Diffusion in Condensed Matter

Paul Heitjans Jrg Krger

Diffusion inCondensed MatterMethods, Materials, Models

With 448 Figures

ABC

Editors

Professor Dr. Paul HeitjansUniversitt HannoverInstitut fr Physikalische Chemieund ElektrochemieCallinstr. 33aD-30167 Hannover, GermanyEmail: [email protected]

Professor Dr. Jrg KrgerUniversitt LeipzigInstitut fr Experimentelle Physik ILinnstr. 5D-04103 Leipzig, GermanyEmail: [email protected]

Library of Congress Control Number: 2005935206

ISBN -10 3-540-20043-6 Springer Berlin Heidelberg New YorkISBN -13 978-3-540-20043-7 Springer Berlin Heidelberg New York

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To Maria and Birge

Preface

Diffusion as the process of migration and mixing due to irregular movementof particles is one of the basic and ubiquitous phenomena in nature as wellas in society. In the latter case the word particles may stand for men orideas, and in the former for atoms or galaxies. In this sense diffusion is atruly universal and transdisciplinary topic.

The present book is confined, of course, to diffusion of atoms and mole-cules. As this process shows up in all states of matter over very large time andlength scales, the subject is still very general involving a large variety of nat-ural sciences such as physics, chemistry, biology, geology and their interfacialdisciplines. Besides its scientific interest, diffusion is of enormous practicalrelevance for industry and life, ranging from steel making to oxide/carbondioxide exchange in the human lung.

It therefore comes as no surprise that the early history of the subject ismarked by scientists from diverse communities, e.g., the botanist R. Brown(1828), the chemist T. Graham (1833), the physiologist A. Fick (1855), themetallurgist W.C. Roberts-Austen (1896) and the physicist A. Einstein (1905).Today, exactly 150 and 100 years after the seminal publications by Fick andEinstein, respectively, the field is flourishing more than ever with about 10.000scientific papers per year.

From the foregoing it is evident that a single volume book on atomic andmolecular diffusion has to be further restricted in its scope. As the title says,the book is confined to diffusion in condensed matter systems, so diffusionin gases is excluded. Furthermore, emphasis is on the fundamental aspects ofthe experimental observations and theoretical descriptions, whereas practi-cal considerations and technical applications have largely been omitted. Thecontents are roughly characterized by the headings Solids, Interfaces, Liq-uids, and Theoretical Concepts and Models of the four parts under which thechapters have been grouped.

The book consists of 23 chapters written by leading researchers in theirrespective fields. Although each chapter is independent and self-contained(using its own notation, listed at the end of the chapter), the editors havetaken the liberty of adding many cross-references to other chapters and sec-tions. This has been facilitated by the common classification scheme. Further

VIII Preface

help to the reader in this respect is provided by an extended common list ofcontents, in addition to the contents overview, as well as an extensive subjectindex.

The book is a greatly enlarged (more than twice) and completely revisededition of a volume first published with Vieweg in 1998. Although the firstedition was very well received (and considered as a must for students andworkers in the field), it was felt that, in addition to the broad coverageof modern methods, materials should also be discussed in greater detail inthe new edition. The same applies to theoretical concepts and models. This,in fact, is represented by the new subtitle Methods, Materials, Models ofDiffusion in Condensed Matter.

The experimental Methods include radiotracer and mass spectrometry,Mobauer spectroscopy and nuclear resonant scattering of synchrotron ra-diation, quasielastic neutron scattering and neutron spin-echo spectroscopy,dynamic light scattering and fluorescence techniques, diffraction and scan-ning tunneling microscopy in surface diffusion, spin relaxation spectroscopyby nuclear magnetic resonance (NMR) and beta-radiation detected NMR,NMR in a magnetic field gradient, NMR in the presence of an electric field,impedance spectroscopy and other techniques for measuring frequency de-pendent conductivities.Materials now dealt with are, among others, metals and alloys, metallicglasses, semiconductors, oxides, proton-, lithium- and other ion-conductors,nanocrystalline materials, micro- and mesoporous systems, inorganic glasses,polymers and colloidal systems, biological membranes, fluids and liquid mix-tures. The span from simple monoatomic crystals, with defects in thermalequilibrium enabling elementary jumps, to highly complex systems, exem-plarily represented by a biomembrane (cf. Fig. 12.3), is also indicated on thebook cover.Models in the subtitle stands for theoretical descriptions by, e. g., correlationfunctions, lattice models treated by (approximate) analytical methods, thetheory of fractals, percolation models, Monte Carlo simulations, molecular dy-namics simulations, phenomenological approaches like the counterion model,the dynamic structure model and the concept of mismatch and relaxation.

Despite the large variety of topics and themes the coverage of diffusion incondensed matter is of course not complete and far from being encyclopedic.Inevitably, it reflects to a certain extent also the editors main fields of inter-est. Nevertheless the chapters are believed to present a balanced selection.

The book tries to bridge the transition from the advanced undergradu-ate to the postgraduate and active research stage. Accordingly, the variouschapters are in parts tutorial, but they also lead to the forefront of currentresearch without intending to mimic the topicality of proceedings, which nor-mally have a short expiry date. It is therefore designed as a textbook or refer-

Preface IX

ence work for graduate and undergraduate students as well as a source bookfor active researchers.

The invaluable technical help of Dr. Sylvio Indris (University of Han-nover) in the laborious editing of the chapters, which in some cases includedextensive revision, is highly acknowledged. We also thank Jacqueline Lenzand Dr. T. Schneider from Springer-Verlag for accompanying this project.As ever, the editors have to thank their wives, Maria Heitjans and BirgeKarger, for their patience and encouragement.

Hannover, Germany Paul HeitjansLeipzig, Germany Jorg KargerAugust 2005

Contents Overview

Part I Solids

1 Diffusion: Introduction and Case Studies in Metals andBinary AlloysHelmut Mehrer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The Elementary Diffusion Step in Metals Studied by theInterference of Gamma-Rays, X-Rays and NeutronsGero Vogl, Bogdan Sepiol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3 Diffusion Studies of Solids by Quasielastic NeutronScatteringTasso Springer, Ruep E. Lechner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4 Diffusion in SemiconductorsTeh Yu Tan, Ulrich Gosele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5 Diffusion in OxidesManfred Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

6 Diffusion in Metallic Glasses and Supercooled MeltsFranz Faupel, Klaus Ratzke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Part II Interfaces

7 Fluctuations and Growth Phenomena in Surface DiffusionMichael C. Tringides, Myron Hupalo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

8 Grain Boundary Diffusion in MetalsChristian Herzig, Yuri Mishin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

9 NMR and -NMR Studies of Diffusion in Interface-Dominated and Disordered SolidsPaul Heitjans, Andreas Schirmer, Sylvio Indris . . . . . . . . . . . . . . . . . . . . . 367

XII Contents Overview

10 PFG NMR Studies of Anomalous DiffusionJorg Karger, Frank Stallmach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

11 Diffusion Measurements by UltrasonicsRoger Biel, Martin Schubert, Karl Ullrich Wurz, Wolfgang Grill . . . . . . 461

12 Diffusion in MembranesIlpo Vattulainen, Ole G. Mouritsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

Part III Liquids

13 Viscoelasticity and Microscopic Motion in Dense PolymerSystemsDieter Richter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

14 The Molecular Description of Mutual Diffusion Processesin Liquid MixturesHermann Weingartner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

15 Diffusion Measurements in Fluids by Dynamic LightScatteringAlfred Leipertz, Andreas P. Froba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

16 Diffusion in Colloidal and Polymeric SystemsGerhard Nagele, Jan K.G. Dhont, Gerhard Meier . . . . . . . . . . . . . . . . . . . 619

17 Field-Assisted Diffusion Studied by Electrophoretic NMRManfred Holz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

Part IV Theoretical Concepts and Models

18 Diffusion of Particles on LatticesKlaus W. Kehr, Kiaresch Mussawisade, Gunter M. Schutz, ThomasWichmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745

19 Diffusion on FractalsUwe Renner, Gunter M. Schutz, Gunter Vojta . . . . . . . . . . . . . . . . . . . . . . 793

20 Ionic Transport in Disordered MaterialsArmin Bunde, Wolfgang Dieterich, Philipp Maass, Martin Meyer . . . . . 813

21 Concept of Mismatch and Relaxation for Self-Diffusionand Conduction in Ionic Materials with Disordered StructureKlaus Funke, Cornelia Cramer, Dirk Wilmer . . . . . . . . . . . . . . . . . . . . . . . 857

Contents Overview XIII

22 Diffusion and Conduction in Percolation SystemsArmin Bunde, Jan W. Kantelhardt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895

23 Statistical Theory and Molecular Dynamics of Diffusionin ZeolitesReinhold Haberlandt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955

Contents In Detail

Part I Solids

1 Diffusion: Introduction and Case Studies in Metals andBinary AlloysHelmut Mehrer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Continuum Description of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Ficks Laws for Anisotropic Media . . . . . . . . . . . . . . . . . . . . . 41.2.2 Ficks Second Law for Constant Diffusivity . . . . . . . . . . . . . 51.2.3 Ficks Second Law for Concentration-Dependent Diffusivity 6

1.3 The Various Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 Tracer Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 Chemical Diffusion (or Interdiffusion) Coefficient . . . . . . . . 81.3.3 Intrinsic Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.1 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.2 Indirect Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Dependence of Diffusion on Thermodynamic Variables . . . . . . . . . 171.5.1 Temperature Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5.2 Pressure Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.6 Atomistic Description of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.6.1 Einstein-Smoluchowski Relation and Correlation Factor . . 191.6.2 Atomic Jumps and Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 221.6.3 Diffusion Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.7 Interstitial Diffusion in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.7.1 Normal Interstitial Solutes . . . . . . . . . . . . . . . . . . . . . . . . . . 271.7.2 Hydrogen Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.8 Self-Diffusion in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.8.1 Face-Centered Cubic Metals . . . . . . . . . . . . . . . . . . . . . . . . . . 321.8.2 Body-Centered Cubic Metals . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.9 Impurity Diffusion in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.9.1 Normal Impurity Diffusion in fcc Metals . . . . . . . . . . . . . . 361.9.2 Slow Diffusion of Transition-Metal Solutes in Aluminium . 391.9.3 Fast Solute Diffusion in Open Metals . . . . . . . . . . . . . . . . . 40

1.10 Self-Diffusion in Binary Intermetallics . . . . . . . . . . . . . . . . . . . . . . . . 42

XVI Contents In Detail

1.10.1 Influence of Order-Disorder Transition . . . . . . . . . . . . . . . . . 431.10.2 Coupled Diffusion in B2 Intermetallics . . . . . . . . . . . . . . . . . 441.10.3 The Cu3Au Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.11 Interdiffusion in Substitutional Binary Alloys . . . . . . . . . . . . . . . . . 491.11.1 Boltzmann-Matano Method . . . . . . . . . . . . . . . . . . . . . . . . . . 491.11.2 Darkens Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.11.3 Darken-Manning Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1.12 Multiphase Diffusion in Binary Systems . . . . . . . . . . . . . . . . . . . . . . 531.13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2 The Elementary Diffusion Step in Metals Studied by theInterference of Gamma-Rays, X-Rays and NeutronsGero Vogl, Bogdan Sepiol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.2 Self-Correlation Function and Quasielastic Methods . . . . . . . . . . . . 66

2.2.1 Quasielastic Methods: Mobauer Spectroscopy andNeutron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.2.2 Nuclear Resonant Scattering of Synchrotron Radiation . . . 732.2.3 Neutron Spin-Echo Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 742.2.4 Non-Resonant Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.3.1 Pure Metals and Dilute Alloys . . . . . . . . . . . . . . . . . . . . . . . . 772.3.2 Ordered Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3 Diffusion Studies of Solids by Quasielastic NeutronScatteringTasso Springer, Ruep E. Lechner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.2 The Dynamic Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.3 The Rate Equation and the Self-Correlation Function . . . . . . . . . . 1023.4 High Resolution Neutron Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 1063.5 Hydrogen Diffusion in Metals and in Metallic Alloys . . . . . . . . . . . 1153.6 Diffusion with Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213.7 Vacancy Induced Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.8 Ion Diffusion Related to Ionic Conduction . . . . . . . . . . . . . . . . . . . . 1263.9 Proton Diffusion in Solid-State Protonic Conductors . . . . . . . . . . . 1313.10 Proton Conduction: Diffusion Mechanism Based on a Chemical

Reaction Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.11 Two-Dimensional Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1433.12 Coherent Quasielastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Contents In Detail XVII

4 Diffusion in SemiconductorsTeh Yu Tan, Ulrich Gosele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.2 Diffusion Mechanisms and Point Defects in Semiconductors . . . . . 1654.3 Diffusion in Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

4.3.1 Silicon Self-Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664.3.2 Interstitial-Substitutional Diffusion: Au, Pt and Zn in Si . 1684.3.3 Dopant Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1724.3.4 Diffusion of Carbon and Other Group IV Elements . . . . . . 1774.3.5 Diffusion of Si Self-Interstitials and Vacancies . . . . . . . . . . . 1804.3.6 Oxygen and Hydrogen Diffusion . . . . . . . . . . . . . . . . . . . . . . 182

4.4 Diffusion in Germanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1834.5 Diffusion in Gallium Arsenide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

4.5.1 Native Point Defects and General Aspects . . . . . . . . . . . . . . 1854.5.2 Gallium Self-Diffusion and Superlattice Disordering . . . . . 1874.5.3 Arsenic Self-Diffusion and Superlattice Disordering . . . . . . 1944.5.4 Impurity Diffusion in Gallium Arsenide . . . . . . . . . . . . . . . . 1964.5.5 Diffusion in Other III-V Compounds . . . . . . . . . . . . . . . . . . 203


5 Diffusion in OxidesManfred Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2095.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2095.2 Defect Chemistry of Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

5.2.1 Dominating Cation Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . 2125.2.2 Dominating Oxygen Disorder . . . . . . . . . . . . . . . . . . . . . . . . . 215

5.3 Self- and Impurity Diffusion in Oxides . . . . . . . . . . . . . . . . . . . . . . . . 2165.3.1 Diffusion in Oxides with Dominating Cation Disorder . . . . 2165.3.2 Diffusion in Oxides with Dominating Oxygen Disorder . . . 222

5.4 Chemical Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2265.5 Diffusion in Oxides Exposed to External Forces . . . . . . . . . . . . . . . 228

5.5.1 Diffusion in an Oxygen Potential Gradient . . . . . . . . . . . . . 2295.5.2 Diffusion in an Electric Potential Gradient . . . . . . . . . . . . . 236

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2425.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

6 Diffusion in Metallic Glasses and Supercooled MeltsFranz Faupel, Klaus Ratzke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2496.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2496.2 Characteristics of Diffusion in Crystals . . . . . . . . . . . . . . . . . . . . . . . 2506.3 Diffusion in Simple Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2516.4 General Aspects of Mass Transport and Relaxation in

Supercooled Liquids and Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

XVIII Contents In Detail

6.5 Diffusion in Metallic Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2596.5.1 Structure and Properties of Metallic Glasses . . . . . . . . . . . . 2596.5.2 Possible Diffusion Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 2626.5.3 Isotope Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2656.5.4 Pressure Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2686.5.5 Effect of Excess Volume on Diffusion . . . . . . . . . . . . . . . . . . 269

6.6 Diffusion in Supercooled and Equilibrium Melts . . . . . . . . . . . . . . . 2706.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

Part II Interfaces

7 Fluctuations and Growth Phenomena in Surface DiffusionMichael C. Tringides, Myron Hupalo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2857.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2857.2 Surface Diffusion Beyond a Random Walk . . . . . . . . . . . . . . . . . . . . 286

7.2.1 The Role of Structure and Geometry of the Substrate . . . 2867.2.2 The Role of Adsorbate-Adsorbate Interactions . . . . . . . . . . 2887.2.3 Diffusion in Equilibrium and Non-Equilibrium

Concentration Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2907.3 Equilibrium Measurements of Surface Diffusion . . . . . . . . . . . . . . . . 297

7.3.1 Equilibrium Diffusion Measurements from DiffractionIntensity Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

7.3.2 STM Tunneling Current Fluctuations . . . . . . . . . . . . . . . . . . 3067.4 Non-Equilibrium Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

7.4.1 Uniform-Height Pb Islands on Si(111) . . . . . . . . . . . . . . . . . 3137.4.2 Measurements of Interlayer Diffusion on Ag/Ag(111) . . . . 320


8 Grain Boundary Diffusion in MetalsChristian Herzig, Yuri Mishin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3378.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3378.2 Fundamentals of Grain Boundary Diffusion . . . . . . . . . . . . . . . . . . . 338

8.2.1 Basic Equations of Grain Boundary Diffusion . . . . . . . . . . . 3388.2.2 Surface Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3398.2.3 Methods of Profile Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 3408.2.4 What Do We Know About Grain Boundary Diffusion? . . . 343

8.3 Classification of Diffusion Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3478.3.1 Harrisons Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3488.3.2 Other Classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

8.4 Grain Boundary Diffusion and Segregation . . . . . . . . . . . . . . . . . . . . 3538.4.1 Determination of the Segregation Factor from Grain

Boundary Diffusion Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Contents In Detail XIX

8.4.2 Beyond the Linear Segregation . . . . . . . . . . . . . . . . . . . . . . . . 3578.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

9 NMR and -NMR Studies of Diffusion in Interface-Dominated and Disordered SolidsPaul Heitjans, Andreas Schirmer, Sylvio Indris . . . . . . . . . . . . . . . . . . . . . 3679.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3679.2 Influence of Diffusion on NMR Spin-Lattice Relaxation and

Linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3699.3 Basics of NMR Relaxation Techniques . . . . . . . . . . . . . . . . . . . . . . . . 3759.4 Method of -Radiation Detected NMR Relaxation . . . . . . . . . . . . . 3809.5 Intercalation Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

9.5.1 Lithium Graphite Intercalation Compounds . . . . . . . . . . . . 3849.5.2 Lithium Titanium Disulfide Hexagonal Versus Cubic . . . 386

9.6 Nanocrystalline Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3909.6.1 Nanocrystalline Calcium Fluoride . . . . . . . . . . . . . . . . . . . . . 3919.6.2 Nanocrystalline, Microcrystalline and Amorphous

Lithium Niobate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3949.6.3 Nanocrystalline Lithium Titanium Disulfide . . . . . . . . . . . . 3979.6.4 Nanocrystalline Composites of Lithium Oxide and Boron

Oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3999.7 Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

9.7.1 Inhomogeneous Spin-Lattice Relaxation in Glasses withDifferent Short-Range Order . . . . . . . . . . . . . . . . . . . . . . . . . . 403

9.7.2 Glassy and Crystalline Lithium Aluminosilicates . . . . . . . . 4059.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4089.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

10 PFG NMR Studies of Anomalous DiffusionJorg Karger, Frank Stallmach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41710.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41710.2 The Origin of Anomalous Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 41810.3 Fundamentals of PFG NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

10.3.1 The Measuring Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42110.3.2 The Mean Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42210.3.3 PFG NMR as a Generalized Scattering Experiment . . . . . 42410.3.4 Experimental Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

10.4 PFG NMR Diffusion Studies in Regular Pore Networks . . . . . . . . . 42710.4.1 The Different Regimes of Diffusion Measurement . . . . . . . . 42810.4.2 Intracrystalline Self-Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 43010.4.3 Correlated Diffusion Anisotropy . . . . . . . . . . . . . . . . . . . . . . . 43110.4.4 Transport Diffusion Versus Self-Diffusion . . . . . . . . . . . . . . . 43210.4.5 Single-File Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

XX Contents In Detail

10.4.6 Diffusion in Ordered Mesoporous Materials . . . . . . . . . . . . . 43710.5 Anomalous Diffusion by External Confinement . . . . . . . . . . . . . . . . 439

10.5.1 Restricted Diffusion in Polystyrene Matrices . . . . . . . . . . . . 44010.5.2 Diffusion in Porous Polypropylene Membranes . . . . . . . . . . 44110.5.3 Tracing Surface-to-Volume Ratios . . . . . . . . . . . . . . . . . . . . . 444

10.6 Anomalous Diffusion due to Internal Confinement . . . . . . . . . . . . . 44710.6.1 Anomalous Segment Diffusion in Entangled Polymer

Melts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44810.6.2 Diffusion Under the Influence of Hyperstructures in

Polymer Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45010.6.3 Diffusion Under the Influence of Hyperstructures in

Polymer Melts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45310.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

11 Diffusion Measurements by UltrasonicsRoger Biel, Martin Schubert, Karl Ullrich Wurz, Wolfgang Grill . . . . . . 46111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46111.2 Diffusion of Hydrogen in Single-Crystalline Tantalum . . . . . . . . . . 46211.3 Observation of Diffusion of Heavy Water in Gels and Living

Cells by Scanning Acoustic Microscopy with Phase Contrast . . . . 46611.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

12 Diffusion in MembranesIlpo Vattulainen, Ole G. Mouritsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47112.2 Short Overview of Biological Membranes . . . . . . . . . . . . . . . . . . . . . 47312.3 Lateral Diffusion of Single Molecules . . . . . . . . . . . . . . . . . . . . . . . . . 477

12.3.1 Lateral Tracer Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . 47712.3.2 Methods to Examine Lateral Tracer Diffusion . . . . . . . . . . . 47912.3.3 Lateral Diffusion of Lipids and Proteins . . . . . . . . . . . . . . . . 482

12.4 Rotational Diffusion of Single Molecules . . . . . . . . . . . . . . . . . . . . . . 49112.5 Lateral Collective Diffusion of Molecules in Membranes . . . . . . . . . 493

12.5.1 Ficks Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49312.5.2 Decay of Density Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . 49412.5.3 Relation Between Tracer and Collective Diffusion . . . . . . . 49512.5.4 Methods to Examine Lateral Collective Diffusion . . . . . . . . 49712.5.5 Lateral Collective Diffusion in Model Membranes . . . . . . . 498

12.6 Diffusive Transport Through Membranes . . . . . . . . . . . . . . . . . . . . . 50012.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

Contents In Detail XXI

Part III Liquids

13 Viscoelasticity and Microscopic Motion in Dense PolymerSystemsDieter Richter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51313.2 The Neutron Scattering Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

13.2.1 The Neutron Spin-Echo Technique Versus ConventionalScattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

13.2.2 Neutron Spin Manipulations with Magnetic Fields . . . . . . 51613.2.3 The Spin-Echo Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

13.3 Local Chain Dynamics and the Glass Transition . . . . . . . . . . . . . . . 51913.3.1 Dynamic Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52113.3.2 Self-Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

13.4 Entropic Forces The Rouse Model . . . . . . . . . . . . . . . . . . . . . . . . . . 52913.4.1 Neutron Spin-Echo Results in PDMS Melts . . . . . . . . . . . . 53113.4.2 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

13.5 Long-Chains Reptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53713.5.1 Theoretical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53713.5.2 Experimental Observations of Chain Confinement . . . . . . . 538

13.6 Intermediate Scale Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54013.7 The Crossover from Rouse to Reptation Dynamics . . . . . . . . . . . . . 54313.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

14 The Molecular Description of Mutual Diffusion Processesin Liquid MixturesHermann Weingartner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55514.2 Experimental Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55814.3 Phenomenological Description of Mutual Diffusion . . . . . . . . . . . . . 55914.4 Thermodynamics of Mutual Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 56414.5 Linear Response Theory and Time Correlation Functions . . . . . . . 56714.6 The Time Correlation Function for Mutual Diffusion . . . . . . . . . . . 56914.7 Properties of Distinct-Diffusion Coefficients . . . . . . . . . . . . . . . . . . . 57114.8 Information on Intermolecular Interactions Deduced from

Diffusion Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57314.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577

15 Diffusion Measurements in Fluids by Dynamic LightScatteringAlfred Leipertz, Andreas P. Froba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57915.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

XXII Contents In Detail

15.2 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58015.2.1 Spectrum of Scattered Light . . . . . . . . . . . . . . . . . . . . . . . . . . 58015.2.2 Correlation Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58215.2.3 Homodyne and Heterodyne Techniques . . . . . . . . . . . . . . . . 587

15.3 The Dynamic Light Scattering Experiment . . . . . . . . . . . . . . . . . . . 58915.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58915.3.2 Signal Statistics and Data Evaluation . . . . . . . . . . . . . . . . . . 594

15.4 Thermophysical Properties of Fluids Measured by DynamicLight Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59715.4.1 Thermal Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59715.4.2 Mutual Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 60015.4.3 Dynamic Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60115.4.4 Sound Velocity and Sound Attenuation . . . . . . . . . . . . . . . . 60415.4.5 Landau-Placzek Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60615.4.6 Soret Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60615.4.7 Derivable Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

15.5 Related Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60815.5.1 Surface Light Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60815.5.2 Forced Rayleigh Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 613


16 Diffusion in Colloidal and Polymeric SystemsGerhard Nagele, Jan K.G. Dhont, Gerhard Meier . . . . . . . . . . . . . . . . . . . 61916.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61916.2 Principles of Quasielastic Light Scattering . . . . . . . . . . . . . . . . . . . . 620

16.2.1 The Scattered Electric Field Strength . . . . . . . . . . . . . . . . . . 62016.2.2 Dynamic Light Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62416.2.3 Dynamic Structure Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 626

16.3 Heuristic Considerations on Diffusion Processes . . . . . . . . . . . . . . . 62816.3.1 Very Dilute Colloidal Systems . . . . . . . . . . . . . . . . . . . . . . . . 62916.3.2 Diffusion Mechanisms in Concentrated Colloidal Systems . 636

16.4 Fluorescence Techniques for Long-Time Self-Diffusion ofNon-Spherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66016.4.1 Fluorescence Recovery After Photobleaching . . . . . . . . . . . . 66116.4.2 Fluorescence Correlation Spectroscopy . . . . . . . . . . . . . . . . . 669

16.5 Theoretical and Experimental Results on Diffusion of ColloidalSpheres and Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67516.5.1 Colloidal Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67616.5.2 Polymer Blends and Random Phase Approximation . . . . . 697


Contents In Detail XXIII

17 Field-Assisted Diffusion Studied by Electrophoretic NMRManfred Holz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71717.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71717.2 Principles of Electrophoretic NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . 719

17.2.1 Electrophoresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71917.2.2 Pulsed Field Gradient NMR for the Study of Drift

Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72017.3 NMR in Presence of an Electric Direct Current. Technical

Requirements, Problems and Solutions . . . . . . . . . . . . . . . . . . . . . . . 72517.4 ENMR Sample Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72717.5 ENMR Experiments (1D, 2D and 3D) and Application Examples 728

17.5.1 1D ENMR Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72917.5.2 2D and 3D Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73417.5.3 Mobility and Velocity Distributions. Polydispersity and

Electro-Osmotic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73717.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741

Part IV Theoretical Concepts and Models

18 Diffusion of Particles on LatticesKlaus W. Kehr, Kiaresch Mussawisade, Gunter M. Schutz, ThomasWichmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74518.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74518.2 One Particle on Uniform Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748

18.2.1 The Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74818.2.2 Solution of the Master Equation . . . . . . . . . . . . . . . . . . . . . . 74918.2.3 Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75118.2.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752

18.3 One Particle on Disordered Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 75318.3.1 Models of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75318.3.2 Exact Expression for the Diffusion Coefficient in d = 1 . . . 75518.3.3 Applications of the Exact Result . . . . . . . . . . . . . . . . . . . . . . 75718.3.4 Frequency Dependence in d = 1: Effective-Medium

Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75818.3.5 Higher-Dimensional Lattices: Approximations . . . . . . . . . . . 76218.3.6 Higher-Dimensional Lattices: Applications . . . . . . . . . . . . . . 76618.3.7 Remarks on Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 769

18.4 Many Particles on Uniform Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 77118.4.1 Lattice Gas (Site Exclusion) Model . . . . . . . . . . . . . . . . . . . . 77118.4.2 Collective Diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77218.4.3 Tracer Diffusion for d > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77318.4.4 Tagged-Particle Diffusion on a Linear Chain . . . . . . . . . . . . 774

18.5 Many Particles on Disordered Lattices . . . . . . . . . . . . . . . . . . . . . . . 778

XXIV Contents In Detail

18.5.1 Models with Symmetric Rates . . . . . . . . . . . . . . . . . . . . . . . . 77818.5.2 Selected Results for the Coefficient of Collective

Diffusion in the Random Site-Energy Model . . . . . . . . . . . . 78018.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78318.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784

18.7.1 Derivation of the Result for the Diffusion Coefficient forArbitrarily Disordered Transition Rates . . . . . . . . . . . . . . . . 784

18.7.2 Derivation of the Self-Consistency Condition for theEffective-Medium Approximation . . . . . . . . . . . . . . . . . . . . . 787

18.7.3 Relation Between the Relative Displacement and theDensity Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790

19 Diffusion on FractalsUwe Renner, Gunter M. Schutz, Gunter Vojta . . . . . . . . . . . . . . . . . . . . . . 79319.1 Introduction: What a Fractal is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79319.2 Anomalous Diffusion: Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . 79719.3 Stochastic Theory of Diffusion on Fractals . . . . . . . . . . . . . . . . . . . . 80219.4 Anomalous Diffusion: Dynamical Dimensions . . . . . . . . . . . . . . . . . 80319.5 Anomalous Diffusion and Chemical Kinetics . . . . . . . . . . . . . . . . . . 80619.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810

20 Ionic Transport in Disordered MaterialsArmin Bunde, Wolfgang Dieterich, Philipp Maass, Martin Meyer . . . . . 81320.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81320.2 Basic Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816

20.2.1 Tracer Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81620.2.2 Dynamic Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81720.2.3 Probability Distribution and Incoherent Neutron

Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81720.2.4 Spin-Lattice Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818

20.3 Ion-Conducting Glasses: Models and Numerical Technique . . . . . . 81920.4 Dispersive Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82220.5 Non-Arrhenius Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83220.6 Counterion Model and the Nearly Constant Dielectric Loss

Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83520.7 Compositional Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83920.8 Ion-Conducting Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843

20.8.1 Lattice Model of Polymer Electrolytes . . . . . . . . . . . . . . . . . 84320.8.2 Diffusion through a Polymer Network: Dynamic

Percolation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84620.8.3 Diffusion in Stretched Polymers . . . . . . . . . . . . . . . . . . . . . . . 849


Contents In Detail XXV

21 Concept of Mismatch and Relaxation for Self-Diffusionand Conduction in Ionic Materials with Disordered StructureKlaus Funke, Cornelia Cramer, Dirk Wilmer . . . . . . . . . . . . . . . . . . . . . . . 85721.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85721.2 Conductivity Spectra of Ion Conducting Materials . . . . . . . . . . . . . 86121.3 Relevant Functions and Some Model Concepts for Ion Transport

in Disordered Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86421.4 CMR Equations and Model Conductivity Spectra . . . . . . . . . . . . . . 86721.5 Scaling Properties of Model Conductivity Spectra . . . . . . . . . . . . . 87121.6 Physical Concept of the CMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87421.7 Complete Conductivity Spectra of Solid Ion Conductors . . . . . . . . 87721.8 Ion Dynamics in a Fragile Supercooled Melt . . . . . . . . . . . . . . . . . . 88021.9 Conductivities of Glassy and Crystalline Electrolytes Below

10MHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88321.10 Localised Motion at Low Temperatures . . . . . . . . . . . . . . . . . . . . . . . 88721.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892

22 Diffusion and Conduction in Percolation SystemsArmin Bunde, Jan W. Kantelhardt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89522.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89522.2 The (Site-)Percolation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89522.3 The Fractal Structure of Percolation Clusters near pc . . . . . . . . . . 89722.4 Further Percolation Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90122.5 Diffusion on Regular Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90322.6 Diffusion on Percolation Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90422.7 Conductivity of Percolation Clusters . . . . . . . . . . . . . . . . . . . . . . . . . 90522.8 Further Electrical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90622.9 Application of the Percolation Concept: Heterogeneous Ionic

Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90822.9.1 Interfacial Percolation and the Liang Effect. . . . . . . . . . . . . 90822.9.2 Composite Micro- and Nanocrystalline Conductors . . . . . . 910


23 Statistical Theory and Molecular Dynamics of Diffusionin ZeolitesReinhold Haberlandt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91523.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91523.2 Some Notions and Relations of Statistical Physics . . . . . . . . . . . . . 916

23.2.1 Statistical Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 91623.2.2 Statistical Theory of Irreversible Processes . . . . . . . . . . . . . 919

23.3 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92223.3.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92223.3.2 Procedure of an MD Simulation . . . . . . . . . . . . . . . . . . . . . . . 923

XXVI Contents In Detail

23.3.3 Methodical Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92523.4 Simulation of Diffusion in Zeolites . . . . . . . . . . . . . . . . . . . . . . . . . . . 925

23.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92523.4.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92623.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928


List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955

Part I

Solids

1 Diffusion: Introduction and Case Studies in

Metals and Binary Alloys

Helmut Mehrer

1.1 Introduction

Diffusion in solids is an important topic of physical metallurgy and materialsscience. Diffusion processes play a key role in the kinetics of many microstruc-tural changes that occur during processing of metals, alloys, ceramics, semi-conductors, glasses, and polymers. Typical examples are nucleation of newphases, diffusive phase transformations, precipitation and dissolution of asecond phase, recrystallization, high-temperature creep, and thermal oxida-tion. Direct technological applications concern, e.g., diffusion doping duringfabrication of microelectronic devices, solid electrolytes for battery and fuelcells, surface hardening of steel through carburization or nitridation, diffusionbonding, and sintering.

The atomic mechanisms of diffusion in crystalline materials are closelyconnected with defects. Point defects such as vacancies or interstitials are thesimplest defects and often mediate diffusion. Dislocations, grain boundaries,phase boundaries, and free surfaces are other types of defects of crystallinesolids. They can act as diffusion short circuits, because the mobility of atomsalong such defects is usually much higher than in the lattice.

This chapter will concentrate on bulk diffusion in solid metals and alloys.Most of the solid elements are metals. Furthermore, diffusion properties andatomic mechanisms of diffusion have most thoroughly been investigated inmetallic solids. On the other hand, many of the physical concepts, which havebeen developed for metals, apply to diffusion in all crystalline solids. Thoseeffects, which are unique to non-metallic systems such as charge effects inionic crystals and semiconductors, are treated in Chaps. 4 and 5.

For a comprehensive treatment of diffusion in solid matter the reader isreferred to the textbooks of Shewmon [1], Philibert [2], Heumann [3], Allnattand Lidiard [4], and Glicksman [5]. A critical collection of data for diffusion inmetals and alloys was edited in 1990 by Mehrer [6]. Recent developments canbe found in the proceedings of a series of international conferences on Diffu-sion in Materials [79]. The field of grain- and interphase-boundary diffusionis described in Chap. 8 and in the book of Kaur, Mishin and Gust [10]. Thebook of Crank [11] provides an excellent introduction to the mathematics ofdiffusion.

4 Helmut Mehrer

1.2 Continuum Description of Diffusion

1.2.1 Ficks Laws for Anisotropic Media

The diffusion of atoms through a solid can be described by Ficks equations.The first equation relates the diffusion flux j (number of atoms crossing aunit area per second) to the gradient of the concentration c (number of atomsper unit volume) via

j = Dc. (1.1)The quantity D is denoted as diffusion coefficient tensor or as diffusivitytensor. The dimensions of its components are length2 time1. Its SI unitsare [m2s1]. Equation (1.1) implies that D varies with direction. In generalthe diffusion flux and the concentration gradient are not always antiparallel.They are antiparallel for an isotropic medium.

For anisotropic media and non-cubic crystalline solids D is a symmetrictensor of rank 2 [12]. Each symmetric second rank tensor can be reduced todiagonal form. The diffusion flux is antiparallel to the concentration gradientonly for diffusion along the orthogonal principal directions. If x1, x2, x3 denotethese directions and j1, j2, j3 the pertaining components of the diffusion flux,(1.1) can be written as

j1 = DIc

x1, j2 = DII

c

x2, j3 = DIII

c

x3, (1.2)

where DI , DII , DIII denote the three principal diffusivities. The diffusioncoefficient for a direction (1, 2, 3) is obtained from

D(1, 2, 3) = 21DI + 22DII +

23DIII , (1.3)

where i denote the direction cosine of the diffusion flux with axis i. Equation(1.3) shows that anisotropic diffusion is completely described by the threeprincipal diffusion coefficients.

For uniaxial (hexagonal, tetragonal, trigonal) crystals and decagonal qua-sicrystals with the unique axis parallel to the x3 axis we have DI = DII =DIII and (1.3) reduces to

D() = DI sin2 + DIII cos2, (1.4)

where denotes the angle between diffusion direction and crystal axis. Forisotropic media such as amorphous metals and inorganic glasses, cubic crys-tals and icosahedral quasicrystals

DI = DII = DIII D. (1.5)

Then the diffusion coefficient tensor is reduced to a scalar quantity.Steady state methods for measuring diffusion coefficients, like the perme-

ation method [3], are directly based on Ficks first law. In non-steady state

1 Diffusion: Introduction and Case Studies in Metals and Binary Alloys 5

situations diffusion flux and concentration vary with time t and position x.In addition to Ficks first law a balance equation is necessary. For particleswhich undergo no reactions (no chemical reaction, no reactions between dif-ferent types of sites in a crystal, etc.) this is the continuity equation

c

t+ j = 0. (1.6)

Combining (1.1) and (1.6) leads to Ficks second equation (also called diffu-sion equation)

c

t= (Dc). (1.7)

1.2.2 Ficks Second Law for Constant Diffusivity

In diffusion studies with trace elements very tiny amounts of the diffusingspecies can be applied. Then the composition of the sample during the inves-tigation does practically not change (see also Sect. 1.3) and D is independentof the tracer concentration. Also diffusion in ideal solutions is described by aconcentration-independent diffusion coefficient. Then for diffusion in a certaindirection x (1.7) reduces to

c

t= D

2c

x2. (1.8)

From a mathematical point of view (1.8) is a second order, linear partialdifferential equation. Initial and boundary conditions are necessary for par-ticular solutions of this equation. Solutions can be found in the book of Crankfor a variety of initial and boundary conditions [11]. These solutions permitthe determination of D from measurements of the concentration distributionas a function of position and time. We consider two simple examples which,however, are often relevant for the analysis of experiments.

Thin-Film Solution

If a thin layer of the diffusing species (M atoms per unit area) is concentratedat x = 0 of a semi-infinite sample, the concentration after time t is describedby

c(x, t) =MDt

exp( x

2

4Dt

). (1.9)

The quantityDt, called diffusion length, is a characteristic distance for

diffusion problems. The experimental determination of diffusion coefficientsby the tracer method discussed in Sect. 1.4 is based on (1.9). It is applicableifDt is much larger than the initial layer thickness.

6 Helmut Mehrer

Error Function Solution

Suppose that at t = 0 the concentration of the diffusing species is c(x, 0) = 0for x > 0. Then, if for t > 0 the concentration at x = 0 is maintained atc(0, t) = cs, the appropriate solution of (1.8) is

c(x, t) = cserfc( x

2Dt

), (1.10)

where the complementary error function is defined by

erfc z 1 erf z (1.11)

and

erf z 2

z0

exp (u2) du (1.12)

denotes the error function. Equation (1.10) describes the in-diffusion of a dif-fuser (or diffusant) into a semi-infinite sample with constant concentrationcs of that species at the surface. It is, e.g., applicable to the in-diffusion ofa volatile solute into a non-volatile solvent, to the carburization of a metalin a carbon containing ambient, and to in-diffusion of a solute from an inex-haustible diffusion source into a solvent with solubility cs.

1.2.3 Ficks Second Law for Concentration-Dependent Diffusivity

Let us consider a case of great practical importance, in which the chemicalcomposition during diffusion varies over a certain concentration range. Dif-fusing particles will experience different chemical environments and hencedifferent diffusion coefficients. This situation is denoted as interdiffusion oras chemical diffusion. We use the symbol D to indicate that the diffusion co-efficient is concentration dependent. D is denoted as interdiffusion coefficientor as chemical diffusion coefficient. Ficks second law (1.7) for diffusion in acertain direction x then reads

c

t=

x

(D(c)

c

x

)= D(c)

2c

x2+

dD(c)dc

(c

x

)2. (1.13)

Theoretical models which permit the calculation of the composition depen-dent diffusivity from deeper principles using, e.g., statistical mechanics arenowadays still not broadly available. Then the strategy illustrated in theprevious section of calculating the concentration field for certain initial andboundary conditions is not applicable. Instead, it is possible to invert thisstrategy and to determine the interdiffusion coefficient from a given concen-tration field by using (1.13). The classical Boltzmann-Matano method forextracting the concentration-dependent diffusivity D(c) from experimentalconcentration-depth profiles will be considered in Sect. 1.11 of this chapter.


1.3 The Various Diffusion Coefficients

Diffusion in materials is characterized by several diffusion coefficients. Inthis section we describe various experimental situations which entail differ-ent diffusion coefficients. We will, however, concentrate on bulk diffusion inunary and binary systems. Diffusion in ternary systems produces mathemat-ical complexities which are beyond the scope of this chapter. We will focuson bulk diffusion since diffusion along grain boundaries and along surfacesis treated in Chaps. 7 and 8 of this book. In this section we will distinguishthe various diffusion coefficients by lower and upper indices. We will drop theindices in the following sections again, whenever it is clear which diffusioncoefficient is meant.

1.3.1 Tracer Diffusion Coefficients

In diffusion studies with trace elements (tagged by their radioactivity orby their isotopic mass) tiny amounts of the diffusing species can be used.Although there will be a gradient in the concentration of the trace element,its total concentration can be kept so small that the overall composition ofthe sample during the investigation does practically not change1. In suchcases a constant tracer diffusion coefficient is appropriate for the analysis ofthe experiments.

Self-Diffusion Coefficient

If the diffusion of A-atoms in a solid element A is studied, one speaks ofself-diffusion. Studies of self-diffusion utilize a tracer isotope A of the sameelement. A typical initial configuration for a tracer self-diffusion experimentis illustrated in Fig. 1.1a. If the applied tracer layer is very thin as comparedto the average diffusion length, the tracer self-diffusion coefficient DA

A is

obtained from such an experiment.The connection between the macroscopically defined tracer self-diffusion

coefficient and the atomistic picture of diffusion is the famous Einstein-Smoluchowski relation discussed in detail in Sect. 1.6. In simple cases it reads

DA

A = fDE with DE =l2

6, (1.14)

where l denotes the jump length and the mean residence time of an atomon a certain site of the crystal2. The quantity f is the correlation factor. Forself-diffusion in cubic crystals f is a numeric factor. Its value is characteristic1 From an atomistic viewpoint this implies that a tracer atom is not influenced by

other tracer atoms.2 Equation (1.14) considers only the simplest case: cubic structure, all sites are

energetically equivalent, only jumps to nearest neighbours are allowed.

8 Helmut Mehrer

of the lattice geometry and the diffusion mechanism (see Sect. 1.6). In sometextbooks the quantity DE is denoted as the Einstein diffusion coefficient3.

In a homogeneous binary AXB1X alloy or compound two tracer diffusioncoefficients for both, A and B tracer atoms, can be measured. A typicalexperimental starting configuration is displayed in Fig. 1.1b. We denote thetracer diffusion coefficients by DA

AXB1X and D

BAXB1X . Both tracer diffusion

coefficients will in general be different:

DA

AXB1X = DBAXB1X . (1.15)

This diffusion asymmetry depends on the crystal structure of the materialand on the atomic mechanisms which mediate diffusion. Both diffusivities,of course, also depend on temperature and composition of the alloy or com-pound and for anisotropic media on the direction of diffusion. Self-diffusionin metallic elements will be considered in Sect. 1.8. Self-diffusion in binaryintermetallics is the subject of Sect. 1.10.

Impurity Diffusion Coefficient

When the diffusion of a trace solute C in a monoatomic solvent A or ina homogeneous binary solvent AXB1X (Fig. 1.1) is measured, the tracerdiffusion coefficients

DC

A and DCAXB1X

are obtained. These diffusion coefficients are denoted as impurity diffusioncoefficients or sometimes also as foreign atom diffusion coefficients.

1.3.2 Chemical Diffusion (or Interdiffusion) Coefficient

So far we have considered in this section cases where the concentration gra-dient is the only cause for the flow of matter. We have seen that such situ-ations can be studied using tiny amounts of trace elements in an otherwisehomogeneous material. However, from a general viewpoint a diffusion flux isproportional to the gradient of the chemical potential.

The chemical potential of a species i in a binary alloy is given by (cf.Chap. 5)

i =(G

ni

)p,T,nj =i

i = A,B (1.16)

In (1.16) G denotes Gibbs free energy, ni the number of moles of species i,T the temperature, and p the hydrostatic pressure. The chemical potential3 This notation is a bit misleading, since the original Einstein-Smoluchowski re-

lation relates the total macroscopic mean square displacement of atoms to thediffusion coefficient (see Sect. 1.6), in which correlation effects are included.


Fig. 1.1. Initial configurations for diffusion experiments:a) Thin layer of A on A: tracer diffusion in pure elements.b) Thin layer of A or B on homogeneous alloy: tracer diffusion of alloy compo-nents.c) Thin layer of C on element A or homogeneous alloy: Impurity diffusion.d) Diffusion couple between metal-hydrogen alloy and a pure metal.e) Diffusion couple between pure end-members.f) Diffusion couple between two homogeneous alloys.

depends on the alloy composition. For ideal solutions the chemical potentialsare

i = 0i +RT lnni

nA + nB, (1.17)

where 0i depend on T and p only. In this case the gradient of the chemicalpotential is directly proportional to the logarithmic gradient of the concen-tration. In non-ideal solutions the gradient of the chemical potential givesrise to an internal driving force. As a consequence the interdiffusion diffu-sion coefficient is concentration-dependent and Ficks equation in the formof (1.13) must be used.

Examples of diffusion couples which entail an interdiffusion coefficient are(see Fig. 1.1):(i) Pure end-member diffusion couples consisting of two samples of pure ele-ments joined together (e.g. Ni/Pd, Cu/Ag, ...).(ii) Incremental diffusion couples consisting of two samples of homogenousalloys joined together (e.g. Ni50Pd50/Ni70Pd30, Ni/Ni70Pd30, ...).(iii) Diffusion couples which involve solutions of hydrogen in a metal (e.g. Pd-H/Pd, Ag1XHX/Ag1Y HY , ...). Binary metal-hydrogen systems are oftenspecial in the sense that hydrogen is the only mobile component.

Interdiffusion results in a composition gradient in the diffusion zone. Inter-diffusion profiles are analysed by the Boltzmann-Matano method or relatedprocedures. This method will be described in Sect. 1.11. It permits to deducethe concentration dependence of the interdiffusion coefficient

10 Helmut Mehrer

D = D(c) (1.18)

from the experimental diffusion profile.

1.3.3 Intrinsic Diffusion Coefficients

The intrinsic diffusion coefficients (sometimes also component diffusion coeffi-cients) DA and DB of a binary A-B alloy describe diffusion of the componentsA and B relative to the lattice planes. The diffusion rates of A and B atomsare usually not equal. Therefore, in an interdiffusion experiment a net flux ofatoms across any lattice plane exists. The shift of lattice planes with respectto a sample fixed axis is denoted as Kirkendall effect, which is illustratedin Fig. 1.27 in Sect. 1.11. The Kirkendall shift can be observed by incorpo-rating inert markers at the initial interface of a diffusion couple. This shiftwas for the first time observed for Cu/Cu-Zn diffusion couples by Kirkendalland coworkers [13]. In the following decades work on many different alloysystems and a variety of markers demonstrated that the Kirkendall effect isa widespread phenomenon of interdiffusion.

The intrinsic diffusion coefficients DA and DB of a substitutional binaryA-B alloy are related to the interdiffusion coefficient D and the marker veloc-ity vK (Kirkendall velocity). These relations were deduced for the first timeby Darken [14] and refined later on by Manning [15]. They will be discussedin Sect. 1.11. If the quantities D and vK are known from experiment theintrinsic diffusion coefficients can be deduced.

We emphasize that the intrinsic diffusion coefficients and the tracer diffu-sion coefficients are different. DA andDB pertain to diffusion in a compositiongradient whereas DA

AB and D

BAB are determined in a homogenous alloy. They

are approximately related (see Sect. 1.11 for details) via

DA = DA

AB and DB = DBAB, (1.19)

where denotes the so-called thermodynamic factor (see Sect. 1.11). In ametal-hydrogen system ususally only the H atoms are mobile. Then the in-trinsic diffusion coefficient and the chemical diffusion coefficient of hydrogenare identical.

1.4 Experimental Methods

Methods for the measurement of diffusion coefficients can be grouped into twomajor categories: Direct methods are based on Ficks laws and the phenom-enological definition of the diffusion coefficients given in Sect. 1.3. Indirectmethods are not based directly on Ficks laws. Their interpretation requires amicroscopic model of the atomic jump processes and then uses the Einstein-Smoluchowski relation to deduce a diffusion coefficient.


1.4.1 Direct Methods

Tracer Method

The tracer method is the most direct and most accurate technique for thedetermination of diffusion coefficients in solids. Since very tiny amounts oftrace isotopes can be applied in tracer experiments, the chemical composi-tion of the sample is practically not influenced by the tracer. In this wayself-diffusion and impurity diffusion can be studied in a material which ishomogeneous apart from the tracer gradient.

As indicated schematically in Fig. 1.2 the tracer is usually deposited ontoa polished, flat surface of the diffusion sample. Normally a radioactive isotopeof the investigated atomic species is used as tracer. Enriched stable isotopeshave also been used in a few cases. Evaporation, dripping of a liquid solution,and electrodeposition of the tracer onto the surface are common depositiontechniques. Sometimes the tracer is ion-implanted as a thin layer below thesample surface in order to overcome disturbing surface oxide hold-up and sol-ubility problems [16]. The sample is usually encapsulated in quartz ampoulesunder vacuum or inert (e.g. Ar) atmosphere and an isothermal diffusion an-neal is performed at temperature T for some diffusion time t. For tempera-tures below 1500 K quartz ampoules and resistance furnaces can be used. Forhigher temperatures more sophisticated heating techniques are necessary.

The best way to determine the resulting concentration-depth profile isserial sectioning of the sample and subsequent determination of the amountof tracer per section.

For average diffusion lengths of at least several ten micrometers mechan-ical sectioning techniques are applicable (for a review see, e.g., [17]). Lathesand microtomes are appropriate for ductile, grinding devices for brittle sam-ples. For extended diffusion anneals and diffusivities D > 1015 m2s1 lathesectioning is sufficient whereas diffusivities D > 1017 m2s1 are accessibleby microtome sectioning. In favourable cases, grinder sectioning can be usedfor diffusivities D > 1018 m2s1.

x2

ln c

slope: 14 D t

Fig. 1.2. Schematic illustration of the tracer method: The major steps depositionof the tracer, diffusion anneal, serial sectioning, and evaluation of the penetrationprofile are indicated.

12 Helmut Mehrer

Diffusion studies at lower temperatures require measurements of verysmall diffusivities. Measurements of diffusion profiles with diffusion lengthsin the micrometer or nanometer range are possible using sputter section-ing techniques. Devices for serial sectioning of radioactive diffusion samplesby ion-beam-sputtering are described in [18, 19]. Such devices permit serialsectioning of shallow diffusion zones, which correspond to average diffusionlengths between several nm and a few m. This implies that for anneal-ing times of about 106 s a diffusivity range between D 1024 m2s1 andD 1016 m2s1 can be examined.

Provided that the experimental conditions were chosen in such a waythat the deposited layer is thin compared with the mean diffusion length,the distribution after the diffusion anneal is described by (1.9). If radioactivetracers are used, the specific activity per section (count rate divided by thesection mass) is proportional to the tracer concentration. The count rateis conveniently determined by nuclear counting facilities (- or -counting,depending on the isotope). According to (1.9) a plot of the logarithm ofthe specific activity versus the penetration distance squared is linear, if bulkdiffusion is the dominating diffusion process. Its slope equals (4Dt)1.From the slope and the diffusion time the tracer diffusivity D is obtained.An obvious advantage of the tracer method is that a determination of theabsolute tracer concentration is not necessary.

Fig. 1.3 shows a penetration profile of the radioisotope 59Fe in the inter-metallic phase Fe3Si obtained by grinder sectioning [20]. Gaussian behaviouras stated by (1.9) is observed over several orders of magnitude in concentra-tion. An example for a penetration profile of 59Fe in the intermetallic phaseFe3Al obtained with the sputtering device described in [18] is displayed inFig. 1.4 according to [21]. From diffusion profiles of the quality of Figs. 1.3and 1.4 diffusion coefficients can be determined with an accuracy of a fewpercent.

Deviations from the Gaussian behaviour in experimentally determinedpenetration profiles may occur for many reasons. We mention two frequentones:

Grain boundaries in a polycrystal often act as diffusion short-circuits withenhanced mobility of atoms. Grain boundaries usually cause a grain-boundary tail in the deeper penetrating part of the profile. In this tailregion the concentration of the diffuser is enhanced with respect to merebulk diffusion.

Evaporation losses of the tracer itself and/or of the diffusion sample willcause deviations from Gaussian behaviour in the near-surface region.

For a more detailed discussion of implications and pitfalls of the tracermethod the reader is referred to [17]. The grain-boundary tails mentionedabove can be used for a systematic study of grain-boundary diffusion in bi-or polycrystals as described in Chap. 8 and in [10].


Fig. 1.3. Penetration profile of the ra-dioisotope 59Fe in Fe3Si obtained bygrinder sectioning. The solid line rep-resents a fit of the thin-film solution ofFicks second law to the data points.

Fig. 1.4. Penetration profile of the ra-dioisotope 59Fe in Fe3Al obtained bysputter sectioning. The solid line rep-resents a fit of the thin-film solution ofFicks second law to the data points.

In some cases several tracer isotopes of the same element are available.Differences between the isotopic masses lead to isotope effects in diffusion.Isotope effects are interesting phenomena although the differences betweendiffusivities of two isotopes of the same element are usually a few percentonly. An exception is hydrogen with its three isotopes H, D, and T, whichhave significantly different masses (see Sect. 1.7.2). Isotope effects of self- andsolute-diffusion in metals can contribute useful information about the atomicmechanisms of diffusion. For a detailed discussion the reader is referred to [2224] (see also Chap. 6, Sect. 6.5.3).

Other Profiling and Detection Methods

Several other profiling and detection methods can be used to measureconcentration-depth profiles. We mention the more important ones:

1. Secondary Ion Mass Spectrometry (SIMS)As already mentioned foreign elements or stable isotopes of the matrixcan be used as tracers in combination with SIMS for depth profiling. SIMSis mainly appropriate for the diffusion of foreign elements. Contrary toself-diffusion studies by radiotracer experiments, in the case of stabletracers the natural abundance of the stable isotope in the matrix limitsthe concentration range of the diffusion profile. Highly enriched isotopesshould be used. An example of this technique can be found in a recent

14 Helmut Mehrer

Fig. 1.5. Interdif-fusion profile of aFe70Al30Fe50Al50couple measured byEMPA.

study of Ni self-diffusion in the intermetallic compound Ni3Al in whichthe highly enriched stable 64Ni isotope was used [25]. Average diffusionlengths between several nm and several m are accessible.

2. Electron Microprobe Analysis (EMPA)In EMPA an electron beam of several tens of keV with a diameter ofabout one micrometer stimulates X-ray emission in the diffusion zoneof the sample. The diffusion profile can be obtained by analysing theintensity of the characteristic radiation of the elements in a line scan alongthe diffusion direction. The detection limit is about 103 to 104 molefractions depending on the element. Light elements cannot be analysed.Because of the finite size of the excited volume (several m3) only fairlylarge diffusion coefficients 1015m2s1 can be measured. EMPA ismainly appropriate for interdiffusion studies. An example of a single-phase interdiffusion profile resulting from a Fe70Al30Fe50Al50 couple isshown in Fig. 1.5 according to [26]. The analysis of interdiffusion profilesis discussed in Sect. 1.11.

3. Auger Electron Spectroscopy (AES)AES in combination with sputter profiling can be used to measure dif-fusion profiles in the range of several nm to several m. It is, however,only applicable to diffusion of foreign atoms since AES only discriminatesbetween different elements.

4. Rutherford Backscattering Spectrometry (RBS)In RBS experiments a high-energy beam of monoenergetic -particles isused. These particles are preferentially scattered by heavy nuclei in thesample and the energy spectrum of scattered -particles can be used todetermine the concentration-depth distribution of scattering nuclei. Thistechnique is mainly suitable for detecting heavy elements in a matrix of


substantially lower atomic weight. Due to the energy straggling of theincident beam the profile depth is limited to less than a few m.

5. Nuclear Reaction Analysis (NRA)High energy particles can also be used to study diffusion of light elements,if the nuclei undergo a suitable resonant nuclear reaction. An example isdiffusion of boron in an alloy. During irradiation with high energy pro-tons -particles are emitted from the nuclear reaction 11B + p 8B + .The concentration profile of 11B can be determined from the number andenergy of emitted -particles as a function of the incident proton energy.Like in RBS energy straggling limits the depth resolution of NRA.RBS and NRA methods need a depth calibration which is based on notalways very accurate data for the stopping power in the matrix for thoseparticles emitted by the nuclear reaction. Also the depth resolution is usu-ally inferior to that achievable in careful radiotracer and SIMS profilingstudies.

6. Field Gradient Nuclear Magnetic Resonance (FG NMR, PFG NMR)Nuclear magnetic resonance (NMR) measurements in a magnetic fieldgradient (FG) or in a pulsed field gradient (PFG, see Chap. 10) providea direct macroscopic method for diffusion studies. In a magnetic fieldgradient the Larmor frequency of a nuclear moment depends on its po-sition. FG NMR and PFG NMR utilize the fact that nuclear spins thatdiffuse in a magnetic field gradient experience an irreversible phase shift,which leads to a decrease in transversal magnetization. This decrease canbe observed in so-called spin-echo experiments [27, 28]. A measurementof the diffusion-related part of the spin echo provides the diffusion co-efficient without any further hypothesis. In contrast to tracer diffusion,FG NMR and PFG NMR techniques permit diffusion measurements inisotopically pure systems. These techniques are applicable for relativelylarge diffusion coefficients D 1013 m2s1 [29].

1.4.2 Indirect Methods

Indirect methods are based on phenomena which are influenced by the dif-fusive jumps of atoms. Some of these methods are often sensitive to one ora few atomic jumps only. Quantities like relaxation times, relaxation ratesor linewidths are measured. Using a microscopic model of the jump processthe mean residence time of the diffusing species is determined and then viathe Einstein-Smoluchowski relation (see Sect. 1.6) the diffusivity is deduced.Indirect methods can be grouped into two categories relaxation methods(mechanical and magnetic) and nuclear methods.

16 Helmut Mehrer

Relaxation Methods (Mechanical and Magnetic)

Mechanical relaxation methods make use of the fact that atomic motion ina material can be induced by external influences such as the applicationof constant or oscillating mechanical stress. In ferromagnetic materials theinteraction between the magnetic moments and local order can give rise tovarious relaxation phenomena similar to those observed in anelasticity. Agreat variety of experimental devices have been used for such studies. Theirdescription is, however, beyond the scope of this chapter.

Some of the more important relaxation phenomena related to diffusionare the following [3032]:

The Snoek effect is observed in bcc metals which contain interstitialsolutes such as C, N, or O. These solutes occupy octahedral or tetrahedralinterstitial sites. These sites have tetragonal symmetry, which is lower thanthe cubic symmetry of the matrix. Therefore the lattice distortions caused byinterstitial solutes give rise to elastic dipoles. Under the influence of externalstress these dipoles can reorient (para-elasticity). The reorientation of solutesgives rise to a strain relaxation or an internal friction peak. The relaxationtime or the (frequency or temperature) position of the internal friction peakcan be used to deduce information about the mean residence time of a solute.A Snoek effect of interstitial solutes in fcc metals cannot be observed, becausethe interstitial sites have cubic symmetry.

The Gorski effect is due to solutes in a solvent which produce a latticedilatation. In a macroscopic strain gradient solutes redistribute by diffusion.This redistribution gives rise to an anelastic relaxation. The Gorski effectis detectable if the diffusion coefficient of the solute is high enough. Gorskieffect measurements have been widely used for studies of hydrogen diffusionin metals [30].

In substitutional A-B alloys the reorientation of solute-solvent pairs underthe influence of stress can give rise to an anelastic relaxation called Zenereffect.

Nuclear Methods

Examples of nuclear methods are NMR, Mobauer spectroscopy (MBS), andquasielastic neutron scattering (QENS). Since MBS, QENS, NMR and PFGNMR are the subjects of the Chaps. 2, 3, 9 and 10 and QENS also of a recenttextbook [33] we confine ourselves here to a few remarks:

The width of the resonance line and the spin-lattice relaxation rate T11in NMR have contributions which are due to the thermally activated jumpsof atoms. Measurements of the diffusional narrowing of the linewidth orof T11 as a function of temperature permit a determination of the meanresidence time of the atoms. NMR methods are mainly appropriate forself-diffusion measurements on solid or liquid metals. In favourable cases (e.g.Li and Na) self-diffusion coefficients between 1018 and 1010 m2s1 are


accessible (see [6]). In the case of foreign atom diffusion, NMR studies sufferfrom the fact that a signal from nuclear spins of the minority component mustbe detected. Nevertheless, detailed studies were conducted, e. g., in the caseof Li-based Li-Mg and Li-Ag alloys via the spin-lattice relaxation of polarizedradioactive 8Li nuclei [34].

The linewidth in MBS and QENS both have a contribution which isdue to the diffusion jumps of atoms. This diffusion broadening can be ob-served only in systems with fairly high diffusivities since must at leastbe comparable with the natural linewidth in MBS experiments or with theenergy resolution of the neutron spectrometer in QENS experiments. Appro-priate probes for MBS must be available. The usual working horse in MBSis the isotope 57Fe although there are a few other Mobauer isotopes avail-able (e.g. 119Sn, 115Eu, 161Dy). MBS has been mainly used to study fast Fediffusion. QENS experiments are suitable for fast diffusing elements with asizable incoherent scattering cross section for neutrons. Examples are hydro-gen diffusion in metals or hydrides and Na self-diffusion (see Chap. 3).

Neither MBS nor QENS are routine methods for diffusion measurements.The most interesting aspect is that these methods can provide microscopicinformation about the elementary jump process of atoms. For single crystals depends on the crystal orientation. This orientation dependence can beused to deduce information about the jump direction and about the jumplength (see Chaps. 2 and 3), which is not accessible by conventional diffusionstudies.

For a more comprehensive discussion of experimental methods for thedetermination of diffusion coefficients we refer the reader to the already men-tioned textbooks on diffusion [13] and to Chap. 1 in [6] as well as to a recentarticle [29] where also an overview of the accessible windows for the meanresidence time are given.

1.5 Dependence of Diffusion on ThermodynamicVariables

So far we have said nothing about the dependence of diffusion processes uponthermodynamic variables, i.e. upon temperature T and hydrostatic pressurep. In binary systems also variations of the diffusivity with the variable compo-sition are of interest. These variations

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Diffusion in Condensed Matter - Methods, Materials, Models