Diffusion ordered nuclear magnetic resonance spectroscopy

Diffusion ordered nuclear magnetic resonance spectroscopy:principles and applications

C.S. Johnson Jr.*

Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599-3290, USA

Received 21 October 1998

Contents

Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2041. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2052. Previous reviews of DOSY and related topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2063. The PFG-NMR experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

3.1. Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2073.1.1. Magnetic field gradients and magnetization helices. . . . . . . . . . . . . . . . . . . . . . . . . . . 2073.1.2. Bloch equations with diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

3.2. Pulse sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2083.2.1. The spin-echo (SE) sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2093.2.2. The stimulated echo (STE) sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

3.3. The FT-PFG-NMR experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2113.3.1. Component analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

4. Diffusion ordered NMR spectroscopy (DOSY). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2134.1. Experimental requirements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

4.1.1. Eddy current reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2134.1.2. Pulse sequences for minimizing effects of eddy currents andJ-modulation . . . . . . . . . . 2144.1.3. Suppression of convection current effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2174.1.4. Dispersion and resolution enhancement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2184.1.5. Data collection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2204.1.6. Utilizing the stray field to obtain large, steady gradients. . . . . . . . . . . . . . . . . . . . . . . . 222

4.2. Data inversion and display. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2224.2.1. Discrete samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2244.2.2. Polydisperse samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2294.2.3. Complete bandshape methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2324.2.4. Analysis recommendations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

5. Effects of chemical exchange. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2345.1. Exchange effects in diffusion spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2345.2. Artifacts from chemical shift encoding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

6. Applications of 1D and 2D DOSY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2396.1. Discrete samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

6.1.1. Biofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Progress in Nuclear Magnetic Resonance Spectroscopy 34 (1999) 203–256

0079-6565/99/$ - see front matterq 1999 Elsevier Science B.V. All rights reserved.PII: S0079-6565(99)00003-5

* Tel.: 11-919-966-5229; fax:11-919-962-2388.E-mail address:[email protected]

6.1.2. Separation by means of hydrophobicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2406.1.3. Equilibria involving sodium dodecylsulfate (SDS) and bovine serum albumin (BSA) . . . 2426.1.4. Mixtures of polymer additives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

6.2. Polydisperse samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2446.2.1. Phospholipid vesicles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2456.2.2. Blood plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2456.2.3. The viscoelastic CTAB/sodium salicylate/water system. . . . . . . . . . . . . . . . . . . . . . . . 2466.2.4. Molecular weight distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

7. 3D DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2497.1. COSY-DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2497.2. HMQC-DOSY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2507.3. NOESY-DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2527.4. TOCSY-DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2527.5. Merged sequences for PFG-DQS, PFG-NOESY, and PFG-TOCSY. . . . . . . . . . . . . . . . . . . . . 253

8. Future prospects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

Keywords:Diffusion ordered NMR; Pulsed magnetic field gradient-NMR experiment; Pulse sequences

Nomenclature

GLOSSARYALS (computer program)ARK* a priori knowledgeBPP* Bipolar Pulse PairsCONTIN (computer program)CORECOSY-DOSYCTPDECRADEPTDISCRETEDLSDSTEDOSY* Diffusion Ordered NMR SpectroscopYEXSYFIDFIDDLE (computer program)GCSTEGCSTESLGPC-NMRGRAM (computer algorithm)HDL (lipoprotein)HMQC-DOSYHR-DOSYHSQC

INEPTILT (inverse Laplace tranform)LDL (lipoprotein)LED* Longitudinal Eddy current Delay or

Longitudinal Encode–DecodeMaxEntMCR (computer algorithm)MOSY* Mobility Ordered NMR SpectroscopYMWDNIPALS (computer program)NLREG (computer program)NOESY-DOSYPFG-NMRPVA (computer algorithm)RDCONSESPLMODSTESTEP (computer program)VLDL (lipoprotein)VMAX (computer algorithm)CSJ is responsible for LED, DOSY, and MOSYand some of the hyphenated forms containingthem. He may have been the first to use theabbreviations BPP and APK. The others are defined inthe text.

C.S. Johnson / Progress in Nuclear Magnetic Resonance Spectroscopy 34 (1999) 203–256204

1. Introduction

One of the most fruitful ideas in NMR spectroscopywas the introduction of a second frequency dimension[1]. This was made possible through the use of pulsesequences having two independent precession peri-ods. In one class of two-dimensional NMR (2D-NMR) experiments, the Hamiltonian is switchedbetween the evolution period and the detection period.As a consequence of the evolution period, resonancesare spread into a second dimension to reveal theirorigins. Examples of such 2D ‘resolved’ spectro-scopies includeJ-resolved where the Hamiltonian isswitched through spin decoupling [2] and NMRimaging where magnetic field gradient directions areswitched [3].

A logical extension of these ideas is the introduc-tion of additional NMR dimensions that depend onmolecular properties such as size, shape, mass, andcharge that are not explicitly included in spin Hamil-tonians. These overall molecular properties are notwell represented in conventional NMR as spin inter-actions tend to be quite local. Therefore, dispersion onthe basis of such properties can provide new informa-tion as well as a means for editing NMR spectra. Theproblem is to identify ways that molecular propertiesinfluence NMR spectra or can be made to affect NMRspectra.

Nuclear relaxation times are obvious candidatesbecause they depend on correlation times for molecu-lar motion, and the correlation times in turn depend onmolecular sizes and shapes. However, relaxationtimes can be quite different for different nuclei inthe same molecule because of site specific magneticinteractions and because local or segmental motionsmay obscure overall molecular motions. In the case oflongitudinal relaxation, high frequency local segmen-tal motion may provide the dominant relaxationmechanism, leading toT1 values that are relativelyindependent of molecular mass. While for transverserelaxation, local motions may contribute motionalaveraging effects comparable to those resulting fromoverall rotation. Even so, there are cases where relaxa-tion resolved spectra of, for example, backbone13Cnuclei in rigid molecules or bilayer1H nuclei invesicles can provide useful information about sizedependent molecular reorientation.

It is clear that new NMR dimensions should be

based on molecular properties that have the sameeffect on all nuclei in a given molecule. Transportproperties of molecules and ions, as determined bydiffusion measurements and electrophoresis meetthis criterion. The connection with structural proper-ties arises because diffusion coefficients (D) dependon friction factors and electrophoretic mobilities (m )depend on both friction factors and effective charges.According to the Debye–Einstein theory [4]:

D � kBTfT

�1�

where kB is the Boltzmann constant,T is the absolutetemperature, andfT is the friction factor. For thespecial case of a spherical particle of hydrodynamicradiusrH in a solvent of viscosityh , the friction factoris given byfT � 6phrH. More realistic models forfTrepresent molecules by ellipsoids of revolution orcollections of spherical subunits [5]. Electrophoresisconcerns the terminal velocityv of a charged particlein an applied electric field,Edc. The relationshipbetween the electrophoretic mobility, defined asm � v=Edc, and molecular properties is not simple;but for small ionsm is proportional to the overallcharge,Ze, and inversely proportional tofT [6].

The implementation of transport ordered NMR ispossible because information about translationalmotion can be encoded in NMR data sets throughthe use of pulsed magnetic field gradient NMR(PFG-NMR) experiments [7]. The idea, as withconventional 2D NMR, is to increment an experimen-tal variable that modulates the detected signal andthen to transform the data with respect to that variableto produce a ‘‘spectrum’’ related in this case to mole-cular translation. Spectra based on electrophoreticmobilities are obtained by incrementingEdc and thentransforming the NMR signal amplitudes with respectEdc. This scheme is realized in mobility ordered NMRspectroscopy (MOSY) [8]. The MOSY method hasnot yet found widespread use because of the unusualinstrumentation requirements, e.g. NMR compatibleelectrophoresis cells, and the restriction to conductingsamples with low ionic strengths. However, thesuccess of the MOSY concept motivated thedevelopment of the more general diffusion basedspectroscopy.

Diffusion spectra can be obtained by incrementingthe areas of the gradient pulses (q) in PFG-NMR and

C.S. Johnson / Progress in Nuclear Magnetic Resonance Spectroscopy 34 (1999) 203–256 205

transforming the NMR signals amplitudes withrespect toq2. The result is diffusion ordered NMRspectroscopy (DOSY) [9]. The three basic DOSYrequirements are (1) distortion free absorption modedata sets acquired with precise gradient encoding, (2)effective data inversion (transformation) procedures,and (3) algorithms for the display of the diffusionspectra. These requirements turn out to be quite severebecause the signal inversion step is extremely sensi-tive to noise and distortions in the signals. This hasnecessitated significant enhancements of the originalPFG-NMR experiments and experimentation withalternative data inversion methods. Even data displayfor DOSY is not straightforward because decisionsmust be made about how to generate the spectra.The contrast with the Fourier transform NMR (FT-NMR) is striking. With FT-NMR, one has a uniquetransformation with an inverse that returns the origi-nal signal. Also, the resulting spectra are ready fordisplay.

This review is concerned with the variousimplementations of DOSY experiments and with

illustrations of the power of this technique. The imple-mentations present solutions to the unique problemsof data acquisition, transformation, and display. Withappropriate instrumentation and software, the user canbe offered menu choices for analysis methods andtypes of display. The result is a convenient NMRmethod for the analysis of mixtures that can revealunexpected components and interactions in mixturesthrough useful and appealing plots.

2. Previous reviews of DOSY and related topics

Transport ordered NMR [10] and diffusionmeasurements by magnetic field gradient methodsincluding DOSY [11] have previously been reviewed.Related reviews of MOSY are also available [12,13].A complete treatment of translational dynamics andits study by NMR can be found in the book byCallaghan [14]. Ka¨rger et al. [15] have reviewed theprinciples and applications of PFG-NMR, and Stilbshas provided a detailed review of FT diffusion studies


Fig. 1. The simple Carr–Purcell spin echo (SE) often called the Hahn echo.

[16]. The recent tutorial articles on PFG-NMR byPrice are also of interest [17,18].

3. The PFG-NMR experiment

3.1. Background

Time domain NMR dates from Hahn’s observa-tions of the free induction decay (FID), the spinecho (SE), and the stimulated echo (STE) [19,20].The effects of molecular diffusion, in the presenceof magnetic field gradients, on echo amplitudeswere evident from the beginning, and Hahn reporteda derivation of diffusion dependent signal attenuation,which he attributed to C.P. Slichter [20]. All NMRdiffusion measurements are based on the fact thatthe diffusion coefficient can be calculated from theecho attenuation if the amplitude and duration of themagnetic field gradient are known. The originalmeasurements were carried out with continuous gradi-ents, but the advantages of pulsed gradients wereconvincingly demonstrated by Stejskal and Tanner[7]. Here we review the principles of PFG-NMR anddisplay selected applications of PFG-NMR to providethe background for DOSY.

3.1.1. Magnetic field gradients and magnetizationhelices

NMR diffusion measurements can be made bymeans of either gradients in the main (dc) magneticfield, B0, or gradients in radio frequency fields (B1). Inthe following only gradients inB0 are considered. Forapplications of RF gradients the reader should consultthe review article by Canet [21] and the monograph byKimmich [22]. Here thez-direction is defined by thedirection ofB0, and we are concerned with gradientsin the z component ofB. Typically, a spatiallyconstant gradient is applied externally by means ofcurrent in a coil set, either of the Maxwell pair [23]or quadrupole type [24,25]. The resulting gradientg isdescribed by:

g� 2Bz

2xi 1

2Bz

2yj 1

2Bz

2zk �2�

where i, j and k are unit vectors in thex, y, and zdirections, respectively. Accordingly, the total

external magnetic field atr is given by

B�r� � B0 1 g·r : �3�In the following we assume that only az-gradient ofmagnitudeg� g·k is present. The purpose of thegradient is to label nuclear spins with phase anglesthat depend on their positions in space, or in thiscase their displacement in thez-direction. This is, ofcourse, possible because spins precess with the angu-lar frequency

v�r� � 2gB�r� �4�and the acquired phase angle depends linearly on bothB(r) and the duration of the gradient. Therefore, az-gradient of durationd produces the position depen-dent phase anglef�z� � 2gB�z�d.

PFG-NMR experiments involving constantz-gradi-ents can readily be visualized by imagining layers ofthe sample perpendicular to thez-axis that are thinenough to experience a uniform magnetic field butthick enough to contain a large number of spins.Each layer is associated with a magnetization vector(isochromat), and these vectors are assigned to thepositions of the layers on thez-axis. We begin theexperiment with a hard 908x RF pulse that rotates allof the vectors into they-direction to create a magne-tization ribbon in the rotating coordinate frame asshown in Fig. 1. The effect of the gradientg is thento twist the ribbon into a helix [26] defined by therelative phase anglesDf�z� � 2�ggd�z. The pitch ofthe helix is given byL � 2p=q whereq� ggd is thearea of the gradient pulse in units of m21. Thus, theeffect of a constant gradient is to produce a magneti-zation pattern with a cosinusoidal projection on theyz-plane. Diffusion in thez-direction will, of course,smear out this pattern, and the smaller the pitch themore rapidly this will happen.

3.1.2. Bloch equations with diffusionThe magnetization of uncoupled spins is well

described by the Bloch equations, and the effects ofdiffusion can easily be incorporated. As gradients inBz only affect the transverse components of magneti-zation, it is appropriate to begin with the Bloch equa-tion for the complex magnetization,M1 � Mx 1 iMy.We assume that the magnetization has been rotated tothey-direction in the rotating frame by a 908x pulse orsome other set of pulses, and that only the main field


Bz is present fort . 0. With Bx � By � 0 the Blochequation becomes [27]:

2M1

2t� 2iv0M1 2

M1

T22 ig�g·r �M1 1 D72M1

�5�where the average precession frequency in the sampleis denoted byv0 � gB0. The precession frequencyand the effects ofT2 relaxation can be transformedaway by means of the substitution:

M1 � c�z; t�exp�iv0t 2 t=T2� �6�to give:

2c�z; t�2t

� 2iggzc�z; t�1 D72c�z; t� �7�

WhenD � 0, Eq. (7) describes the free precessionof transverse magnetization in the gradientg. At theend of a gradient pulse of durationt, the isochromatsdefine a helix as described earlier. The effect of diffu-sion on the amplitude (diameter) of the helix,C(t),can be obtained by substituting

c�z; t� � c�t�exp 2igzZt

0g�t 0�dt 0

� ��8�

into Eq. (7). The result is:

ln�c�t�� 2DZt

0q2�t 0�dt 0

� ��9�

where

q�t 0� �Zt 0

0gg�t 00�dt 00 �10�

Of course, this is only an attenuation factor, and for itsobservation the isochromats must be refocused in thexy-plane to form an FID or echo. It should be notedthat, while gradients only affect the transverse compo-nents of the magnetization, the attenuation by diffu-sion as described by Eq. (9) also applies to sinusoidalpatterns in thez-component of magnetization, i.e.stored magnetization.

3.2. Pulse sequences

Immediately after a 908x pulse, the signal detectedalong they-axis in the rotating frame decays in ampli-tude. The primary cause of the decay in liquid stateNMR is dephasing that results from magnetic fieldinhomogeneities. This decay is rapid in the presenceof the applied constant gradientg because the volume


Fig. 2. The Hahn stimulated echo (STE) with pulsed field gradients.

integral of the helix of isochromats is zero when thepitch is much smaller than the length of the sample.

In order to detect a signal, the isochromats must berefocused, i.e. the helix must be unwound, so that thevolume integral of the magnetization gives a nonvan-ishing component in thexy-plane. This requires asequence with two matched gradient pulses. Thefirst pulse encodes nuclear positions through positiondependent phase angles, and thus sensitizes thesample to diffusion and flow. The second gradientreverses the encoding and brings the isochromatsback into theyz-plane, thus forming an echo.

In the following we consider only PFG-NMRexperiments where the gradients are applied in theform of relatively short pulses. The major advantagesof an echo sequence with pulsed gradients are (i) thegradient pulse areas can be controlled independentlyof the time for the echo and (ii) the signal can be readout in a homogeneous magnetic field. The disadvan-tage is that relatively large currents must be switchedon and off. This produces mechanical forces, Jouleheating, and transient eddy currents.

3.2.1. The spin-echo (SE) sequenceA simple Carr–Purcell [28] sequence is shown in

Fig. 1. Also shown is the ribbon of isochromatsproduced by the 908x pulse and the ‘‘Saarinenhelix’’ produced by the first gradient pulse [26]. Theeffect of the 1808y pulse is to reverse the effect of theprevious gradient pulses to give the effective gradientsequence (g*) [14]. This is equivalent to defining theeffective gradient at any time byg* � pg wherep isthe coherence order. Straightforward application ofEq. (9) to this sequence shows that the echo amplitudeat 2t is [7]:

S�2t� � M0exp�22t=T2�exp�2Dq2�D 2 d=3�� 11�where M0 is the equilibrium magnetization and therelaxation factor containingT2 has been reintroduced.The correction termd /3 is a consequence of therectangular shape of the gradient pulses. Other shapes,such as the sine lobe, can easily be incorporated [29].The Stejskal–Tanner attenuation factor [7] fordiffusion can be isolated asc�2t� � S�2t�=S0�2t�whereS0�2t� is the echo amplitude in the absence ofa gradient. Continuous background gradients havebeen neglected in deriving Eq. (11). This is usually

adequate for experiments in modern, high homogene-ity magnets; but a more complete expression is avail-able for the case where a background gradientg0 mustbe considered [7].

The Carr–Purcell echo has important advantages.In particular, the maximum possible signal is recov-ered, in the absence of relaxation effects, andchemical shifts are refocused at the echo. The disad-vantages result from the long period that the magne-tization is transverse, i.e. in thexy-plane. Transversemagnetization is subject to both transverse relaxationandJ-modulation effects.T2 can be short for slowlytumbling macromolecules, and this can lead to asevere loss of signal.J-modulation refers to signalmodulation resulting from hard RF pulses thatexchange the spin states of nuclei that are coupledto the nuclei of interest thus preventing complete refo-cusing. These effects present special problems forstrongly coupled spin systems [30].

3.2.2. The stimulated echo (STE) sequenceSequences containing two and three 908 RF pulses

were investigated by Hahn in his classic paper on spinechoes [20]. He found that the three-pulse sequencewith a steady (cw) gradient can generate up to fiveechoes. The first echo after the third RF pulse, the so-called stimulated echo (STE), is of particular interesthere. The effects of diffusion on the STE with bothsteady [31] and pulsed gradients [32] have beencomputed. We show the standard PFG-STE diffusionexperiment in Fig. 2. An instructive three dimensional(spherical polar) model of the first part of this experi-ment including the formation of the primary echo at2t (not shown) was presented by Hahn and attributedto E.M. Purcell [20].

The amplitude of the PFG-STE is given by

S�T 1 2t� � �M0=2�exp��22t=T2�

2 �T=T1��exp�2Dq2�D 2 d=3�� 12�where the Stejskal–Tanner factor has again beenderived by the application of Eq. (9). It should berecognized at the outset that the STE is quite differentfrom the SE. First, we see that the amplitude isreduced by a factor of two. This results from twofeatures of the sequence. The second 908x pulse storesthe magnetization by rotating only they-componentsinto the ^z-directions. Thex-components remain


transverse and can contribute to the primary andsecondary echoes. Then, after the storage periodT,the third 908x pulse returns thez-components to the^y-directions where the action of the second gradientpulse refocuses the isochromats so that the STE signalappears att � T 1 2t. However, the isochromats arenot coplanar at the time of the echo, and in fact theirprojection onto thexy-plane defines a circle that istangential to thexz-plane.

The advantages of the STE sequence arise becausethe evolution time for transverse magnetization can belimited. With 2t p T, spin relaxation depends primar-ily on T1 rather thanT2, and witht p 1/J, J-modula-tion is not significant. The advantage ofT1 relaxationrelative toT2 relaxation depends on the ratiosR�T2=T1 and X � T=T1. Assuming that T ù D andT q t in the STE experiment, we find that the STE/SE signal ratio is�0:5�exp�R�X 2 1�=X�. For examplewhenR� 0.5, X must be greater than ln(2) to breakeven, but withR� 0.1 andX� 0.5 the enhancementfactor is greater than 200. In general the advantages ofSTE more than compensate for the 50% smallercoefficient.

Two other points need to be mentioned. Firstthe reduction oft to a value only slightly largerthan the gradient pulse durationd means that theSTE is very close to the trailing edge of thesecond gradient pulse. Unless appropriate provi-sions are made, the signal will be distorted bygradient pulse induced eddy current effects. Thesecond point concerns the encoding of chemicalshifts after the first RF pulse. For a spin withoffset frequencyvA and positionz, the componentcos�vAt 1 ggzd� will be stored in thez-directionby the second 908 pulse. The third 908 pulsebrings this component back into theyz-planeand, after an additional timet , the component in they-direction is cos2�vAt 1 ggzd� neglecting the effectsof diffusion. The echo signal, obtained by integrationof the y-components over the sample volume,gives 1/2 as expected wheng is present, but inthe absence ofg we encounter the modulationfactor �1=2�cos2�2vAt�. Therefore, in STE experi-ments the echo amplitudeS0�T 1 2t� for g � 0,i.e. the q � 0 point in plots of echo amplitudeversusq2, is dependent on the chemical shift and


Fig. 3. A 1H FT-PFG-NMR stack plot obtained at 99.6 MHz for a microemulsion sample containing sodium octylbenzenesulfonate (SOBS),n-butanol, toluene, and water (D2O). Reproduced with permission [16].

should be avoided in data analysis (see Section5.2) [33].

3.3. The FT-PFG-NMR experiment

For analysis of spin relaxation in complex mixturesit is essential that we make use of the complete

spectral information that is contained in FID’s andhalf-echoes. The FT method for accomplishing thiswas described by Vold et al., in connection withtheir study of frequency resolved inversion recovery[34]. This idea can also be extended to NMR diffusionmeasurements when pulsed gradients are used as theFID’s can be acquired in the absence of applied


Fig. 4. 400 MHz1H NMR spectra for human blood plasma: (a) normal spectrum with 10% maximum gradient strength, (b) spectrum obtainedwith 50% strength, and (c) the difference between (a) and (b). Reproduced with permission [39].

gradients. In the following we focus on features andapplications of FT-PFG-NMR that are relevant to thelater discussion of DOSY.

3.3.1. Component analysis

3.3.1.1. Complex mixturesJames and McDonalddemonstrated the determination of diffusion coeffi-cients for each component in a multicomponentsystem by means of FT-PFG-NMR [35]. In theirexperiment, a Carr–Purcell echo (Fig. 1) was acquiredas a function of the gradient durationd with t ù d,and the second half of the echo was Fourier trans-formed to produce a set of NMR spectra. The attenua-tion factor,c�2t� � S�2t�=S0�2t�; for each line wasthen used with Eq. (11) to determine the correspond-ing value of D. The gradientg was calibrated in aseparate experiment on a compound with a knowndiffusion coefficient.

This groundbreaking experiment, carried out on astandard commercial NMR spectrometer, establishedthe basic NMR diffusion measurement still in use. Theauthors recognized the analytical implications,because each component of a mixture is revealed byits unique diffusion coefficient, and possibilities forthe study of dynamics in solutions. In particular, theeffects of rapid chemical exchange were considered,and the determination of binding constants for smallmolecules with large molecules by means of theweighted average diffusion coefficient was suggested.

The major limitation of the James and McDonaldexperiment was the small gradient amplitude avail-able and the necessity of using long gradient pulses.Kida and Uedaira remedied the problem by designinga gradient driver that permitted the use of narrowgradient pulses that were compatible with the opera-tion of a field stabilized NMR system [36]. They alsointroduced the stack plot display of spectral intensityversusq2�D 2 d=3�. Other features of this early paperare the analysis of the apparent hydroxyl group protondiffusion coefficients on the basis of rapid exchangebetween water and methanol, and the correlation ofmolecular diffusion coefficients with realistic modelsfor translational friction factors (see Eq. (1)).

The FT-PFG-NMR method, primarily in the spinecho version, has been applied to a wide variety ofchemical systems. We note the studies of complexmixtures, especially those containing surfactants, by

Stilbs and coworkers. That work has been reviewed,and an illustration involving a microemulsion isshown in Fig. 3 [16]. Note that the stack plots arearranged with the gradient pulse durationd increasingfor the lower (front) spectra, and the effects ofJ-modulation are evident for the inverted toluene reso-nances.

3.3.1.2. Spectral editing Stilbs has emphasized thefact that, in a PFG-NMR diffusion measurement for amixture, the complete spectrum for each component isattenuated as the quantityq2�D 2 d=3� is increased;and the non-overlapping signals can be classified bytheir measured diffusion coefficients [37]. He alsosuggested that spectra with different amounts ofattenuation could be scaled and subtracted to zeroout a component and in favorable cases to isolate acomponent. A demonstration of this procedure with1H spectra of a 50:50 mixture of decane and 1-decanolwas presented, and potential problems withJ-modulation andT2 effects were noted. This type ofspectral editing was characterized as ‘‘size-resolvedNMR spectrometry.’’

The pseudo-separation of different molecules in acomplex mixture by spectral editing on the basis ofnuclear relaxation times and molecular diffusion coef-ficients has recently been pursued with the aid ofsophisticated, modern PFG-NMR sequences. Forexample, an STE experiment including the WATER-GATE water elimination sequence [38] has been usedto assign spectral resonances of slowly diffusingmolecules in human blood plasma [39]. This experi-ment is illustrated in Fig. 4. The spectrum in Fig. 4(a)was acquired with a small gradient (g� 59 mT/m,D�500 ms) and shows essentially no attenuation. Thespectrum in Fig. 4(b), acquired withg � 295 mT/memphasizes large molecules and permits the assign-ment of peaks between 3.4 and 3.9 that havepreviously been obscured by amino acids and carbo-hydrates. Finally, Fig. 4(c), the difference betweenFig. 4(a) and (b), shows the small rapidly diffusingmolecules and their assignments.

These spectral editing experiments and numerousother experiments involving diffusion filters takeadvantage of strong diffusion based discriminationagainst small rapidly moving molecules withoutrequiring sophisticated data transformations. Editingand filtering techniques can be very valuable; but, of


course, do not provide complete diffusion spectra ofmixtures.

3.3.1.3. Affinity NMR Another variant of diffusionedited NMR, known as affinity NMR, deservescomment [40,41]. In the pharmaceutical industrycombinatorial chemistry methods are now producinglarge numbers of compounds in mixtures for testing indrug discovery programs. It is important to be able todetect the presence of molecules with desiredproperties without resorting to physical separation ofthe mixtures. If the desired property involvescomplexation with a partner in solution, differencesin diffusion coefficients can be important indicators.

In the affinity NMR experiment, the gradient ampli-tude and duration are adjusted so that all signals fromsmall molecules in a mixture just vanish. Then apotential complexing agent is added and the experi-ment is repeated. In a demonstration experiment, ninecomponents with molecular weights in the range 200-400 were attenuated to the noise level by adjusting thegradient duration in a PFG-NMR experiment. Hydro-quinine 9-phenanthryl ether was then added and thespectra were found to contain signals from two of thesmall compounds in addition to the added ether [40].The combination of PFG-NMR and TOCSY wassufficient to identify the two complexing compounds[42].

4. Diffusion ordered NMR spectroscopy (DOSY)

4.1. Experimental requirements

For the standard DOSY experiment we envisionautomated data collection with a programmed set ofgradient areas. This is followed by data inversion by

means of one or more user selected transformationsand the generation of one, two, or three-dimensionalspectra. This scheme can only be successful if datacollection and initial processing, i.e. Fourier transfor-mation, phasing, baseline correction and deconvolu-tion, yield undistorted absorption mode NMR spectra.

DOSY requires high quality gradient probes thatincorporate active shielding [43,44] and are designedto provide constant (flat) gradients over the NMRactive sample volume. DOSY also requires computercontrolled gradient drivers that can provide gradientpulses with reversible polarity and pulse areas that arematched to within at least 10 ppm [45]. The latterfeature is essential for automated DOSY experiments.We assume that such equipment is commerciallyavailable or will be available in the near future anddo not discuss it further. However, readers should beaware that much commercial gradient equipmentcurrently in use does not meet these criteria.

As previously discussed, the pulse sequences ofchoice are based on the STE sequence (see Fig. 2).The immediate problem is that gradient pulses tend toinduce eddy currents in the surrounding metal struc-tures of the probe and the magnet. These eddy currentsin turn produce slowly decaying magnetic fields at thesample that lead to spectral distortions resulting fromtime dependent phase changes. Therefore, experi-ments must be designed that avoid or at least mini-mize the effects of eddy currents. There is also therelated requirement that the NMR resolution be maxi-mized to avoid overlap of peaks from differentcomponents in a mixture as data transformations,required to produce diffusion spectra, fail whenNMR peaks for similar sized molecules overlap.Here we consider current hardware, software, andexperimental designs that address these requirements.

4.1.1. Eddy current reductionThe best way to avoid the effects of eddy currents is

to prevent the formation of eddy currents in the firstplace. There are some easy but not completely effec-tive solutions. For example eddy currents in the probecan be reduced by means of special RF coil designs,and eddy currents in the inner bore of the magnet canbe reduced by using widebore magnets. Also, the rateof change of the magnetic field when a gradient pulseis switched on or off can be reduced by shaping thegradient pulse. Shapes that have been investigated


Fig. 5. The LED pulse sequence [48].

include sine functions, sine squared functions, andnearly rectangular functions with modified rise andfall times [29]. Gradient pulse shaping is helpfuland this capability is now available on some com-mercial instruments.

The most effective means for avoiding eddycurrents is to reduce the magnetic fields of the gradi-ent coil set outside the probe to such low values thatsignificant disturbances do not occur. This reductioncan be achieved through active shielding of the gradi-ent coils. Imagine a wire carrying a time-dependentcurrent close to a conducting metal sheet. A surfacecurrent distribution is induced in the sheet that screensthe magnetic field and reduces it to zero inside thesheet. The idea of active shielding is to introduce amesh of wires with an externally generated currentpattern that mimics the induced surface current distri-bution in the conducting sheet. Mansfield and Chap-man have reported an iterative procedure fordetermining the positions of wires in a discrete cylind-rical mesh to approximate the continuous currentdistribution required to screen the field of a currentloop [43]. Another strategy is to use a multidimen-sional minimization program to determine the axialcoordinates and radii of a predetermined number ofMaxwell pairs that will shield most effectively aspecified gradient coil set. A small volume outsidethe shielding coils is chosen as the indicator of shield-ing efficiency [44]. Active shielding is now so wellestablished that commercial imaging and gradientprobes can be expected to include efficient shieldingcoils.

4.1.2. Pulse sequences for minimizing effects of eddycurrents and J-modulation

The pulse sequences described in the following textall require phase cycling for coherence transfer path-way (CTP) selection. The principles of CTP throughphase cycles are, of course, well known [1,46]; but inpractice the construction of cycling procedures thatare both efficient and effective is not straightforward.As complete phase cycles for coupled spin systemsmay be very time consuming, it is common to reducethe number of steps by removing what are thought tobe the least important parts. However, combinationsof incomplete phase cycles with gradient pulses arenot unique and cycling procedures are often deter-mined by trial and error without optimization. Theoptimum phase cycle depends on the nature of thesample but the weighting of the various CTP’s alsodepends on the properties of the RF pulses and thepresence of transport phenomena. Jerschow andMuller have recently developed a method for evaluat-ing the latter effects by simulating the CTP selection


Fig. 6. The BPP-LED pulse sequence. The phase cycle for the 908

pulses is: P1: 016, P2: (0022)4, P3: 04 24 14 34, P4: 0202 2020 13133131, P5: 04 24 14 34, Rec: 0220 2002 3113 1331, and the 1808 pulsesare 1 x or 2 throughout [53].

Fig. 7. NMR spectra of protons ona-carbons of alanine in D2O: (a)5 ms after a bipolar gradient pulse pair (g� ^ 1.50 T m21, d /2�1 ms,t � 1.5 ms), (b) 5 ms after monopolar gradient pulse (g �1.50 T m21, d � 2 ms), (c) 100 ms after a monopolar gradient pulse(g� 1.50 T m21, d � 2 ms). The frequency origin is arbitrary, andthe dashed line (b) shows the baseline. Reproduced with permission[53].

process and have implemented this method in acomputer program named CCCP [47]. For most ofthe pulse sequences illustrated here, the phase cyclesactually used are listed in the figure captions.

4.1.2.1. Longitudinal eddy current delay (LED)sequence In spite of the best efforts, eddy currenteffects are still significant, and they depend on thestrength of the gradient pulses. This, of course, canbe disastrous for experiments that automaticallysample a wide range of gradient values. Thestimulated echo (see Fig. 2) is primarily affected byeddy currents induced by the final gradient pulse, andthe problem is exacerbated by the need to keept shortin order to minimize transverse relaxation andJ-modulation. This conflict motivated Gibbs’smodification of the STE sequence as illustrated inFig. 5 [48]. The major change is the addition of afourth 908 pulse at the center of the stimulated echofor the purpose of storing the magnetization in thelongitudinal direction while the eddy currents decay.After the eddy current settling periodTe, themagnetization is recalled with a fifth 908 pulse andthe FID is acquired. The effectiveness of thissequence is further enhanced by adding threegradient pre-pulses (not shown) to make a chain offive equally spaced pulses. This arrangement ensures

that the transient magnetic fields resulting fromprevious gradient pulses have the same effect duringthe transverse evolution periods after the first andthird RF pulses.

Either phase cycling or homospoil pulses can beused to eliminate transverse components duringT.However, phase cycling of the last two RF pulses isessential to remove the effects of longitudinal relaxa-tion during Te. This can be accomplished by alter-nately storing the STE in the1z and 2z directionsand then taking the difference between the associatedsignals that are returned to thexy-plane by the fifth RFpulse.

We note that the LED sequence is analogous toB1

gradient experiments. The first 908x–g–9082x compo-site pulse sandwich whereg � gz is equivalent to asingleB1 gradient pulse in encoding the spatial posi-tion in longitudinal magnetization [21,49]. Then afterthe storage periodT, the second sandwich decodesposition and again stores longitudinal magnetization.Hence LED can stand for either LongitudinalEncode–Decode or Longitudinal Eddy-current Delay.

The LED sequence significantly improves the qual-ity of spectra obtained in FT-PFG-NMR experimentsbut still suffers from the slowly decaying eddy-currents. The consequence is that theTe period,required to obtain undistorted spectra in experimentswith large gradient pulses, can be unacceptably long.There is also the problem that the gradient pre-pulsesintroduce additional heat.

4.1.2.2. Bi-polar LED (BPP-LED) sequenceOne ofthe best ways to diminish the eddy currents inducedby a short gradient pulse (g) is to replace the pulsewith two pulses of different polarity separated by a1808 RF pulse, i.e. the composite bipolar gradientpulse combination (g–1808–( 2 g)). As previouslynoted, the effective gradientg* is equivalent inthese two cases; but the composite pulse providesself-compensation of the induced eddy currents [50].Gradient pulse sequences with alternating polaritywere introduced into PFG-NMR by Karlicek andLowe [24] to take advantage of the fact that the1808 RF pulses refocus static gradients. Also, Cottset al. [51] proposed a number of STE sequences withalternating grading polarities to minimize the effect ofbackground gradients. More recently Fordham et al.[52] replaced all of the gradient pulses in the LED


Fig. 8. Pulse sequences (a) GCSTE [Phase: P1: 0213, P2: (0828)2

(1838)2, P3: (0424)2 (1434)2, Rec:f2 1 f3 2 f1] and (b) GCSTESL[Phases: P1: 0213, P2: (0828)2 (1838)2, P3: (0424)2 (1434)2, P4: 131

064264 1 f2 1 f3 2 f1, Rec:.f2 1 f3 2 f1] [54].

sequence with bipolar pulse pairs (BPPs) to permitdiffusion measurements in the presence of largebackground gradients.

The LED experiment with BPPs shown in Fig. 6was investigated for DOSY applications by Wu et al.[53] With short gradient pulse separations (t , 1 ms)the BPPs were found to cancel more than 95% of theeddy currents, and undistorted signals could beobtained withTe reduced by a factor of 20. The effec-tiveness of the BPPs in reducing signal distortions isillustrated in Fig. 7. Further, this improvement couldbe obtained without the need for gradient prepulses,thus reducing undesirable heating effects.

The use of Eq. (9) with the effective gradientg* forthe BPP-LED sequence gives the corrected attenua-tion factor:

c�D 1 d 1 2t� � exp�2Dq2�D 2 d=3 2 t=2��: �13�

Of course, eddy current compensation is morecomplete when bothd and t are short. The extra1808 pulses introduced here cause some loss of signalbecause of the greater sensitivity to inhomogeneitiesin the RF pulses. However, this turns out to be anadvantage because signal acquisition is limited tothe region where the gradient is constant and higherquality data result. Also, the refocusing effect of the1808 pulses does eliminate the effect of steady

background gradients, and more importantly forDOSY it refocuses chemical shifts. The latter can bevery important when chemical exchange [33] or spindiffusion is present [54].

At present the BPP-LED sequence is the sequenceof choice for many DOSY experiments, especiallythose requiring maximum gradient strengths withsmall temperature rises. It, of course, can be combinedwith relaxation filters, water elimination sequences,etc. as desired.

4.1.2.3. New sequences and comparisonsThechoice of a pulse sequence for DOSY depends onboth the capabilities of the available instrumentationand the nature of the system under study. When verysmall diffusion coefficients are involved, largegradient amplitudes are required to obtain adequatesignal attenuation. As disturbances to the local fieldand the lock signal increase with the amplitude of thegradient pulse, the LED feature is often required.However, when only modest gradients are requiredand an efficient, well designed probe with activeshielding is available, the settling periodTe may notbe necessary. In such cases the BPP-STE sequenceshould be considered [54,55]. The direct detectionof the STE, without storage and recall, is especiallyimportant when the phase of the echo must bedetermined as in MOSY experiments [8].

The self-compensating feature of BPPs is extre-mely important in eddy current reduction, but thereare other benefits from the (g)–1808–(2g) compositepulse as well. The refocusing and cancellation ofsteady gradients in inhomogeneous systems was theinitial motivation for the introduction of alternatinggradients in PFG-NMR. In high resolution NMRtwo other effects are encountered that are also sensi-tive to refocusing. In the absence of a 1808 pulse thechemical shifts are encoded along with the positiondependent phase information during the first trans-verse interval of the STE sequence. Wheng ± 0chemical shift information usually does not appearin the STE; however, when spin exchange or chemicalexchange interchange chemical shifts during thestorage intervalT, the chemical shifts do affect theamplitude of the STE [33,54]. With coupled spinsystems there is another consequence of unrefocusedchemical shifts because the second 908 RF pulsegenerates zero-quantum coherences (ZQCs) in


Fig. 9. Velocity insensitive gradient pulse sequences: (a) A gradientsequence with zero first moment (see text), and (b) the double STEsequence (DSTE) [59].

addition to z-magnetization [22,56]. In cases wherethe ZQCs decay more slowly than the longitudinalmagnetization, phase errors become evident withlong values of the storage timeT [54].

Fortunately, the (g)–1808–(2g) sandwich elimi-nates both the exchange and ZQC problems, and asdiscussed earlier the extra 1808 pulses increase thequality of the collected data because of their tendencyto restrict the excitation volume. The negative aspectis that the BPP-STE and BPP-LED sequences requireconsiderable phase cycling. This is not a problemwhen signal-to-noise ratios are low and signal aver-aging is already necessary, but with strong signals theexperimental time must be increased to accommodatethe phase cycle sequences. Another potential disad-vantage of sequences incorporating BPPs is that thetotal amount of time required to complete the compo-site gradient pulse pair will exceed the time requiredfor a single gradient pulse. WhenT2 is very short, theextra amount of time with transverse magnetizationwill lead to loss of signal.

Pelta et al. [54] have recently compared the STEbased PFG-NMR sequences in use (STE, BPP- STE,LED, and BPP-LED) and have suggested two addi-tional sequences with self-compensating gradientpulse pairs. The new sequences, known as gradientcompensated stimulated echo (GCSTE) and gradi-ent compensated stimulated echo spin lock(GCSTESL), are illustrated in Fig. 8(a) and (b), respec-tively. Both of these sequences feature BPPs in which

one of the pulses is placed in the storage intervalT. Thisarrangement avoids the extra 1808 pulses required whenthe BPPs are placed in the transverse intervalst andprovides a homospoil effect as well. The major advan-tage is a reduction in the required phase cycling. Thedisadvantages are (1) double the amount of heating forthe sameq values, (2) potential phase anomalies asso-ciated with ZQCs for coupled spin systems, and (3)amplitude modulation effects associated with spin-exchange and chemical exchange. The GCSTESLsequence incorporating the spin locking intervaltST

corrects the line shapes for ZQCs but not for exchangeeffects, and provides somewhat better resolution byrestricting detection to the region of constant gradient.

We conclude that all of the BPP-STE sequences areuseful under appropriate circumstances. In theabsence of spin coupling and exchange effects andwhen only small gradients are required, the BPP-STE and GCSTE sequences are reasonable choices.When exchange effects are not present and modestgradients suffice, GCSTESL becomes a strong candi-date, but in the general case requiring chemical shiftrefocusing, background gradient compensation, andstrong gradients the BPP-LED sequence is indicated.These conclusions are, of course, based on currenttechnology.

4.1.3. Suppression of convection current effectsConvection currents are easily induced in nonvis-

cous samples by temperature gradients. The wellknown Rayleigh–Benard instability results fromtemperature inversions where higher temperature,lower density layers lie below lower temperature,higher density layers. The resulting fluid flowproduces a distribution of velocity components paral-lel to the z-gradients in typical PFG-NMR experi-ments, and a corresponding attenuation of the STEthat interferes with diffusion measurements [57].This effect is characterized by a downward curvaturein plots of the logarithm of the STE amplitude versusq2. However, the deviation may not be easy to distin-guish from a simple increase in the diffusion coeffi-cient. Of course, the velocity distributions found ingravity driven mass convection have less effect ondiffusion measurements made with gradients in thexor y directions.

The first protection against convection currents is awell designed temperature control system that can


Fig. 10. Block diagram of the stop-and-go sample spinning system.Reproduced with permission [60].

minimize temperature gradients in the sample. Suchsystems usually require a large flow rate for the heatexchange gas. Also, sample tubes with smallerdiameters are less susceptible to convection. Anextreme example is the stabilization of solutions ofpolymers in liquid and super-critical CO2 in fusedsilica capillaries having inner diameters of approxi-mately 100mm [58].

When one must deal with low viscosity solvents attemperatures far from ambient, mass convection isoften difficult to avoid. In such circumstances it isadvisable to use pulse sequences that are insensitiveto constant velocities. Such sequences can beconstructed by requiring that the first moment ofeffective gradient sequence,g*, vanish [14], i.e.Zt

0g*�t 0�t 0dt 0 � 0: �14�

A gradient pulse sequence satisfying Eq. (14) isshown in Fig. 9(a). The velocity insensitive doubleSTE diffusion sequence (DSTE) shown in Fig. 9(b)was proposed by Jerschow and Mu¨ller [59]. (Hereselection of the proper coherence pathway is essentialbecauseg* � pg.) They found that the diffusion coef-ficient measured for test a sample of 80% glycol inDMSO-d6 (Bruker temperature-calibration sample) at347 K with an uncompensated BPP-LED sequenceshowed a deviation of a factor of 9 from the correctvalue. This deviation was eliminated when thecompensated DSTE sequence or a BPP version of itwas used.

4.1.4. Dispersion and resolution enhancementA hard lesson for most experimentalists to learn is

that sums of exponential functions with added noiseare extremely difficult to resolve into unique sets ofcomponents. Even with two components having wellseparated decay constants, one pays a considerableprice in accuracy and computing time comparedwith the analysis of a single exponential decay. There-fore, for mixtures of monodisperse components thehighest accuracy in DOSY analyses will be obtainedfor components that are already completely resolvedin the chemical shift dimension. The realization of thisfact has motivated attempts to obtain resolution ofNMR peaks through experimental refinements andsignal processing. In the following we consider

resolution enhancements for 1D-NMR. 3D DOSYbased on 2D NMR is covered in Section 7.

4.1.4.1. Stop-and-go spinnerSample spinning inNMR is often the final step in getting the maximumpossible resolution. When line widths are dominatedby magnetic field inhomogeneities, spinning at evenmodest frequencies, e.g. 20 Hz, can offer significantresolution enhancement. However, it is commonknowledge that conventional sample spinning is notcompatible with PFG-NMR diffusion measurementsbecause of unavoidable sample movement in thegradient direction. Here the restrictions are severe asPFG-NMR experiments can detect displacements inthe gradient direction of the order of 1mm.

With modern shim stacks, the requirement fornon-spinning samples may not be a seriousproblem. However, if spinning is a necessity, theoptions are limited. It is probably possible to design aprecision spinner that will limit sample excursions inthez-direction to acceptable bounds. But this is likelyto be a costly venture and to require much less conve-nient sample tubes and handling procedures. Fortu-nately, a low cost alternative, based on the fact thata stationary 5 mm NMR sample tube can be spun upto the required 20 Hz speed in approximately 10 ms,has been demonstrated.

Fig. 10 shows a block diagram of a stop-and-gospinner system that is compatible with high-resolutionNMR spectrometers [60]. A computer controlled DCservo motor is directly coupled to the sample tube sothat the sample tube can be arrested during the motionsensitive part of the experiment and can be restartedduring the delay periodTe when the magnetization isstored in thez-direction. The spinning sample isstopped in about 10 ms by an ‘‘active brake’’ afterdata acquisition, but the liquid in the sample requires1–2 s to reach a quiescent state that is compatible witha diffusion measurement.

With a standard resolution test sample in a 10 mmgradient probe mounted in the narrow bore magnet ofa Bruker AC-250 spectrometer, the best nonspinninglinewidth was about 2 Hz. This width was reduced toabout 0.2 Hz when spinning was controlled by the DCmotor. This apparatus has few disadvantages and hasbeen used routinely for data collection in DOSYexperiments requiring the best resolution [61].


4.1.4.2. Reference deconvolutionEven with the bestefforts at field shimming, eddy current avoidance,sample spinning, etc., one is left with line shapes inFT-PFG-NMR experiments that suffer fromexperimental artifacts. If the remaining linewidthsinterfere with the resolution of components, it isworthwhile to consider postprocessing of the data toenhance the resolution or to improve the line shapes.

Consider a single NMR line with a frequency offsetof V. The corresponding FID has the formf �t� �g�t�exp�2iVt� with g�t� � s�t�b�t� where s(t) is theideal response obtained with a perfect instrumentand b(t) accounts for all deviations from ideality.The line shape in the frequency domain,I(v ), isgiven by the Fourier transform of the products�t�b�t�

I �v� � FT�b�t�·s�t�� 15�

and according to the convolution theorem this shape is

also equal to the convolution ofB�v� � FT�b�t�� withS�v� � FT�s�t�� or I �v� � B�v�*S�v� [1,62].

The ideal spectrum with a shape depending only onthe intrinsicT2 can in principle be extracted from theexperimental shape by simply dividingf(t) by b(t)prior to performing the Fourier transformation. Onecan even remove some of the intrinsic linewidth byadditional filtering of the FID at the risk of addingnoise and distorting the frequency domain signal.The tradeoffs between sensitivity and resolutionhave been investigated in detail [63], and it is nowcommon practice to multiply the FID with a weightingfunction that provides some resolution enhancementwithout introducing unacceptable noise. Commonexamples are the sine–bell and Lorentz–Gaussweighting functions.

The demands on postprocessing in DOSY may beespecially severe as the FIDs are acquired in thepresence of some level of transient magnetic fieldsand the strengths of the transient fields depend onthe amplitude and duration of the gradient pulses.Therefore, the free induction decays may not betruly free, and the Fourier transforms of decaysacquired in the presence of changing frequenciescannot produce symmetrical line shapes. A reasonableapproach to this problem is to use the experimentalFID itself for deconvolution. This appealing idea, nowknown as reference deconvolution (RDCON), hasbeen rediscovered many times in different contexts[64]. In practice one locates a signal from a singlet


Fig. 11. The STE-INEPT pulse sequences for heteronucleardetected DOSY with coherence transfer [68].

Fig. 12. The pulse sequence for shuttle based fringe field 2D-DOSY [phase cycle: P1: 08, P2: 08, P3: 08, P4: 0202 0202, P5: 0220 1331, Rec:0022 1133] [74].

component in an NMR spectrum, zeros the remainerof the spectrum, and performs an inverse FT to obtainthe reference FID,g(t). Division of the experimentalFID by g(t) corresponds to removing all linewidth anddoes not lead to useful results. However, multiplyingthe FID bys�t�=g�t�, wheres(t) is a guess at the idealFID for a single line, is practical and useful.

The implementation of the RDCON methodrequires that a complex reference signal be isolatedand that a spectral region be available that containssufficient ranges of both absorption and dispersionmode signals. The magnitude of the reference FIDmust be greater than zero and the baseline must bechosen carefully. These and many other practicalconsiderations are discussed in detail in a recentreview article by Morris et al. [64]. Also, a softwarepackage namedfiddle has been prepared for certainspectrometers.

Impressive demonstrations of RDCON in DOSYdata processing have been reported, and it is claimedthat with modern actively shielded diffusion probesthe improved diffusion fit gives an increase in accu-racy by about a factor of three [64,65]. Differences indiffusion coefficients of 0.5% are distinguished in thiswork.

4.1.4.3. Heteronuclear NMRStill another approachto the problem of peak overlap is to increase thespectral dispersion. The simplest way to do this is toincrease the chemical shifts by working at the largestpossible magnetic fields. Unfortunately, the chemicalshift range for protons is inherently small and spectralcrowding may be a problem with commonly available

500 and 600 MHz spectrometers. Chemical shifts are,of course, much larger for nuclei with larger atomicnumbers and heteronuclear NMR may be analternative.

Consider for example13C NMR. In addition to thewider range of chemical shifts, resolution for13C isaided by longer values ofT2, the virtual absence ofhomonuclear spin-spin coupling, and lower sensitivityto magnet field inhomogeneities and eddy currenteffects relative to1H because of the smaller gyromag-netic ratio. The problem with13C for DOSY applica-tions is that the sensitivity is quite low. One mustcontend with low natural abundance, and the lowgyromagnetic ratio and the long relaxation timesthat enhance resolution also limit signal-to-noiseratios. Further, the all importantq factor is propor-tional to the gyromagnetic ratio, and this can be aproblem when available gradient amplitudes aresmall.

Coherence transfer experiments can solve theq-factor problem while improving sensitivity. Theidea is to transfer polarization information from1Hto 13C prior to detection. This can be accomplished bycombining a STE based PFG sequence with INEPT[66] or DEPT [67]. A simple INEPT-DOSY sequenceis shown in Fig. 11 [68]. After the first 908 protonpulse, a composite BPP encodes spatial positionsand the magnetization is then returned to thez-direc-tion by the second 908 pulse. The optional homospoilgradient pulses during the storage periodT eliminateresidual magnetization on thexy plane. Then detec-tion is carried out with a refocussed-decoupled INEPTsequence [69] where an initial 908 heteronuclear RFpulse has been added to remove the natural hetero-nuclear spin polarization. An example of the13CINEPT-DOSY is shown in Fig. 14 of Section 4.2.1.1for a test sample containing sucrose, glucose, andsodium dodecyl sulfate (SDS).

4.1.5. Data collectionAssuming that provisions have been made to obtain

maximum resolution and undistorted absorption modesignals for each value ofq, there are still choices to bemade concerning data collection. It is important toselect the minimum number ofq2 values that willpermit the fastest and slowest decaying componentsto be analyzed with sufficient accuracy. Minimizingthe number ofq2 values will, of course, increase the


Fig. 13. Comparison of FFT and ILT transformations.

efficiency of signal averaging and permit sampleshaving limited lifetimes, e.g. blood plasma, to bestudied. Also, the judicious choice of sampling valuescan avoid the collection of useless data sets that do notprovide enough information for the determination ofdiffusion coefficients of all the components.

We assume that the attenuation of NMR peaks withincreasing gradients strengths can be represented byfunctions of the formS�x� � exp�2Gx�, or weightedsums of such factors, whereG � D; x� q2D 0 andD 0

is the diffusion time appropriately corrected for theshape and duration of a gradient pulse. As theG valuesrequired to fit diffusion data for all the peaks in anNMR spectrum may differ by orders of magnitude, itis seldom appropriate to use linear spacing ofx values.In fact, when the ratios ofG values exceed three,linear spacings lead to very large errors.

Logarithmic spacing ofx values can be a very goodchoice if the parameters are properly chosen [70]. Aconvenient form for computingxn is shown in Eq. (16)

xn � x12�n21�=m: �16�

We take the smallest decay ‘time’ to betmin � 1=Gmax

and the largest to betmax� 1=Gmin, and it is reason-able to setxN � 5tmax whereN is the maximum valueof n. The choice of the first data point,x1, is a bit more

tricky because it is easy to waste data points at smallxvalues while undersampling the larger values. Wesuggest choosingx1 � 0:2tmin and then computing,or determining by trial and error, the value ofm thatwill permit N data points to fit in the desired range(according to Eq. (16) the appropriate value ism� �N 2 1�ln�2�=ln�xN=xl�).

Labadie et al. have investigated errors in biexpo-nential data analysis with various sampling schemes[71]. They find that geometric spacing with

xn � x1�an 2 1��a 2 1� �17�

can give low errors and a reasonable distribution ofdata points along the decay. Herea is the constantratio of successive interval lengths. The reported erroranalysis suggests that with logarithmic spacingx1

should not be smaller thantmin/5.8, while forgeometric spacing the optimum value ofx1 is tmin/33.4.

In some applications it is also important to collect adata point with a very small value ofq. For example,systems involving chemical exchange may show largeinitial slopes in plots of ln(Signal) versusq2D 0 withstrong curvatures that make the extrapolation to thezero point difficult. However, one must bear in mindthat with a zero gradient, the echo amplitude in a STE


Fig. 14. 13C INEPT-DOSY for a mixture containing glucose, sucrose, and SDS in D2O. Dotted lines show average diffusion coefficients of eachcomponent. The 1D13C INEPT spectrum of the mixture is shown at the top. Reproduced with permission [68].

experiment will depend on the chemical shift unlessbipolar gradient pulse pairs are used. Also, very smallq values introduce two additional problems. For asample of lengthL in the direction of the gradient ifq is not much larger than 2p /L, the magnitude of thestored magnetization will depend on the phase of thesecond 908 pulse. Also, restricted diffusion will intro-duce errors unlessDD=L2 , 0:001 [72].

4.1.6. Utilizing the stray field to obtain large, steadygradients

NMR measurements of very small diffusion coeffi-cients require large gradient amplitudes, but in PFG-NMR large gradients invariably produce heatingeffects, vibrations, and eddy currents. These problemswere avoided by Kimmich et al. in an elegant experi-ment making use of the steady gradient in the fringefield on a superconducting magnet [73]. Experimentsanalogous to PFG-NMR can be run in the fringe fieldwith an STE sequence as encoding/decoding occursonly when the magnetization is transverse to the mainfield. There are, however, two major problems withfringe field diffusion measurements. First, data collec-tion in the presence of a large gradient eliminatesresolution and permits only the echo amplitude to berecorded. Thus, all structural information is lost, and itis not even possible to distinguish between the reso-nances of1H and19F. If resolution is not essential forsome sample, there is still the problem that availableRF pulses excite only a thin slice of a sample in astrong gradient field.

DOSY experiments with resolution in both thechemical shift and diffusion dimensions require thatthe FIDs be acquired in a homogeneous magneticfield. This was accomplished in stray field DOSY[74] with a shuttle system similar to that used inzero-field NMR experiments [75]. The LED pulsesequence used in this experiment and the location ofthe sample as a function of time (dotted line) areshown in Fig. 12. The STE is created by the firstthreep /2 RF pulses, the fourthp /2 pulse stores theecho magnetization in thez-direction for the shuttlingperiodTs, and the fifthp /2 pulse recalls the magneti-zation for detection in the homogeneous field.

A widebore (89 mm) magnet was equipped with ahomebuilt shuttle probe containing electronics forexcitation in the fringe field at 140 MHz and detectionin the homogeneous field at 360 MHz. A Kel-F

(36ml) sample cell was shuttled pneumatically in aprecision bore quartz tube a distance of about 0.3 mbetween the two RF coils. The RF pulse widths in thehomogeneous field and the fringe field were 10 and1.4ms, respectively. Accordingly, only a 0.3 mmthick slice was excited in the fringe field (g �53 T m21) and special phase cycling was necessaryto select only the spins excited and encoded withdiffusion information in the fringe field from the back-ground of all protons excited in the homogeneousfield.

A DOSY experiment was performed on a testsample containing SDS, glycerol, and H2O (1:2:4 byweight) withT� 5 ms, shuttle timeTs� 150 ms, andd incremented from 10 to 300ms in 18 steps [74].This provided satisfactory resolution for the testsample even though the NMR linewidths exceeded20 Hz because of susceptibility differences betweenthe sample and the short cylindrical cell. This experi-ment demonstrates that fringe field DOSY experi-ments are practical for samples having sufficientlylongT1s and well resolved spectra, and the advantagesare considerable. Experiments can be performed withvery short diffusion times and with gradient ampli-tudes not easily obtained in other ways. Further, thiscan be done without gradient drivers and coils andwith relatively inexpensive, though unorthodox,instrumentation.

4.2. Data inversion and display

The ability of DOSY to provide accurate distribu-tions of diffusion coefficients for 1D analysis or for theconstruction of 2D and 3D DOSY displays depends onthe inversion of data sets that consist of NMR spectracollected with predetermined values ofq2D 0. In thefollowing we assume that FT-PFG-NMR experimentsof the STE or BPP-STE types provide 2D data sets ofthe form:

I �q; nm� �Xn

An�nm�exp�2DnD0q2� �18�

whereAn�nm� is the amplitude of the 1D-NMR spec-trum of thenth diffusing species wheng is small butnot zero (see Section 4.1.5), andDn is the associatedtracer diffusion coefficient. Also,D 0 � �D 2 d1�where1 depends on the shape of the gradient pulse.The goal of DOSY analysis is to transform 2D data


sets as shown in Eq. (18), which are basically stackplots of attenuated spectra, into 2D spectra withchemical shifts on one axis and the distribution ofdiffusion coefficients on the other.

We define a ‘discrete’ system as one that can bedescribed by small set of values ofn in Eq. (18). Themaximum value ofn may be more than ten, but for aparticular peak in a high-resolution spectrum weexpect that the attenuation can be described by afew values ofn, i.e. usually single exponential fits.In the event that a polydisperse component contri-butes to the peak at frequencyn , a continuum ofvalues ofD must be considered. The 1D data setI(s)describing attenuation of this peak must then bedescribed by:

I �s� �Z∞

0a�l�exp�2ls�dl �19�

where l � D�D 2 d1� and s � q2. (An alternativechoice isl � D and s� q2�D 2 d1�.) In Eq. (19)we recognize thatI(s) is the Laplace transform ofa(l) and thata(l ) is the Laplace spectrum of diffusioncoefficients. When only discrete components with thedecay constantsl i are present,a(l ) is a weighted sumof delta functions,d�l 2 li� and Eq. (18) is recovered.

The hope of new comers to this field is that a uniquetransform, akin to the FT, exists that can produceunique diffusion spectra and can invert them torecover the decay curves. This problem is illustratedin Fig. 13. At the top an FFT converts the FID to aunique NMR spectrum including the line shapes. Atthe bottom the decay on the left contains two compo-nents with diffusion coefficients and amplitudesdiffering by a factor of two. A perfect transformwould produce the Laplace spectrum of delta func-tions shown on the right and the inverse transforma-tion would exist. In fact, there is no perfect transform,and in the presence of noise it may be impossible toobtain any useful spectrum. The dotted curves in thediffusion spectrum indicate broadening associatedwith errors in an approximate transformation thatwould be acceptable.

Gardner et al. [76] took a significant step towardfinding the Laplace spectrum for discrete sums ofexponential components by introducing the transfor-mation s � exp(x) and constructing the functionexp(x)a[exp(x)]. This function turns out to be theconvolution of the desired spectrum of decay

constants with the shape function exp(x)exp[2exp(x)]. Thus the solution is at hand if theshape function can be removed by deconvolution.The catch is that the removal of the large line widthsby means of Fourier deconvolution or any othermethod usually introduces unacceptable errors, e.gtruncation artifacts, and this scheme is not practical,though it can serve as the starting point for moreextensive analyses. Actually, the method of Gardneret al., as modified by Swingler [77] was investigatedfor DOSY applications but was abandoned in favor ofthe more robust schemes described in the followingtext [78].

Here we confront the fact that the desired spectruma(l ) is the inverse Laplace transform (ILT) of thedecay functionI(s). Computinga(l) is an ill-condi-tioned problem and one that may be intractable [79].Actually, solutions fora(l ) can usually be found thatagree withI(s), but they are often not unique. This isthe source of much wasted computer time and muchnonsense in the literature. The same data inversionproblem has been faced for many years by thedynamic light scattering (DLS) community and exten-sive reviews are available in the literature [80–82].

The success of DOSY analysis hinges on the moremodest goal of computing the most likely spectrum ofdiffusion coefficients by means of an approximate ILTor some appropriate fitting algorithm. It is usually truethat FT-PFG-NMR data sets do not contain enoughinformation to permit an exact analysis. Reasonableassumptions must be made and additional informationmust be supplied. We know for example that NMRabsorption mode signals are positive and that theirLaplace spectra are also positive as are the associateddecay constants, and in dealing with polydispersesamples it may be reasonable to assume that thedistribution functions are in some sense smooth.There is also the possibility of combining informationfrom peaks at different chemical shifts, and of coursethere are physical limits on the maximum possiblediffusion coefficients and the minimum values thatcan be detected in any given experiment. Priorknowledge concerning non-negativity, parsimony,and other features may be essential for obtaining themost likely diffusion spectrum from an experimentaldata set.

DOSY analysis begins with a set ofN absorptionmode NMR spectra each havingv frequency points or


channels. In the following we consider various dataanalysis schemes that have been implemented toprocess the channels, groups of channels, or completespectra. The analyses that are readily available to theNMR community in the form of software packagesare emphasized.

4.2.1. Discrete samplesSamples are characterized as discrete if they

contain N monodisperse components and can becompletely described byN pairs of amplitudes(concentrations) and decay constants. The decisionto use Eq. (18) as the model function involves abso-lute prior knowledge (APK) and assumptions becausethe data set alone may not suffice to distinguishbetween a continuous distribution and a number ofdiscrete components at the same chemical shift.Even the knowledge that a solution was made withN monodisperse solutes does not assure thatN discretehydrodynamic entities are present because aggrega-tion and chemical exchange may be present. Whenthere is doubt about the nature of a solution, an analy-sis method that can handle continuous distributionsshould be applied first.

The analysis programs for discrete samples report alist of diffusion coefficients and amplitudes for eachchemical shift channel. In principle diffusion spectracan be constructed from these lists with delta func-tions having appropriate amplitudes at the specifiedpositions on the diffusion axis. However, this

prescription is not satisfactory in practice becauseanalysis errors cause channel to channel fluctuationsin the computed diffusion coefficients, even within asingle absorption peak. This means that a projectionof the diffusion spectra from all the channels ontoa single axis will now show clusters of peaks foreach diffusing species rather than a single line. Abetter procedure is to construct the DOSY spectrumwith normalized Gaussians having center positionsand intensities equal to the diffusion coefficients andamplitudes, respectively [9,83]. This gives a DOSYdata set of the form:

F�D; n� �XNl

j�1

Aj�n�Gj�D� �20�

where

Gj � 1��2ps2

j

q exp 2�D 2 Dj�2

2s2j

" #�21�

Here s j can be set equal to the standard deviationreported by the analysis program, a fixed value, orsome combination of the two to take into accountthe local error and the estimated systematic error forthe complete data set. The Gaussian componentmethod is now widely accepted for the constructionof 2D and 3D DOSY displays.

4.2.1.1. Levenberg–MarquardtThe Levenberg–Marquardt (L–M) algorithm is the standard


Fig. 15. Early DOSY display for a sample containing tetramethyl ammonium ions (TMA) and mixed micelles in D2O. The1H data set wasanalyzed with DISCRETE. Reproduced with permission [9].

non-linear least squares method [84,85]. L–Mrequires a data set with standard deviations, themodel function to be fit, and initial guesses for theparameters. The standard deviations are seldomknown, but they can be estimated; and the modelfunction for discrete samples is known in principle.However, it is our experience with typical NMR datasets that fits to sums of two or more exponentialcomponents are unreliable unless the signal-to-noiseratios and the ratios of decay constants are large. Amajor problem is that estimates of the parametersmust be supplied, and this is a very bad idea forDOSY analysis. Subjectivity in the analysis must beavoided as far as possible.

We recommend that the L–M method be reservedfor situations where the model function is a singleexponential, and the data set itself can be used toprovide estimates of the amplitude and decay factor.Also, it is important that the data be fitted to the expo-nential function directly rather than fitting the loga-rithm of the data to a straight line. The latterprocedure incorrectly weights the standard deviationsand even when corrected standard deviations aresupplied appears to give inferior results.

With special techniques, for example using high-fields, high-resolution spectra can often be obtained

that lend themselves to single component analysessuch as L–M. An example is provided by13CINEPT-DOSY of a mixture of glucose, sucrose(Aldrich Chemical Company), and sodium dodecylsulfate (SDS) (Aldrich Chemical Company) in D2O(500 mM each) [68]. The spectrum shown in Fig. 14was obtained with a Bruker AC-250 spectrometer, thegradient driver was home-built [45], and the gradientprobe was a modified Bruker dual probe. The pulsesequence shown in Fig. 11 was used withD� 0.103 s,and theq-values ranged from 11 364 to 681 880 m21

in 15 steps. For analysis, the area of each peak wasfitted to the single exponential decay in Eq. (13) bynon-linear regression with the L–M method, and the2D DOSY was displayed and plotted by Felix (HareResearch, Version 1.1).

The 1D 13C INEPT spectrum at the top of Fig. 14shows that most of the peaks are well resolved, andthe 2D DOSY display permits the diffusion coeffi-cients of all components to be easily read: glucose(3.60 × 10210 m2/s), sucrose: (2.92× 10210 m2/s),and SDS: (7.78× 10211 m2/s). Note that SDS isinvolved in rapid exchange between monomers andthe dominant micelles. The line widths for the Gaus-sian components (diffusion dimension) were taken tobe the fitting errors reported by the L–M routine.


Fig. 16. DOSY display of the SPLMOD analysis of simulated data. All diffusion peaks are correctly positioned. Reproduced with permission[83].

There turns out to be one significant overlap(sucrose and SDS) in the13C NMR spectrum, andthis produces a stray peak in the L–M analysis.Multi-component programs such as DISCRETE andSPLMOD can resolve the diffusion peak as thediffusion coefficients of the components differ by afactor of 3.8, but at the cost of increased errorselsewhere.

4.2.1.2. DISCRETE The Fortran programDISCRETE [86,87] has been freely distributed formany years. DISCRETE makes use of datatransforms to estimate the parameters and thenproceeds with a non-linear least squares analysis. It,therefore, does not require initial guesses, and itprovides best fit results for a range ofN values.DISCRETE was, in fact, used in the first reportedDOSY analysis to determine amplitudes and decayrates for every frequency point (channel) [9]. Adrawback of this procedure is that the resultsobtained for different frequencies within a singleNMR peak seldom agree exactly.

An early example of DISCRETE analysis is thestack plot DOSY display in Fig. 15 for a samplecontaining 10.0 mM tetramethylammonium chloride

(TMA) with mixed micelles[4.00 mM SDS and8.00 mM octaethylene glycol dodecylether (C12E8)].The 1H spectra were obtained at 295 K with a BrukerAC-250 spectrometer and custom built probe andgradient driver. A LED pulse sequence (Fig. 5) wasused withD � 100.0 ms,t 2 d � 0.500 ms, andTe�100.0 ms; and NMR spectra were collected withq-values ranging from 144 to 8.19× 103 cm21. The2D data set was analyzed channel by channel usingDISCRETE, and the widths of the Gaussian compo-nents were chosen to be the average of thes i

0 valuesfor the complete data set.

4.2.1.3. SPLMOD SPLMOD [88,89], the successorto DISCRETE, is also freely distributed. SPLMODalso analyzes sums of exponentials withoutrequiring initial guesses, but is not restricted toexponential components, and permits the analysis ofNd 1D data sets simultaneously. The user supplies thedata sets and specifies the maximum number ofcomponents to be searched for in an analysis. Themajor advantage for DOSY applications is that allof the frequency points in an NMR peak ormultiplet can be fit in parallel with the restriction


Fig. 17. DOSY display of the SPLMOD analysis of simulated data. The ‘‘cross-talk’’ artifact is illustrated. The correct (input) diffusioncoefficients are indicated by dotted lines. Reproduced with permission [83].


Fig. 18. DOSY displays for a mixture containing D2O, TEA, and SDS micelles. (a) 2D spectrum generated directly from SPLMOD ‘‘best fit’’parameters. (b) 2D spectrum generated from SPLMOD parameters after processing with rejection criteria (see text). Reproduced with permis-sion [83].

that all of the points must share the same small set ofdecay constants [83].

Successful SPLMOD analysis of PFG-NMR datarequires decisions concerning the setup, the applica-tion of APK, and the screening of potential solutions[83]. An input program must be prepared to take a setof spectra processed by FELIX or some other NMRsoftware package and to arrange the data according touser specifications in a form that can be read bySPLMOD. In the implementation of Morris and John-son [83], the input program prompts the user for athreshold value and then divides the spectrum intoregions containing the data columns to be analyzed.The user also specifies the form of the decay kernelsand other restrictions. For DOSY, exponentials withpositive amplitudes and decay constants are selected;and the maximum number of discrete components isspecified to be three. This means that SPLMOD willautomatically carry out analyses with one, two, andthree components.

The SPLMOD output consists of a set of decayconstants, a set of amplitudes, and the associated stan-dard deviations for each region of the spectrum. Theseresults are repeated for one to three component fitsand the ‘‘best fit’’ is determined. However, theremay be reasons to select one of the other results.For example, experimental artifacts such as residualeddy currents and phase errors may distort the decaysin ways that cannot be represented with exponentialkernels. In such cases, fits with larger numbers ofcomponents tend to give smaller deviations in spiteof the lack of any physical justification. A pragmaticoperating procedure is to reject fits on the basis of alist of reasonable criteria. If the solution withNl

components is rejected for any reason, the solutionwith Nl 2 1 components is evaluated, and so ondown to one component. This procedure may losesome authentic peaks, but at least it does not introducespurious results.

On the basis of experience the following criteriawere selected for the rejection filter [83].

1. Diffusion coefficients must be feasible and experi-mentally accessible.

2. Standard errors in diffusion coefficients must beless than 30%.

3. Pairs of diffusion coefficients must differ by afactor of two or more.

Numerous simulated data sets with added noise havebeen analyzed with SPLMOD to investigate the rangeof applicability of this program. Fits to the simulateddata sets were quite successful; but, of course, thesehave the advantage that the decay kernels are trulyexponential. For example, a data set was synthesizedfor two isolated Lorentzian peaks and three overlap-ping Lorentzian peaks with a maximum S/N of 20 andhalf widths 5.0 Hz as shown at the top of Fig. 16. Theset contained 50 spectra withq-values ranging from556 to 2.78× 104 cm21 and the decay constants weregiven bylj � Dj�D 2 d=3� with D � 0.1 s andd �0.0 s. In the SPLMOD analysis the NMR channelswere divided into three sets and a maximum of threecomponents were allowed for each set. The result ofthe analysis is shown in Fig. 16 where the 2D spec-trum is projected onto the left hand diffusion axis toshow the complete diffusion spectrum. The diffusionpeaks reproduce quite accurately the input values ofthe diffusion coefficients (dotted lines).

Fig. 17 illustrates a problem encountered in theanalysis of a synthesized data set that was not caughtby the listed criteria. The simulation contains threeLorentzian peaks with a maximum S/N of 20 andhalf widths of 10.0 Hz, and the central peak has twodistinguishable components. In the analysis 50 spectrawere used and 85 frequency points were analyzedsimultaneously with a maximum of four components.Unfortunately, the SPLMOD generated 2D-spectrumshows three diffusion components for the center peakinstead of the correct two components, and the peaksfor the valid components are displaced from thecorrect values (dotted lines). The extra peak at515 Hz is a result of ‘‘cross-talk’’ with the peak at475 Hz. This artifact is associated with the numberand magnitude of decay constants in one analysisblock and does not require overlapping peaks. Wefind that when a decay constant is required in somefrequency channels it will be used elsewhere through-out the block to improve the fit. One strong lesson isthat three components at the same chemical shift withdecay constants differing by factors of less than tenalways lead to considerable errors.

Real data may differ from simulated data becauseexperimental artifacts can affect the decay kernels.One consequence is that the decays of isolated singlecomponents may be better fitted by mixing in a smallamount of a second or third component. The effect of


the rejection filter in this situation is illustrated in Fig.18 for the SPLMOD analysis of a mixture containing40 mM tetraethylammonium chloride (TEA) and20 mM SDS [83]. An LED experiment was performedwith 40 q-values ranging from 167 to 7.79×103 cm21, d from 1 to 2 ms,D � 100.0 ms,Te �50.0 ms, andt � 2.50 ms. Each spectral region wasthen fitted with a maximum of two components(NNL � 2). The DOSY display in Fig. 18(a) wascomputed directly from the SPLMOD ‘‘best fits’’without the benefit of the post analysis rejection filter.The real overlap of TEA and SDS peaks at 1.11 ppmis properly handled, but the other major peaks allshow errors in position and artificial companionpeaks here enclosed by dotted ellipses. The ampli-tudes of the peaks and the shifts in positions areevident in the projected diffusion spectrum on theleft hand side.

The remedial effect of the rejection filter is clearlyshown in Fig. 18(b). Here the spurious two-compo-nent fits were rejected in favor of single-componentfits on the basis of standard deviations and peak

separations. Thus, all artifacts were eliminated andconsistent diffusion spectra were obtained. From thespectrum the diffusion coefficients for HOD, TEA,and SDS were found to be 1.74× 1025, 4.54 ×1026, and 8.84× 1027 cm2 s21, respectively. Theconclusion is that filtered-SPLMOD analysis of realdata, where two components occasionally overlap,can give reasonable results.

It is easy to imagine extensions of the rejectioncriteria to take into account data from other spectralregions and our knowledge of the features of NMRspectra of complex spin systems. For example, withcomplex molecules it is unlikely that a decay constantwould appear at only one chemical shift. Ultimately,artificial intelligence based software may provide thebest route to the ‘‘most likely’’ diffusion spectra.

4.2.2. Polydisperse samplesMany systems of interest are characterized by

continuous distributions of diffusion coefficients,e.g. polymers and vesicles. In such cases Eq. (18)describing the 2D FT-PFG-NMR data set must be


Fig. 19. DOSY display processed by CONTIN for a mixture containing D2O, glucose, and methyl cellulose. Reproduced with permission [83].

replaced with:

I �q; nm� �Z

R�T1;T2�a�D; nm�exp�2q2D�D 2 d1��dD

�22�whereR�T1;T2� accounts for nuclear spin relaxationduring intervals of the pulse sequence anda(D) is themass weighted distribution of tracer diffusion coeffi-cients D. The Laplace spectrum in this case is theproduct R�T1;T2�a�D; nm� rather than the desireddistributiona(D). If the relaxation times are correlatedwith the diffusion coefficients, the separation ofa(D)may not be possible. Fortunately, in high molecularweight polymers the relaxation rates often depend onsegmental motion and are approximately independentof the molecular weight thus permittingR�T1;T2� tobe replaced with a constant [90,91].

An interesting situation arises with phospholipidvesicles whereT2 values for the bilayer protonsdepend on tumbling rates of the vesicles and henceon their sizes. The result is that thea(D) distributionobtained for bilayer protons contains an excessivecontribution from the smaller vesicles that have smal-ler line widths. This produces a larger apparent diffu-sion coefficient for the bilayer protons than ismeasured for protons entrapped in the aqueouscavities of the vesicles [70]. In the following weassume that the relaxation factor is constant and canbe removed from the integral.

As analysis of polydisperse components producesdistributions of diffusion coefficients rather than peaksand positions, the plotting of diffusion spectra is morestraightforward. However, two points must be consid-ered. First, the programs usually give a small numberof points that barely suffice to define spectra whenmore than one peak is present, especially if one isnarrow. Additional points can be generated by inter-polation but the line shapes are likely to be distorted.The second, less obvious point, has to do with appar-ent peak heights and areas in plots of intensities versusD or ln(D). There is really no problem when spectraare plotted with a linearD axis as the peak areas aregiven byZ

a�D�dD �23�

as expected. Confusion arises when the dispersionaxis is still based onD but logarithmic spacing is

used because the apparent peak widths have verydifferent meanings in different parts of theD axis. Ifthe diffusion axis is relabeled to be linear in the loga-rithm of D, e.g. to show2 4, 2 5, 2 6 in place of1024, 1025, 1026, it is implied that the integration islinear in ln(D). Therefore, the integration in Eq. (23)must be replaced withZ

Da�D��dD=D� �Z�Da�D��d�ln�D�� 24�

and it makes sense to plotDa(D) rather thana(D)versus ln(D) or log(D).

4.2.2.1. CONTIN with extensionsThe first programused for DOSY analysis of polydisperse data wasCONTIN [83]. This program has been availablesince the early 1980s and has been extensivelytested [80,92,93]. CONTIN uses constrainedregularization to fit experimental data,yk and tk, tofunctions of the form

yk �XNg

m�1

cmf �lm; tk�a�lm�1XNL

i�1

biLki�tk� �25�

where thecm are weights of the quadrature formula,f �l; tk� � exp�2ltk� are the known decay kernels,Ng

is the number of grid points, anda(l ) is determined bythe analysis. The second term in Eq. (25) permits aconstant background to be included, e.g.NL � 1 andLk1 � 1 gives the constant backgroundb1.

The problem with solving Eq. (25) fora(l ) is thatan infinite number of oscillatory solutions fora(l ),that have no physical meaning, are consistent with thedata setyk. CONTIN attempts to eliminate oscillatorysolutions through the use of constraints based on (a)APK, (b) statistical prior knowledge, and (c) parsi-mony [94,95]. What this means to the user is thatnon-negativity is enforced fora(l ) and that, of thesolutions not eliminated by (a) and (b), the simplestone is chosen. In this context the simplest solution isthe one with the least detail, i.e. the smoothest onewith the minimum number of peaks. Smoothness isselected by penalizing solutions on the basis of theintegrated squared second derivatives, and the extentof the penalty depends on the regularization parameterchosen.

It is straightforward to use CONTIN in DOSYapplications because the user only has to supply a


threshold value, and the program output provides theLaplace spectruma(l ) essentially ready for display.The major complaint is that the essential smoothingfeature broadens all of the peaks so that even mono-disperse components show considerable linewidths.For many applications this is acceptable as broaderdistributions are usually well represented, and insamples having monodisperse and polydispersecomponents the monodisperse components areusually distinguishable [96].

An example of the CONTIN analysis for a mixturecontaining both monodisperse and polydispersecomponents is shown in Fig. 19. Data was obtainedwith an LED sequence for 75.0 mM glucose and1.00 wt.% methyl cellulose in D2O at 228C. The 26q-values ranged from 278 to 2.50× 104 cm21, withd � 1 to 5 ms,D � 350.0 ms,Te� 50.0 ms, andt �5.80 ms. CONTIN analysis produced 50 point diffu-sion coefficient distributions for each of the 500 chan-nels in the 2D data set, and an interpolation routine


Fig. 20. DOSY display of a mixture of SDS, ATP, and glucose, all at 0.1 M in D2O. The diffusion axis is in units ofmm2 s21. (a) The DOSYspectrum, (b) central region of the same spectrum. Reproduced with permission [98].

was used to generate 512 data points from each 50point distribution. No user input other than a thresholdvalue was required.

This experiment permitted the glucose and methylcellulose peaks to be easily distinguished and thedistribution of diffusion coefficients for the polymerto be measured; but it also demonstrated the unnaturalbroadening and distortion of the glucose peak and themisleading area ratios arising from logarithmicspacing on theD axis. As expected, the treatment ofthe monodisperse component is marginal becausesmoothing by the regularization procedure coupledwith the small number of points produces strangepeak shapes. We note that the HOD peak in this spec-trum was deleted when the DOSY display wasconstructed.

For broad distributions in the presence of noiseCONTIN tends to give smaller average diffusioncoefficients kDl and reduced standard deviationskSDl/kDl relative to the true values. These errorsresult from the tendency of CONTIN to over-smooth a(D) at the small D end while under-smoothing for largeD values [93]. The effect,which is clearly evident in distributions thathave a small secondary peak on the highD side,arises because the penalty for oscillatory solutionsis based on a logarithmic axis. The identitya�D� dD � a�D�Dd ln D suggests thata(D)D ratherthan a(D) should be analyzed on the logarithmicaxis. This change is easily accomplished inCONTIN by setting ‘‘integration off’’ with thecontrol parameter selection IQUAD� 1 so thatcm � 1. With this change most of the ‘‘noise’’peak disappears, but in order to obtain the greatestaccuracy in the computeda(D) function a moresophisticated adjustment of the penalty weightingfactor is required [97] (see Section 6.2.4).

4.2.2.2. MaxEnt A DOSY processing module basedon the maximum entropy method (MaxEnt) is nowincluded in the GIFA software package for NMRanalysis [98,99]. In MaxEnt the idea is to determinethe ‘‘most probable’’ Laplace spectrum bymaximizing the entropy of the spectral distributionsubject to certain constraints. Accordingly, theLaplace spectruma(l i) in Eq. (19) is associatedwith the most probable distributionPi, and the

entropy of the distribution is defined by

S� 2Xni�1

�Pi =F�ln�Pi =F� �26�

where F is a normalization constant. Thenormalization of thePi and the relationship betweenPi and the experimental data setIi specified in Eq. (19)serve as constraints on the allowedPi values. InMaxEnt the functionQ� S2 lx2 is maximized forvarious values ofl wherex 2 represents the Euclideandistance between the data and the Laplace transformof Pi.

MaxEnt is a general purpose analysis method thatcompetes with CONTIN for polydisperse samples anddoes a reasonable job for discrete samples as well. In acomparison, Levenberg-Marquardt (L–M), CONTIN,and MaxEnt were applied to 100 synthetic data setscontaining four exponential components with decayconstants ranging from 0.1 to 13.0 and amplitudesbetween 1000 and 8000 [98]. It was found that L–Mwas much less accurate than either CONTIN orMaxEnt and that MaxEnt was better than CONTINat dealing with weak components. Also, CONTINfrequently failed to find the correct number of compo-nents because of over-smoothing.

MaxEnt analysis for DOSY is illustrated in Fig. 20for a mixture containing SDS, adenosine 50-tripho-sphate, and glucose all at 0.1 M in D2O at 358C[98]. A BPP-LED experiment was performed withd � 3 ms,D � 100 ms, and 28 geometrically spacedgradient values ranging from 1 to 44 G/cm. TheLaplace inversion was calculated by MaxEnt on 100points with a maximum of 1000 iterations, a jobrequiring 50 min on an R5000 SGI workstation. Theonly user input was to select 50:1 as the S/N thresholdfor the maximum signal in each channel. The zoomedinsert in Fig. 20(b) shows impressive resolution forsuch a robust scheme.

4.2.3. Complete bandshape methodsPFG-NMR has a major advantage over scattering

methods, e.g. dynamic light scattering, because of theadditional information obtained from chemical shifts.The problem of resolving exponential componentscan often be avoided by finding isolated NMR peaksfor individual species. The most desirable case is, ofcourse, to have sufficient resolution to avoid any


overlap. But in the real world overlap often occurs,and it is desirable to have analysis methods that canresolve components with arbitrary amounts of over-lap. The methods reviewed to this point have involvedanalysis at a single chemical shift or a limited range ofchemical shifts and have not made use of the totalinformation available. In this section we considermethods that can analyze the complete bandshape.

4.2.3.1. Multivariate analysis Kubista has proposeda method for the analysis of correlated data sets [100].In particular he imagined two types of spectra beingobtained for a mixture with varying concentrations,and he showed that exact solutions for the ‘pure’spectra of the components could be obtained.Schulze and Stilbs implemented the Kubista scheme(NIPALS) by designing a type of double PFG-NMRexperiment with different gradient amplitudes in thetwo parts [101]. Unfortunately, the two data sets werecompromised by this design and the resulting spectrawere distorted. The idea was revived by Antalek andWindig who pointed out that, because the decays inPFG-NMR are exponential, the required pair of datasets can be obtained from a single experiment, e.g.spectra 1–10 and spectra 2–11 provide twosatisfactory sets [102,103].

Consider experiments givingc spectra havingvpoints each for a mixture withn components. Thecorrelated data sets are represented by:

A � CP; B � CbP �27�whereA andB are data matrices of sizecv, C(cn) isthe concentration matrix,P(nv) is the matrix of purespectra, andb(nn) is a diagonal matrix. (Note that thespectra and the concentrations are completely corre-lated in the two data sets but that there is a scalingfactor defined byb.)

The Kubista scheme, reformulated in terms of thegeneralized rank annihilation method (GRAM)[104,105], has been applied to FT-PFG-NMR datasets for mixtures with considerable success. Thismethod, now known as the Direct ExponentialCurve Resolution Algorithm (DECRA), is direct,does not require a threshold, can deal with any amountof overlap, is not very sensitive to noise, and further,is fast. The requirement of exponential decays may bea limitation in some cases, and tests with mixturescontaining large numbers of components and both

monodisperse and polydisperse components arenecessary; but DECRA appears to be well suited forgenerating data for DOSY displays. The softwareroutines for DECRA, developed for the MatLab envir-onment, are available for independent testing [103].

Recently, an analysis method known as multivari-ate curve resolution (MCR), another descendent ofNIPALS, has been reported for DOSY and GPC-NMR experiments [106]. MCR does not require apriori knowledge except that the spectral intensitiesand diffusion decay profiles must both be non-nega-tive. In particular, the decay profiles are not restrictedto exponentials. This method is, therefore, moregenerally applicable than GRAM; however, if expo-nentials are known to be present, it is expected thatbetter answers can be found by making use of thatknowledge.

Impressive analyses of DOSY data sets by MCRhave been reported, and this method deserves muchwider use. The implementation involves principalvalue analysis (PVA) [107] to obtain eigenvectorsand eigenvalues of a data matrix followed by conver-sion of the abstract vectors into chemical factors. Thegeneration of ‘‘pure’’ spectra and concentrationprofiles is accomplished by a Varimax rotation(VMAX) [107–109] followed by alternating least-squares optimization (ALS) [110]. Software for theMCR analysis was written in GRAMS/386 (GalacticIndustries).

4.2.3.2. Component resolved NMRA global least-squares analysis method labeled Component-Resolved FT-PGSE NMR spectroscopy (CORE) hasbeen reported by Stilbs et al. [111]. Their procedure isto perform a total fit of the 2D raw PFG-NMR data set.This has been implemented in a FORTRAN programthat includes two copies of a direct-searchminimization routine (STEP). One global (higher-level) minimization optimizes the diffusioncoefficients and a second (lower-level) minimizationfits the component amplitudes in each frequencychannel using the higher-level results.

CORE is based on modeling each frequency chan-nel with a number of discrete exponentials (1–5), or ifit is decided that a channel has contributions frompolydisperse components, a sum of Kohlrauch–Williams–Watts (KWW) distributions (1–2) in addi-tion to discrete exponentials (1–3). The operation of


this program implies considerable knowledge of themixture by the user and interactive processing.

CORE has been applied to simulated data and tovarious test samples, and it is reported that globalminimization ‘‘may increase the signal/noise ratioby a factor or more than 10.’’ The global analysis iscomputer intensive, but with the current rate ofincrease in affordable computing power this shouldnot be a problem. The more serious cost is user timeand effort in preparing the input model. Unfortu-nately, the CORE software is not generally available.

4.2.4. Analysis recommendationsThe philosophy of DOSY from the beginning has

been to obtain the most likely distribution of diffusioncoefficients at each chemical shift using any availablemethods or software. As unique solutions cannot beguaranteed when signals from different species over-lap, it is desirable to make use of a variety of analysismethods with complementary strengths and weak-nesses. This is easy to accomplish with available soft-ware and computer workstations. A reasonableanalysis suite might contain the following:

1. single exponential analysis (Levenberg–Marquardt);

2. biexponential fitting routine with rejection rules(perhaps SPLMOD);

3. continuous distribution analysis (CONTIN or simi-lar program).

However, it is essential that the user of such programsbe well informed about artifacts and potential pitfalls.The fact that a spectruma(D) is computed does not byitself prove that it has any physical validity. Therecently introduced MaxEnt package appears to be agood addition to the aforementioned list. It can beused to screen samples and may be able to competewith CONTIN in the analysis of broad distributions.

Choices 1–3 plus MaxEnt will serve very well forhigh-resolution spectra of multi-component mixturesof discrete, low molecular compounds. In contrast,when there is severe overlap and a relatively smallnumber of components, the multivariate analysismethods DECRA and MCR may be good alternatives.Also, brute force analysis by totally fitting 2D datasets with user defined models such as CORE can beuseful in special cases.

5. Effects of chemical exchange

Chemical rate processes have been studied by NMRmethods for decades [112]. In the standard model forchemical exchange, nuclei or groups of nuclei explorea number of sites by means of a Markovian randomprocess. These sites are characterized by spinHamiltonians that may be associated with a chemicalenvironment, e.g. a basic group that can accept aproton, or a molecular configuration, e.g.cis andtrans isomers. Slow exchange is defined as the situa-tion in which spectra, characteristic of the individualsites, can be observed; and fast exchange implies theobservation of some kind of average spectrum. Thepath from slow exchange to fast exchange consists ofline broadening, coalescence, and motional narrowingas the mean lifetimes for occupation of the sitesdecrease. Time is always the variable, and theexchange rates, i.e. inverse lifetimes, are manipulatedby changes in temperature, or perhaps concentrationsfor intermolecular reactions.

In diffusion NMR new possibilities arise because ofthe importance of sites that differ in hydrodynamicproperties with or without differences in spin Hamil-tonians. Also, in PFG-NMR for DOSY applications,the variable isq2 rather than time, and there is nothinganalogous to lifetime broadening in the diffusiondimension [113]. Time does enter through the storagedelayT, and the variation ofT sometimes permits theobservation of slow and fast exchange limits withoutchanging the physical condition of the sample.

5.1. Exchange effects in diffusion spectra

Here we consider the effects of chemical exchangeon longitudinal magnetization during the diffusionsensitive intervalT in STE type experiments (Fig. 2)with t; d p T so that T < D. Taking into accountdiffusion, flow, and chemical exchange, the rate ofchange of the nuclear magnetizationMn(r,T) for thenth species with spatial coordinatesr is given by[1,27]:

dMn�r ;T�dT

� 21

T1nMn�r ;T�2 7·Jn�r ;T�

1Xm

KnmMm�r ;T� �28�

where the rate constants arekn � 2Knn and


knm� Knm, and the fluxJn�r ;T� is defined by:

Jn�r ;T� � 2Dn7Mn�r ;T�1 Mn�r ;T�vn�r ;T�: �29�In Eq. (29)Dn andvn�r ;T� are the diffusion coefficientand velocity for thenth species, respectively.

The calculation of the STE amplitude is simplifiedby taking the spatial Fourier transform of Eq. (28) toobtain the linear algebraic equations:

dMn�q;T�dT

� �iVn 2 Rn�Mn�q;T�1Xm

KnmMm�q;T�

�30�where

Mn�q;T� �Z

exp�2iq·r�Mn�r ;T�dr : �31�

In Eq. (30)Rn � �1=T1n�1 Dnq2 describes relaxationandVn � qvn is the frequency resulting from uniformflow in thez-direction. These equations and their solu-tions have been discussed by Ka¨rger, et al. [15], andsimilar equations for nuclear magnetic relaxation inmultiple phase systems were previously treated byZimmerman and Brittin [114].

Here we solve Eq. (30), neglecting 1/T1nandVn, for

the Fourier componentsMn(q,T) that determine theamplitude of the STE. Real applications often involve

two-site exchange, and for this case analytical solu-tions can be obtained. Thus Eq. (30) reduces to

dM �K;T�dT

� LM �K;T� �32�

where

L �LAA LAB

LBA LBB

!�

2kA 2 RA kB

kA 2kB 2 RB

!:

�33�The matrix elements [exp(LT)]AB are available in theliterature [1], the required combinations, e.g. AA1AB for MA, for 1D NMR spectra are:

MA � MA0

22�dMA0 2 kBMB0�

2D

� �exp��2s 1 D�T�

1MA0

21�dMA0 2 kBMB0�

2D

� �exp��2s

2 D�T�; �34a�

MB � MB0

21�dMB0 2 kAMA0�

2D

� �exp��2s 1 D�T�

1MB0

22�dMB0 2 kAMA0�

2D

� �exp��2s

2 D�T�; �34b�

where MA0 � MA(q,0), MB0 � MB(q,0), and thesymbolss , d , andD are defined by:

s � 12�kA 1 kB 1 DAq2 1 DBq2�

d � 12�kA 2 kB 1 DAq2 2 DBq2�

D ��d2 1 kAkB

q(The functionsd andD should be distinguished frompreviously defined time intervals that are associatedwith the same symbols.)

If the chemical shift difference between the sites ismuch larger than the exchange rate, then Eqs. (34a)and (34b) describe the dependence of the individualpeak areas on exchange rates and diffusion coeffi-cients. We consider the special case in which the


Fig. 21. Echo amplitude versusq2T for two-site exchange calcu-lated with Eq. (35) withDA � 2.0, DB � 0.1, kA � 10, andMA �0.40.

chemical shift difference is zero or is much less thanthe exchange rate so that a single line is observed inthe NMR spectrum. In this situation the diffusionspectrum is calculated with the sumMA�q;T� � MA�q;T�1 MB�q;T�:

M � M0

21

L

2D

� �exp��2s 1 D�T�

1M0

22

L

2D

� �exp��2s 2 D�T� �35�

whereM0 � MA0 1 MB0 and

L � d�MB0 2 MA0�1 MA0kA 1 MB0kB:

It is should be noted that the right side of Eq. (35) isnot a simple sum of exponentials because the variableq appears in the coefficients as well as the exponents.

In this simple example the probability of occupa-tion of thenth (n� A,B) site ispn and the mean life-time of a spin in thenth site is tn � 1=kn. Also,k � kA =pB � kB=pA � 1=t, pA � tA =�tA 1 tB�, andfor convenience we letMn0 � pn. Another importantparameter isN � kT/2 that can be interpreted as themean number of times that a spin changes sites duringthe intervalT. Also, the storage time can be written asT� tA 1 tB wheretA andtB represent the total amountsof time that a spin occupies sites A and B, respec-tively. It should be clear thattA and tB are related tothe equilibrium constantKeq� PB=PA � tB=tA but

give no information about the magnitude of theexchange rate or the magnitudes of the mean life-times.

In attempting to understand the two-site exchangeproblem, it is useful to plot ln(M) versusq2T for arange of T values as shown in Fig. 21. The slowexchange limit is captured in the upper most curvewhereT , 2/k and N , 1. It is essentially the sumof two exponentials, one associated withDA and theother withDB. The slope of this curve in the limit oflargeq2T values isDslow, which we take to beDB, andthe initial slope is given byDav� pADA 1 pBDB. If theS/N ratio is high, it is possible to estimate thepA andpB values as follows. The intercept, i.e. theq2T � 0value, of the line with slopeDav is taken to be ln(MA 1MB) while the intercept of the slow exchange lineextrapolated from largeq2T values is equal toln(MA). It is also possible to use the latter interceptfor a set of curves near the slow exchange limit toestimate the exchange rate as intercept� constant2kBT. The lowest curve, actually a straight line withslopeDav, represents the fast exchange limit obtainedby settingT . 20/k so thatN . 10.

A DOSY experiment on this two site system would,under favorable conditions, yield a diffusion spectrumwith two peaks in the slow exchange limit. Similarly,we would expect the fast exchange limit to give adiffusion spectrum with a single peak. In the


Fig. 23. Stack plot of1H spectra of Vancomycin and DDFA mixtureas t increases: (a) LED pulse sequence (t increases from 0.3 to6.2 ms) and (b) BPP-LED pulse sequence (t increases from 0.4 to6.4 ms).T� 100 ms,g� 5.8 G/cm, andTe� 5 ms in both experi-ments. Reproduced with permission [33].

Fig. 22. Diffusion spectra for two-site exchange with equal popula-tions (pA � 0.5). The effective rates are indicated by values ofN � 0:5�kA 1 kB�T. Reproduced with permission [113].

intermediate exchange regime DOSY analysis islikely to fail or to indicate extreme polydispersity.However, the exact diffusion spectra can be derivedfrom Eq. (35) either by directly performing the ILT toobtain analytical expressions [115] or by substitutingiv for q2 and performing the inverse Fourier transfor-mation of the resulting expressions with respect tovto obtain the diffusion spectra [113].

A two-site spectrum computed withpA � pB � 1/2is shown in Fig. 22 forN values ranging from 2 to 50[113]. ForN , 1, delta functions appear atDA andDB

with intensities proportional to exp(2kAT) andexp(2kBT), respectively. The line widths for thesharp peaks in Fig. 22 result from the numericalcomputation and are not meaningful. ForN $ 5,intensity with a Gaussian profile grows at the positionof the average diffusion coefficient (DA 1 DB)/2, andas N continues to increase this peak narrows whilemaintaining a standard deviation that is proportionalto N21/2. The calculation forpA ± pB is similar exceptthat for large values ofN the intensity is centered atpADA 1 pBDB.

5.2. Artifacts from chemical shift encoding

It was previously mentioned that chemical shiftencoding can seriously affect echo amplitudes inFT-STE and FT-LED experiments when chemicalexchange is present [33]. To see how this comes

about consider an STE experiment (Fig. 2) for spinswith the frequency offsetvA. After the first 908 RFpulse, the magnetization precesses with the frequencyvA plus a contribution from the field gradient depend-ing on the displacement in thez-direction. The second908 RF pulse stores the y-components of magnetiza-tion in the z-direction and the remainingx and ycomponents are eliminated by phase cycling or homo-spoil pulses. For spins in the layer atz to z 1 dz thestored magnetization is described by cos�vAt 1 qz�where q � ggd . The third 908 RF pulse returnsmagnetization to theyz-plane and precession reducesthe y-component by an additional factor ofcos�vAt 1 qz� at the echo. For a sample of lengthLthe amplitude of the peak atvA in the absence ofrelaxation is given by

1L

ZL=2

2 L=2cos2�vAt 1 qz�dz

� 12

112

cos�2vAt�sinc�Lq=p�:�36�

The last factor on the right hand side,sinc(Lq/p ) �sin(Lq)/Lq, is unity when the gradient vanishesthus permitting the amplitude of the echo todepend on the frequency offsetvA. As previouslynoted, theq � 0 point should be avoided in diffu-sion measurements. However, as soon asLqexceeds 2p , i.e. the pitch of the magnetization


Fig. 24. DOSY display for protons on alpha carbons in a mixture containing alanine (A), glutamine (Q), and lysine (K) in D2O. Reproduced withpermission [53].

helix becomes less than the sample length, themodulation term becomes very small and vanishescompletely with significant gradients.

When chemical exchange occurs, intensitymodulation reappears for the exchanging groupseven in the presence of gradients. The previousdiscussion is easily extended to two-site chemicalexchange. Suppose that the sites have differentchemical shifts and that exchange can beneglected during the encode and decode intervals.During the diffusion delayT, exchanges do occurso that the spins in site A at the time of the third 908RF pulse may have occupied either site during theencode interval. In terms of the offset frequencies(vA, vB) and the mole fractions (xA, xB), the peak

amplitude atvA is then given by

1L

ZL=2

2L=2�xAcos�vAt 1 qz�

1xBcos�vBt 1 qz��cos�vAt 1 qz�dz

� 12�xA 1 xBcos�vAt 2 vBt��:

�37�

The modulation predicted by Eq. (37) has beenobserved in affinity NMR experiments where thepulse durationd was set equal to encode intervalt ,and spectra were obtained as a function oft . Anexample (Fig. 23) is provided by a mixture ofVancomycin (Sigma) and the tetrapeptide


Fig. 25. 500 MHz DOSY display of a perchloric acid extract of gerbil brain in D2O. Selected assignments are: ac� acetate: ala� alanine: cho�choline: cr� creatine: cre� creatinine: etn� ethanolamine: GABA� g-aminobutyric acid: glu� glutamine: GPC� glycerophosphocholine:lac� lactate: m-ino� myo-inositol: NAA� N-acetylasparate: succ� succinate: and tau� taurine. Reproduced with permission [118].

Asp–Asp–Phe–Ala (DDFA) in D2O [33]. The signalsfor the free DDFA (1.30 ppm) and the boundDDFA (0.55 ppm) that are involved in exchangedecay with increasingt because of diffusion andtransverse spin relaxation, but also oscillate at 373 Hz,their chemical shift difference in the 500 MHz spec-trometer. The other (non-exchanging) peaks show nooscillation.

Intensity oscillations can be useful in identify-ing exchanging pairs and editing spectra, but canalso be deceptive if not properly identified. Fortu-nately, the oscillations can be completely avoidedby using BPP based sequences that refocus chemicalshifts.

6. Applications of 1D and 2D DOSY

6.1. Discrete samples

Most DOSY applications to date have concerneddiscrete mixtures. Selected examples are presentedin this section. The reader is referred to the literaturefor additional DOSY examples including studies ofchloroaluminate melts [116] and perturbed cis-transisomerization of phenylalanylproline [117].

6.1.1. BiofluidsBiofluids here refer to multicomponent mixtures of

amino acids and other molecules commonly found in


Fig. 26. (a) DOSY display processed with SPLMOD for a mixture containing 5.0 mM methanol, 10.0 mMiso-propanol, 10.0 mMt-butanol,and 10.0 mMneo-pentanol in D2O, and (b) DOSY display for a mixture with the same solute concentrations plus 0.150 M DTAB. Reproducedwith permission [61].

tissue extracts. These examples emphasize the optimi-zation of resolution in the chemical shift dimension tominimize overlap and the incorporation of experimen-tal design features to maximize the accuracy of theextracted diffusion coefficients to enhance resolutionin the diffusion dimension.

A mixture of alanine (A), glutamine (Q), and lysine(K) in D2O provides a good illustration of the neces-sity of high resolution in the DOSY experiment as thediffusion coefficients are much too similar to be sepa-rated when there is overlap. The display in Fig. 24shows part of the 2D spectrum (protons ona -carbons)for this mixture [53]. As previously described, thespectrum was constructed with Gaussian componentscentered on the computed values of the diffusion coef-ficients and having widths equal to the errors reportedin the data analysis.

Sample spinning with the ‘‘stop-and-go’’ systemand the BBP sequence were used to achieve highresolution (Dn , 0.5 Hz), and good line shapes inthe chemical shift dimension were obtained withoutspecial filtering of the FIDs. With no overlap and S/Nratios greater than 100, data analysis with SPLMODprovided diffusion coefficients with errors of a fewpercent, but even better results would be expectedfrom an L–M analysis limited to single components.For this work a Bruker AC250 spectrometer with acustom built 10 mm diffusion probe was used. Theother experimental parameters were:d /2� 1 ms,t �

1.5 ms, 27 gradient amplitudes (g � ^ 0.03 to ^

0.62 T m21) andTe� 20 ms.Impressive examples of mixture analysis have also

been reported under the label high-resolution DOSY(HR-DOSY) [54,64,65,118]. An example is shown inFig. 25 for a perchloric acid extract of gerbil brain[118]. Fifteen spectra were acquired with the LEDsequence on a 500 MHz spectrometer using 512 tran-sients per spectrum. Post processing with referencedeconvolution (FIDDLE) was based on the referenceline for DSS (sodium 3-trimethylsilypropanesulfo-nate), and the DOSY display was constructed frommeasurements on 116 peaks. The contour lengths(errors) in the diffusion dimension are somewhatlarger than in test cases because of poorer S/N ratiosin the 1D NMR spectra. The assignments of a numberof metabolites are shown in Fig. 25, and as expectedthe diffusion coefficients are well correlated withmolecular sizes.

6.1.2. Separation by means of hydrophobicityFT-PFG-NMR has proved a powerful tool for the

study of solubilization [119]. DOSY based on high-resolution 1D NMR spectra extends this work andprovides ‘‘at-a-glance’’ recognition of the compo-nents in complex mixtures and their interactions. Ina number of such cases complete resolution of peakshas been obtained even at 250 MHz through the use ofthe ‘‘stop-and-go’’ spinner. The example shown in


Fig. 27. DOSY display for a solution containing 2 g/dl BSA, 2 g/dl SDS, and 0.01 Mb-mercaptoethanol in phosphate buffer (pH� 7.2, ionicstrength� 0.020 M), The unlabeled line represents the reaction product, HOCH2CH2SSCH2CH2OH. Reproduced with permission [145].

Fig. 26(a) for a mixture of 5.0 mM methanol,10.0 mM iso-propanol, 10.0 mM t-butanol, and10.0 mMneo-pentanol in D2O does not require espe-cially high chemical shift resolution, but does sufferfrom accidental overlap in the diffusion dimensionbecause of the near equality of the diffusion coeffi-cients oft-butanol andneo-pentanol [61].

Fig. 26(b) shows that the effective diffusion coeffi-cients can be manipulated by adding the micelle form-ing surfactant, dodecyl trimethylammonium bromide(DTAB). This is an example of fast chemicalexchange where solute molecules partition into themicelles according to their solubilities but undergoexchange between the interior of the micelle and thebulk solution. In the fast exchange limit DOSY onlyreports the time average diffusion coefficientDj forthe jth solute

Dj � pjDmicj 1 �1 2 pj�Dfree

j �38�

wherepj is the degree of solubilization, andDjmic and

Djfree are the tracer diffusion coefficients for solubi-

lized and free molecules, respectively. It should berecognized thatDj

mic is equal to the diffusion coeffi-cient of the micelle as the average displacement of amicelle during the diffusion timeD is much greaterthan the radius of the micelle.

The experiment was performed at 250 MHz withthe LED sequence. Twenty-eight spectra wereacquired withq-values ranging from 208 to 8.34×103 cm21, d � 1 ms,t � 2.0 ms,D � 100 ms, andTe� 50 ms. SPLMOD (N� 2) was used for the dataanalysis. Complete resolution of the diffusion peakswas obtained by addition of 0.150 M DTAB, and thevalues ofpj shown for the solutes in Fig. 26(b) werecalculated with Eq. (38) usingDj

free values from theDTAB free solution (Fig. 26) and allDj

mic valuesset equal to the diffusion coefficient measured forDTAB.


Fig. 28. 1H DOSY display for a mixture containing ‘‘Tinuvin P’’ (1 ), ‘‘Irganox 1330’’ (*), and ‘‘Irganox 1098’’ (W). The solvent is labeled Sand silicon grease contaminant is labeled Sg. Reproduced with permission [146].

6.1.3. Equilibria involving sodium dodecylsulfate(SDS) and bovine serum albumin (BSA)

SDS–protein interactions are important in anumber of fields. For example protein/SDS-polyacryl-amide gel electrophoresis (PAGE) is routinely used todetermine protein molecular weights and to separateproteins. Also, SDS is known to be a potent proteindenaturant. However, the protein–SDS complexesand their electrophoretic mobilities are not completelyunderstood, and this has provided motivation forphysical studies. The aim of a DOSY study of BSA-SDS equilibria was to determine the binding isotherm,i.e. the fraction of surfactant bound to a protein versusthe surfactant concentration.

The DOSY display for a D2O solution containing

2 g/dl BSA, 2 g/dl SDS, and 0.01 Mb -mercaptoetha-nol in phosphate buffer (pH� 7.2, ionic strength�0.020 M) at 298^ 1 K is shown in Fig. 27. Thisspectrum was obtained from 20 to 40 FIDs in anLED experiment withq-values ranging from 2.1×104 to 1.9× 106 m21 andD values typically between50 and 80 ms. The diffusion dimension reveals BSAsignals from the slowly diffusing BSA–SDS complex,a single SDS peak, and weak peaks from the denatur-ing compound and the unlabeled reaction product,HOCH2CH2SSCH2CH2OH. The single SDS peakindicates the fast exchange limit for the SDS exchangeamong monomers, micelles, and the BSA–SDScomplex. This limit reduces the amount of informa-tion available and requires that the analysis begin with


Fig. 29. Structures for polymer additives. Reproduced with permission [146].

an expression for the observed average diffusion coef-ficient kDl for SDS:

kDl � PproDpro 1 PmonDmon 1 PmicDmic �39�with

Ppro 1 Pmon 1 Pmic � 1 �40�where Ppro, Pmon, and Pmic are the fractions of SDS

molecules on the protein, as monomers, and inmicelles, respectively; andDpro, Dmon, and Dmic arethe associated self-diffusion coefficients. The calcula-tion of the desired quantityPpro requires the determi-nation of all the diffusion coefficients and eitherPmon

or Pmic. Both kDl andDpro can be read directly fromFig. 27, butDmon andDmic must be determined fromadditional experiments and the use of the mass action


Fig. 30. DOSY display for medium unilamellar vesicles in a mixture containing 30 mM total lipid (POPC) with 100 mM sucrose in D2O.Reproduced with permission [70].

Fig. 31. The1H DOSY display (250 MHz) for whole plasma at 408C. The reference is TMA at 3.22 ppm [126].

model [120]. The conclusion of this study was a plotof the binding ratio [bound SDS]/(SDS) versus [totalSDS]/[BSA] in the range 0–6 (g/g) showing a satura-tion binding of 1.9 g bound SDS per gram BSA that isreached at about 4 g total SDS per gram of protein.Also the effective hydrodynamic radius of thecomplex was determined as a function of the numberof bound SDS molecules.

6.1.4. Mixtures of polymer additivesDOSY has been used to analyze mixtures of poly-

mer additives [146]. The diffusion coefficients facili-tated spectral assignments and were especially helpfulfor isolated protons where no J-couplings could beresolved. 2D-DOSY of a mixture containing ‘‘Tinu-vin P’’ (Mn 225) ‘‘Irganox 1098’’ (Mn 637), and‘‘Irganox 1330’’ (Mn 775) all from Ciba in 1,1,2,2-tetrachloroethane-d2 (TCE-d2) is shown in Fig. 28,and the associated molecular structures are shown inFig. 29. The data were acquired with BPP-LED on aBruker AM360 spectrometer equipped with a 5 mmz-gradient Bruker inverse probe. The gradient pulseduration was 2 ms, the diffusion delay was 0.5 or1.0 s, andTe was 15 ms. Typically, 15–20 gradientvalues were used and the values ranged from 0.026to 0.30 T m21.

The diffusion information in Fig. 28 correlates withthe structures of the additives. ‘‘Irganox 1330’’ is star

shaped while ‘‘Irganox 1098’’ has a more extendedlinear shape. The shapes compensate for the differ-ences in molecular weights and produce quite similardiffusion coefficients. Also, it is observed that theexchangeable OH protons (4.96, 5.08, and11.15 ppm) have larger diffusion coefficients thanthe other protons in their respective molecules. Appar-ently the protons are exchanging with protons of waterin the solvent. Note also the well resolved resonancesfor the solvent and silicon grease in the diffusiondimension.

6.2. Polydisperse samples

DOSY is especially useful for samples containingpolydisperse components. With widely available soft-ware, diffusion spectra can be obtained that permitcomponents with broad distributions of diffusioncoefficients to be distinguished from those withnarrow distributions, i.e. the monodisperse compo-nents. Also, with no additional assumptions beyondsmoothness, the shape of distributions of diffusioncoefficients can be measured. In this section DOSYexamples are displayed that have been processed byCONTIN. Similar results can be obtained with otheranalysis programs. In particular we note that a recentDOSY study of polymer mixtures successfully used


Fig. 32. DOSY display processed with CONTIN for a mixture containing CTAB, PVME, and NaSal (see text). Reproduced with permission[96].

both CONTIN and the analysis program NLREG[121].

6.2.1. Phospholipid vesiclesThe first DOSY paper showed a diffusion resolved

spectrum of a sample containing sucrose and phos-pholipid vesicles in D2O [9]. Two sucrose spectrawere evident in the 2D display at diffusion coefficientsof 3.00 × 1026 and 1.86× 1028 cm2 s21, and it wasreasonable to assign the spectra to free sucrose andsucrose entrapped in vesicles, respectively. Dataanalysis in this case was performed with DISCRETEand, consequently, information concerning polydis-persity of the vesicles was lost.

A much more extensive CONTIN based analysiswas performed on palmitoyl-2-oleoyl phosphatidyl-choline (POPC) vesicles with diameters rangingfrom 30 to 100 nm [70]. A DOSY display for sucrosewith medium POPC vesicles is shown in Fig. 30. Atthis level of gain the signals from the vesicle bilayerare almost invisible, the exception being the smallpeak at 3.1 ppm. The bilayer proton signals aremore pronounced for smaller vesicles and in factdominate the spectrum for 30 nm vesicles. From theDOSY peaks it is evident that the effective diffusioncoefficients of the bilayer protons are larger that thediffusion coefficients of the entrapped sucroseprotons. This fact became a focus of the study.

The vesicle study made use of dynamic light scat-tering (DLS) and electron microscopy (EM) as well asCONTIN analysis of PFG-NMR data. As the largestvesicles have diffusion coefficients in the vicinity of2 × 1028 cm2 s21, it is necessary to include largeq-values in the analysis. Here theq-values rangedfrom 2000 to 32 000 cm21 giving grating spacingsof 30 to 2mm. The LED sequence was used becauseBPP-LED was not available and theTe values tendedto be long. Also, at the largest gradient values convec-tive flow was a problem in 10 mm sample tubes butnot in 5 mm tubes. This probably resulted from poortemperature control in the probe.

The major conclusions were the following: (1) Thedistribution of particle diameters for large unilamellarvesicles obtained from the CONTIN analysis of PFG-NMR data agrees quite well with distributionsobtained from DLS and electronmicroscopy eventhough the width from the CONTIN analysis containsa contribution from the smoothing effect. (2) The

apparent diffusion coefficients obtained for phospho-lipids in the bilayers are heavily weighted by contri-butions from the smaller vesicles in the distributionthat tumble more rapidly and give narrower NMRsignals.

6.2.2. Blood plasmaPlasma lipoproteins are microemulsions-like

assemblies consisting of a protein–phospholipidshell with a core of triglycerides and cholesterol.The compositions of the lipoprotein particles arerelated to their densities and sizes, and in generalthe lowest density particles have the largest radii[122]. Lipoproteins have received a lot of attentionin recent years because their concentrations in bloodhave been correlated with the risk of coronary heartdisease. Unfortunately, standard clinical proceduresfor measuring the low density fraction are inconveni-ent and subject to experimental errors.

An alternative NMR method for rapid analysis hasbeen proposed that takes advantage of differences inchemical shifts for methyl1H signals exhibited by thevarious lipoprotein fractions: very low density(VLDL), low density (LDL) and high density (HDL)[123,124]. This is also a potential application forDOSY as the chemical shifts appear to be correlatedwith particle diameters. A DOSY CONTIN generateddisplay for whole plasma with added D2O and tetra-methylammonium chloride (TMA) as a referencecompound is shown in Fig. 31 [125,126]. The experi-ment was performed on a 250 MHz spectrometer with‘‘stop-and-go’’ sample spinning and the LED pulsesequence. Typically, 23 exponentially spaced valuesof q (84–12 540 cm21) were used withd � 2 ms,t �3.5 ms,D � 100 ms, andTe � 50 ms. Water signalswere eliminated in all experiments by continuoussaturation except during data acquisition.

The diffusion coefficients associated with the lipo-protein lipid signals [methyl, methylene, vinyl, andallylic] are all the same order of magnitude, and asexpected the smaller molecules (lactate, EDTA,TMA) have much larger diffusion coefficients [127].As the methyl and methylene signals for the largerlipoproteins are shifted downfield from those of smal-ler particles, we expect the diffusion coefficients tovary with the chemical shifts in the DOSY display.There is a trend toward smaller diffusion coefficientsat the low end but not as pronounced as expected.


This unimpressive result prompted DOSY analysisof VLDL, LDL, and HDL fractions prepared by agar-ose gel chromatography and kindly supplied by J.D.Otvos. CONTIN reports momentsMn of peaks in thediffusion spectrum:

Mn �Zlna�l�dl �41�

Here the moments for the methyl peaks were used tocalculate average diffusion coefficients, i.e.kDl � �D 2 d=3�21M1=M0. The resulting mean diffu-sion coefficients yield uncorrected Stokes–Einsteinradii of 106 A (VLDL), 46 A (LDL), and 23 A(HDL). The ratios of these numbers are in reasonableagreement with the literature [122], but the calculatedradii are considerably smaller than expected. Thismay indicate the presence of breakdown products orthe operation of relaxation effects that emphasize thecontribution of smaller particles. Also, chemicalexchange cannot be ruled out. Successful completionof this work demands the dispersion and sensitivity ofa high field spectrometer.

6.2.3. The viscoelastic CTAB/sodium salicylate/watersystem

The well-known viscoelastic micellar system hexa-decyltrimethylammonium bromide (CTAB)/sodiumsalicylate/NaSal/ water has been found to undergo aremarkable polymer-induced non-Newtonian to

Newtonian transition [128]. Upon the addition ofpoly(vinyl methyl ether) a striking decrease in visc-osity occurs that is thought to result from the breakingup of long rod-like micellar aggregates. A proposedmodel for the transition includes the formation ofspherical micelles ‘‘wrapped around’’ by the polymerand the retention of bound Sal2 ions. A DOSY studyof this interesting system included the CONTINprocessed display shown in Fig. 32 for a Newtonianmixture containing 25 mM CTAB, 15.0 mM NaSal,and 7.40 mg/ml PVME at 308C [96].

The spectrum shows PVME peaks at 3.28 and1.69 ppm with a broad distribution of diffusion coeffi-cients. At approximately the same average diffusioncoefficient, we find CTAB peaks (2.97, 1.28, and0.87 ppm) with much narrower distributions. In fact,the diffusion distribution for these peaks resultsentirely from the smoothing effect of CONTIN. Thisis consistent with the model if the CTAB ionsexchange rapidly between different polymer chains.Also, we find Sal2 at about 8× 1026 cm2 s21 whichis about 10 times slower than the diffusion coefficientfor Sal2 in the absence of CTAB. With the helpof Eq. (38) it is estimated that the fraction ofbound Sal2 is about 0.95, again in agreementwith the proposed model. The diffusion coefficientof CTAB in the presence of PVME is greatlyincreased and at 408C is consistent with sphericalCTAB micelles.


Fig. 33. The simulated distributiona(D)D with M0 � 105 ands � 3.2 (solid line), computedcm values (dashed line), and the effectivedistribution to be estimated by CONTIN with the computedcm values (dotted line). Cubic splines were used to obtain smooth displays from the31 cm values. Reproduced with permission [97].

6.2.4. Molecular weight distributionsDOSY analysis for polydisperse samples produces

a diffusion or Laplace spectruma(D) that is actually adiffusion coefficient distribution. The question here iswhether DOSY data can also be used to obtain mole-cular weight distributions (MWD). Other popularphysical methods for the determination of MWDs(M . 25 000) include gel permeation chroma-tography (GPC) and DLS. As with PFG-NMR, thesemethods measure transport rates that depend onmolecular size and require calibration. Also,hydrodynamic methods such as DOSY and DLSyield accurate parameters for individual moleculesonly in the limit of low concentrations [129]. Inthe following we assume that concentrations aresufficiently low that microaveraging effects are negli-gible [130,131].

The only reported DOSY MWD determination wasbased on a diffusion spectrum,a(D), computed withCONTIN [97]. In that work it was found that thestandard application of CONTIN leads to unaccepta-ble errors ina(D) because ofD dependent under-smoothing of the distribution as discussed in Section4.2.2.1. With CONTIN the inversion of the data setyk

is handled by solving the set of linear algebraic equa-tions shown in Eq. (25). A major improvement of theanalysis for broad distributions is obtained by settingcm� 1 so thata(D)D rather thana(D) is analyzed onthe logarithmic axis as previously discussed. The factthat a scale change improves the results suggests thatcm be varied continuously to improve the analysis inregions where the amplitude ofa(D) is small. Theprocedure adopted was to replacecm with

�Dm=Dmax�xm

wherexm was incremented from1 2 to 2 2 as log(D)ranged from 2 12 to 2 9. In general the range oflog(D) should be set as narrow as possible while stillcovering the distribution to zero amplitude at bothlimits. With reasonable limits the amplitude ofG(D)D is typically about 10% of the maximumvalue at thexm � ^1 points. This givescm � 1 nearthe maximum of the distribution (whereD � Dmax)and provides appropriate amplitude enhancementwhere the amplitude is small. The idea is that thecm

values can be chosen to emphasize the region of inter-est, and CONTIN will then return the distributiona�Dm�Dm=cm from which a(Dm) can be extracted.

The mass weighted distribution,W(M), can becomputed directly with the equationW�M� �a�D�udD=dMu if the relation betweenD and M isknown for the particular system. Also, the numberweighted MWD, n(M), can be obtained becausen�M� �W�M�=M. For the special case of the Gaussianrandom coil the scaling relation has the form

D � AMa �42�but this cannot be assumed for real polymers.

These ideas were tested by computer simulation.The MWD, W(M), was represented by log normaldistributions with the reference molecular weightM0

ands values ranging from 1.25 to 3.25. The diffusiondistributionsa(D) were obtained from theW(M) bymeans of the scaling law in Eq. (42) withA � 1027.5

anda � 2 0.6. Finally, PFG-NMR dates sets werecomputed with Eq. (19) for sets ofq-values and Gaus-sian random noise with RMS deviation of 1023 wasadded. The simulated data sets were then processedwith standard CONTIN, CONTIN withcm � 1, andCONTIN with computed cm values to obtain


Fig. 34. Pulse sequence for 3D DOSY-COSY experiment [Phases: P1: 0, P2: 0, P3: 016 216, P4: 0, P5: 0, P6: 08 28, P7: 04 14, P8: 0213 1321, Rec:0022 1133 2200 3311 0033 2200 3311 0022 1133]. Reproduced with permission [133].

‘‘experimental’’ diffusion distributionsa(D). Fig. 33shows a simulated distributiona(Dm)Dm, thecomputedcm values, and the functiona�Dm�Dm=cm

processed by CONTIN.From the computed a(D) distributions the

quantities kDl, standard deviation (SD), the MWDW(M), mass average molecular weightMw, andnumber average molecular weightMn were computedas functions ofs . It was found that these quantitiesobtained with all versions of CONTIN agreed almostperfectly with values computed directly from theinitial log normal distribution fors , 2.0. However,for larger values ofs , standard CONTIN showedlarge and erratic errors, the errors were moderatewith cm� 1, and with computedcm values the agree-ment was very good. CONTIN with computedcm

values has been successfully used to determineMn

and Mw values for commercial samples of polydis-perse poly(ethylene oxide) [97].

Recent work by Jerschow and Mu¨ller also demon-strates that 2D DOSY can be effective in resolvingmixtures of polymers [121]. They have obtainedimpressive DOSY displays for mixtures of high andlow molecular weight polypropylene and mixtures ofpolypropylene and polystyrene from experimentsperformed on a 600 MHz spectrometer by means ofthe BPP-LED sequence. The ILTs were performed


Fig. 35. A 3D view of the DOSY-COSY data. In the F1 and F2dimensions the chemical shifts range from 1.0 to 4.0 ppm, and thediffusion axis ranges from 3.0× 10210 to 9.0× 10210 m2 s21. Repro-duced with permission [133].

Fig. 36. (a) 2D COSY display obtained with DOSY-COSY pulse sequence having the lowestq value, and (b)–(d) the 2D COSY planes (slices)from the 3D DOSY data set that correspond to the diffusion coefficients of alanine, glutamine, and arginine, respectively. Reproduced withpermission [133].

with both CONTIN and NLREG [132]. Theseprograms make use of regularized least squares proce-dures but use different methods to determine the regu-larization parameter. In the examples presented, theNLREG provided somewhat better separations thanCONTIN.

7. 3D DOSY

Here we describe 3D DOSY experiments in whicha diffusion coordinate is added to conventional 2D-NMR. The initial motivation for the development of3D DOSY was to obtain additional dispersion ofNMR peaks in order to avoid overlap as the highestaccuracy in DOSY analysis is obtained when singlecomponent decays can be assumed. The analysis ofdata cluster intensity (cross-peaks in the 2D-NMRspectrum) versusq2 then yields diffusion peaks thatlie in layers corresponding to the diffusion coefficients

of the various species in solution. Here as elsewherewe emphasize that the acronym DOSY implies thatthe third dimension consists of peaks on a diffusionaxis and not simple signal attenuation versusq2.

The creation of 3D DOSY requires (a) the design ofa combined 2D-NMR/DOSY pulse sequence and (b)the development of algorithms for grouping datapoints in clusters corresponding to cross-peaks,inverting cluster intensity versusq2 data sets, andconstructing 3D spectra. In the original report it wasemphasized that any 2D-NMR experiment can beused in this combination and 3D COSY DOSY wassimply used as an illustration [133]. Also, HMQC-DOSY was reported and NOESY-DOSY andEXSY-DOSY were suggested. In the following avariety of 3D DOSY experiments from differentlaboratories are reviewed. Some of the followingexamples demonstrate only the PFG-NMR step ofobtaining q-dependent attenuation of cross-peaks,and thus do not fully implement 3D-DOSY.

7.1. COSY-DOSY

A pulse sequence for COSY-DOSY constructed bylinking BBP-LED and COSY sequences is shown inFig. 34 [133]. After the eddy current delayTe, theCOSY evolution time t1 and acquisition timet2provide, through Fourier transformation, the twochemical shift dimensions. A 32-step phase cycle(Fig. 34) was devised to select the coherence transferpathway for phase modulated 2D COSY [1] and toencode diffusion information.

A FORTRAN program (DOSY3D) was used withthe data set having the smallest (non-zero) value ofq to assign the pixels, above a preselected thresholdvalue, to cross-peaks. This assignment grid was thenused for the remainder of the data sets. The analysiswas performed by inverting either pixel intensityversusq2 decays with SPLMOD, or integrated cross-peak integrals (volumes) versusq2 by means ofthe L–M algorithm. The latter method is preferablewhen S/N ratios are modest and there is little cross-peak overlap.

COSY-DOSY was illustrated with a mixture ofthree amino acids, alanine (A), glutamine (Q), andarginine (R), each at a concentration of 100 mM inD2O. Eight q values (184–1966 cm21) with D �0.104 s were used in the DOSY preparation period


Fig. 37. A pulse sequence for X-H COSY-DOSY or HMQC-DOSY[136].

Fig. 38. A pulse sequence for gHMQC-DOSY. Delayst � 1/4JCH,gradient pulse length isd , separation of midpoints of outer gradientpulses isD. The simple, time saving phase cycle was: first13C 908

pulse follows (1 x, 2 x) or 02 on successive transients, second1H908 pulse follows 0022 1133, and for the receiver 0220 3113. Repro-duced with permission [137].

to generate eight magnitude mode 2D spectra with512 × 256 data points each. These were processedwith DOSY3D and the L–M algorithm to obtain 64points in the diffusion dimension and 512× 256× 64total data points. Then Felix (Hare Research, Version1.1) was used to produce the 3D display shown in Fig.35. This view is not very useful, but the box can berotated on a workstation screen to reveal the planes ofpoints for the various species.

Fig. 36(a) shows the 2D COSY spectrum of themixture (lowestq value) and the COSY planes forthe individual amino acids, obtained by summingover a few data points in the diffusion dimension,are shown in Fig. 36(b)–(d). The circle in Fig. 36(a)indicates the resolved cross-peaks associated with Qaand Ra that overlap exactly in the 1D spectrum. Notethat the 2D planes appear at the correct positions inthe diffusion dimension for alanine, glutamine, andarginine. Also, the COSY planes are identical withCOSY spectra obtained for the separated compoundsexcept for the diagonal peak ofa -protons of

glutamine and alanine. This overlapping peak, thatcannot be resolved in the diffusion dimension, wasincluded in the planes for both glutamine and argininethrough expanded integration ranges in the diffusiondimension.

As expected 3D DOSY has higher effective chemi-cal shift resolution than 2D DOSY and single expo-nentials are usually adequate. Resolution of planes inthe diffusion dimension, of course, depends on stan-dard deviations of the single components fits, which infavorable cases for1H NMR are less than 2%. Thereis, however, a cost in time for 3D DOSY. With 124FIDs for each of the eightq values, data acquisitionrequired 36 h on an old 250 MHz spectrometer. Witha modern 500 MHz system, the same S/N ratio couldbe obtained in less than an hour

7.2. HMQC-DOSY

We have seen that both the use of heteronuclear(13C) chemical shifts and cross-peaks in 2D-NMRreduce peak overlap and enhance DOSY analysis.Even better resolution is possible through the combi-nation of DOSY with 1H–13C shift correlationspectroscopy. The problem, of course, is low sensitiv-ity resulting for13C signals compounded by extensivedata acquisition required for 3D DOSY. X-H shiftcorrelation with inverse detection, e.g. HSQC [134]or HMQC [135], may provide the best chance ofsuccess because these experiments benefit from the


Fig. 39. Pulse sequence for NOESY-DOSY. h1 and h2 are homo-spoil type gradients andtm and t1 are the NOESY evolution andmixing times, respectively. Reproduced with permission [139].

Fig. 40. TOCSY-DOSY sequence. The narrow and wide filled rectangles represent 908 and 1808 pulses and the rectangles with diagonal linesare ramped gradients. The phase cycle used was: P1: 0022; P2: 1230; P3: 222; rec: 0220 2002 2002 0220. Reproduced with permission [140].

large magnetic moment of the proton for both spatialencoding and data acquisition.

HMQC-DOSY as implemented by Wu et al. [136]combined BPP-LED and X-H COSY is shown in Fig.

37. The gradient pulses after the second1Hp /2 RFpulse are optional homospoil pulses to minimizephase cycling requirements. This experiment wasconsidered to be impractical on our 250 MHz spectro-meter, and was demonstrated on a 500 MHz spectro-meter in the University of North Carolina MedicalSchool. A sample containing alamine, glucose,sucrose, and caffeine all at 500 mM in D2O wasanalyzed in the chemical shift ranges 3.0–5.5 ppmfor 1H and 20–100 ppm for13C to produce a 3Ddisplay similar to the one shown in Fig. 35. However,the S/N ratios obtained with an acquisition time of40 h were less than 10, and the resulting errors indiffusion coefficients were approximately 10%.Unfortunately the limited resolution in the diffusiondimension did not permit complete separation of the2D spectra for all of the compounds.

A successful implementation of HMQC-DOSYwas recently reported by Barjat et al.[137] Thatwork made use of the combination of GCSTE [54]and conventional gHMQC [138] shown in Fig. 38.The gradient pulses during thetD delay provideeddy current compensation, but otherwise areoptional. As experimental time is crucial, the number


Fig. 41. 3D1H TOCSY-DOSY display for a mixture of propylene and decane. Reproduced with permission [140].

Fig. 42. (a) The PFG-DQS sequence for merged DOSY, (b)sequences for NOESY/ROESY and TOCSY diffusion experiments[141].

of increments for the diffusion dimension and thenumber of phase cycling steps were necessarily keptsmall. Gradient pulses were used in the coherenceselection part of the HMQC sequence to reducet1noise and to reduce the need for phase cycling.Another, important experimental point concernsproton decoupling. The broadband adiabaticWURST method with low power was selectedbecause the heat input from standard decouplingsequences, e.g. WALTZ, and the resulting convectioncurrents could not be tolerated.

The DOSY-gHMQC experiment was applied to atest mixture containing quinine (30 mg), geraniol(20ml), camphene (19 mg), and TMS in deuteratedmethanol room temperature. Data were collectedwith a Varian INOVA 400 spectrometer equippedwith a 5 mm indirect detect PFG probe. The storagedelaytD was kept short (43 ms) to avoidT1 relaxation,and only five increments of the gradient pulse ampli-tude were used. Even so, data acquisition required17 h.

This experiment and the homonuclear COSY-DOSY of the previous section, while impressive inresolving complex mixtures, are very time consumingeven with test samples containing relatively highconcentrations.

7.3. NOESY-DOSY

Gozansky and Gorenstein combined LED andNOESY sequences as shown in Fig. 39 to obtain diffu-sion labeling of NOE cross-peaks [139]. They did notactually implement 3D DOSY as the data were notinverted and diffusion spectra were not generated. Thediffusion labeling of cross-peaks was demonstratedfor a D2O sample (0.6 ml) of 1.6 mg d(AG) dinucleo-tide and 5 mg 14-mer duplex d(ACAATATA-TATGR)2 in a phosphate buffer (pH� 7.4). Thedata were collected with a 400 MHz spectrometerwith a wide bore magnet and an actively shieldedgradient probe. The gradient amplitudes rangedapproximately 4–40 G/cm in eight equal steps,t �6.6 ms,T� 20.0 ms,Te� 50 ms,h1� 3.0 ms,h2�97.0 ms, andtm � 100 ms. Diffusion coefficientsobtained from cross-peak attenuation were found tolie in the proper ranges, but there was some depen-dence on the delayT.

7.4. TOCSY-DOSY

A TOCSY-DOSY experiment was implementedby Jerschow and Mu¨ller for the study of poly-disperse polymer samples [140]. The sequence


Fig. 43. A representation of 3D-DOSY obtained from a series of 10 PFG-TOCSY experiments on a mixture of ATP/g-cd. A 65 mn MLEV-17supercycle was used for spin locking. Reproduced with permission [141].

(linked BBP-LED and TOCSY) is shown in Fig. 40,and it should be noted that the homospoil pulses areorthogonal to the encoding and decoding gradients. Itis clearly appropriate here to use an STE based experi-ment becauseT1 q T2. This 3D DOSY experimentdiffers from the others described here because datainversion was performed by CONTIN on data setsdescribed by Eq. (22) with appropriate correctionfor BPP-pulse pairs (Eq. 13).

The demonstration experiment was performed on asample of 0.2% propylene (Mw � 44 000,Mw/Mn �3.3) plus 0.8%n-decane in tetrachloroethane-d2 at298 K with a 600 MHz spectrometer equipped withan actively shielded gradient probe. The parameterswereT� 120 ms,d � 2.6 ms,Te� 24 ms,t � 1 ms,tm � 30 ms, and the gradient amplitudes were incre-mented linearly in 32 steps to a maximum of 63 G/cm.The 2D intensity data for discrete coordinate pairs(v1, v2) were fed into CONTIN for analysis, andthe resulting 3D data set is displayed in Fig. 41. Inspite of the large peak volumes, the components areseparated quite well into their respective diffusionplanes.

7.5. Merged sequences for PFG-DQS, PFG-NOESY,and PFG-TOCSY

Birlirakis and Guittet [141] have reported‘‘merged’’ pulse sequences that introduce variablegradient pulses into 2D-NMR without the elongationfound in the linked schemes described previously.Examples are shown in Fig. 42(a) for PFG-DQS andFig. 42(b) for PFG-(NOESY, ROESY, or TOCSY).With PFG-DSQ no phase cycling is required and thespectra are obtained in the magnitude mode. Thisexperiment was demonstrated with a mixture contain-ing ATP/g -cyclodextrin (50:7 mM) in D2O at 298 Kon a 600 MHz spectrometer. A series of 11 gradientvalues were used and data acquisition required 14 h.The resolution permitted monoexponential fittingyielding Dg -cd � 2.39 × 10210 m2 s21 and DATP �3.08 × 10210 m2 s21 with a dispersion smaller that7% and the quality of exponential fittingR . 0.997.

The 2D TOCSY experiment was implemented forthe same sample with ten gradient values and a totaldata collection time of 43 h. In this case the dispersionincreased to 10%. The DOSY idea was carried out bymanual construction of 2D spectra to obtain the

separation of components shown in Fig. 43. Themerged sequences described here can produceincreased senstivity in some cases, but the advantagesof STE based experiments should be kept in mindwhen shortT2s andJ-modulation are problems.

8. Future prospects

The concept of diffusion spectra and their incor-poration as a new ‘‘frequency’’ dimension in NMRis now well established. DOSY has been fully imple-mented in several laboratories, and interesting exam-ples of diffusion ordered spectra are available asdemonstrated in this review. With the advent of bothfree and commercial DOSY analysis packages and theavailability of high quality diffusion accessories frommajor vendors, the benefits of DOSY analysis arecertain to spread to a much larger audience. The resultwill be another NMR tool to select from a menu. Weexpect that DOSY will become transparent andnatural to users and will no longer attract specialattention any more than FT-NMR does. The successof DOSY will tend to make it invisible.

The question here is whether there will be majoradvances in instrumentation or analysis methods thatwill significantly affect the power and ease of use ofDOSY. We do not expect exciting advances in dataanalysis. The analysis problems have been around fora long time, and the available techniques are able toextract most of the information from experimentaldata sets. Therefore, we expect the future to holdrefinements, slicker user interfaces, and perhaps theincorporation of artificial intelligence to protect usersfrom themselves. Of course, pulse sequences andexperimental design are certain to evolve. For exam-ple the use of multiple quantum NMR to enhance theeffects of gradients may become more common [142].

The question of hardware is much more open.Higher sensitivity through higher frequencies andbetter probes is certain. One obvious area not yetexplored for DOSY is the use of RF gradients inplace of or in addition to the pulsed dc gradients.We note the recent development of toroid cavitydetectors that permit RF imaging and diffusionmeasurements with respect to the radial (transverse)direction [143]. These devices have important advan-tages in special cases such as the measurement of


diffusion in thin films and in high pressure samples[144]. DOSY applications of toroid cavities may alsobe possible, but there is the immediate problem ofreduced resolution resulting from magnetic suscept-ibility mismatch that becomes serious in the shortcylindrical cavities.

Acknowledgements

This work was supported in part under a grant fromthe National Science Foundation (CHE-9528530). Itwas facilitated by a Reynolds Research Leave fromthe University of North Carolina. The author thanksAidi Chen, Stephen J. Gibbs, Kevin F. Morris,Konstantin Momot, and Donghui Wu for reading themanuscript and providing helpful comments. Also,the author thanks Alexej Jerschow, Cynthia K. Larive,Gareth A. Morris, and Norbert Mu¨ller for sendingrelevant preprints.

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Diffusion ordered nuclear magnetic resonance spectroscopy