liquid. In particular, it is now possible to refocus the proton signal into a spin echo and to use pulsed magnetic field gradients to probe the motion of the lipids as well as the water in the sample [8, 15] (Fig. 18.3). In the NMR experiment the translational diffusion is measured in the direction of the magnetic field gradient which normally is directed parallel to Bo. For a lipid membrane the observed diffusion coefficient, D, depends on the orientation and two diffusion coefficients D Land D 1.. D L is the lateral diffusion coefficient for motion parallel to the membrane and D 1. for the motion perpendicular to the bilayer. Then D is

(18.11)

For a bilayer oriented at the magic angle sin2 (hD = ~ and, since it is reasonable to assume that D 1. is orders of magnitude smaller than D L so that the second term in Eq. (18.11) can be neglected, D L is given by D L = l.5D.

In our laboratory, two goniometer diffusion probes are presently operational for studies of lateral diffusion. A 10 mm 1 H probe for a 100 MHz NMR spectrometer with maximum gradient strength of 3 Tim is routinely used for lipid diffusion measurements (D ~ 10-11 _10- 12 m2/s) anda5 mmdual l H/Xprobefora400MHzsystem, capable of gradient strengths up to 10 Tim is used for more demanding measurements, such as slow diffusion (D ~ 10-13 - 10-14 m2/s), fast relaxing samples, and isotopically enriched samples. For measurements in which the gradient direction can be varied with respect to Bo we use a microimaging system with a specially built goniometer sample holder (c.! section 5.3).

4. Preparation of macroscopically oriented lamellar systems

Several methods have been employed in order to obtain macroscopically oriented lipid bilayers, of which four different categories can be mentioned: Langmuir-Blodgett films; magnetic or electric field orientation; biological alignment; and glass plate assisted orientation. Only the last method will be discussed here; for a brief review of the other methods see [16] and references cited therein.

The glass plate assisted method for orienting bilayers depends on the ability of the glass to interact with the lipid molecules so that a layer oflipids will form at the surface. Silicon has also been used as an interacting surface. This layer will then act as a support for adjacent bilayers and, at least for short distances from the glass surface, a stack of bilayers will form with the normal parallel to the glass plate normal. This method has a history dating from the early 1970's when the first pfg NMR diffusion measurements were performed. In those days hydrated lipids were applied to the glass plates and a procedure of simultaneous pressing and shearing of the plates, often at elevated temperatures, produced the desired degree of orientation for the pfg experiments [17].

The method has since been improved by the application of the lipids dissolved in suitable solvents to the glass plates followed by solvent evaporation. This produces a


thin film of lipids on the glass plates and the subsequent hydration results in a higher degree of orientation. The choice of solvent can be critical [16] and in some cases the glass surface has to be modified in order to obtain a suitable degree of hydrophilicity for the lipids to adhere to it. Recently a method using a sublimable solid as an aid for orientation has been reported [18]. The degree of orientation is also affected by the way the dry lipid layers are hydrated. Addition of liquid water to the lipid disrupts the bilayers and results in the formation of vesicular structures, while hydration in a humid atmosphere generally gives better results [19]. This is especially true for cases where more than one glass plate is needed in order to obtain a sufficient signal/noise ratio in the measurements. If the glass plates are stacked before hydration, addition of liquid water will disrupt the bilayers as water is sucked in between the plates by capillary forces. Attempts to stack prehydrated plates often results in mechanical disruption of the lipid bilayers. If only one supporting surface is sufficient, methods has been reported that give well oriented samples with excess water [20].

In our laboratory good results have been obtained by the following procedure. Lipids dissolved in a 1:4 mixture of methanol: I-propanol is deposited onto thoroughly cleaned, but otherwise untreated, glass plates to a concentration of about 4 Ilg/mm2. The solvent is evaporated and the plates are placed into high vacuum for at least 4 hours to remove traces of solvent. This choice of solvent mixture gives a good adhesion to the glass surface and results in thin films covering the glass plates. The plates are then stacked on top of each other and placed into a glass tube with square cross section. Typically, 35 or 60 plates are used for one sample, depending on which NMR probe is to be used, (Fig. 18.4).

The tube is placed in humid atmosphere at a slightly elevated temperature, compared to the gel to LOt transition temperature, for several days. During this time the lipids become hydrated and oriented bilayers are formed. Finally, after obtaining the desired water content (checked by weighing the sample tube), the tube is sealed and the sample is given another day or two for final equilibration. This procedure gives samples with large dark areas when observed between crossed polarizers, since the sample

Figure 18.4. Left: Picture of the arrangement of glass plates in a glass tube for the the 10 mm goniometer probe. The sample is 14 mm long and the glass tube has an inner quadratic 4.7 x 4.7 mm2 cross section. The tube typically takes 60 glass plates and 20 mg lipid. Right: Picture of the sample arrangement for the 5 mm goniometer diffusion probe. The sample is 14 mm long and situated in the centre of the glass tube which has a quadratic 2.5 x 2.5 mm2 cross section. The sample typically consists of 35 glass plates and 6 mg of hydrated lipid.


becomes optically isotropic when viewed at a glancing angle parallel to the bilayer normal (Fig. 18.5). The degree of orientation is mostly found to be more than 85% as determined from 31 P NMR. The hydration of the samples can also be monitored by the quadrupole splittings of 2H20 (Figs. 18.6 and 18.7).

a)

b)

c)

Figure 18.5. This picture shows oriented samples viewed in crossed polarized light. Top: A sample for the 10 mm probe consisting of 57 glass plates and 20 mg of lipid. In the dark areas in the right part of the sample all plates contain more or less perfectly oriented bilayers while the left part of the sample shows the effect of less perfect orientation on the polarized light. Middle: A well oriented sample for the 5 mm probe consisting of 35 glass plates and 5 mg of lipid. The bright areas are caused by the teflon spacers on each side of the sample. Bottom: Same sample as in the middle picture but with the glass plate normal tilted slightly in order to demonstrate the birefringence when the glancing angle is not parallel to the bilayer normal.

In order to be able to orient the bilayer normal of the oriented sample with respect to Eo, a special goniometer stage has to be built into the NMR probe. Fig. 18.8 shows


Hydration time (hour.)

Figure 18.6. 2H20 NMR spectra demonstrating the hydration of lipids stacked between glass plates oriented at 90· relative to Bo. At time zero excess water was added and the sample was sealed. 2H NMR spectra were then acquired at the indicated times. As time passes the free water signal decreases and the signal from water between the bilayers increases in integrated intensity and gradually sharpens as the bilayer hydration equilibrates. The process is fully completed after 6 days.

-1500 -1000 -~oo o

Hz

'tIr--t+-++-A---- 50%

--r-t-+-H-+---- 30% .T" .... IoMo-__ 23%

16%

''WJII '.JP\o 10%

~oo 1000 I~OO

Figure 18.7. 2H20 NMR spectra showing the dependence of the quadrupole splitting on the water content in a sample of dioleoyiphosphatidylcholinePH20 at various water contents. The bilayer normal is oriented at 90· relative to Bo. From bottom to top: 10, 16, 23, 30 and 50 weight-% 2H20. Note that maximum hydration is attained between 30 and 50% as indicated by the central signal from free water in the top trace.


Figure 18.8. This picture shows the If-coil of the 10 mm probe with the goniometer sample holder that enables the sample to be rotated along an axis perpendicular to Bo. The glass tube that holds the sample is rotated by twisting the white teflon rod visible in the lower right corner of the picture. The anti-Helmholtz coils used to produce the field gradients have been removed from the probe in order to show the goniometer stage.

the goniometer stage of the 10 mm probe used in our laboratory. By twisting a rod connected to the goniometer stage the sample can be rotated so that the bilayer normal makes the angle 54.7" with respect to Bo. thereby canceling the dipole interaction and producing an isotropic-like spectrum (Fig. 18.9).

5. Examples 5.1 Lateral diffusion in total lipid extracts

The regulation of the membrane lipid composition by the bacterium Acholeplasma laidlawii strain A which does not have a cell wall has been extensively studied in our laboratory [21]. We have also studied the metabolic regulation and the phase equilibria of the membrane lipids from wild-type cells of the Gram-negative bacterium Escherichia coli [22]. It was conclusively shown that for these two prokaryotic organisms the cells strive to maintain a certain balance between the lipids constituting a


8 6 4 2 o -2

ppm

Figure 18.9. This figure shows the different sensitivity of the proton NMR line broadening as (J is moved away from the magic angle MA. The broadening due to the dipolar interaction will be proportional to P2 (cos (J) and the averaged dipole coupling constant (Eq. (18.9». It can be inferred from the figure that the signal from the hydrocarbon chain protons at 1.1 and 0.7 ppm has a higher dipole coupling constant than that of the head group protons at 3.1 ppm. Finally, the water signal at 4.6 ppm has a very small dipole coupling constant and is, therefore, much less sensitive to the deviation of (J from the MA.

bilayer and those forming non-lamellar liquid crystalline structures. The adjustments performed by the cells are able to keep the membrane lipids in a "window" between a gel crystalline phase and reversed non-lamellar phases. Consequently, it was concluded that the regulation of the composition of the membrane lipids is based on their lipid phase behaviour.

Recently, it was reported that the diffusion of a fluorescent probe studied by FRAP in multilayers of total lipid extracts from E. coli shows an anomalous maximum at the growth temperature [23]. It was proposed that the sudden increase in the lipid diffusion is a consequence of the adjustment of the membrane lipid composition to the growth temperature. These new data prompted us to measure the lateral diffusion for total lipid extracts from E. coli and A. laidlawii membranes by pfg NMR at different temperatures [24]. We found that the lipid lateral diffusion in the total lipid extracts increases monotonously with temperature and no anomaly is observed at the growth temperatures (Fig. 18.10), strongly indicating that the lipid dynamics is not involved in the adjustment of the membrane lipid composition. With the existing data it is difficult to explain the discrepancy seen between NMR and FRAP. In both the NMR and FRAP experiments the lipid bilayers are fully hydrated, i.e. they are in equilibrium with excess water. However, the NMR diffusion is performed on a single-phase system, namely a lamellar phase, while the FRAP experiment is carried out on a fluorescent probe molecule in a lamellar phase in contact with an excess water reservoir (i.e. a two-


-25.0

-25.2

-25.4

-25.6

~ e -25.8

.s -26.0

-26.2

-26.4

-26.6

-26.8 3.15

• DMPC V E. coli • A. laldlawli30 o A. laidlawil37

3.20

..........

3.25 3.30

413

•••

3.35 3.40

Figure 1B.1 O. Arrhenius plots oflipid lateral diffusion coefficients of total lipid extracts of Acholeplasma laidlawii grown at 30'C (squares) and 37"C (open circles), total lipid extracts of E. coli grown at 27'C (triangles) and DMPC/2H20 (filled circles). In all samples the lipids were fully hydrated in 2H20. The plus signs mark the growth temperatures of E. coli and A. laidlawii. (Adapted from [24]).

phase system). This difference in experimental set-up might be one possible source for the observed discrepancy, another could be the different time scales utilized in the experiments.

5.2 Lateral organization in lipids

The role of lateral organization in lipid bilayer systems has recently been the subject of intense study [25-28]. Evidence of lateral lipid asymmetry has been gathered in a wide variety of systems, ranging from the so called rafts [28] and caveolae [26] formed by sphingomyelin, saturated phosphatidylcholines and cholesterol, to peptide aggregates [29,30] and gel phase patches formed in two-phase lipid systems [31]. The influence of the presence of domains on the translational motion of membrane lipids enables the use of pgf NMR diffusion to investigate domain formation. Lipids will either diffuse in and out of the two phases in an exchange mechanism or, provided that the border between different domains presents an obstacle to lipid diffusion, lipids will encounter restrictions in the diffusional motion. Both these processes will influence the observed diffusion coefficient, and the following examples will illustrate some of our recent results in domain systems.

5.2.1 DMPC/cholesterol. Studies of the role played by cholesterol in membranes have attracted scientists over more than 70 years and still no final consensus has been reached. There have been many suggestions over the years. A recent one suggests its involvement in the formation of domains in the membrane, usually referred


-24.5

-25.0

-25.5

'"";:) ~ -26.0 --.s

-26.5

-27.0

-27.5 2.95 3.00

o 6FCH V DMPC

3.05 3.10 3.15 3.20 3.25 3.30 3.35

lOOOff (1("1)

Figure 18.11. An Arrhenius plot of the lateral diffusion of 6FCH (circles) and DMPC (triangles) obtained from an oriented sample of 38 mol% cholesterol (6FCH)IDMPC in excess water. The apparent energy of activation obtained is 58 kJ/mol. (Adapted from [36]).

to as lipid rafts and caveolae [26,27,32]. In these latter flask-like invaginations in the plasma membranes various signalling functions of the cell are taking place, such as for example the signals transmitted into the cellular network by the insulin receptor which is localized in caveola microdomains [32,33], and which is fully dependent on the presence of cholesterol in the membrane. Depletion of cholesterol results in the disappearance of the caveola structures [32]. Another very interesting idea was put forward by McConnell [34] that the chemical activity of cholesterol (which was suggested to form "condensed complexes" with phospholipids in the plasma membrane) may serve a regulatory function with respect to intracellular cholesterol transport and biosynthesis [25]. With the aim of probing some of the functions of cholesterol in lipid bilayers we have started an NMR spectroscopy project, where we investigate the lateral diffusion of both the lipids and cholesterol in macroscopically aligned bilayers. Pig. 18.11 shows the results of a direct determination of the lateral diffusion of cholesterol in a lipid bilayer. The great advantage with the pfg NMR method [8] is that it is non-perturbing and in most cases does not need any labelling, although for the study of cholesterol we have had to perform a negligible 19p labelling on the steroid ring. Thus, in order to obtain an NMR signal from cholesterol separately we have used a singly fluorinated cholesterol (6PCH) in which the proton in the 6-position in the nucleus has been exchanged for a 19p [35].

As can be inferred from Pig. 18.11, DL(DMPC) and DL(6PCH) are equal with an apparent energy of activation of 58 kJ/mol in the temperature interval 30-60·C. This can be compared with the lipid lateral diffusion in DMPC/water bilayers, in which the activation energy is lower and DL is about 4 times larger [24]. The most


12._------------------------------------------~

10

4

___ 25 wt-% water --v- 30 wt-% water ___ 35 wt-% water

2~--.-------_.--------~------~--------._--~

o 10 20 30 40

Cholesterol content (mol-%)

415

Figure 18.12. Lateral diffusion of sphingomyelin at 333 K as a function of the cholesterol content. The transition from single-phase to two-phase regions takes place between 10-20 mol% cholesterol. (Adapted from [37]).

interesting observations here are: (i) cholesterol and DMPC exhibit the same lateral diffusion coefficients over the whole temperature interval investigated; (ii) the addition of cholesterol to the lipid bilayer results in a decrease in the lipid translational diffusion; and (iii) the apparent energy of activation of the diffusion process is the same for both components in the mixed system but larger than that for DMPC in a pure lipid/water system [24].

5.2.2 SphingomyelinlcholesteroIPH20. An example of the effect of domains on the lateral diffusion in bilayers can be seen in the system with mixtures of cholesterol with sphingomyelin. In this system the proximity to the main phase transition of the sphingomyelin (39°C) gives rise to a two-phase area and NMR studies show that samples with less than 15 mol% cholesterol are in a liquid ordered single phase, while the samples with higher cholesterol content are in a two-phase region of the liquid ordered and the liquid disordered phases [37]. As seen in Fig. 18.12, the lipid lateral diffusion is effectively reduced by the domains of liquid ordered phase formed at cholesterol contents above 15 mol%.

5.2.3 DMPC/gramicidinPH20. The lipid lateral diffusion has been studied in the vicinity of the main phase transition temperature of fully hydrated DMPC samples containing various amounts of gramicidin D. DSC studies have shown that the

416

..l ~ .s

-25

-26

-27

-28 0 DMPC

\l DMPC:Gd=99:1 0 DMPC:Gd=97:1

-29 <) DMPC:Gd=95:5


\l

<)

0

& <) o

-30+---_r----~--_r----~--_r----~--_r----~~

0.00324 0.00326 0.00328 0.00330 0.00332 0.00334 0.00336 0.00338 0.00340

Iff

Figure 18.13. Arrhenius plots of the lipid lateral diffusion of DMPC with various amounts of gramicidin D. The onset of two-phase formation, according to 31 P NMR is marked by a filled symbol. The lines, all with the same slope, are merely drawn as a guide for the eye. (Adapted from [38]).

main transition for pure DMPC at 23·C gradually broadens with increasing amount of gramicidin. Atomic force microscopy has indicated gramicidin aggregation in the gel phase of similar systems and we wanted to investigate whether lipid lateral diffusion could give further evidence for such aggregates. The NMR studies indicate that within the temperature interval of the phase transition the lipids are still in a disordered phase and, moreover, the lateral diffusion of the lipids is largely reduced at the onset of the two-phase area. The diffusion is thus shown to be sensitive to gel/liquid crystalline phase coexistence and/or the presence of linear peptide aggregates (Fig. 18.13).

5.3 Anisotropic water diffusion in macroscopically oriented bilayers

A limitation of the pfg NMR method is that the bilayer normal has to be directed at 54.7· with respect to Bo while the field gradient is usually parallel to Bo. This means that the observed diffusion coefficient will be a superposition of the diffusion parallel (DL) and perpendicular (D -d to the bilayer according to Eq. (18.11). By using NMR imaging hardware it is possible to produce magnetic field gradients in an arbitrary direction with respect to Bo (Fig. 18.14). This method has been used to measure the translational diffusion of water as a function of the angle between the magnetic field gradient and the normal of the lipid membrane [39]. The observed diffusion coefficient is found to depend strongly on this angle (Fig. 18.14) and the anisotropy, D L/ D 1., of


10 .-----------,

n

100 200 JOG

Figure 18.14. Water diffusion in an oriented sample of DMPCIH20 measured as a function of cp at 29'C. The left picture shows the oriented sample with the bilayer normal oriented at 54.7' with respect to Bo. The right figure shows the diffusion coefficient obtained as a function of the angle, cp, between the gradient and Bo. The diffusion coefficients obtained from the fit are DL = (8.01 ± 0.07) x 10-10

m2/s and Dol = (0.04 ± 0.07) x 10-10 m2/s. (Adapted from [39]).

water diffusion in fully hydrated bilayers of DMPC at 29°C could thus be measured by pfg NMR. The anisotropy cannot be accurately determined due to the very small value of D 1., but a lower limit of about 70 can be estimated from the observed diffusion coefficients. This method is currently under development in the study of membrane permeability.

Acknowledgments

This work was supported by the Swedish Research Council and the Knut and Alice Wallenberg Foundation. We also wish to thank Dr. Tobias Sparrman for critically reading the manuscript.

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Scotland, 1996.

[2] Stemin, E., Bloom, M., and Mackay, A.L. (1983),1. Magn. Reson., 55:274.

[3] McCabe, A.M., and WassaIl, S.R. (1995),1. Magn. Reson. 8,106:80. [4] Schafer, H., Miidler, B., and Volke, F. (1995),1. Magn. Reson. A, 116:145.

[5] Sternin, E., Schafer, H., Polozov, I.v., and Gawrisch, K. (2001),1. Magn. Reson., 149:110.


[6] Seelig, J. (1978), Biochim. Biophys. Acta, 515:105. [7] WennerstrCim, H. (1973), Chem Phys. Lett., 18:41.

[8] Lindblom, G., and OrMd, G. (1994), Prog. NMR Spectrosc., 26:483.

[9] Stilbs. P. (1987), Prog. NMR Spectrosc., 19:1.

[10] Lindblom. G .• and WennerstrCim. H. (1977). Biophys. Chem .• 6:167.

[11] Callaghan. P.T. Principles of Nuclear Magnetic Resonance Microscopy. Clarendon Press. Oxford. 1991.

[12] Stejskal. E.O .• and Tanner. J.E. (1965). J. Chem Phys., 42:288.

[13] Tanner. J.E. (1970).1. Chem. Phys .• 52:2523.

[14] Roeder. S.B.W, Burnell. E.E .• Kuo. A.-L., and Wade. C.G. (1976), J. Chem. Phys .• 64:1848. [15] Lindblom. G .• and Ortldd. G. Encyclopedia ofNMR, (D.M. Grant. and R.K. Harris, Eds.). John

Wiley & Sons. Ltd., Chichester, 1996. [16] Moll III. E. and Cross. T.A. (1990), Biophys. J., 57:351.

[17] Asher. S.A., and Pershan. P.S. (1979), Biophys. J .• 27:393.

[18] Hallock. KJ .• Henzler Wildman. K.. Lee. D.-K.. and Ramamoorthy. A. (2002). Biophys. J .• 82:2499.

[19] Kurze. v., Steinbauer. B .• Huber, T., and Beyer. K. (2000). Biophys. J .• 78:2441. [20] Katsaras, J. (1997). Biophys. J .• 73:2924.

[21] Andersson, A.-S., Rilfors. L., Bergqvist. M .• Persson, S., and Lindblom. G. (1996). Biochemistry, 35:11119.

[22] Morein. S., Andersson, A.-S .• Rilfors, L.. and Lindblom. G. (1996). J. Bioi. Chem .• 271:6801.

[23] Jin, AJ., Edidin, M .• Nossal, R., and Gershfeld, N.L. (1999), Biochemistry. 38:13275.

[24] Lindblom. G .• OrMd. G., Rilfors, L., and Morein, S. (2002), Biochemistry, 41:11512.

[25] Anderson, T.G., and McConnell, H.M. (2001), Biophys. J., 81:2774.

[26] Andersson, R.G.W (1998), Annu. Rev. Biochem., 67:199.

[27] Simons, K., and Ikonen. E. (1997). Nature. 387:569.

[28] Vaz. WL.. and Almeida. P.EE (1993). Curro Opin. Struct. Bioi .• 3:482.

[29] Rinia. H.A .• Kik. R.A., Demel. R.A., Sneel. M.M.E., Killian, lA., van der Erden, J.P.J.M .• and de Kruijff. B. (2000). Biochemistry. 39:5852.

[30] Mou, J., Czajkowsky, D.M., and Shao, Z. (1996). Biochemistry. 35:3222.

[31] Dolainsky, C, Karakatsanis, P., and Bayerl, T.M. (1997), Phys. Rev. E,55:4512.

[32] Gustavsson, J., Parpal. S .• Karlsson, M., Rarnsing, C., Thorn. H .• Borg. M., Lindroth, M .• Holm-gren Peterson, K.. Magnusson, K.-E., and StrMfors, P. (1999), FASEB 1.,13:1961.

[33] Parpal. S., Karlsson. M., Thorn. H .• and Strlilfors. P. (2001),1. Bioi. Chem .• 276:9670. [34] Radhakrishnan, A., and McConnell, H.M. (2000). Biochemistry, 39:8119.

[35] Kauffman, J.M .• Westerman, P.W. and Carey, M.C (2000).1. Lip. Res .• 41:991.

[36] OrMd. G .• Lindblom, G., and Westerman, P.W (2002). Biophys. I., 83:2702. [37] Filippov. A., Oradd. G., and Lindblom, G. (2003), Biophys. I., in press. [38] OrMd. G. (2003), in preparation. [39] Wasterby, P., OrMd. G .• and Lindblom, G. (2002),1. Magn. Res., 157:156.

Chapter 19

DYNAMIC NMR IN LIQUID CRYSTALS AND LIQUID CRYSTALLINE SOLUTIONS

ZeevLuz Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel

1. Introduction Nuclear Magnetic Resonance spectroscopy is a most powerful tool for the study of

dynamic processes in condensed phases. The dynamic range of NMR spectroscopy can roughly be divided into three categories. One corresponds to the extreme fast regime where the characteristic rates, 1/ T, are of the order of the Larmor frequencies, W L = 27rlJL, of the nuclei studied. Under these conditions the structure of the spectrum corresponds to the average Hamiltonian over the exchanging states, but dynamic information is still contained in its relaxation parameters. Relaxation time measurements are thus most useful for studies of fast molecular reorientation and conformational equilibria in the range 107 s-l and faster. At the other extreme of the dynamic range lies the regime of the ultraslow motions, where the rate is too slow to affect the lineshape, but still faster than the longitudinal relaxation, 1/T1 < l/T < 1/T2. In this regime the exchange is measured by one or another modification of the magnetization transfer methods, including the various two-dimensional exchange experiments. Depending on T1, correlation times of the order of less than a millisecond upwards can be measured by these techniques.

The third category corresponds to the intermediate dynamic range were the exchange rates are of the order of the interactions responsible for the spectral structure, and at any rate faster than T2. In this regime the spectrallineshape depends on the exchange rates and the latter can accurately be determined by fitting simulated line shapes to the experimental spectra. The term "dynamic NMR" refers to this regime and in the present Chapter we review some aspects of this field that can be related specifically to liquid crystals and liquid crystalline solutions.

Dynamic effects on the NMR lineshape showed up soon after the discovery of the fine structure (chemical shift and spin-spin coupling) in high-resolution spectra. The first theoretical formulation of the effect is due to Gutowsky, McCall and Slichter [1] (GMS) and involves a complicated statistical averaging procedure to follow the pathway of the reactions. The formalism was greatly simplified when McConnell [2] introduced his modified Bloch equations (now commonly referred to as the Bloch-McConnell



equations) in which the effect of the exchange is added as a classical rate term to the regular Bloch equations. Both the GMS and the Bloch-McConnell equations are "classical" and strictly apply only in the limit of weak interactions, where each peak in the high-resolution spectrum can be associated with a transition of a particular spin. In the strong coupling case, where the spectrum consists of "mixed transitions", this approach does not apply and the more general density matrix formalism must be used. This was first worked out by Kaplan [3,4] and subsequently generalized by Alexander [5,6] and again by Binsch [7-9]. High-resolution proton NMR of solutes dissolved in liquid crystals essentially always corresponds to the strong coupling case due to the non-zero average intramolecular dipolar interactions between the nuclei. Analysis of dynamic effects in such systems therefore requires the use of the density matrix formalism. On the other hand the dipolar interactions of deuterons are usually much smaller than the average quadrupolar interaction (in fact they usually fall within the linewidth). The spectral peaks in this case can readily be associated with particular nuclei in the molecule and the Bloch-McConnell equations almost always apply.

We distinguish between two types of applications of dynamic NMR in liquid crystalline systems. In one the liquid crystal merely serves as a solvent in much the same way as water and organic liquids do in regular high-resolution dynamic NMR. For this application non-viscous liquid crystals, such as calamitic nematic mesophases, are generally used. This type of application will be discussed in sections 2.2 and 3.2 below. In the other application the method is used to study dynamic properties of the mesogens in the neat liquid crystalline phases. This is only practical in the less fluid liquid crystals, such as columnar mesophases, where molecular reorientation or even the conformational isomerization of the side chains are sufficiently hindered to fall within the range of dynamic NMR. The situation is then similar to dynamic solid state NMR. We discuss such applications in sections 3.3 and 4.2. The purpose of this Chapter is not to provide an encyclopedic summary of the area. Rather, it aims to demonstrate the power of the method using selected examples with a rather detailed explanation of the theoretical background. Not surprisingly the selected examples are mainly from the author's work.

The simulation ofNMR spectra in liquid crystalline systems requires the knowledge of the average orientation of the molecules and of their average magnetic interactions. We refer to Chapter 1 in this book for the basic equations describing the molecular ordering and the spin Hamiltonian interactions in liquid crystalline systems.

2. Dynamic proton NMR of solutes in nematic solvents In general, dynamic processes have similar effects on the line shape of solutes

dissolved in liquid crystals as they do in isotropic solvents, in particular the sequence of line broadening, coalescence, and narrowing with increasing rate. By comparing such spectra with simulated traces, the reaction mechanisms and kinetic parameters can be determined. In isotropic solutions these simulations are usually quite simple, involving isolated groups of spin-spin coupled nuclei. The couplings are often small allowing the use ofthe Bloch-McConnell equations. In liquid crystalline solutions the

Dynamic NMR in liquid crystals and liquid crystalline solutions 421

situation is usually quite different because the weak coupling condition rarely applies and the direct dipolar interactions essentially couples all spins in the molecule. In such situations the more general density matrix formalism must be applied. The extra effort of using this approach pays off only in some special cases. For example, this is the case when the isotropic spectrum is not sensitive to the dynamic process (no modulation of the chemical shifts and / or scalar couplings) or when measurements over a wide dynamic range are required. It is therefore not surprising that only a few systems have been studied by this method [10--12]. As a demonstration we shall discuss in section 2.2 one such example, namely the bond-shift rearrangement of cyclooctatetraene [11, 12]. Before doing so a brief introduction to the density matrix formalism as applied to dynamic high-resolution NMR is given in the next section.

2.1 The density matrix method

We refer to text books [13] and to Chapter 2 of this book for a general introduction to the density matrix formalism. For the application to dynamic NMR we follow the derivation of Alexander [5], starting with the general expression for the expectation value of the transverse magnetization of an ensemble of spins,

M+(t) = Tr[p(t)I+] = L[P(t)]kl(I+)lk, (19.1) k,l

where p is the density matrix, p(t) is its value following an excitation pulse, and I+=2:i Ii,+, where i labels the nuclei in the molecules. We restrict the discussion to intramolecular rearrangements such as conformational equilibria, tautomerism, or hindered rotation. The exchange then corresponds to permutation of the nuclei in the molecule and can formally be described in terms of a permutation, P, of the nuclear spin coordinates in the nuclear spin wave function,

(19.2)

where 7jJ(t) are basis functions ofthe Hamiltonian, essentially linear combinations of product single-spin functions. The + j - superscripts indicate "just after" and "just before" the exchange. In the following we suppress these indices. The operator P obeys the relation pn = 1, where n is the order of the exchange process. For a two (three) site exchange n = 2(3), etc. The corresponding effect of the exchange on the density matrix is

(19.3)

while the spin Hamiltonian remains the same before and after the exchange. In analogy with the Bloch-McConnell equations we write for the density matrix

!p(t) = i[p(t), 'H] + k[Pp(t)p-l- p(t)]- p(trd jT2 + [po - p(t)diagl/T1, (19.4)

where 'H is the spin Hamiltonian, p(t)diag and p(t)od are the diagonal and off-diagonal parts of p(t), Po is its thermal eqUilibrium value and k is the specific rate constant of


the reaction. The spin Hamiltonian includes terms due to the isotropic and anisotropic chemical shifts (CS), indirect spin-spin couplings, dipolar interactions and for spins I ~ 1, also quadrupolar interactions. Only secular terms need be considered for calculating the dynamic line shapes. Furthermore, from Eq. (19.1) it follows that only off-diagonal elements, p(t)kl, corresponding to f:lMz = Mz(l) - Mz(k) = 1 are required for the simulations. The last term in Eq. (19.4) can therefore be discarded and the equation for a particular off-diagonal element, p(thl, becomes

:,P.I ~ i (~P"'1i" -~1i"P'.) + k [~/kmpmn(P-l)nl- pnl] + Pki/T"

(19.5)

where the time variable, t, in p(t) is suppressed. Since the Hamiltonian is secular and since the permutation operator preserves the total M z of the basis function, Eq. (19.5) only couples Pkl with Pk'l' for which Mz(k) = Mz(k') and Mz(l) = Mz(l'). Thus, the problem reduces to solving blocks of coupled differential equations, each block corresponding to a set of Mz(k) ~ Mz(k) + 1 transitions. They can be cast into a compact Liouville equation,

d dt [p]k,k+l = i[p, H]k,k+1 + k[ppp-l - p]k,k+1 + (T2- 1 ) [p]k,k+1

(19.6)

Lk,k+1 [pJk,k+l,

where the indices, k, k + 1, label the Mz values of the basis functions. The formal solution ofEq. (19.6) is

(19.7)

where the initial values of [p(O)Jk,k+l are proportional to [L Jk,k+l' These equations can now be solved by standard matrix diagonalization methods. In practice, for proton spectra of solutes dissolved in liquid crystals the size of the Liouville blocks may become quite large even for moderate size molecules. The largest blocks that need be solved correspond to the O~ +1 or -! ~ ! transitions, depending on whether the number, N, of protons in the molecule is even or odd. For these transitions the sizes ofthe blocks are, respectively [14],

( N ) (N!)2 (N + 1)2 (N!)2 N + 2 [(N/2)!J4 and -2- [((N + 1)/2)!J4' (19.8)

In the example below we discuss the bond-shift process in cyclooctatetraene (COT), a molecule with just eight hydrogens. The total number of possible f:lM z = 1 transitions in the spectrum of this molecule is 11,440 (many of them are forbidden and of the rest many are degenerate), with the dimension of the largest (O~ + 1) block being 3920. It is therefore desirable to seek methods for factorizing the problem further. One


way to do so is to use group theoretical methods that take into account the symmetry of the molecule and the nature of the dynamic process. This has been done in the original COT study leading to significant factorization, with the largest block consisting of "only" 224 equations [15].

2.2 The bond-shift rearrangement in cyclooctatetraene (COT)

As a specific example of dynamic proton NMR in nematic solvents we discuss the cyclooctatetraene (COT) molecule. The compound is one of the more interesting examples of cyclic polyenes. Being non-aromatic, it takes on a non-planar structure and, in fact, acquires a tub-shape (D2d symmetry, see Fig. 19.1) with alternating single and double bonds. All protons in the molecule are equivalent and the spectrum in isotropic solvents therefore consists of a single line. From earlier proton NMR studies of the carbon-13 satellites in this compound [16] and from measurements of substituted COT [17] it is known that the molecule is dynamic on the NMR time scale. Two types of processes can be imagined for the molecule. One involves inversion, in which the tub folds into its mirror image. This process, however, does not modulate any of the magnetic interactions in the molecule and is therefore unobservable by dynamic NMR. The other possible process is the bond-shift rearrangement (see Fig. 19.2, top) which can be visualized as pseudorotation resulting in cyclic permutation of the atoms around the COT ring. This process clearly modulates the direct (and scalar) interactions and should therefore affect the spectrum in liquid crystalline solvents.

In the bottom trace of Fig. 19.1 is shown an experimental proton NMR spectrum of a COT solution in a nematic solvent recorded at -25°C [12]. At this temperature the bond-shift process is too slow to affect the line shape and the spectrum reflects the static structure of the molecule. The COT molecule has an S4 axis. Its orientational order can therefore be described using a single motional constant. Taking this axis as the z-coordinate in the molecular frame (MF) yields the following average secular spin Hamiltonian,

'H = 'HD + 'HJ = 2:{Szz (- 2~) [l:l.ij,zz - ~ (l:l.ij,xx + l:l.ij,yy)] i<j TtJ

x [J. z 10z - ~(J.+10_ -J._10+)] t, j, 4 1., j, 'I., J, (19.9)

+ J,; [I"zI;,z + ~ (1,,+1;,- - I,,_I;,+)]},

where the first and second terms correspond to the direct dipolar interactions and the scalar indirect spin-spin interaction between the protons in the molecule, and we have neglected the anisotropic part of the latter. Since the CS is the same for all protons, it is set to zero (in the rotating frame). The quantity Szz in Eq. (19.9) is Saupe's order parameter [18, 19], K is a constant that depends on the gyromagnetic ratios of the interacting nuclei, being 120067 Hz A 3 for a pair of protons, flij,a{j = ((}i - (} j) (f3i - f3j), where ai, f3j, are the x, y, z coordinates of the i and j nuclei


2

3 8

4 7

Figure 19.1. Top: Structural formula of cyciooctatetraene (COT). Trace 1 is an experimental proton NMR spectrum of COT dissolved in a nematic solvent at -25°C. Trace 2 is a best fit simulated spectrum. The separation between ticks in the frequency scale is 100Hz. (Reproduced with permission from [12]).

in the MF, and Tij are the corresponding internuclear distances. The only important scalar couplings in COT are the 3 Jij for which i, j are vicinal protons separated by a double bond [16]. This coupling was determined from the carbon-13 satellite signals in isotropic solvents to be 11.8 Hz and was assumed to be positive. All other Jij's

are negligibly small. Due to the high molecular symmetry of COT, knowledge of the geometrical parameters of one hydrogen in the MF provides the coordinates of all hydrogens. Thus the spectrum can be fitted to just two geometrical parameters, x/z and y/z, and a scale factor Szz. As may be seen in Fig. 19.1, a perfect fit to the experimental spectrum could indeed be obtained [12]. Incidentally, based on the (assumed) positive sign of the vicinal 3 J coupling, the sign of Szz was found to be


'&=r.' 7 8

,~ 4 I

--

o o

Figure 19.2. Top: The bond-shift rearrangement process in cycJooctatetraene. Bottom: Experimental (left) and simulated (right) dynamic proton NMR spectra of nematic solutions of COT. The spacing between the frequency markers corresponds to 500 Hz for the bottom four traces, 250 Hz for the next three, and 125 Hz for the topmost trace. Note the three exchange-invariant pairs of peaks, indicated by asterisks in the 70°C spectrum. (Reproduced with permission from [11, 12]).


negative, indicating that the COT molecules prefer to align with their symmetry axes perpendicular to the director.

As the temperature is raised to above -25°C line broadening sets in, reflecting the bond-shift rearrangement process. Examples of spectra are shown in Fig. 19.2 [12]. To allow measurements over as wide a temperature range as possible, different nematic solvents were used at different temperature intervals according to their range of stability. This made it possible to perform kinetic measurements over approximately 200K. To derive kinetic parameters for the reaction the experimental spectra were compared with simulations, using the theory described above. In the fitting the rate constant for the bond-shift process and a scale factor served as the only free parameters. The resulting Arrhenius curve is shown in Fig. 19.3 [11].

It is interesting to note that symmetry considerations are helpful, not only for simplifying the calculations, but also because they may directly provide most of the essential information on the system. The COT results in Fig. 19.2 provide an excellent example. It may be seen in the dynamically broadened spectra that out of more than a thousand transitions there are three pairs of lines that remain sharp throughout the dynamic range. They are most clearly observed in the 70°C spectrum, where they are indicated by asterisks. A fourth pair of nearly invariant lines is also apparent; its weak broadening appears to be accidental. The origin of the invariant lines can readily be understood by using group theoretical considerations [11,12,15]. We recall that for an NMR transition to be invariant to exchange both its initial and final states must be invariant to the process. For the bond-shift in COT this amounts to identifying basis functions that remain invariant under cyclic permutation of the carbon atoms around the ring. This is easy to check on symmetry adapted basis functions, since transitions are allowed only between states having the same symmetry. For example the basis func-

1 tions, aaacw!aaa (Mz =4) and 8-2" (,Baaaaaaa + a,Baaaaaa + .... aaaaaaa,B) (Mz =3), both belong to the completely symmetric representation (AI) of the COT symmetry (D2d) group and both are invariant under permutation. They also happen to be eigenfunctions of the COT spin system. Therefore the (AI) Mz=4 to M z=3 and the corresponding Mz = -3 to Mz = -4 transitions are invariant to the bond-shift process. By straightforward inspection of all symmetry adapted basis functions it was shown that there are just two more pairs of such invariant transitions. Their identification in the experimental spectra provide strong support for the assumed dynamic model. In fact it turned out that, with very little additional considerations, they provide full structural information on the COT molecule [12].

The above example demonstrates the power of high-resolution NMR of solutes dissolved in liquid crystals for studying their structure and dynamics. At the same time it also shows the limitation of the method. Even for a medium size molecule such as COT, the spectra are barely resolved and their simulation has already become a major computational job. The complexity of the spectra of such molecules can be considerably reduced using multiple quantum (MQ) spectroscopy [201 (see also Chapter 4). In fact this method was used to record dynamic MQ spectra [21], but experimentally there is a considerable loss of intensity and the computations are as complex as for single quantum spectra.


t (OC)

Figure 19.3. Arrhenius plot for the bond-shift rearrangement in COT. The circles are experimental results derived from dynamic line shapes of the type shown in Fig. 19.2 [11]. The square symbols are results derived using slow and fast exchange limit equations. (Reproduced with permission from [12]).

In principle carbon-13 NMR could also be used for dynamic studies in liquid crystals. However, fully enriched samples (with or without proton decoupling) will suffer from the same limitations as 1 H NMR, while in natural abundance the spectra are similar to those in isotropic liquids. The method was nevertheless successfully used to study the effect of liquid crystal ordering on the dynamics of hindered rotation in substituted aminopyridine and aminofulvene [22-24].


3. Dynamic deuterium NMR spectra Considerably simpler spectra are obtained for deuterium NMR of fully deuterated

(or in natural abundance with proton decoupling) solutes. Deuterium is a quadrupolar nucleus with I = 1. Its quadrupole coupling constant in C-D bonds is typically of order 170 kHz, and consequently it is the dominant term in the spin Hamiltonian of the deuterons. The deuterium spectrum in liquid crystalline solutions thus consists of superpositions of doublets with spacings given by the average quadrupole coupling of the various deuterons over the molecular distribution. Except for special cases when the dipolar or scalar interactions are particularly large (and / or the lines especially narrow) these interactions fall within the experimental line width. As an example we show in Fig. 19.4 the low-temperature (-5°C) deuterium spectrum of perdeuterated cis-decalincis-Decalin dissolved in a nematic solvent [25]. The molecule has C2 sym-

e

e

1,20 5e,2e e --

I III II I I II II I I

Figure 19.4. Top: Structural fonnula and numbering system of cis-decalin. Bottom: Deuterium NMR spectrum of perdeuterated cis-decalin in a nematic solvent at -5°C. The separation between ticks in the frequency scale corresponds to 950 Hz. The stick diagram above the scale is a best-fit to the spectrum using three order parameters. (Reproduced with permission from [25]).


metry and contains eighteen deuterons, arranged in nine symmetry-related pairs. The deuterium spectrum thus consists of nine doublets which could readily be assigned as indicated in the spectrum, by fitting the splittings to three orientational order parameters. The corresponding proton spectrum of an isotopically normal cis-decalin would be completely structureless. Small chemical shift differences between the various types of deuterons can also be discerned by the fact that the centres of the various doublets do not completely coincide. However, no splittings due to dipolar or scalar couplings between the deuterons are observed. These interactions are reduced by a factor ('YH i'YD)2 ~ 42 compared to the interproton interactions and fall within the range of the spectral line width (",80 Hz). This line width may contain contributions from slight misalignment or local disorder which occurs even in well oriented liquid crystalline samples or single crystals. Experimentally it results in loss of signal during the pulse dead-time before signal acquisition. This inhomogeneous broadening results in fast decay of the free induction decay (fid) signal and corresponding loss of signal intensity during the dead time following the excitation pulse. To overcome this effect it is customary to detect the deuterium signal by the quadrupole echo method [26]. For static systems the quadrupole echo signal is essentially identical to that expected for a sharp pulse and zero dead-time. However, in dynamic systems an extra distortion may result due to the effect of exchange during the quadrupole echo sequence. For accurate simulation of dynamic quadrupole echo spectra, this effect must be included in the calculations [27-29]. We therefore start the discussion with a brief account of simulating dynamic deuterium lineshapes in quadrupole echo experiments. The use of the quadrupole echo method is even more important for non-aligned samples, because of their very short fid decays in such samples. In fact the dynamic distortions of the quadrupole echo signal in such systems can be used to better characterize the motion and more accurately determine the kinetic parameters. Following the theoretical section we discuss two examples. The first concerns the double-ring inversion of cis-decalin in liquid crystalline solutions, while the second concerns the motion of mesogen molecules in an unaligned columnar discotic mesophases.

3.1 The dynamic quadrupole echo spectrum The basic pulse sequence for the quadrupole echo experiment is

7r /2)x ~ 7r /2)y ~ acquisition, (19.10)

with appropriate phase cycling. We first consider the effect of the above sequence on the density matrix of a single type of deuteron in the absence of exchange. The Hamiltonian in the intervals between the pulses and during the acquisition is then given by

(19.11)

where wQ is the (average) quadrupolar frequency. During the pulses we consider only the effect of the rf field with the Hamiltonian

(19.12)


where WI = ,BI . Starting with the equilibrium density matrix, Po = Clz, following the first 7r /2 pulse it becomes p(O+) = -C ly, where the constant C is proportional to the original magnetization and will henceforth be set to unity. Under the quadrupole Hamiltonian the Y -magnetization undergoes "precession" in much the same way that it precesses under the chemical shift Hamiltonian. However, rather than the familiar precession in the X, Y plane, under 'H.Q the precession of a spin I = 1 occurs in the plane defined by axes given by Y and the double quantum coherence, J X Z

lzIx + Ix Iz· Thus, just before the second 7r12 pulse [28],

(19.13)

where a phenomenological transverse decay term, assumed to be the same for the ly and J x z coherences, has been added. Any distribution in wQ will result in the spreading out of p( T-) in this plane. The second pulse is applied in quadrature to the first, i. e. along Y, in the present case. It has no effect on I y, but being a double quantum coherence, Jxz undergoes a (2x7r/2) 7r-ftip about Y and thus reverses its sign. Consequently just after the second pulse we have,

(19.14)

and during the following period the spread in the magnetization is rephased, yielding an echo at 2T,

p(2T) = [- COSWQT(ly COSWQT - Jxz sinwQT)

- sinwQT(JxZ CoSWQT + Iy sinwQT)]e-2T/T2

_ Iye- 2T/T2 .

From this point on the density matrix evolves as after the first pulse,

p(2T + t) = e-2T/T2 ( -Iy coswQt + Jxz sinwQt)e- t / T2 ,

so that the signal detected during the acquisition period becomes,

fid(t) = Tr[p(2T + t)I+]e-(2T+t)/T2 "" coswQte-(2T+t)/T2

=~e-2T/T2 [(e+iwQt + e-iwQt)e-t/T2] 2

(19.15)

(19.16)

(19.17)

where in the last term the effect of transverse relaxation has been divided into attenuation, e-2T/T2 , and line width, e-t/T2 , factors. Fourier transformation of fid(t) yields the familiar quadrupole doublet with components at ±wQ.

We note from the above discussion that the density matrix of a spin I = 1 during the quadrupole echo pulse sequence can be written as the sum of two coherences with time dependent coefficients,

p(t) = y(t)ly + b(t)Jxz. (19.18)


In analogy with the transverse magnetization for spin I = ~ it is convenient to introduce a complex magnetization [28],

G(t) = y(t) + ib(t), (19.19)

where y(t) is the expectation value of Iy, corresponding to the observed fid(t) [fid(t) "" ~G(t)], while b(t) corresponds to an unobservable single-quantum coherence J x z. From the discussion above it is clear that under the effect of 'H.Q, G (t) obeys a Bloch-type equation of the form

:t G(t) = -[iwQ + 1/T2]G(t), (19.20)

while the effect of a 7r /2)y pulse is to transform G(t) into G(t)*. To extend these results to include exchange between different deuterons, i, j, we

introduce in Eq. (19.20) exchange terms in much the same way as in the usual BlochMcConnell equations,

! G(t)i = -[iw~ + 1/T2 + kii]G(t)i + LkijG(t)j Hi

(19.21)

where kij are the specific jump rates from site j to i and kii = 2:.kji. For the comHi

putation it is convenient to define a column vector G(t) whose components are the G(t)j's. Eq. (19.21) then becomes

where

:t G(t) = -RG(t),

Rjj = iwb + 1/T2 + kjj . ~i = -kij i -=f j .

(19.22)

(19.23)

Thus integrating Eq. (19.22) during the first T-period of the quadrupole echo sequence yields, G(T-) = (e-R'T)G(O), where the vector G(O) is proportional to the (real) initial (equilibrium) population vector, P, with components pi. The second 7r12 pulse transforms G(T-) into its complex conjugate, so that G(T+) = (e-R'T)*p and at the top of the echo G(t) becomes G(2T) = (e-R'T) (e-R'T)*P. Finally, during the acquisition it becomes, G(2T+t) = (e- Rt ) (e-R'T) (e-R'T)*P and the observed fid(t) signal is

(19.24)

where the scalar product with the row vector 1 = (1,1,1, .... ) serves to sum over all the components of G(2T + t). Fouriertransformation ofthis expression then gives the absorption spectrum. The equations are conveniently solved by standard eigenvalue / eigenvector methods and the spectrum is obtained as a sum of complex Lorentzians. The above derivation assumes identical CS for all sites. For the more general case we refer to Poupko et al. [29]


3.2 Interconversion of cis-decalin As an example of the application of deuterium NMR to study dynamics of solute

molecules we present results obtained on solutions of perdeuterated cis-decalincisDecalin in nematic solvents. The spectrum of "static" cis-decalin (recorded at -5°C) was already shown in Fig. 19.4. Like cyclohexane this molecule undergoes (double) ring inversion (see top of Fig. 19.5) resulting in pairwise exchange of deuterons, for example the pairs (3e,4a), (5a,2e), etc. The only doublet that remains invariant to the exchange is due to deuterons 1 and I' which are bonded to the ternary carbons. The effect of the ring inversion is clearly demonstrated by the experimental spectra depicted on the left hand side of Fig. 19.5 [25]. Using the theory described in the previous section the various experimental spectra could be faithfully simulated as a superposition of exchange broadened pairs of doublets with a single rate constant at each temperature (see right column in Fig. 19.5). The reSUlting rate constants so obtained yield the Arrhenius plot shown in Fig. 19.6. The plot covers a dynamic range of five decades, demonstrating the power of deuterium NMR to study rate processes in nematic solvents. For comparison, results from earlier measurements of dynamic carbon-13 NMR in isotropic solvents are also shown (open circles in the Arrhenius plot) [30]. As may be seen they cover a much smaller dynamic range.

To conclude this section we show in Fig. 19.7 a 2D-exchange spectrum of a solution of cis-decalin in a nematic solvent, recorded at -5°C [25]. As expected the doublet due to deuterons 1 and I' shows no cross peaks, while all others are pairwise linked by exchange cross peaks. The cross-peak pattern not only serves to identify the exchanging peaks, but also discloses the relative sign of their quadrupolar interactions. It may be seen that one exchanging pair of deuterons (3e,4a) has quadrupolar couplings of opposite sign (cross-peaks located on opposite sides of the spectrum centre), while all other pairs of exchanging deuterons have couplings of the same sign. These results are consistent with the analysis of the static and dynamic spectra which showed that the sign of the quadrupole coupling of all deuterons in the cis-decalin solution is the same (negative), except for deuteron 3e whose sign is positive. As an aside, we note that the method of high-resolution deuterium NMR in liquid crystalline solutions has been used to study ordering, structure and dynamics in a number of related systems [31-38].

3.3 Restricted motions in columnar liquid crystals of discotic dimers.

So far we have discussed dynamic processes involving solutes dissolved in nematic solvents. In more ordered liquid crystals, such as columnar mesophases, dynamic processes involving the mesogen molecules may lie within the dynamic NMR regime. Deuterium NMR may then be an ideal tool for studying these motions provided specifically labeled mesogens are available. In the present section we discuss one such


129

31

II

-5

o o Frequency (kHz)

Figure 19.5. Top: The double ring inversion of cis-decalin. Bottom: Experimental (left) and simulated (right) deuterium NMR spectra of perdeuterated cis-decalin dissolved in nematic solvents. The intervals between ticks in the frequency scale correspond to 10kHz for the bottom three traces and 5 kHz for the top four. (Reproduced with permission from [25]).


o

Figure 19.6. Arrhenius plot for the double ring inversion of cis-decalin. The solid symbols are from deuterium NMR spectra in liquid crystals [25]. Open symbols are from carbon-13 NMR in isotropic solvents [30]. (Reproduced with permission from [25]).

example. It concerns the dynamic properties of twin discotic molecules in columnar mesophases [39].

Discotic liquid crystals consist of molecules with (usually) flat rigid cores to which flexible side chains are linked at the periphery. Typical examples are the various hexasubstituted triphenylenes (see Fig. 19.8(a». Most often discotic liquid crystals form columnar mesophases (see part c of the figure) with the molecules stacked on top of each other to form rod-shaped aggregates which, in tum, form two-dimensional arrays of different symmetries. In the conventional columnar mesophases the molecules reorient rapidly about the columnar axes and the side chains are dynamically disordered. This is clearly demonstrated by deuterium NMR studies of such mesophases [40]. Under certain conditions the reorientation motion of the core is slowed down, leading to dynamic line shapes from which kinetic parameters of the motion can be derived. Examples of such mesophases are those derived from substituted cyclotri-


3e

J

20,1 2e,5e - .....

tk 'ti l l==:I

r-- ---- --- ------------1 I I I I I

I -20 : r~::,,-r

_------- __ ~----------~~.I I I I I I I" I I I I"'_~I I I ~--1'--1 I I I I I I

I I I I I I 0 I I

I I I I I

r-T--~ I I I ""r-~ I I :

: :~=l.~----------:---------~ I ~--11 I "'---.J :

: 20 I I I I

I I ~ _____________________ J

20 o -20 kHz

Figure /9.7. Deuterium two-dimensional exchange spectrum of cis-decalin in a nematic solvent at -SoC. Mixing time is 0.1 s. The dashed lines indicate the exchange connectivities. (Reproduced with permission from [25]).

and cyclotetraveratrylenes [41,42]. We discuss below a system belonging to this category in connection with the application of dynamic carbon-13 MAS NMR. Here we apply dynamic deuterium NMR to a special columnar system consisting of "twin discotic" molecules [39]. The molecule (see Fig. 19.8b) can be viewed as a dimer of hexa-pentyloxy triphenylene in which the ends of one chain in each unit are covalently bonded to each other, thus forming a spacer linking the two cores of the dimer. We refer to the monomer and dimer compounds as THE5 and DTHE5, respectively. Both compounds are mesogenic, exhibiting isomorphic hexagonal columnar mesophases with similar structural parameters and similar stability ranges (65-121°C and 67-136°C for the monomer and dimer, respectively). They differ, however, in their dynamic


(a) OR (b) OR

OR

OR RO OR

RO

RO OR RO c0CHz'zn ......... 0

OR' OR OR

OR

R = -C5 HII

(c) (d) Zc

Z~B.)

Figure 19.8. Top: Fonnulae of hexa-pentyloxytriphenylene (THE-5) and its twin dimer (DTHE-5). Bottom: Schematic representations of the columnar phase of the twin dimer and the coordinate system used to describe the librationaI motion.

properties due to the fact that in DTHE5 the two linked parts of the dimer occupy neighbouring columns. Thus, while in THE5 the molecules reorient freely and rapidly about the columnar axis, in the mesophase of DTHE5 their motion is restricted by the spacer. Instead, each unit undergoes high amplitude librations and as it happens the characteristic time for this motion lies within the range of dynamic NMR. We describe below the effect of this motion on the deuterium NMR spectra of DTHE5 deuterated in the unsubstituted aromatic site. Before describing the results it should be emphasized that, unless special procedures are applied, columnar phases usually do not align by the spectrometer magnetic field as do, for example, the nematic phases described above. The sample then consists of an isotropic distribution of domains and the observed spectrum consists of a superposition of signals from all domains. In the simulation one therefore needs to integrate (sum) over the entire distribution of the liquid crystalline domains.

Examples of deuterium NMR spectra of DTHE5 (deuterated in the aromatic sites) at different temperatures in the solid and mesophase are shown in the left column of Fig. 19.9. In the solid the spectrum shows a typical axially symmetric "Pake doublet" with "parallel" shoulders at vQ ~ ±140 kHz and "perpendicular" features at vQ ~ ±70


T/"C A Phase

l07~ LM

J J

200 o -200 100 o -100 100 a -100

Figure 19.9. Deuterium NMR of aromatic deuterated DTHE-5. Left column: Spectra in the solid and mesophase as a function of the temperature. Middle: Spectra in the mesophase at 97°C as a function of the quadrupole echo delay time as indicated. Right: Corresponding simulated spectra for the indicated diffusion constant. (Reproduced with permission from [39]).

kHz. Upon transition to the mesophase the spectrum changes discontinuously to that of a biaxial tensor with reduced overall width with one of the "perpendicular" features remaining unchanged. With increasing temperature there are slight changes in the lineshape and the biaxiality increases. Also shown in the figure (centre column) are spectra recorded in the mesophase at a fixed temperature (97°C), but different quadrupole echo delay time, T. Clear quadrupole echo distortions are observed as would be expected for dynamically broadened spectra. These results have been interpreted in terms of a model in which the two parts of the dimer undergo high-amplitude planar librations.

For the analysis of these results [39] it is convenient to define for a particular domain a coordinate system, Xc, Y c, Zc, with Zc parallel to the columnar axis. It is assumed that under the effect of the spacer each unit of the dimer experiences a local potential,

(19.25)

where Vo is a (positive) parameter and cp is the deflection of the C-D bond in the Xc - Y c plane from its equilibrium (minimum energy) orientation. This yields a


planar equilibrium distribution of the form

pO(<p) = J f3:0 exp( -f3Vo<p2) , (19.26)

where f3 = l/kT. The librational motion under the restriction effect of the potential of Eq. (19.25) obeys the following diffusion equation [43,44],

d d2 dV(<p) d d2V(<p) dtP(<P,t) [DR d<p2 +f3DW-J:P- d<p +f3DR d<p2 JP(<p,t)

d2 d (19.27) [DR d<p2 + (2Vof3<pDR d<p + 2Vof3<pDR)JP(<p, t)

= r ",P(<p, t),

where (in the second row) the first term corresponds to free diffusion, and the second to potential driven diffusion, with DR the diffusion constant. To calculate the dynamic NMR line shape we add the above diffusion operator to Eq. (19.20) and substitute G(t, <p) for P(<p, t)

!G(t,<P) = -[iwQ(<p) + 1/T2 - r",JG(t,<p), (19.28)

where

3 e2qQ 1 2 WQ =4-n-"2[3cos e - 1J

3 e2qQ 1 . 2 2 =4-n-"2[3sm 00 cos (<po - <p) - 1],

(19.29)

G(O, <p) = pO(<p), and the angles are defined in Fig. 19.8(d). One common way to solve such diffusion equations is to partition the space of <p into sectors of width 8<p and view the diffusion as jumps between neighbouring sectors. For the above case a set of Bloch-McConnell equations are obtained for each segment <pi (bound by <pi + ~8<p and <pi - ~8<p) of the form,

- [iwQ(<pi) + 1/T2 + (ki+1,i + ki-l,i)JG(t, <pi)

+ ki,i-I)G(t, <pi-I) + ki,i+1)G(t, <pi+l) , (19.30)

where the ki,j are related to DR, 8<p and f3Vo by the following sets of recursion formulae [39],

kO,±1 DR

(8<p)2

ki,i-l exp[-f3Vo(8<p)2J (19.31) ki-l,i

ki,i+l 2DR

= (8<p)2 - ki,i-l


and the subscript 0 corresponds to cp = O. These relations allow one to mimic the diffusion to any desired accuracy by proper scaling of 8cp. The sets of equations now have the same form as those ofEq. (19.21) and can be solved as discussed in the previous section with the effect of the quadrupole echo delay included. For comparison with the experiments, spectra must be calculated and summed over the domain distribution.

For the DTHE5 it turned out that the calculated line shapes were especially sensitive to the quadrupole echo delay parameter. By fitting calculated spectra to the experimental ones for different T values (see centre and right columns in Fig. 19.9) the diffusion parameters could be derived. The results demonstrate the power of the dynamic quadrupole echo method to analyze quite intricate motions in condensed phases. The method has been extensively used also to study dynamics in disordered solids, glasses and polymers [45].

4. Dynamic carbon-13 MAS NMR Columnar liquid crystals provide examples of highly ordered mesophases, inter

mediate between solids and nematic liquids. We have seen how dynamic deuterium NMR can be used to study motions in such phases. The method has, however, several drawbacks. It requires specific deuteration, and since the homogeneous T2 of the deuterons is often in the milliseconds range, only dynamic processes with specific rates faster than about 103 s-1 can be studied by this method. An alternative approach to studying such systems is carbon-I3 MAS NMR. These experiments can be performed on naturally abundant materials. Due to the sample spinning the resolution is high and since the natural line width of the 13C resonances is often only a few tens of Hz, much slower process can be studied by this method. The high resolution of such spectra allows dynamic measurements of different parts of the molecules. The use of sample spinning, however, complicates the simulation, since the (anisotropic) magnetic parameters are now time dependent. In the next section we show how this problem is overcome. This is followed with an example, again from the field of columnar liquid crystals.

4.1 Basic equations for the dynamic MAS NMR method The MAS NMR method, in particular as applied to carbon-13, has gained much

impetus since it was demonstrated that cross-polarization (CP) [46] enhancement can be effectively applied under MAS conditions in much the same way as in the absence of sample spinning [47]. Dynamic effects in MAS spectra show up in a similar way as in high-resolution liquid or single crystal NMR, exhibiting line broadening, coalescence, and narrowing. However, the quantitative solution of the corresponding Bloch-McConnell equations is complicated by the fact that the magnetic parameters (essentially the chemical shifts) are (periodically) time dependent due to the sample spinning. They can, therefore, not be solved by simple exponentiation. The problem was first solved by Schmidt and Vega [48], using the Floquet theory. Here we provide a simplified version of their theory [49].


We consider an ensemble of non-interacting I = 1/2 spins with anisotropic chemical shifts as the only magnetic interactions. This applies, in particular, to natural abundant carbon-13 nuclei in organic solids or mesophases, where homonuclear interactions are negligible and dipolar interactions with the protons are decoupled by a secondary rf irradiation. In the presence of exchange the transverse magnetizations, Mi (t) = M'i(t) + M~(t), of the various sites, i, can be described by a set of BlochMcConnell equations of the type,

:t Mi (t) = -[iwL<5~(t) + 1/T2 + kiiJMi (t) + L kijM{ (t), (19.32) Ni

where 8~ (t) = 8i,iso +8tniso (t) and the superscript iso and aniso refer to the isotropic and anisotropic parts of the CS tensor. The latter is rendered time dependent by the sample spinning. The dynamic process may involve jumps between similar or even symmetry related sites. In that case 8iso is the same for all i, while 8ZisO differs from site to site, reflecting their different orientations. Alternatively, the dynamic process may involve chemical exchange between inequivalent sites for which both 8iso and 8rzniso are different. For diffusion processes it is possible to partition the orientation scale into sectors as was done for the last example.

Eqs. (19.32) are written for a particular domain in the sample and the 8~(t) are thus functions of the Euler angles a, {3 and, that describe its orientation (referred to henceforth as the molecular frame - MF) relative to that of the rotor coordinate system (RF). To calculate 8~(t) the chemical shift tensor must be transformed from the MF via the RF to the laboratory frame (LF) (see Fig. 19.10),

1,j.L,v,k (19.33)

x R(a, {3, ,tt R['lj;(t), OMJk"i,

where the R's are rotation matrices, 8~[:IF are the elements (/-L, v = 1,2,3) of the chemical shift tensor in the MF, OM is the (magic) angle between the rotor axis and the magnetic field direction (OM =54.7°) and 'lj;(t) is the azimuthal angle of the rotor. Since the sample is rotating about the rotor axis, this angle is time dependent, 'lj;(t) = 'lj;o + W R (t), where 'lj;o is a phase angle that can be set equal to zero and W R (t) = 27fv R is the sample spinning frequency. Carrying out the products in Eq. (19.33) yields,

8~(t) = 8i,iso + C~ COS(2WR(t) + 2,) + S~ sin(2wR(t) + 2,)

+ C~ COS(WR(t) +,) + sf COS(WR(t) + ,), (19.34)

where the c~, S~, are constants depending on the Euler angles a, {3 and on the elements 8~[:IF (see, for example Luz et al. [49]). Using these relations the term wL8~(t) in Eq. (19.34) can be written as a sum of five Fourier components,

+2

wL8~(t) = L wteikwRt, (19.35) k=-2


Figure 19.10. Coordinate systems for the MAS experiment. XMF, YMF, ZMF are the coordinates of the molecular frame (MF), a, (3, "Y, are the Euler angles that transform the MF to the rotor (R) frame XR, YR, ZR, and ZL is the unique axis of the laboratory (L) frame along the magnetic field direction, B o.

The angle c; is the magic angle (54.7°) and w(t) is the phase angle of the rotor.

where

and Wi - w 1 (Ci ,;Si)ei2'Y - (wi )* 2 - L"2 2 -. 2 - -2 .

Substituting into Eqs. (19.32) yields

:tMi(t) = -[iLwieikwRt + 1/T2 + kiiJMi + LkijMi(t). k Hi

(19.36)

(19.37)

(19.38)

Since the coefficients in these equations are periodic in time with a characteristic frequency w R. we attempt for the Mi (t)'s solutions of the form

+00 Mi(t) = l: a~(t)einwRt, (19.39)

n=-oo

so that if the a~(t) are known (in principle infinite in number. in practice of the order of the number of observed spinning side bands) so are the Mi (t). Substituting the


expansion in Eqs. (19.38) and assuming that the coefficients of einwRt on both sides of the resulting equations are equal [49] leads to a set of coupled differential equations for the a~(t) with time-independent coefficients,

d· "'''' . dta~(t) = ~.w - i[?-iij-m + nwRc5(i,j)c5(n, m)]a~(t), (19.40) J m

where c5(k, l) are delta functions and

(19.41)

Eqs. (19.40) can be cast into matrix form

(19.42)

where A( t) is a N x oo-dimensional column vector ofthe a~ (t) eN being the number of sites), and ?-iF is the so-called Floquet Hamiltonian with elements,

(in I?-iFI jm) = [w~_m + nwRc5(n, m)]c5(i,j) + ikij c5(n, m). (19.43)

A formal solution of Eq. (19.42) gives

A(t) = e-i'HFt A(O), (19.44)

where the initial values, A(O), can be related to the equilibrium population, a~(O) "" p i c5(n, 0), so that,

a~(t) = 2: (in le-i'HFtl jO) pj. j

(19.45)

Substituting into Eq. (19.1) and summing over the Mi (t) finally yields for the free induction signal for a single domain

fid(t) V\ LMi (t) = LLa~(t)einwRt i i n

= ~~ [L(inle-i'HFtljo)einwRt] pj. t J n

(19.46)

The overall signal is obtained by repeating the calculations and summing over the angles a, {3 and 'Y. The solution of these equations are obtained using similar eigenvalue methods as employed in the previous examples. As indicated the matrices that need to be diagonalized are in principle of infinite dimension, N x 00, but in practice the index n is truncated according to the number of observed spinning side bands.


4.2 Molecular reorientation in pyramidic liquid crystals

As an example of applying MAS NMR to study dynamics in liquid crystals we discuss the carbon-13 spectra in the columnar phase of nona-alkanoyloxy-cyclotriveratrylene (CTV-n, where n is the number of carbons in the side chains) [50]. The core of these compounds can exist both in the crown and in the saddle form (see top part of Fig. 19.11), but their interconversion is slow and they can therefore be separated (chromatographically) and studied individually. When the side chains are sufficiently long (n 2 5 for the crown and n 2 4 for the saddle isomer) both homologous series exhibit columnar hexagonal phases. The mesophases of the two series have

R R

R'~R R R R R R

R R

.~ . . ~.~ R~ R~Rk'f'-R

R~R .~. ~:Z:r;V(Rkr

R~ .~: It R It It It

Figure 19.11. Top: Fonnulae of nona-alkanoyloxy-cyclotriveratrylene (CTV-n). From left to right, structural fonnula (R = R' = -OC(O)Cn- 1H2n-l), crown confonnation, and saddle confonnation. Bottom: Schematic representation of the hexagonal columnar mesophase (middle), and the stacking of the crown (left) and saddle (right) fonns.

444

CROWN

tf'C

57

20

25

,.1, ,. .... , .... "-.-". , , I ,

I .. I' Ii I 170 160 150 140

" .... nn"u·

M

In COCh

., . ., . ' ..... " . .,. ..... ",r

""'I" '"'''1''''' '''' '''I'''' 130 ppm 170

,


SADDLE

tf'c

107

20

25

, n'

I ,

""I"'" I , ,

I" I"'" 160 150 140 130 ppm

Figure 19.12. Carbon-13 spectra (aromatic and carboxylic regions only) of the crown (left) and saddle (right) forms of CTV-S. Bottom traces: Room-temperature solution spectra. Middle traces: Roomtemperature MAS spectra. Top traces: MAS spectra within the mesophase, at the indicated temperature. The MAS spectra were recorded at a spinning frequency of 5 kHz.

similar thermodynamic properties, but they differ in their dynamic characteristics. The molecules of the saddle form stack on top of each other in a locking position that prevents them from reorienting. On the other hand the crown molecules in the mesophase can easily reorient relative to each other (see bottom part of Fig. 19.11). This difference is clearly manifested in the carbon-13 MAS spectrum of the two isomers.

In Fig. 19.12 are shown carbon-13 MAS spectra of the neat saddle and crown isomers of CTV-8 at room temperature (solid or supercooled mesophase) and at higher temperatures within the mesophase region (top traces). Also shown are the regular high-resolution spectra in isotropic solutions (bottom traces). Only the low-field region exhibiting the signals of the aromatic and carboxylic carbons are shown. The solution spectra appear similar, but they reflect different situations. The crown isomer is rigid and possesses C3v symmetry, while the saddle isomer is highly flexible and undergoes fast pseudorotation that results in an average D3h symmetry. Both situations result in six distinct aromatic carbon peaks and three carboxylic lines. On the other hand the MAS spectra in the mesophases (and solids) are quite different for the two isomers. The crown form shows six aromatic and three carboxylic carbon signals as in solution. The spectrum of the saddle isomer is considerably more complicated with eighteen

Dynamic NMR in liquid crystals and liquid crystalline solutions

i i i i

180 160 140 120

trc 135

127

120

106

ppm

445

Figure 19.13. Carbon-13 MAS spectra (5 kHz) of the crown isomer of crV-8 as a function of temperature within the mesophase region. Only the centre bands of the aromatic and carboxylic carbons are shown [50].

aromatic peaks and a multitude of unresolved carboxylic signals. This indicates that the saddle form in the mesophase and in the solid is frozen in its conformation. In fact, its spectrum remains unchanged up to the clearing temperature, indicating that the conformation remains frozen throughout the mesophase region.

The MAS spectrum shown for the crown isomer in Fig. 19.12 corresponds to the low-temperature region of the mesophase (57°C). On heating within the mesophase (see Fig. 19.13), line broadening sets in, followed by smearing out of the spectrum, and eventually coalescing to essentially the same spectrum (peak position) as in the low temperature region. This behaviour is characteristic of molecular reorientation and can

446

N 1.0 :l2 € 'C . ~

~ 0.5

0.0 •

..... 0

III :;;;:

2 3 6


143.7 111.6 84.1 60.3 tfc 106

• •

105

104

103 •

102

2.4 2.6 2.8 3.0

(103rn/K·1

Figure 19.14. Left: Plot of the calculated full width at half maximum intensity of the aromatic C-H carbon (centre band) under MAS conditions. The calculations correspond to l/R = 5 kHz and l/L = 75 MHz. The solid symbols are experimental. Right: Arrhenius plot of the three-fold jump process in the crown mesophase of CTV-8 as obtained from spectra of the type shown in Fig. 19.13 [50].

be quantitatively interpreted in terms of molecular reorientation about the columnar axis. We assume that the reorientation follows a three-fold jump mechanism and use the theory discussed in the previous section for the analysis.

To do so it is necessary to identify and determine the CS tensor for at least one of the rigid (aromatic) carbons. This is most straightforward for the unsubstituted aromatic (C-H) carbon which has been identified as the isolated high-field peak at 123.6 ppm. The principal values of its CS tensor were determined from the intensities of its spinning side bands [51] to be 80, 20 and -100 ppm relative to its isotropic value. Their direction in the MF could be fixed by comparison with earlier chemical shift data to lie along the C-H bond direction, in the plane of the benzene ring perpendicular to the C-H bond, and perpendicular to the benzene ring. Taking the angle between the latter and the molecular C3 axis to be 43° [52], the chemical shift tensor components in the MF were obtained. Inserting into Eq. (19.46), Fourier transformation and powder averaging provided calculated MAS spectra of the C-H carbon for different rates of the three-fold jump process. The full line width at half maximum intensity of the centre band signal so obtained is plotted on the left hand side of Fig. 19.14. This curve served as a reference for determining the reorientational jump rate from the experimental width of the C-H peak. In practice, only rates in the slow and fast jump extremes could be determined. Excessive overlap precluded accurate analysis in the intermediate regime. The results are plotted on the right hand side of Fig. 19.14, yielding the following Arrhenius parameters for the process, A = 3.1 X 1022 s-l, Ea = 31.1 kcal / mol. These values are considerably higher than expected for simple


molecular reorientation within the columnar structure. They suggest that the barrier to the reorientation may be temperature dependent, decreasing somewhat with increasing temperature. Neglecting such a temperature dependence will tend to increase the slope and intercept of the Arrhenius plot. A temperature dependent barrier for reorientation is not unexpected for the "soft" columnar structures and may be the reason for the unusual kinetic parameters obtained for the pyramidic mesophase.

These results demonstrate the potential of applying carbon-13 MAS NMR for studying columnar liquid crystals, a possibility largely ignored so far.

5. Concluding remarks We have described several applications of dynamic NMR in the field of liquid

crystals. The dynamic range dealt with here refers to the intermediate regime where the spectral line shape is affected by exchange processes, i.e., when the characteristic rates are of the order of the interactions. This range thus falls between the ultraslow regime, suitable for magnetization transfer and 2D-exchange experiments, on one hand, and the extreme fast exchange, where dynamic information can be obtained from relaxation data on the other.

Dynamic NMR serves researchers in the field of liquid crystals in two ways. One type of application uses highly fluid liquid crystals, particularly nematics, as anisotropic solvents to study the dynamic properties of solutes. In such solvents the spectra of the solutes exhibit features, due to the anisotropic interactions, which may provide more detailed information than in isotropic solvents. Also the spectral splittings are usually much larger than in isotropic solvents, thus increasing considerably the dynamic range of the measurements. The second type of application concerns the dynamic properties of mesogen molecules in neat liquid crystals. It applies to the more viscous mesophases such as certain smectic and columnar phases. Such measurements provide information on molecular reorientation and rearrangement within the mesophase in much the same way as they are used in solids. The price tag for performing detailed line shape analysis is often the necessity to apply sophisticated simulation procedures, but the extra work is rewarding and often provides information unattainable by other means.

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Index

2D-exchange spectrum, 432 5CB,52-54,59-62,378,381,383-384,387-396 5CB confined, 390 7CB,58,60 8CB, 381, 392-393 A2 spin system, 19 AB spectrum, 17 Acetylene, 338, 341-342 Acoustic ringing, 110, 122 Activation energy, 129 Additive potential, 292, 311, 331, 359 Adiabatic condition, 379 Aggregation, 170 Alkanes, 286,288,296

confonnation of, 76, 83 ALPHA,52 a-helix, 198,200-201 Analysis of simple solutes, 96 Anchoring free energy, 241, 243 Anhannonic terms, 143 Anisole, 315 Anisotropy

chemical shift, 22, 47,89, 138, 140, 149, 163, 174, 179, 193

diffusion, 132,416 indirect spin-spin coupling, 22, 89, 138, 144 interaction, 89 intermolecular forces, 325 molecular polarisability, 273 shielding, 111 solvents, 21 tumbling, 13

A. laidlawii lipids, 411 Anopore

cavities, 383 membrane, 392

Anthracene, 249-250, 338 Anthraquinone, 249-250, 338 Anti-ferroelectric, 371 Anti-symmetric part of the J -tensor, 9 Aromatic-aromatic interactions, 155 Arrhenius equation, 131 Asymmetry parameter, 141 Atom-block representation, 297-298 Atomistic force field, 280 Atomistic potentials, 327

Attractive interaction, 252 Attractive potential, 253 Automatic analysis, 91, 141, 150 Average Hamiltonian theory, 70 Axes transformation, 11 AX spin system, 19 Backbone structure determination, 184 Bacteriophage, 163, 171 Bacteriorhodopsin, 203-204 Barrier to reorientation, 447 Benzaldehyde, 313 Benzene, 56,60-61, 137-138, 155,329-330,341 Benzyl bromide, 81 ,a-barrel channel, 200 ,a-strand,198,2oo,202 BHOf-4,56 Biaxiality, 244, 249,437 Biaxial solute, 328 Bicelle, 156, 163, 169 Bifuryl,313 Biological function, 191 Biological membrane, 191 Biomolecular structure determination, 163 Bioran glasses, 384, 388 Biphenyl, 83,150-151,313,336-337 Biselenophene, 313 Bithienyl, 313 BLEW-48,55 Bloch-McConnell equations, 419 Boltzmann, 287 Bond angle, 92 Bond length, 92 Bond order parameter, 291, 293 Bond representation, 297 Bond-shift rearrangement, 423 BR-24,55 BR-52,55 Bromobenzene, 80 Bromonaphthalene,81 Bromotriftuoroethylene, 152 Butane, 83,100-102,150,289,314 Canonical ensemble, 251 Cartesian tensor, 7 Cavities treated with surfactants, 392 Cavity, 247 Cellulose crystallites, 163, 172

449

450

Charge, 221 Charged particle, 169 Chemical shift,S, 7, 11 0 Chemical shift tensor, 195 Chiral discrimination, 153 Chiral nematic, 130 Chloroiodobenzene, 73 Chlorotoluene, 97 Cholesterol,413-415 Chord model, 268, 299-301, 311, 331 Chord segment representation, 297 cis-Decalin, 428, 432 Cluster expansion method, 301 Coherence transfer echo, 72 Collective molecular fluctuation, 384 Collision complex, 285 COMARO,52 Commutator table, 38-39 Complexity of the NMR spectrum, 153 Compression, 173 Computer simulation methods, 325 Configurational phase space, 287 Confined liquid crystal, 388-389, 391-392, 395 Confinement, 383, 393 Conformation, 287 Conformational behaviour, 330

conformational problem, 307 conformation structure tensors, 262 conformer order tensor, 25 conformer orientation, 287 conformer probability, 25, ISO, 288, 292-293, 334,

336 distribution, 335 equilibria, 421 flexible molecules, 292 flexible solute, 333

Conformational change, 25,143,185,376 Conformer, 330 Continuum models, 267, 275 Correlation functions, 350, 356, 360 Critical mixture, 112, 121, 130, 133 Critical spinning frequency, 46 CROPSY, 209-211 Cross polarization, 199,210,439 Cross relaxation, 377, 382, 386 Crown isomer, 444 CS tensor, 446 CW magic-angle decoupling, 55 Cyanobiphenyl, 150 Cyclohexane, 145-146 Cyclooctane, 145 Cyclooctatetraene, 423 Cylindrical cavity, 392 D2 molecule, 20, 125,224,227

ortho,20 para,20

DAISY, 91 Damping, 388 DANSOM,91 DANTE,75


DCE (see dipolar-correlation effect) Decoupled model, 359-360 Decoupling, 51, 55 Density functional theory, 251 Density matrix, 27, 421, 429 Density wave, 126 Deuterated, 145 Deuteration, 76, 145 Deuterium

decoupling, 145, 289 isotope effects, 153 NMR,289 NMR relaxometry, 391 spectra, 145

Diagonalization, 17 Diamagnetic anisotropy, 149 Diamagnetic susceptibility, 46, 111 Dibromobenzene, 339 Dichlorobenzene, 97,339 Dichloroethenylbenzene, 81 Dielectric anisotropy, 245, 248 Dielectric continuum, 247 Dielectric permittivity, 248 Diffusion, 84

axial, 192, 205-206 examples of, 411-417 measurement of, 404-407 signal attenuation due to, 406 temperature dependence of, 413-416 xenon, 130, 133

Difluorobenzene, 339 Dihedral angle, 287 Dihedral-angle energy, 288 Dilute liquid crystals, 163 Dimethylfuran,74 Dimethylmaleic anhydride, 150 Dimyristoylphosphatidylcholine (DMPC), 413-417 Dipolar-correlation effect, 378, 384-385, 387 Dipolar coupling, 7,10,47,89,138-139,193,308-309 Dipole, 221, 248 Dirac notation, 27 Director, 244

fluctuations, 394 manipulation, 46

Discotic liquid crystal, 434 Distance restraint, 183, 192 DMPC, 210-211 DMPC I DHPC mixture, 169 Domain-domain motion, 186 Double-quantum filtered COSY, 76 Double spin echo (DSE), 130 Dynamic

carbon-13 MAS NMR, 435, 439 deuterium NMR, 428, 435 distortions of the quadrupole echo, 429 fluctuation, 203 MASNMR,439 NMR,419 process, 419 properties, 447

INDEX

proton NMR - solutes in nematic solvents, 420 quadrupole echo spectrum, 429

Effective density, 270 Elastic continuum, 243 Elastic restoring force, 296 Electric dipole, 169,338 Electric field, 22, 148 Electric field gradient, 121,221,227,338 Electric polarizability, 338 Electric quadrupole, 338 Electrostatic free eneIgy, 248 Electrostatic induction, 249 Electrostatic interactions, 245-246, 249,273,278,281,

337 Electrostatic repulsion, 170 Ellipsoid, 168 End-to-end vector model, 302 Energy levels, 17 E. coli lipids, 412 Ensemble average, 12 EIgodic, 287 ESR,289 Ethoxybenzene, 319 Ethylbenzene, 320 Euler angles, 181, 242, 244, 441 Exchange, 393

broadened, 432 induced relaxation, 393 processes, 419 rate, 419

Excluded volume, 252, 254, 297 External ejg, 125 Extra molecular eneIgy, 292 Ferroelectric, 371 Field cycling, 378, 380

bulk liquid crystals, 380 confined liquid crystals, 382

Field gradient tensor, 9, 11 Filamentous phage, 171 Fitting calculated spectra, 439 Fitting simulated line shapes, 419 Flexible, 290

molecule, 92, 246, 293, 305, 330 solute, 285, 289, 296 tail,291

Flexoelectric effect, 246 Floquet theory, 439 Fluctuation, 205 Fluctuation mode, 388 Fluid bilayer signature, 290 Fluorinated liquid crystals, 321 Folds, 202 Force field, 163, 184 Force-field minimization, 182 Free eneIgy, 251 Frequency selective excitation, 94 Frequency shifted Lee-Goldburg sequence, 198 FSLG-2,56 Fullerene, 155 GARP,52

Gauche, 288 GAUSSIAN 98, 229 Gay-Berne potential, 327 Generalised Van der Waals theory, 272 Geometrical restriction, 388 Goniometer probe, 409 Gramicidin A, 205, 210-211, 415 H2,227 Haller function, 121 Hamiltonian, 138

NMR,6 Hard ellipsoids, 327 Harmonic corrections, 143 Harmonic force field, 143 Helical irregularities, 204 Helical tilt, 203 Helical tilt angle, 200 Helical twisting power, 246 Helium, 109, 115 Heteronuclear dipolar decoupling, 51 Hexafluorobenzene, 155,341 Hexagonal columnar mesophase, 435 Hexane,288,301,330-331,335-336 Hexa-pentyloxy triphenylene, 435 High-field approximation, 9 Hindered rotation, 421

451

Histogram-based determination of order parameter, 178 HNIN experiment, 199,202 Homology model, 184 Homonuclear dipolar decoupling, 55 Hooke's law, 232 H-PDLCs (holographic polymer dispersed LCs), 395 HSQC, 163 Human ubiquitin, 154, 156 Hydrogel matrix, 173 Hydrogen molecule, 223, 226, 338 INADEQUATE,76 Indene,79,82 Independent bond model, 293 Indirect spin-spin coupling, 5, 7, 9, 138, 149 Inertial frame (IF) model, 295 Inertia tensor, 243 Initial spectral parameters, 93 Interaction tensor, 311 Interface, 391 Intermediate dynamic range, 419 Intermolecular forces, 242 Intermolecular interaction, 243, 292 Intermolecular motion, 286 Intermolecular potential, 221, 306 Internal dynamics, 164, 177, 185 Internal motion, 24, 208 Internal potential, 337 Internal rotation, 92 Internal rotational potential, 306 Internal torsional motion, 335 Internuclear distance, 142 Interpore fluctuation, 389 Intramolecular eneIgy, 287-288 Intramolecular motion, 142,286

452

Intramolecular rearrangement, 421 Inversion, 423 IPAP,I63 Isomerization, 287, 291 Isotopic labeling, 192 Jeener-Broekaert, 38 Jump rates, 360-361 Kronecker delta, 7 Krypton, 109, 123 Laboratory-fixed axes, 7 Ladder operators, 9 LAOCOON,77 LAOCOONOR,141 Lattice, 349 Lattice models, 326 Least squares, 91, 144 Lee-Goldburg decoupling, 55 Legendre polynomial, 293, 327, 330 Lennard-Jones, 253, 329 LEQUOR, 77, 91, 141 Libration, 185,205,207 Linear molecule, 20 Line broadening, 426 Lineshape, 419 Linkages

Arylalkoxy, 319 Arylalkyl, 320 Aryloxycarbonyl,321

Liouville equation, 28, 422 Lipid,163 Liquid crystal mixtures, 149 Liquid crystals in Anopore membranes, 392 Liquid-crystal solvents, 67, 223 Liquid crystals with low order parameters, 155 Local field effects, 381 Local field spectroscopy, 57 Local solvent effects, 149 Local structure refinement, 156 Long molecular axis, 291 Low-frequency relaxation, 377 Lyotropic, 168, 170-172,371 Macromolecular order tensor, 168 Macroscopically oriented lamellar systems

deuterium spectra of, 410 lateral diffusion in, 411-417 phosphorous spectra of, 403 preparation of, 407-409 proton spectra of, 406, 412 viewed through crossed polarizers, 409

Magic angle, 146, 199,210-211,404,412 Magic angle spinning, 192, 207 Magic mixture, 223, 245, 341 Magnetic

field gradient, 72 field orientation, 153-154 fields, 23 ordering of biomolecules, 163, 166 potential energy, 46 susceptibility. 23, 166 susceptibility tensor, 154


Maier Saupe, 243, 272-273, 310 MAS, 439 MAS of small membrane proteins, 209 Master rate equation, 360 Maximum entropy, 302, 312, 337 Mean field, 222, 242-243, 249, 286

attractive potential, 255 model, 292, 331 potential, 253, 327

Membrane, 191 fluidity, 289 protein, 191 protein structure, 191

Merz-Kollman-Singh procedure, 249 Methane, 25,226,228,338 Methyl fluoride, 341 Mixtures, 231 Model

CI,234 elastic distortion, 232 integral, 232 membrane, 191 prediction, 239 surface, 232

Modular models, 296 Modular potential, 267 Molecular

diffusion, 390 dynamics, 205-208,392 dynamics simulations, 326 field theory, 328 fixed axes, II frame, 244, 286 geometries, 21, 149 hydrogen, 338 mechanics, 288 motion, 191 polarizability, 243 properties, 13 quadrupole moment, 229 reorientation, 376 reorientation in pyramidic liquid crystals, 443 shape, 295 structure, 89, 137 surface, 243 symmetry,15 vibration, 142

Moment of inertia, 205-206 tensor, 168, 200, 335

Monte Carlo simulations, 326 Motif recognition, 183 Motional averaging, 208, 286 Motional constants, 15 Motionally narrowed, 286 Motion

fast limit, 205 MREV-8,55 MSHOT-3,56 Multinuclear techniques, 145 Multiple co-dissolved solutes, 97

INDEX

Multiple quantum, 91, 150 3DNMR,94 filter, 75 NMR,42-43,67,93

non-selective detection, 69 non-selective excitation, 69 relaxation, 84 selective detection, 70, 73 selective excitation, 73

transitions, 67-68 Multipole expansion, 221, 248 Multipulse techniques, 145 NaphtluUene, 155,339 Naphthoquinone, 80 Natural abundance 2H NMR, 153 Near magic angle spinning, 150 Nematic, 6, 137 Neon, 109, 114, 123-127 Neu protein transmembrane domain, 205-206, 208 Nitrobenzene, 155 Nitrobenzene-lis, 23 Nitroxide spin label, 289 NMR of noble gases, 109 NMR relaxometry, 376 NMR spectroscopy, 5, 29 nOCB,318-319 Nona-alkanoyloxy-cyclotriveratrylene, 443 Non-bonded steric energy, 288 Non-Polarised (NP), 378 NTFEB,316 Nuclear Overhauser effect, 192, 386 Nuclear quadrupole interaction, 8, 11 Nucleic acids, 163 Odd-even effect, 290, 295 ODF, 380, 387-388 Off-lattice model, 326 Onsager formulation, 247, 272 Optical pumping, 110 Order director fluctuations, 349, 356, 376, 383 Ordered boundary layer, 394 Ordered surface layer, 394 Order parameter, 78, 89, 92,116,119-122,125-126,

140,142,146,163,177,181,223,332,386,401, 423

Order parameter profile, 290 Order tensor, 177, 222 Orientational

distribution function, 242, 327, 330, 332 fluctuation, 205

Orientationally ordered solute, 137 Orientational

order, 21, 221, 243 electric field, 22 magnetic field, 23

order parameter, 327 potential, 241 reference, 224 restraint, 192, 205 wetting, 395

Orientation-conformaticn correlation, 287

Orientation dependent NMR interactions chemical shift, 401 dipole-dipole, 402 quadrupole, 400

Orientation parameter, 12, 15 Orienting biological macromolecules, 156 Orienting media, 168, 170 Overlap function, 270-271 Pair correlation function, 252 Pake doublet, 436 Parallelepiped, 243 Paramagnetic, 167, 183 Partially oriented phase, 174 Partially oriented protein, 156 Particle distribution function, 120 Pauli matrices, 31 PDLC, 382, 393, 395-396 PDLF, 60 Permutation, 426 Perturbational analysis, 252 Pfg experiment

spin-echo (SE), 404 stimulated spin echo (STE), 405-406

Phage, 163 Phase cycling, 70 Phase microsegregation, 273

453

Phase transition, lll, ll3-ll5, 117-118, 121, 127-128, 130,132

Phenanthrene-dlO , 24 Phenomenological model, 243 Phenomenological potential, 266 Phenylacetylene, 80 Phenyl alcohol, 154 Phenylpropyne, 82 PISA wheel, 200 PISEMA,199 Pitch,246 Planar Iibration, 437 Po!arizability, 133, 223 Polarization, 242, 247 Polarization inversion spin exchange, 199 Polarization transfer, 383 Polyacrylamide, 163, 173

hydrogels, 156 Polyalanine, 200-201 Polymer dispersed liquid crystal (PDLC), 394 Polymer-stabilized liquid crystal, 173 Polypeptide helices, 200 Pore size, 390-391 Potential barrier, 92 Potential of mean torque, 265-268, 285, 292, 296-297,

328,357-358 Powder sample, 400-404 Precession, 29-30 Pre-Polarised (PP), 378 Probability density, 252 Probability function, 15 Propyne 3-phenyl, 82 Protein, 191

data bank, 197

454

folding, 156 folds,I83 molecular structure, 191

Proton-detected local-field, 57,60 Proton pump, 203 Pseudo-contact paramagnetic shift, 167, 183 Pseudorotation, 444 Pulsed field gradient, 72 Pulsed gradient spin echo (POSE), 130 Pulsed NMR, 30 Purple membrane, 163, 171 Pyramidic liquid crystal, 443 Pyrene-dlO, 24 Quadrupolar order, 42-43 Quadrupole, 121, 155,221,251

constants, 22 coupling, 49, 89 coupling constant, 141, 146 echo, 429 interaction, 20, 341 quadrupole interaction, 327 splitting, 141, 226, 290

Ramachandran plot, 163 Reaction field, 246-249 Reduced temperature, 115,224 Reentrant nematic, 129 Regression, 91 Reintroduction of dipolar interaction, 192 Relaxation, 84

quadrupolar noble gas nuclei, 130 xenon, 127

Reorientation mediated by translational displacement (RMTD),377

Residual dipolar coupling, 156, 163-166, 174, 179, 181, 183

Resolution enhancement, 198 Restoring Force model, 310 Restraints in biomolecular structure determination, 181 Restricted diffusion, 132 RHODIUM, 57-58 Rhombicity of the order tensor, 177 Richards-Connolly model, 249 Rigid cores, 321 Ring inversion, 432 RMTD, 383, 395 Rod-shaped virus, 171 Rolling sphere, 243 Rotameric model, 358 Rotating frame, 33 Rotational diffusion, 168 Rotational isomeric state (RIS), 287, 309, 335 Ryckaert-Bellemans potential, 335 Saddle isomer, 444 SAS, 45 Saupe order matrix, 165,311 Scaled dipolar couplings, 148 Scaling, 224 Segmental order parameter, 260, 293, 358 Segment order tensor, 287 Selective magnetization inversion, 383

NMR OF ORDERED UQUIDS

Separated local field, 57, 193, 197 SHAPE,I44 Shape anisometry, 274, 282 Shape anisotropy, 168 Shape model, 333 Shielding, 110 Shielding hyperpolarizability, 119 Shielding tensor, 140 Short-range forces, 242 Short-range interactions, 222 Short-range repulsive forces, 339 Short-range repulsive interaction, 338 Sign of indirect coupling, 22, 228 Simulated annealing, 163, 181 Simulation, 301, 326 Singlet orientational distribution function, 306, 327 Singlet orientational energy, 306 Singular value decomposition, 179 Size and shape, 222, 229, 332 Size anisotropy, 232 SLF,57 Small-step rotational diffusion model, 357 Smectic A, 146 Smectic ordering, 393 Solid state NMR, 191 Solubility, 153 Solute, 89, 221

as a parallelepiped, 339 probes, 222

Solute-solvent interaction, 254 Solute-solvent radial distribution function, 340 Solvent density, 335 Solvent electrostatic polarization, 241 Solvent mean field, 296 SPARC-16,52 Spatially constrained thermotropic LC, 377 Specific deuteration, 145 Spectral analysis, 90, 141, 145 Spectral densities, 350, 360 Spectral subtraction, 100 Spherical coordinates, 10, 12 Spherical harmonics, 15 Spherical solute, 340 Sphingomyelin, 415 SPINAL,54 Spin echo

measurement of diffusion, 84 Spin exchange, 386 Spin-label, 289 Spin-lattice relaxation, 350, 355 Spinning,46,149 Spinning side band, 446 Spin relaxation, 349 Spin-spin relaxation, 350, 355 Spin-state editing, 176 Spontaneous polarization, 303 Stabilization energy, 253-254 Steric effects, 168, 173 Sterie interaction, 288 Sternheimer antishielding, 122

INDEX

Stimulated echo, 384 Strained polyacrylamide gel, 173 Straley, 335, 339 Structural

constraint, 177 detennination, 142 genomics, 163, 177, 183 imperfection, 203 interpretation, 179 parameter, 92

Substituted benzene, 338 Surface

anchoring, 383 charge, 172 charge density, 247 contact model, 253 induced order, 39(}-392, 394 integral, 332 interaction, 388 normaI,244 order parameter, 395 tensor, 243,249 tensor model, 241, 255, 302, 310

Surfactant / hexanol mixtures, 170 Switched angle correlation, 51 T2,227 Tautomerism, 421 Taylor series, 22, 142 TCB (see trichiorobenzene) Tensor

spherical, 35 Tensor transfonnation, 12 Thermotropic lipid mixtures, 169 Three dimensional NMR, 73 Three-fold jump, 446 Tilt angle, 134, 264 Tilted smectics, 264 TPPM,52 Trans, 288 Trans - gauche energy difference, 292

Translational diffusion, 376, 395 Translational diffusion anisotropy, 169 Translational order parameter, 126 Translational-orientatilnaI order parameter, 126 Transmembrane helix, 203 TREV-8,55 Trichiorobenzene, 97 Twisting ability of chiral molecules, 246 Two-dimensional dynamic director correlations, 50 Two-parameter CI model, 333 Ubiquitin, 163 Ultraslow collective molecular motion, 387 Ultra-slow motion, 378 Undulation, 207 United-atom model, 330 Van der Waals, 243, 252, 254, 256 Variable angle spinning, 58-60 VAS, 45 VASS, 321 Vibration, 185 Vibrational corrections, 142, 318 Vibrational motion, 25 Vibration-reorientation coupling, 25, 228 Vibration-reorientation interaction, 143 Virial coefficients, III WAHUHA,55 WALTZ-16,52 Wave functions, 18 Whole body motion, 208 Wigner function, 249, 327 Wigner rotation matrices, 15,242,244,351 WIN-DAISY, 141 Wobble, 205 Wobbling, 207

455

Xenon, 109, 111-113, 115, 117-118, 122-123, 126, 128-133

X-ray crystallography, 205 Xylene, 97 Zeeman interaction, 6, 9, 138 Zero field NMR, 153

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NMR of Ordered Liquids