Fourier transform pulsed-gradient spin-echo studies of molecular diffusion

Progress in NMR Spt~mscopy, Vol. 19. pp 14% 1987. Printed in Great Britain. All rights reserved.

FOURIER

Copyright 0

00794565/87 so.00 + sll 1986. Pergamon Journals Ltd

TRANSFORM PULSED-GRADIENT SPIN-ECHO STUDIES OF MOLECULAR DIFFUSION PETERST8LBs

Institute of Physical Chemistry, Uppsala University, P.O. BOX532, S-75121 Uppsala, Sweden* (Receioed 11 December 1985)

CONTENTS 1. Introduction 2. NMR Diffusion Measurements in Retrospective 3. Previous Reviews on NMR Diffusion Measurement8 4. Scope of The Present Review 5. Diffusion 5.1. Self-diffusion 5.2. Mutual diffusion 5.3. Multicomponent diffusion in non-equilibrium systems 6. Methods for Measuring Self-Diffusion by NMR 6.1. Longitudinal and transverse spin relaxation 6.2. The nuclear spin-echo (SE) method: basic principles 6.3. The static field gradient spin-echo method 6.4. Continuous wave NMR methods for measurement of self-diffusion 6.5. The pulsed field gradient spin-echo (PGSE) methods 6.5.1: The basic-90-180”kxperiment 6.5.2. Three-pulse sequences; the stimulated echo method 6.5.3. Carr-Purcell echo train PGSE techniques 6.5.4. Multiple quantum spin echoes 6.5.5. PGSE sequences using alternating or non-rectangular field-gradient pulses 6.5.6. The diffusional timescale in field gradient NMR SE experiments 6.5.7. Notes on measurements of restricted diffusion and flow by PGSE techniques 6.5.8. Maanetic field gradient SE experiments on ions in the presence of an electric current 6.5.9. The-frequency-%solved PGSE extension (FT-PGSE) . 6.5.9.1. Instrumental requirements 6.5.9.2. Field gradient coils 6.5.9.3. Gradient driver arrangements 6.5.9.4. Gradient calibration 6.5.9.5. T2 and J-modulation effects 6.5.9.6. Temperature effects 6.5.9.7. &,-effects 6.5.9.8. Notes on the use of absolute-value frequency-domain spectra 6.5.9.9. Notes on IT-PGSE pulse programming on commercial instruments 6.5.9.10. FT-PGSE based on the stimulated echo sequence 6.5.9.11. Evaluation procedures 6.5.9.12. Interpretation of PGSE experiments on polydisperse or heterogeneous systems 6.5.9.13. Survey of different nuclei 6.5.9.14. Comments on the accessible range of diffusion coefficients in FT-PGSE NMR 7. Applications of FI-PGSE 7.1. Isotropic systems 7.1.1. ‘Miscellaneous applications of FT-PGSE spectroscopy 7.1.1.1. Analvtical utilization: ‘Sizeresolved NMR 7.1.1.2. Solvent signal suppression 7.1.2. Diffusion in simple liquid mixtures 7.1.3. Diffusion in solvent-polymer systems: solute/solvent interactions 7.1.4. Solute diffusion in gels *Present address: Department Sweden. 3PNMRS 19/1-a

2 2 3 3 4 4 6 6 7 7 7 8 11 11 11 13 14 14 14 15 16 17 17 19 20 21 22 23 26 26 26 27 27 29 30 31 32 34 34 34 34 34 34 34 34

of Physical Chemistry, Royal Institute of Technology, S-10044 Stockholm, 1

P.

2

STILBS

7.1.5.

Counterion diffusion in polyelectrolyte systems in solution 7.1.5.1. Polymeric polyelectrolytes: counterion binding 7.1.5.2. Ionic surfactant aggregates: counterion binding 7.1.6. Diffusion in binary surfactant/water systems: surfactant aggregation 7.1.7. Diffusion in ternary surfactant/water systems: solubilization and mixed micelle formation microemulsion structure 7.1.8. Diffusion in microemulsions: monitoring of miscellaneous aggregation processes 7.1.9. Diffusion-based in aqueous solution 7.2. Restricted and anisotropic diffusion in heterogeneous systems 7.2.1. Solute diffusion in liquid crystals and emulsions-phase structure 7.2.2. Monitoring of transport through cell membranes 8. Conclusions Acknowledgements References Note Added in Proof

35 35 35 37 37 37 38 38 38 40 41 41 42 45

1. INTRODUCTION

The interest in studies of diffusional processes in solution and in the solid state has grown rapidly during the past few years. This is in part due to an increased number of available techniques for the purpose and the realization that self-diffusion data provide uniquely detailed and easily interpreted information on molecular organization and phase structure. Self-diffusion rates are quite sensitive to structural changes and to binding and association phenomena, in particular for colloidal or macromolecular systems in solution. As an additional benefit, experimental self-diffusion coefficients need no further interpretation; their values are directly related to lateral molecular displacement in the laboratory frame. From an NMR-spectroscopic point of view, one should note that the interpretation of spin relaxation rates, on the other hand, relies on model frameworks for molecular reorientation and spin relaxation. A general trend in the description of matter in recent decades is to turn to computer simulation of structure and dynamics, for example in terms of Monte-Carlo and molecular dynamics methods. It is straightforward and natural to evaluate self-diffusion coefficients from the latter type of simulations and the comparison between predictions of theory and experimental results thus becomes particularly meaningful. 2. NMR DIFFUSION

MEASUREMENTS

IN RETROSPECTIVE

The traditional way to measure self-diffusion coefficients is through radioactive tracer techniques; self-diffusion doefficients tire also frequently called tracer diffusion coefficients in the literature. Tracer methods, at their best, are still the most accurate techniques for the purpose. For anything but simple liquids, they require difficult synthetic work and measurement periods that may be of the order of days or weeks for a single component. A further disadvantage of the technique is the inherent system perturbation by isotope substitution. As will be demonstrated in the present review, NMR techniques can provide indioidual multicomponent self-diffusion coefficients with good precision in a few minutes, without the need for isotopic labelling. Self-diffusion measurements by NMR have been utilized in numerous studies ever since the discovery of spin echoes by Hahn. (I) In that pioneering study, several effects on spin echoes were discovered and correctly interpreted, one of which was the diffusional effect on echo amplitudes in an inhomogeneous magnetic field. In its basic form, the (SE) technique for measuring diffusion entails monitoring of spin-echo amplitudes obtained in the presence of a linear gradient in the &-field. The SE experiment was significantly improved in the mid-sixties in the form of the pulsedfield gradient spin-echo (PGSE) technique. The basic idea was evidently put forward in a paper by McCall et a/.(*’ and the methodology, first experiments and the detailed analysis were later presented by Stejskal and Tanner. (3.4) Several modifications to the PGSE technique were later suggested and tested (mostly designed for the purpose of extending the measurement range to slower diffusion rates and to solids and heterogeneous systems) and the technique has been heavily utilized for investigations of molecular transport in matter during more than three decades.

3

Molecular diffusion studies 3. PREVIOUS REVIEWS ON NMR DIFFUSION

MEASUREMENTS

Apart from the pioneering work mentioned so far, a number of review articles on the basic methodology have appeared. (5-‘3i The most complete discussion so far is that of Reeves,“’ which gives a literature survey of the methodological developments up to the early seventies. Very knowledgeable presentations of the practical aspects of ‘large-gradient PGSE’ along with an outline of the underlying theory were presented quite recently by Hrovat and Wade in two papers.‘i4*i5) and by Fukushima and Roeder in their unique NMR textbook approach.“@ In combination with Reeves’ review, these three presentations cover most methodological developments of time-domain SE and PGSE techniques. The basic techniques will therefore not be covered in too much detail here. The physico-chemical applications of normal, singlacomponent, self-diffusion measurements are also well established; excellent reviews’ by Callaghan on the use of NMR-based diffusion data for probing liquid state molecular organization” ‘i and by von Meerwall on the subject of investigating molecular transport in polymeric systems (la) have been published recently. Self-diffusion studies on solids have been reviewed by Gordon and Strange!ig’ and by Stokes. (“) Studies on plastic crystals have been reviewed by Britcher and Strange,‘21i mesophase structure of membrane lipids by Lindblom!**’ colloidal systems by Stejskalt23) and microemulsions by Lindman and Stilbs.‘24’ The utilization of NMR techniques for studies of ionic diffusion in solid electrolytes has also been summarized recently.‘25’ One should also note the overlap in methodology between PGSE and NMR imaging techniques. Some recent reviews and monographs in the latter field contain much reference material which is also useful in the present context, (26.27) for example chapters on gradient coil design, diffusional effects on NMR images’*‘) and the best available review so far on NMR measurements of flop.

4. SCOPE OF THE PRESENT REVIEW SE and PGSE methods in their original form are unresolved techniques in the frequency domain; in the SE technique, the frequency resolution is intrinsically worse than normal because of the intentionally poor B, homogeneity. In the common implementation of the PGSE technique, frequency resolution is hampered through characteristics of the traditional instrumental design, which involve field homogeneities at the sample location that are not of ‘high-resolution magnitude’, and rather primitive techniques for the actual registration of the spin echo (tide inJra). In the time domain, one finds that unless isotopic labelling is made of all but one component, one cannot distinguish between individual contributions to the multiexponential echo decay of SE or PGSE experiments on multicomponent systems unless rather restrictive criteria are fulfilled with regard to relations between individual diffusion rates and tranverse (T2) spin relaxation rates. As an example, it is quite easy to study (rapid) solvent and (slow) polymer diffusion separately in a polymer solution, through straightforward selection of the appropriate PGSE pulse parameters. Boss et al. have used a technique where the PGSE experiment is preceded by an inversion pulse of selected decay, so as to null the contribution of one of the two components in a mixture.‘3oi In most multicomponent self-diffusion studies this approach will fail, since NMR bands from the same molecule have different spin relaxation rates. An extension of this experiment varies both the inversion-recovery and PGSE time delays. In the FT mode one can then achieve a simultaneous determination of T and D in a single experiment (T. Wlmheim and U. Henriksson, private communication). The concept offrequency-resolved FT-PGSE was evidently first put forward by Vold et al. in their classic paper on spin-relaxation measurements in the pulsed Fourier transform mode(3’) and the technique (FT-PGSE) was first tested by James and McDonald on the system waterdimethylsulphoxide,‘32’ The present review will, after a brief introduction to the general subject and the older literature in the field, focus on the methodology needed to perform accurate, quantitative, frequency-resolved, diffusion measurements of the type illustrated in Fig. 1, and on the recent physico-chemical

P. STILBS

4

i=A=lLOms ppm m 1.0

7.0

6.0

1.0

3.0

a.0

1.0

0.0

FIG. 1. Diffusional effects on the Fourier transformed 300 MHz absorption-mode ‘H-PGSE echo signal from a 12-component solvent mixture (about l.drop each in 0.3 ml deuterioacetone). The signal amplitudes are shown as a function of the duration of the 0.64 x 10m4 T cm-’ field gradient pulses (the previous record, 8 components, was set in Ref. 82). From left to right an evaluation of the experiment leads to D-values (10m9 m* SC’) of 2.76. 3.10, 3.25, 2.84, 1.99. 2.69, 2.30, 3.35, 3.07, 2.78, 2.10 and 2.65; all determined with about 1 % accuracy (90:;, confidence intervals in a statistical analysis).

applications of such multicomponent self-diffusion measurements. The utilization of the combined information in multicomponent self-diffusion data and the selective measurement of individual selfdiffusion coefficients in complex systems are new tools in chemistry. For obvious reasons, high-resolution FT-PGSE applications almost exclusively concern liquid systems References will be included on the merit of: (a) basic theoretical or methodological interest for PGSE (as well as FT-PGSE) techniques, (b) basic theoretical or methodological interest for multicomponent self-diffusion studies, (c) actual experimental utilization of multicomponent selfdiffusion data, as obtained by frequency-resolved FT-PGSE techniques. References to applications of traditional. time-domain PGSE or SE techniques will not be included unless (to the present reviewers judgement) one or more of the above criteria are met. Emphasis will be on NMR-methodology, also in the context of applications. The physico-chemical aspects of the applications will only be briefly touched upon. 5. DIFFUSION When a liquid system is heated, the energy added increases the internal kinetic energy, leading to overall increased rates of molecular and particle motion. Apart from macroscopic convection and convection-like phenomena (for example caused by thermal or density gradients in the system) one usually considers motional partitioning into internal molecular motions (rotation about bonds and vibration), and overall reorientation and translational (or lateral) diffusion of molecules and aggregates. The theory of diffusion and molecular transport in solution is highly developed and extensive. There exist several excellent monographs on diffusion processes, to which the reader is referred for a more complete presentation.‘33-37’ 5.1. Self-Diffusion

Self-diffusion is the net result of the thermal motion-induced

random-walk

process experienced by

Molecular diffusion studies

5

particles or molecules in solution. In an isotropic homogeneous system the conditional probability P(r,, r,t) of finding a molecule, initially at a position ro, at a position c after a time t is equal to P(r,,r,t)=(47rDt)-3’2

exp(-(r-r,)*/4Dt).

(1)

The radial distribution function of molecules with regard to their original positions in an infinitely large system with regard to an arbitrary reference time is thus a Gaussian one, increasingly growing in width as time increases. The single parameter D, the self-diffusion coeficient characterizes this radial distribution completely. The situation differs for obvious reasons under certain circumstances. Statistical considerations must be taken into account if the system has very few particles or if the observation time is short. Different manifestations of restricted diffusion (wall effects and effects of diffusion through semipermeable barriers) are often significant in micro- and macro-heterogeneous systems and anisotropic diffusion rates are a characteristic feature of many liquid crystalline systems. In an isotropic system, whithout thermal or concentration gradients, the average molecule/ particle displacement in all three directions is zero; the mean square displacement is non-zero, however, and is given by the Einstein relation: (r2)=6Dt.

(2)

In these equations the angular brackets symbolize a time average. There are as many self-diffusion coefficients in a system as there are distinct components during the time of observation (the observation time may depend on the method used, however). For NaCl in water there are, in principle four, one each for Na+, Cl-, hydrogen and OH-. Although Hz0 is a distinct species, the latter two self-diffusion coefficients in principle differ because of water autoionization. The actual values for proton and OH- self-diffusion are almost the same, however.“@ Proton migration mechanisms in water have been thoroughly discussed in two papers by Halle and Karlstriim.“9) Typical self-diffusion coefficients in liquid systems at room temperature range from about 10e9 m’s_’ (small molecules in nonviscous solution), to lo-i2 m’s_’ (high polymers in solution). With regard to the actual magnitudes of displacement, the figure 10m9 m’s_’ corresponds to a root mean square displacement during one second in the three-dimensional space of 7.7 x 10m5 m (slightly less than 0.1 mm). For a 100 times smaller self-diffusion coefficient the displacement is consequently 10 times smaller. Within a given phase boundary, self-diffusion coefficients are not dramatically temperature- or concentration-dependent; the temperature coefficient for proton diffusion in water, for example, is approximately 3 % per degree at 25°C a rather typical value. Formally, the magnitude of the diffusion coefficient is given by

D=kaT,K where T represents the absolute temperature, kB Boltzmann’s constant and f the so-called frictional factor. For a sphere of radius r in a continuous medium of viscosity r~,f is given by the Stokes equation: f = 6nqr which, when combined with eqn. (3), leads to the familiar Stokes-Einstein D = k,T/6nqr.

(4) relation: (5)

For other geometries, and when the diffusing particle is of similar size to the solvent molecules, more complex theories and equations describe f (see e.g. Refs 35-36). A representative collection of typical self-diffusion coefficients is given in Table 1.

P. STlLBS

6 TABLE

1. Some representative self-diffusion coefficients at infinite dilution at 25°C (10e9 m2 s-‘) Gases:

Oxygen in air

18000

Solids:

Helium in Pyrex

4x10-6

oxygen Benzoic acid Sucrose Haemoglobin Typical surfactant micelle Li + Na+ ClIacetate-

2.1 1.0 0.52 0.069 O.OlJ3.05 1.03 1.33 2.03 2.05 1.09

Organic .sohtion:

Benzene in ethanol Cyclohexane in benzene

1.81 2.09

Polymer systems:

Polystyrene (M = 105) in Ccl, n-octane in cis-polyisoprene

0.05 0.06

Aqueous

solution:

Data taken from Refs 35 and 18.

5.2. Mutual Diffusion In a non-equilibrium two-component system; for example, solvent/solute layered upon (or below, depending on the relative densities) pure solvent, the mutual diffusion coefficient characterizes the relaxation of concentration gradients in the system according to Fick’s Law: J = -

D’(dC’/du)

(6)

where dC/dx represents the solute concentration gradient, D’ the mutual diffusion coefficient and J the flow of solute molecules per unit cross section area. The net molecular motion, of course, originates from the same thermal motion that causes self-diffusion. Eventually the once sharp boundary becomes blurred and the concentration difference disappears and a single homogeneous solution (of lower concentration) remains. In a two-component system there is only one mutual diffusion coefficient. Depending on the composition, it may approach either of the self-diffusion coefficients of the components. In an infinitely dilute ‘two-component’ solution the (single) mutual diffusion coefficient is equal to the self-diffusion coefkient of the solute and not directly related to that of the solvent. At intermediate concentration ranges, the distinction between mutual and self-diffusion coefficients is important.‘33-37’ With regard to self-diffusion in the NaCl/water example just given, one should also note that although individual chloride ions diffuse about 50% faster than individual sodium ions in water, their net transport rates must equal those of the sodium ions to preserve electroneutrality when relaxing a concentration gradient; the overall chloride ion transport is thus to some extent ratedetermined by the net transport of sodium ions, the net transport rate of which is somewhat enhanced by the parallel transport of chloride ions. The formalism for describing mutual diffusion and self-diffusion in ideal associated systems has been discussed by Hall in a recent paper.‘40) 5.3. Multicomponent Diflusion in Non-Equilibrium This is, of course, the most common situation

Systems

in nature. The formal treatment of multicomponent

Molecular diffusion studies

7

is made in the framework of irreversible thermodynamics, and the reader is referred to Refs 35 or 41 for an introduction to the subject. In general, there are (~1)~ different diffusion coefficients characterizing the relative fluxes of species in an n-component mixture. These multicomponent diffusion coefficients are difficult to measure or interpret physically unlike the multicomponent self-diffusion coefficients measured by FT NMR techniques. Experimental values exist to any appreciable extent only for rather simple threocomponent systems. (See Ref. 35 for a partial, but recent, compilation.) The quantitative treatment of non-equilibrium multicomponent diffusion in heterogeneous or partly organized systems is even more complex than that of homogeneous solution phases (see for example Ref. 42 and further papers in that publication volume for an introduction). diffusion

6. METHODS

FOR MEASURING

SELF-DIFFUSION

BY NMR

6.1. Longitudinal and Transverse Spin Relaxation

Intermolecular contributions to spin relaxation processes originate in part from modulation of dipole-dipole interactions through translational diffusion. They are significant for protons in solution and for electron-nuclear dipole relaxation in paramagnetic systems. The quantitative treatment of translational diffusion contributions to spin relaxation is presented in a classic paper by Torrey’43) and can also be found in Abragam’s book. (44)Several extensions and refinements of the theory and transport models have been made, and a good summary of the pertinent literature has been presented by Kowalewski et aLC4’) In the present context it should be noted that the relation between spin relaxation rates and selfdiffusion coefficients relies on model frameworks. The diffusion contribution to spin relaxation is difficult to disentangle from other relaxation contributions and the separation of contributions from different species in a multicomponent solution is even more difficult. Also, the self-diffusion coefficient which is obtained from spin relaxation refers to a much shorter time span (of the same order of magnitude as the rotational correlation times in the system) than that measured by PGSE techniques (which monitor diffusion during millisecond to second intervals). In microheterogeneous systems (colloidal, macromolecular . . .) the distinction between the two is significant. One can note that the local (spin relaxation inducing) mobility of a high polymer in solution, for example, may be high, while the (overall molecular transport) translational diffusion rate remains low. In general, it is , difficult to achieve the accuracy in measuring self-diffusion coefficients from relaxation rates that is needed for physico-chemical considerations. An extension to liquid crystals of Torrey’s treatment of nuclear spin relaxation by translational diffusion has been presented by Zumer and Vilfan.‘46) Some years ago, Stokes and Ailionr4’) suggested an interesting technique for studying slow, multicomponent diffusion in the solid state. It is based on a new dipolar relaxation time T;, which characterizes the spin-lattice relaxation of secular dipolar interactions in the presence of a large rf field. Measurements of T’,, are claimed to be particularly useful for studying slow atomic motions in multispin systems, since such measurements enable one to vary the relative contributions of the motion of particular spin species, thus enabling one to identify the diffusing species. 6.2. The Nuclear Spin-Echo (SE) Method: Basic Principles In the classic paper by Hahn/” published only a few years after the first successful NMR experiments by the Purcell and Bloch groups, effects of multiple radiofrequency pulses in the NMR induction experiment were explored. Unlike the situation only fifteen years ago (when most NMR measurements were made on continuous-wave instruments) the majority of NMR spectrometers now use the pulsed mode, as did the instrument used by Hahn. The basic principles are simple; after an initial 90-degree rf pulse, spins at different precession rates dephase in the.x’$ plane The origin of different precession frequencies may differ, such as chemical shift effects, effects of spin-spin couplings and also effects of an inhomogeneous magnetic field. The general useful feature of

8

P.

Excitation ( 901 - pulse) la)

STILBS

Oephasing

Inversion (180> -pulse)

2’

& -_.

‘\

\I

%

Y’

X’

Refocussing

Echo peak

Dephasing

FIG 2. The basic 90”,-180”, spin-echo experiment.

NMR experiments is some form of spin refiussing, i.e. the individual magnetic spin vectors regain phase coherence to some extent at some time after the second or higher numbered pulses, leading to one or more ‘spin-echoes’. Such experiments, of course, do not refocus the normal random spin dispersion which is due to transverse (T2) spin relaxation processes, nor do they completely refocus effects related to homonuclear spin-spin couplings. The reader is referred to Turner’s recent review on multipulse NMR in liquids(48) and two papers by Kaiser et a1.‘49.s0)for lucid discussions of complex echo effects in multi-pulse situations. (Ref. 50 actually discusses diffusion and field gradient effects in NMR Fourier spectroscopy quantitatively, but in the context of suppression of undesirable echo effects, rather that for the measurement of self-diffusion.) The original experiments by Hahn were made with two or more 90” pulses, the effects of which are more difficult to visualize (see Refs 1 and 48, for example). Some refocussing occurs with any pulse angle, but in the most common form in use today, the basic spin-echo experiment utilizes a 90” pulse, followed by a single 180” pulse of the same phase (or shifted 90” with regard to the first). The effects of that pulse sequence are particularly simple to visualize of both pulses have the same phase (Fig. 2). Refocussing occurs along the negative y’-axis and the echo amplitude is consequently negative. For a 90” phase shift of the second pulse refocussing occurs instead along the positive y’-axis as shown in Fig. 3. The 90”-180” echo sequence was originated by Carr and Purcell/s) but the 90”-180” echo (as well as the 90”-90” echo) is commonly named a Hahn echo in the literature. The ‘Carr-Purcell sequence’ (CP), or its Meiboom-Gill (CPMG) modification (‘I) comprises a 90” preparation pulse, followed by a train of 180 -pulses in between yhich refocussing and dispersion of the spin vectors occurs indefinitely in the absence of transverse relaxation processes. The CPMG technique is the traditional method for measuring transverse relaxation rates in NMR,“” and also provides an alternative method for studying diffusion and chemical exchange.“) Frequency-resolved FT-CPMG capability is a standard feature on the present generation of NMR research spectrometers. multipulse

6.3. The Static Field Gradient Spin-Echo

Method

The SE experiment on a sample in an inhomogeneous magnetic field provides the basis for NMR self-diffusion experiments. The basic technique is of considerable methodological interest but has several disadvantages and is rarely used today.

Molecular diffusion studies

DEPHASING

1 ‘REFOCUSSING

9

ECHO

z iii oz ECHO

FIG. 3. An illustration of the basic concept of diffusion measurements through the SE technique (after Singer”‘). The diffusional echo attenuation results from randomization of positions and phases during the time of the experiment. In an inhomogeneous magnetic field the nuclei at different spatial locations are labelled by different precession frequencies (In quantitative diffusion measurements, the magnetic field is generally linearly inhomogeneous and inhomogenous in one direction only.) The approach to diffusion measurements is as follows: the 90”-180” spin-echo experiment refocuses resonant spins, but on/y if the precession frequency is constant during the experiment. Therefore, if the nuclei move in an inhomogeneous magnetic field, their frequencies vary as well. For incoherent random motions like those in self-diffusion, a random (i.e. Gaussian) phase shift of individual spin vectors occurs, leading to more or less incomplete refocussing at the time of the echo (Fig 3). One should clearly note that the quantity monitored by field gradient SE techniques is not self-diffusion directly, but rather the general probability distribution for lateral displacement in the direction of the gradient. For a linear field gradient in the z-direction (G), there is a direct correspondence between a displacement in the z-direction (AZ) and the spin phase change, i.e. the extra rotation after a further time r amounts to A@=yG(Az)t. For a Gaussian diffusion process in the direction of the linear gradient, the probability distribution in the phase space [c.f. eqn. (I)] is:(5*7)

P(m)= (4ny2G2Dt/3)- ‘I2 exp( -302/4y2GzDt3), and the remaining step is then to calculate the amplitude of the echo. In general, the amplitude is given by a normal probability average integral (s.7) for the y’-component of the magnetization A(t)= A(O)

7cosw(o)d@,

-02

(8)

10

P. Sl-iLBS

where P(m) is the appropriate probability function for the displacement process. The concept of selfdiffusion in spin-echo NMR thus enters within a Gaussian model for P(m). Coherent motion in an inhomogeneous magnetic field, for example, changes the directions in which the spins refocus (ideally completely), leading to a phase shift in the observed echo. (This provides a basis for measuring flow and similar phenomena by NMR.) Singer, in his review’*’ presents a very illustrative graphs of the dephasing-refocussing processes for different transport conditions in the sample (like that in Fig. 3). The recent review by Hemminga ‘29’ should be consulted for an upto-date presentation of NMR techniques for measuring flow. Magnetic field gradient SE studies of restricted diffusion are discussed later in this review. Several computational procedures for achieving the integration implied in eqn. (8) have been suggested, following the original analysis by Hahn. “) The reader is referred to the papers by Stejskal and Tanner’3.4) Carr and Purcell”’ and the reviews by Reeves”’ and Callaghan”‘) for the detailed procedures in the general case, StepiSnik has also presented a density matrix formalism for a more general analysis of the NMR field-gradient spin-echo experiment.‘53r The simplest computational scheme appears to be that of Karlicek and Lowe!9q54’ where the experiment is brought from the rotating frame to the rotating ‘flipflop’ frame; i.e.. the spin inversion by the 180” pulses is ignored and instead replaced by an inversion in the direction of the gradient itself (the justification for that procedure is found in the Appendix of Ref. 54). It is then quite easy to calculate the quantities sought, even for complex pulse and gradient schemes. The key equation in the case of unrestricted diffusion, characterizable by a single self-diffusion coefficient, is(54’

(9) where y represents the gyromagnetic ratio of the nuclei and G the magnitude of the linear magnetic held gradient. When applied to a field-gradient SE experiment with an interval between the two radiofrequency pulses of T, the evaluational scheme leads tocS4) jG(t’)dt’= - Gt

for

t
(104

0

for

jG(t’)dt’= G(t -2r)

t> T;

UW

0

and In [A”$:]=

-y2~[~2t2dt+7G’(r-2r)2dt]

= ~2y2~G2T3.

Including the T, term and taking the exponent, the echo attenuation diffusion in the static-gradient experiment thus becomes /t(2T)/,4(0)= eXp(-2r/T,

- 2y2DG2r3/3),

(11)

effect in the case of self-

(12)

in agreement with. previous results.“.” The basic restriction in all spin-echo diffusion experiments is evident in equation”‘); the irreversible transverse spin relaxation always competes with the field-gradient/diffusion-induced echo attenuation. Diffusion experiments are thus very much facilitated if the transverse relaxation times of the nuclei in question are long and their gyromagnetic ratio is high. Excessively rapid transverse spin relaxation actually makes diffusion measurements difficult or impossible for many nuclei. Although the underlying effects are not directly related, there is also a positive correlation between slow diffusion and rapid spin relaxation. In other words, in systems where the effect of

Molecular diffusion studies

11

diffusion on the echo attenuation is particularly small, the transverse spin relaxation effect is often large In the static gradient method, the only remedy is to shorten the rf pulse interval and further increase the strength of the field gradient. One should also note that it is not entirely straightforward to experimentally isolate the effect of transverse relaxation from that of diffusion on the attenuation of the echo amplitude in the static-gradient spin-echo experiment. Further disadvantages are that the lower limit of accessible diffusion is rather high, and that the technique is inherently contradictive; it is impossible to uniformly excite with a single pulse a spectrum that is spread out over several kHz or more under the influence of a large field gradient.““)

6.4. Continuous Wave NMR Methods for Measurement of Self-Diffusion

A continuous-wave (CW) analogue of the SE experiment for measuring self-diffusion was presented some years ago by StepiSnik et al.(55*56) The essential feature of this method involves supplementing the usual homogeneous magnetic field modulation in a CW experiment with a magnetic field gradient (MFG) with modulation frequency (oG/2n) larger that the resonant line width (expressed in frequency units). Line widths of the observed centre-band and sideband as a function of the peak MFG then depend in a simple way on the self-diffusion constant of the nucleus under investigation. Multi-component self-diffusion studies are claimed to be feasible by this CW technique. Since CW spectrometers exist in very few research laboratories today, the technique has lost much of its original interest and the two references mentioned appear to be the only work presented so far.

6.5. The Pulsed Field Gradient Spin-Echo (PGSE) Methods 6.5.1. The Basic 90”~180” Experiment. The static-gradient

SE experiment was significantly improved by Stejskal and Tanner in the mid sixtiesJ4*3) in the form of the pulsed-gradient technique (Fig. 4). Here the basic magnetic field is (essentially) homogeneous throughout the experiment. The effective dispersion and refocussing of the spins occurs in two identical (or as nearly identical as possible) field gradient pulses. The latter are almost always (but need not be) rectangular pulses of the same sign, separated in time so as to fit into the chosen rf pulse interval. The advantages over the static gradient method are twofold The echo attenuation effect from diffusion can be separated from the transverse relaxation effect by performing the experiment at a fixed rf pulse interval (r) (not eliminating, but keeping T,-effects constant) and varying either the gradient pulse interval (A), the gradient strength (G), or the duration of the gradient pulses (8). Also, the detection of the spin-echo is made in a relatively homogeneous magnetic field, eliminating the need for very rapid, broadband, electronic circuitry in the spectrometer. Thus, for strong static gradients the signal is essentially a ‘spike’ in the baseline, which is increasingly sharp as the strength of the gradient increases Broadband electronic circuits are inherently noisy by NMR standards and have an adverse effect on the signal/noise ratio in the detection of the echo.

9o”

180’

ECHO FIG. 4. The basic Stejskal-Tanner

pulsed field-gradient experiment.

12

P. STILBS

In a perfectly homogeneous static field, the quantitative relation between the attenuation pulse field gradient parameters for a singlecomponent sample becomest3*4) r4(2T)= A(o)exp( -25/T’

-(yGs)‘D(A - b/3)).

and the

(13)

In the presence of significant gradients in the static field (symbolized G,), the relation (even in a strictly linear and orthogonal two-dimensional case) is considerably more complicated. The echo attenuation is described by:13’ A(2t)=

A(O)exp( -2r/T’

-(rG#D(A

- h/3)) x exp( -2y2G@r3/3)

x exp( - yZG G,D&t: + ti + s(ti + t2)+ 26’/3 -2~‘))

(14)

where t, represents the time between the first rf pulse and the leading edge of the first gradient pulse and f2 the time span from the trailing edge of the second gradient pulse to the echo at 2T. Thus when G, is large (and which would normally be far from linear) it becomes necessary to increase the pulsed gradient strengths by one or two magnitudes above G, for adequate and correctable conditions of measurement and data evaluation. It would thus appear highly desirable to reduce the gradients G, as much as possible. Traditionally, however, the PGSE measurements are made in fields which are inhomogeneous by high-resolution NMR standards. One reason for this is that; the normal shim coils are usually replaced by a pair of gradient coils; the resultant inherent magnet field inhomogeneity may then correspond to a frequency resolution in the kHz range The other factor is related to the actual detection of the echo; for larger gradients G, the echo becomes sharper and more easily observable in the time-domain. (This may actually have discouraged the process of improving the magnetic field inhomogeneity and ultimately detecting the echo through its frequency-domain equivalent.) In common practice, the actual pulsed magnetic field gradient strength in PGSE experiments is of the order of 10-1000 Gauss cm-‘, corresponding to several tens of Amperes through the field gradient coils. (Without exception, the PGSE literature expresses field gradients in the traditional c.g.s. units. The conversion relation to SI units is 1 Gauss cm- ’ = 10 mT m-l.) Rather specialized electronic circuitry is needed to provide clean and truly rectangular gradient pulses of this magnitude, and with a duration typically 0.1 to 50 ms (see the references of Section 659.3). Imperfections in the balance between the two field gradient pulses lead to partial echo attenuations and displacement of the echo in the time domain, effects which can actually be partly suppressed through the intentional presence of a background field gradient of controlled magnitude and direction. Hrovat and Wade have discussed these and other related problems in considerable detail.(i 4*1‘) When rapidly switching high currents through electrical wires and coils, it is inevitable that mechanical effects arise (analogous to those of a common loudspeaker), leading to mechanical vibrations and displacement of the sample during measurement. This, of course, can have a disastrous effect on certain experiments, since PGSE techniques are supposed to monitor r.m.s. diffusional displacements that are of the order of 0.001-0.1 mm. The situation is particularly troublesome when measuring very slow diffusion and T2 is very short too; one is then forced to use very powerful and short gradient pulses. Coherent motion of the sample is momentarily equivalent to ‘plug flow’ (see Section 6.5.7), and the echo distortion to first approximation takes the form of a phase error. Eddy currents and their associated magnetic fields in aluminium probe sidewalls or magnet poles caused by rapid changes in the magnetic field are a further complication. It is instructive to study Refs 14, 15, 57 and 58 to fully appreciate the considerable problems in this con text. Common practice in ‘large-gradient PGSE’ is to ‘reject’ poorly reproducible or mislocated echoes by manually adjusting the width or amplitude of one of the field gradient pulses and the phase of the second rf pulse to obtain a maximum echo at the ‘right’ location and with the ‘right’ phase, in partial compensation for the inevitable overall imperfections of the experiment. Contrary to the impression conveyed by the common visualization of the SE experiment as shown in Fig. 2, the spin vectors actually precess hundreds to thousands of turns back and forth around the

Molecular

diffusion

studies

13

z-axis during a typical PGSE sequence, and it is thus actually difficult, at first thought, to understand how refocussing can work at all! 6.5.2. Three-Pulse Sequences; The Stimulated Echo Method. The combination of the inversionrecovery and the PGSE experiment t3’) has already been mentioned. Hahn, in his original paper, (I) discussed the concept of stimulated echoes and their utilization for self-diffusion measurements. As many as five spin echoes may result from a three-pulse rf sequence. The new concept which is relevant for studying diffusion is that the attenuation of the stimulated echo from diffusion competes with Tl rather than with T,-relaxation. When chemical exchange occurs, for example, T2 may be much less than T, and it may then be more advantageous to use the PGSE-based stimulated echo technique. In a 90”-90”-90” three-pulse sequence, the first pulse (at time zero) rotates the magnetization into the x’y’-plane, after which the spins in various volume elements lose phase coherence because of precession at different rates, and acquire various phase angles in the rotating frame. The second pulse (at time or) stores the memory of the current phase angles in the z-direction, where they are unaffected by field gradients and relax in the longitudinal direction. The third pulse (at time r2) restores the phase angles with reversed sign, so that they now precess to form an echo at time r’= r1 +r, (Fig 5). Woessner has given an expression for the echo attenuation by molecular diffusion in a static field gradient for three- and four-pulse sequences, including the stimulated echo/“’ and Tanner has given expressions for the pulsed-gradient modification of that basic sequence.‘4*60) As shown in Fig. 5, the height of the stimulated echo when the static field-gradient is zero is at most 50% of the initial magnetization (hence the factor 0.5). The dependence of A (71 +r,), the stimulated echo amplitude, on Tl and T2 is

A(7, +r,)=

A(0) 0.5 exp(-(7,-7,)/T,

-2r,/T2-

(rG8)‘D(A-8/3)).

(15)

Three recent papers concerning NMR imaging discuss the stimulated echo technique in considerable detai1,‘61.62.63)also in particular discuss diffusion and flow.

RF FIELD GRAOIENT

e

FIG. 5. (a) Echo formation as a result of three 90” pulses. The echo phases are ignored for simplicity; the relative echo amplitudes depend on T, and T2 and are indicated only schematically. (b) PGSE, as based on the stimulated echo.

P. STILES

14

6.53. Curr-Purcell Echo Train PGSE Techniques. In the original paper by Carr and Purcell’s) it was pointed out that a 90‘-(r180”)” sequence is actually an excellent method for suppressing the diffusional effect on echo amplitudes when determining T2. By making t very short and n large, one can essentially eliminate the diffusional term. Good discussions of the experiment can be found in Reeves’ review”) and also in the review by Freeman and Hill. (“) The technique has no advantages over the standard PGSE method for measurements of self-diffusion on normal liquid systems. Modifications of it are very valuable for investigations on heterogeneous systems, or when there is a significant background gradient present in the PGSE experiment. Williams er al. have demonstrated that the cross terms of eqn. (14) vanish in the case of a twin field gradient pulse superimposed on a Carr-Purcell train and that the echo attenuation is simply given by the Stejskal-Tanner relation [eqn. (13)].‘64’ Also, large (random) background gradients often exist at surfaces or in solids, and modified CPMG-PGSE techniques can be designed so as to neutralize their effect on the echo attenuation. (64*s4)As an example, Karlicek and Lowe demonstrate that it is possible to measure proton diffusion in water by specialized CPMG-PGSE techniques under the influence of a 1.6 T m-l background gradient,(s4) and Williams et al. successfully measured self-diffusion in metal hydrides like NbHo.ss, where random gradients of 208 mT m-l exist. (64) Applications of these and the stimulated echo technique to diffusion of small molecules at the gas-solid interface have been described.(6s) 6.5.4. Multiple Quantum Spin Echoes. The Vold group(66) and also Zax and Pine@‘) have pointed out that multiple quantum spin echoes’6*’ can be utilized for self-diffusion measurements (Fig. 6). This new approach has the particular advantage that the effective precession frequency in the rotating frame is multiplied by the order of the coherence; in other words, the effective field gradient at the nucleus is doubled for a two-quantum transition, as shown in Fig. 7. Also, multiple-quantum spectra can be made selective and generally are simpler and better resolved than their single-quantum counterparts, and hence a higher degree of selectivity and frequency resolution in the investigation of complex systems might be achieved. The combined concepts are potentially very useful for studies of slow diffusion in partially organized systems or solids, but the methodology is at such a high level that only tests of the method and no routine applications have been presented so far. Pulsed multiple quantum NMR spectroscopy requires complex transmitter/receiver phase cycling schemes and highperformance instrumentation. 6.5.5. PGSE Sequences Using Alternating or Non-Rectangular Field-Gradient Pulses. Different schemes for producing alternating field gradient pulses have been discussed by Gross and Kosfeld,‘@ Karlicek and Lowe’s4) and Hrovat et al.w’) Advantages over the traditional twin-pulse PGSE experiment are found for heterogeneous systems which have random field gradients localized in the

900 leoo 900

lW

ECMHQo 90'

ECHO

b)

FIG. 6. Simplified schemes

for multiquantum

PGSE according to Ref. 66 and 67, respectively. phase cycling is required.

Radiofrequency

15

Molecular diffusion studies I

I

n=l

0

10

5

6’tA-6/3

I&-

15 x10

-7

FIG. 7. Results of proton NMR diffusion experiments for all n-quantum orders of benzene (25 mol %) dissolved in Eastman liquid crystal No. 15320. The plots show the normalized echo amplitude on a logarithmic scale vs the gradient pulse timing parameter. The straight lines are linear least-squares fits to the accumulated data whose slopes vary as n2 (from Ref. 67).

material itself, and for the case of significant background gradients (see Section 6.5.3.), and when one wishes to achieve the shortest possible diffusion time in the experiment (c.f. the next two Subsections). In general, the diffusional echo attenuation effect from alternating gradient pulse schemes is greater than that of twin-pulse PGSE. In practice, however, the full realization of these techniques is difficult. Electronically, the design is more complex; generating identical gradient pulses of opposite sign is by no means trivial and double sets of gradient coils may be needed as well (see Section 659.2). Gross and Kosfeld in 1969 tested sinusoidal gradient pulse shapes that were generated with analogue electronics.t6i With present-day digital technology (i.e. using a digitally stored pulse shape, a D/A converter and a suitable gradient driver circuit), such experiments are becoming realistic, even in the high-resolution FT mode. One can foresee advantages with respect to a smaller perturbation to the spectrometer lock system for an equivalent gradient strength with regard to the standard PGSE experiment.

6.56. The Difusional Timescale in Field Gradient NMR SE Experiments. The timescale (the actual period during which motion is monitored) of the normal pulsed-gradient spin-echo experiments is equal to (A-S/3)J6~70*71) which is typically in the l-1000 millisecond range. The actual value can be changed at will, provided that other experimental conditions (such as diffusion rates and gradient strength) provide large enough effects so as to allow an adequate determination of the echo attenuation from diffusion processes. With alternating rectangular or sinusoidal gradient shapes (c.f. previous Subsection), the effective diffusion time is considerably smaller for the same gradient magnitude. Gross and Kosfeld have discussed the practical aspects of non-rectangular and sinusoidal pulse schemes in some detail.‘@ In physico-chemical studies on isotropic liquids, one should note that even macromolecular displacements over a millisecond time span correspond to distances that are several magnitudes larger than typical molecular, supramolecular or macromolecular radii [cf. eqns. (1) and (2)]. The selectable time-scale of NMR diffusion experiments and the magnitude of diffusion coefficients in general is in such a range that restricted diffusion can conveniently be investigated over a large span of submillimetre domain geometries (c.f. next Subsection).

16

P. STlLBS

FIG. 8. Echo attenuation (R) versus diffusion time (A-6/3) at a fixed value for G (6 x lo-” T cm-‘) for water diffusion perpendicular to the leaves of a mica stack. Note the asymptotic, non-zero, echo amplitude at long diffusion times. The residual echo amplitude is related quantitatively to the interleaf spacing (about 0.016 mm in the present case) (from Ref. 71).

6.5.7. Notes on Measurements of Restricted Diffusion and Flow by PGSE Techniques. One should remember that the NMR field-gradient spin-echo technique basically monitors motional processes in the direction of the field gradient. Concepts such as diffusion are introduced into the derivations of echo attenuation and echo phase via motional models. PGSE techniques are actually very powerful tools for investigations of many other transport situations as well. The first discussion of flow effects on spin echoes was given by Carr and Purcell in their classic paper’5) and the reviews on the subject by Singer”’ and Hemminga (2g)have already been mentioned in the context of Section 6.2. Woessne+“) first noted the effects of restricted diffusion on echo amplitudes in steady-gradient SE experiments on benzene in rubber; the (apparent) diffusion coefficient varied with the rf pulse interval in a linear fashion. With barriers present during the time of observation (t), the net displacement ((x2)) becomes less that that of Gaussian diffusion ((x2)=2Dt) and thus the apparent D-value will decrease. The PGSE technique is much better for the investigation of non-Gaussian diffusion, in that the observation time is more easily varied (c.5 previous Subsection). Stejskal”” has provided a detailed analysis of the PGSE echo attenuation for several types of system organization and the ideas were successfully tested on model systems like water in mica stacks, emulsion droplets and yeast cells.‘4,71) In general, a positive deviation from the Stejskal-Tanner relation or an asymptotic decay towards a non-zero echo amplitude is observed if effects of restricted diffusion (or system polydispersity, see Section 659.11.) are significant during the time of observation. From an analysis of such data one can, in principle, obtain information on the geometry of the domains in which restricted diffusion occurs, i.e. on quantities like cell sizes or domain size distributions (Fig. 8). The experimental problems are, of course, considerable, requiring high-quality data, collected with PGSE parameters carefully selected in the right range for the geometry to be investigated. Analytical solutions to the problem of echo attenuation as a function of system geometry and dynamics are in most cases quite complicated, or do not exist. The equation describing echo decay in the relatively simple case of two-dimensional diffusion within planar impermeable barriers is, for exampIe!4*71.7’

17

Molecular diffusion studies R=exp(-y2S2G,,ZAD)

2( 1 - cos(ySG,a)) ----~ WC,@*

+4(ydG,a)*

(16)

where R is the ratio A (2r)/A(O) and G,, and G, designate vector components parallel and perpendicular to the barriers. The sum converges relatively rapidly. The self-diffusion coefficient within the barriers is represented by D and the barrier spacing by u. A result of particular interest is the asymptotic behaviour when the field gradient is oriented normally to the obstructing layers. The result is R, =lim R= A
2[1 -cos($Ga)] p-Y_ (@Cc)*

(17)

where G = G,, and G, =O. Jn the case of semipermeable barriers one can, in principle, obtain the rate of permeation quantitatively from the echo attenuation as function of the diffusion time. The calculation is by no means simple and no analytical solutions have been found for any geometry. The first numerical analysis of the PGSE experiment in the case of semipermeable planar barriers was presented by Tanner as late as 1978.‘73’ A highly interesting paper was published a few years ago by Zientara and Free.d/74r as a new approach for analyzing PGSE experiments in terms of the microscopic details and chemical properties of heterogeneous systems. The suggested numerical method for analyzing PGSE experiments is based on the stochastic Liouville equation (SLE) and includes the discontinuities in transport and solubility properties which are due to the different spatial regions. A double step computational algorithm, which takes advantage of the different time scales of diffusive and spinquantum phenomena, is then introduced to obtain an approximate solution for the time dependent SLE. This method was applied to the calculation of spin echo amplitudes. Von Meerwall and Ferguson tested this computational technique and were able to derive numerically echo attenuations for the planar barrier case of semipermeable barriers to diffusion. (“) They also reinterpreted older results in the literature with the aid of this new computational tool. The review by Callaghan”” is highly recommended for a summary of later experimental PGSE investigations of similar kind. References to data evaluation procedures for echo attenuation analyses for cases of non-Gaussian diffusion or system polydispersity are given in Section 6.5.9.11. 6.5.8. Magnetic Field Gradient SE Experiments on Ions in the Presence of an Electric Current. Holz et a/. have successfully applied field gradient SE techniques to the investigation of classic electrochemical phenomena in solution, i.e. for the determination of ionic velocities under the influence of a potential difference, and of transference numbers. The construction of a special NMR probe for investigations of this kind is also described.(76.77) 6.5.9. The Frequency-Resolved

PGSE Extension (FT-PGSE). The FT extension of the PGSE experiment is straightforward as a concept in Fourier transform pulsed NMR.‘31) A paper by Chan (“) which discussed time-domain high-resolution echo shape fitting to resolve individual echo contributions should be mentioned in this context. No actual experiments were made, however, and no obvious advantages over FT techniques are seen either. The use of Fourier transformation in the PGSE experiment was first tested in the absolute-value mode by James and McDonald on the DMSO-water system in 1973!32) (and later applied to vesicle diffusion in water by McDonald and Vanderkooi’79’). The full high-resolution absorption-mode Fourier transform (FT) modification of

18

P. STILBS

PGSE experiment has only recently been developed into a practical tool, and been extensively applied to physico-chemical problems. The key step is to take full advantage of the current design, sensitivity, timing stability, frequency resolution and reproducibility of modern high-resolution pulsed FT spectrometers and to use field gradients as weak as 10 mT m-r (corresponding to a few 100 mA through the gradient coils). With such weak gradients, one can still cover more than two decades (possibly up to four) in the diffusion constant. The limiting factor is the T,-value of the particulaisignal under investigation. The use of internal field/frequency lock is still feasible and the inevitable overall spectrometer field-gradient perturbation can be neutralized by proper adjustments, making possible high-stability, unattended and automatized measurements of the type presented in the application section. Much of the desirable features and the associated problems have been discussed in 1977 by Kida and Uedaira!“’ a paper we had not noticed when we started our work along similar lines in 1978.‘81.82.83)The work of Kida and Uedaira was made with a high-resolution probe, but with manual pulse timing and with rather high gradients (up to 400 mT m-l). Callaghan et .1.‘8s~86’ and also Cosgrove’87’ designed their computer-controlled PGSEinstrumentation for moderate resolution, medium to high gradient absolute-value Fourier transform capability at about that time too. Under medium to high pulsed field-gradient conditions the spectrometer system becomes so perturbed that the use of absorption mode spectra becomes impossible and one has to rely on the often undesirable absolute-value presentation (this is also the case when the pulse timing is non-ideal; uide supra). The basic idea of frequency-resolved FT-PGSE is that the second half of the echo can be Fourier transformed, to separate individual contributions to the echo in the frequency domain, very much like in the normal basic pulsed FT-NMR experiment (c.f. Fig. 9). It is evident that one gains significantly in signal/noise through the Fourier transformation as well. For the detection of wellresolved spectra to be feasible in practice, the background magnetic field gradient must be of ‘highthe

-S+N 0 IL% PEO M = 73000 in (One

020 transient I

II.3 Hz IIS*Nl/N=lO) FREQUENCY

S+N

6.99 ms-

DOMAIN

1.53 Hz

((S+Nl/N.2.71 HALF

‘H half-echo from a dilute solution of poly(ethyleneoxide) in heavy water and its Fourier transform. Note the high sensitivity of the experiment which was done on a

FIG. 9. An experimental

absorption-mode

ECHO

narrowband

high-resolution

spectrometer.

Molecular diffusion studies

(ot TOLUENE

BUTANOL* SOBS lah0 -

EUTANDL -OXH2-

1 H 996 MHz

19

HDO-OH

tnormall

-

I\

x/-J>

A

4

70 60

1

-xl0

A 4

ppm

6

6

c

2

110ms 0

FIG. 10. An illustration of a Fourier transform

pulsed-gradient spin-echo experiment. A sequence of 99.6 MHz ‘H absorption-mode PGSE spectra is shown from a microemulsion sample composed of sodium octylbenzenesulfonate (SOBS),n-butanol, toluene and water (D,O) as a function of the magnetic field gradient duration. 6. Peak amplitudes reflect the values of the self-diffusion coefficients of the individual constituents according to the Stejskal-Tanner relation.

resolution NMR quality’ on the non-spinning sample (corresponding to a contribution of at most 10 Hz to the line width in the majority of applications). Under adequate field-homogeneity conditions, it is straightforward to resolve the contributions to the combined spin-echo. Apart from ‘phase-distortion’ effects originating from homonuclear spin-spin-couplings (so-called J-modulation effects; Section 6.5.9.5) and ‘amplitude distortion’ effects originating from the fact that spin-echo spectra are partially relaxed and that different signals are associated with different transverse relaxation rates (amplitudes in the spectrum thus deviate significantly from estimates based on the relative numbers of nuclei), spin-echo spectra look almost like normal NMR spectra and the interpretation is not very much more difficult (c.f. Fig. 10). 659.1. Instrumental requirements. A high-resolution instrumental factors.

(1) A high-resolution (2) (3) (4) (5) (6) (7)

FT-PGSE experiment requires the following

magnet and high-resolution high-sensitivity narrowband probe (preferrably a tuneable multinuclear one, or a proton-only). with integral field gradient coils. Computer-controlled (or possibly externally triggered rf pulses) pulse timing of microsecond accuracy for rf and field gradient pulse timing. A stable field/frequency lock, that is reasonably immune to the application of field gradient pulses. A stable field gradient driver, capable of supplying 10-1000 ms gradient pulses of typically 300-3000 mA current, with rise/fall times better than 1 ms. Time-averaging and absorption mode Fourier transform capability. Provision for digital registration of individual echo amplitudes. Adequate on-line and off-line data handling evaluation procedures.

20

P.

STlLBS

In essence, these are the inherent characteristics of modern pulsed FT NMR instrumentation. The exception is normally the field gradient coils, the PGSE pulse sequence generation capability, the gradient driver and the full data handling procedures. In the following, some advice with regard to the actual method implementation of the experiments is given. 659.2. Field gradient coils. The classic paper in this respect is of course the one by Golay on NMR shim coiIs.‘a*) For quantitative diffusion measurements, the field gradient coils should produce a gradient that is as uniform as possible at the sample location. Tanner has discussed anti-Helmholz type coil design in considerable detail, giving simple criteria for optimum coil geometries.‘4*8gi A different coil type is the quadrupole coil(goi (Fig. 1l), the detailed analysis of which has been described by i)dberg and bdberg.“‘) The main advantages this coil has over the anti-Helmholz type is that it gives field gradient uniformity over large sample volumes and much lower inductance. The latter property is important for better performance in high-gradient PGSE. The quadrupole coil can also be rotated in the main field to change the direction of the gradient (Fig. 12).

A FIG. 11. Different field gradient coil geometries (after Hutchison (Q1’).(a) The Maxwell or the anti-Helmholz pair in two geometries, (b) the quadrupole set, and (c) Golay set.

ii

N

S

S @

N Ouadrupole Field

s +

Uniform Field

Resulting Field Gradient

FIG. 12. Illustrating the addition of a quadrupole field to a uniform field to produce a field gradient in two orthogonal directions (after Hutchison”“).

Molecular diffusion studies

21

Karlicek and Lowe’54i have described a twin winding quadrupole coil, suitable for generating positive and negative gradient pulses. An interesting coil type was suggested and analyzed by Sobol and Blicharski.‘g3i It comprises two anti-phase coils would on pyramids; the triangular pyramidal shaped type, in particular, may offer advantages over the traditional anti-Helmholz type in terms of gradient/current performance and field gradient uniformity. The paper by Zupancic and Pirs(g4) also contains much useful information on coil design. The great activity in NMR imaging techniques has revived the interest for advanced highlinearity gradient coils. Three.very valuable papers that also review the older literature in the field were published recentIy.‘g5-g7) With regard to FT-PGSE techniques and probe/coil design, there must be a compromise between design criteria for (a) good field gradient linearity, (b) good rise and fall times of the actual gradients at the sample location, with minimum gradient shape distortion from Eddy-current or inductionrelated effects, (c) good field homogeneity at the sample location with gradients switched off (i.e. good resolution in general), and (d) good rf tuning properties and good probe sensitivity. Bangerte+‘ai has described the modification of the Bruker HFX-90 shim circuitry for pulsed field gradient operation in the context of homospoil pulse generation. A similar approach can be adopted on most other spectrometer systems. It is not advisable to use a standard high-resolution probe and to utilize parts of the normal shim coil circuity for quantitative FT-PGSE experiments, however, since the background homogeneity may become degraded or unstable and most shim coils have unacceptably poor performance anyway with regard to criterion (b) above. In practice, the great majority of present and future users will probably try to add gradient coils to an existing probe, rather than to design a field-gradient probe from scratch or to ask spectrometer manufacturers to offer one commercially. In this case, the easiest way is to remove from a spare probe the spin decoupling coils and wind new gradient coils on the dewar insert. Anti-Helmholz designs are often feasible only on supercon probes, because of internal space in the probe housing. Moseley and Loewenstein have presented a twin-gradient (x and/or z) add-on quadrupole coil design, suitable for standard iron magnet probes. (g9) In general, one should anticipate a moderate to severe degradation of both the probe resolution and sensitivity after adding extra gradient coils inside the probe walls. Of standard commercial instruments, only the JEOL FX-100 series has suitable (anti-Helmholz) gradient coils already installed in the (plastic) sidewalls of the probe (originally designed for field modulation purposes on a sweep instrument). These are normally used for homospoil pulse generation. The FX-60 still has these coils connected in-phase, but it is trivial to change their relative polarities. The FX-90 or FX-200 gradient coil arrangements are not optimal for quantitative PGSE experiments. Varian and Bruker currently offer custom-built supercon probes with integral (zdirection, anti-Helmholz) field-gradient coils on special order, and the trend is that field gradient probes my be a standard option in the near future. Our own custom-built, field-gradient Smm switchable multinuclear Varian XL-300 probe is completely within resolution and sensitivity specifications for the equivalent standard probe. One potentially troublesome problem with supercon probes in the context of FT-PGSE is that the active sample volume is much larger than in iron magnet probes; it may be difficult to achieve a uniform gradient linearity over the whple sample, which could cause severe systematic errors in quantitative FT-PGSE measurements. Quadrupole coils are superior to anti-Helmholz coils with regard to field gradient uniformity over large volumes, and would therefore be the preferred type in most occasions, provided that internal probe space permits. The simplest solution to the problem of uniformity of the field gradient is to reduce the sample volume; the sample should never be spun in PGSE experiments.

6.5.9.3. Gradient drioer arrangements. As mentioned earlier, the gradient driver electronics of ‘highgradient PGSE’ is rather elaborate. It should be capable of generating identical pairs of truly rectangular millisecond current pulses of up to 100 Amperes, without droop or any other nonlinearity effect. Usually a trim circuit is added to the main circuitry to reduce the effects of the

22

P. SllLBS

PULSE IN TO COILS

d

FUSE 1A

FIG. 13. A simple field gradient driver suitable for FT-PGSE work.

pulse mismatch of the two gradient pulses of the high-gradient PGSE technique. Some modem high-performance driver designs are discussed in Refs 54.69,85, 100 and 101. FT-PGSE does not require such high currents; the gradient driver can (and should be) be made trivially simple. Provided that one has a good 5V/l-2A supply (adequate commercial units are priced below 100 dollars), one only needs a buffer TTL gate driving a power transistor in series with a highquality resistor (typically in the range 5-25 ohms), a fuse and the gradient coil. It is convenient to include a provision for switching between different series resistors to select an appropriate precalibrated field gradient for a particular experiment. (One should, of course, convince oneself by regular checks that the calibrated values do not drift with time.) We originally used the standard JEOL homospoil driver in our FT-PGSE experiments. It is adequate for the purpose in question, but has the disadvantage that the gradient cannot be changed without the need for recalibration. Our own present driver (the basic design of which was suggested by J. F. Martin, personal communication) is outlined in Fig. 13. It performs very well in the typical FT-PGSE field gradient pulse range (lo-300 ms) when driving a switch-selected current value of 100-1000 mA into a lowresistance field-gradient coil. Moseley and Loewenstein (99)have described a similar gradient driver. inevitable

6.5.9.4. Gradient calibration. It is difficult to calculate the actual field gradient inside an NMR probe in a magnet to the desired accuracy directly from the coil shape and current. Traditionally, gradient calibration in SE or PGSE experiments is made either through band shape analysis of the spin echo in a steady field gradient, or indirectly through analysis of the echo decay of a PGSE experiment on a singlecomponent solution with a known molecular self-diffusion rate. The latter procedure is the preferred one. The echo shape reflects quantitatively the (background) gradient strength, and is also strongly dependent on the sample geometry (which is of course accurately known in most cases). For a long, cylindrical sample in an iron magnet, and a linear gradient in the z-direction (that is, along the static field), the echo shape (on-resonance and in the absence of J-modulation effects, uide injra) is given bq5’ S(t)= 2J~,[rRG,(t -t,)]/yRG,(t

- t,).

(18)

where J1 is a first order Bessel function, R is the radius of the sample and t, is the position of the spin echo maximum. Similar expressions apply for other sample geometries and gradient conditions”41 As discussed by Fukushima and Roeder’16’ one should note, however, that it is intrinsically impossible to have the whole sample on-resonance in a steady-gradient, and that the echo shape calibration procedure is consequently only approximate in the outlined form. Furthermore, a ‘long, cylindrical sample’ in the above context means infinitely long; effects of ‘nonideal’ sample shapes may introduce other systematic errors in the echo shape gradient calibration procedure.“) The most popular PGSE gradient calibration liquid has apparently been glycerol, which is characterized by relatively slow, temperature-dependent molecular diffusion, and short and

Molecular diffusion studies

23

temperature-dependent T,-values. The choice is somewhat difficult to understand, except perhaps when studying of very slow diffusion with short rf pulse intervals or in attempts to locate the echo. Possibly, it originates from the use of glycerol in other classic investigations in the early days of NMR spectroscopy. For the mere purpose of locating and adjusting the echo, D,O solution of poly(ethyleneoxide) of molecular weight of the order of 100,000 is an excellent alternative, in that the T,-values are much longer and the diffusion attenuation is very small. Strong temperature dependence in D and short (also very temperature-dependent) T’.-values are undesirable features in the context of gradient calibration (see Section 6.5.9.6). Tabular values for glycerol diffusion, as well as references to older work in the field, are found in Refs 14 and 102. In the former paper an interesting technique (lo3i for absolute gradient calibration is discussed in some detail. It is based on an analysis of the echo position/amplitude relation in experiments with a systematically varied intentional mismatch between the two rectangular gradient pulse durations. For calibration in low-gradient high-resolution FT-PGSE experiments we have found that indirect calibration using Hz0 diffusion in H,O/D,O mixtures containing between l-5 y0 Hz0 is the best choice. (It is actually impossible to study glycerol self-diffusion with the low gradients of high-resolution FT-PGSE, because of the short proton T,-values). The prime advantages of the H,O/D,O method of calibration are (a) accurately known and tabulated diffusion coefficients,13*~‘04~105i (b) relatively low temperature dependence of D, (c) long and relatively temperatureindependent T,-values, and (d) the deuterons provide an excellent field/frequency lock signal. In our own laboratory the FT-PGSE experiment works ‘ideally’ and we believe that although gradient calibration is made on a liquid like water which has a value of D of the order of 2 x 10m9 m2s-i, this can be inserted into eqn. (13) and utilized for studies of much slower diffusion (with appropriate changes in 7, A and 6). Also, we believe that the gradient calibrated in this way is correct for ‘other’ nuclei with much lower gyromagnetic ratio than protons, when using a multinuclear probe and a common sample size. Ideal FT-PGSE behaviour will, of course, not be the case in all current and future applications of the techniques and water diffusion may be too fast for appropriate calibration procedures when using stronger gradients than 20-30 mT m-i. A secondary multicomponent calibration procedure can then be applied. First, the gradient is calibrated with a low gradient on H,O/D,O, secondly a sample is prepared with say three components with diffusion rates that differ by one magnitude each, and the diffusion rates measured with the previously determined gradient. (Convenient choices are micellar 0.01/20/l/0.2 H,O-D,O-butanol-surfactant system or a mixture of HsO-D,O-crown ether-poly(ethyleneoxide), for example.) Thirdly, the gradient is increased and the gradient recalibrated on the now known constituent diffusion coefficients, in one or more steps. 6.5.9.5. T, and J-modulation effects. These are the basic sources of potential artifacts when measuring the self-diffusion in multi-component samples by m-PGSE experiments. In the. reviewers opinion, they have been overlooked as a source of artifacts in many of the time-domain PGSE measurements reported in the literature. Firstly, it is important to note that T2 in a spin coupled system can be much shorter than T,, even in the absence of slow motional processes or chemical exchange. This was first clearly pointed out by Vold et al.,“‘@ and is also discussed by Kaiser et al.(‘O)The quantitative treatment of spin relaxation in coupled systems has been reviewed by the Volds.‘lo’i In the present context we will not concern ourselves with the detailed theory (which rapidly becomes formidable above three coupled spins), but rather illustrate the effects and focus interest on the consequences for PGSE and FT-PGSE echo amplitudes and phases. Figure 14 probably will be familiar to the majority of NMR spectroscopists. It is evident that the carbon-13 T2 is much shorter in the proton spin coupled case than when the protons are broad-band decoupled. In Fig. 15 we show the spin-echo spectrum of the protons in n-propylbenzene, and one notes first the familiar J-modulation effecti “ai of the phases (not the amplitudes directly) of the components of the propyl group. The origin of these effects is that different spin states are interchanged by the 180” pulse of the SE sequence; the precession frequencies then change as well and spin vectors will not refocus at the negative y’-axis. Depending on the spin-echo rf pulse interval,

P. STlLB.9

24

No decoupling

I

13 C

( Ethyl benzene)

25 MHz Proton noise decoupled

I 200 rns, FIG. 14

13C free

induction

decays from

ethyl

benzene at 25 MHz decoupling.

with

Propyl benzene L

‘H .JzlJt’

and without

-CH2-

proton

broad-band

-CH2-

. nonspinrkg limp.1

l

L ,....,....,‘.“,..“‘~~~‘,‘~~“‘~~~,””””~I”-~I~~..~~“~I~‘~‘~~~’~l~~~~‘~~“~’ ..D ..e ..L 7.0 ..O

spin- echo

3.0

1.0

1. @I

0.0

f = 140 ma A

FIG. 15. Normal

and spin-echo ‘H NMR

spectra of n-propyl benzene at 99.6 MHz.

the components of first-order multiplets undergo a cosinusoidal phase modulation in periods of integer fractions of l/J. (A good overview on spin echo effects in high-resolution NMR was presented some years ago by Freeman and Hi11152)and the subject is also discussed in Turner’s review.‘4*‘) It is evident that signal amplitudes in spin-echo spectra may be either positive or negative, depending on the pulse interval. This corresponds to different phases of the signals in the timedomain at the echo maximum of unresolved POSE techniques (and thus for the contribution to the echo maximum). A further effect is clearly seen in the spin-echo spectra; the shape of the phenyl band is highly distorted and the overall amplitude is much less than that of its relative counterpart in

25

Molecular diffusion studies

sQ5,a-cnf PGSE

s

- CHJ

‘,I

1,1,1.11,‘1’

,‘11,

,

5.0

ppm

3.0

4.0

FIG. 16. Normal and absorption-mode

IT-PGSE

.?.O

1.0

‘H-spectra on a 0.2 M sodium dodecylsulphate heavy water.

solution in

A further example is shown in Fig. 16. Here the main alkyl -CH,-band is essentially nulled in spin-echo spectra. A general empirical observation of frequency-resolved spin-echo spectroscopy of homonuclear coupled spin systems is that there is a gradual change in signal appearance as a function of the order of the spin system; from simple cosinusoidal J-modulation effects with very little effective amplitude distortion in first-order systems through severe amplitude distortions for intermediate coupling and then back to ‘normal’ effective amplitude for very strongly coupled systems. The general amplitudedistortion effects should appear familiar to anyone who has attempted to run a two-dimensional COSY experiment on a partly non-first order spin system; the cause is the same. Bandshapes, effective amplitudes and T2 effects in general in a particular spin system combine into a single term which depends solely on the rf pulse interval of spin-echo spectroscopy (r) and are completely unrelated to the time intervals A or 6 in the PGSE field gradient extension. By performing the experiment at a fixed 7, the diffusion contributions to the echo decay can thus be completely isolated from the combined T,-effects. This procedure, of course, is the common practice in PGSE experients. It should be evident, however, that relative echo contributions in a homonuclear spin coupled system normal spectra.

bear no direct relation to the relative number of nuclei involved; individual echo contributions need not even have the same sign! This applies in FT-PGSE, as well as in unresolved PGSE. In general, with

increasing r-values, echo amplitude contributions become less and less correlated to the actual number of nuclei giving rise to the echo, as a collective result of the T2 and J-modulation effects. One can easily convince oneself of these conditions through direct observation of frequency-resolved spin-echo spectra but they may not be immediately apparent in time-domain SE or PGSE experiments. Failure to recognize the complex relations between signal amplitudes and signs and the pulse interval conditions may be the major potential source of evaluation artifacts in multicomponent self-diffusion studies by FT-PGSE as well as PGSE techniques (vide supra). As discussed further below, the difficulties can easily be overcome through the use of correct evaluation procedures.

26

P. SllLBS

6.5.9.6. Temperature effects. Since PGSE experiments are performed under conditions of constant T, one might overlook that there is a further source of artifacts that originate from the T,-term. T2values are, of course, temperature-dependent. If experiments for some reason are performed in such a way that the diffusional term in the PGSE echo attenuation is not entirely dominant (such as when there are low gradients, low gyromagnetic ratio, or slow diffusion), then temperature drift or fluctuations may cause the variations in the T,-contribution to become significant or even dominant in extreme cases. The effects on the echo decay because of short term fluctuations in temperature will degrade the precision of the experiment. A long term drift in temperature increases or decreases the apparent echo decay because of diffusion effects. The indirect effect on the echo decay caused by temperature or T2 variations is potentially much larger than the actual temperature effect on the (real) self-diffusion coefficients (which does not usually call for better precision in temperature control than *(OS-1)“C to be compatible with the overall precision of the experiment). Good temperature control is therefore a necessity for adequate PGSE experiments. When running FT-PGSE experiments on heavy nuclei such as ‘33Cs (as cesium ions in water, for example) with inadequate temperature control, there may be direct effects on the echo and potential evaluation artifacts from the large temperature dependence of the chemical shift; the signal may become artificially broadened by timeaveraging, and the ‘peak-picking’ routines of automated spectrometer software may fail to follow the peak maximum properly during the course of the experiment. A further temperature-related effect should be mentioned at this point; temperature gradients in the sample may induce convective flow, particularly for low-viscous liquids. When the sample is at elevated temperatures, in particular, the sample volume should be reduced as much as possible. It is advantageous to work at elevated temperatures in low-gradient FT-PGSE applications when diffusion is slow; in this case T2 and D will both almost always increase with temperature, thus making the diffusional contribution to the echo decay more dominant [c.f. eqn. (13)], and hence improving the precision with which D can be measured.

6.5.9.7. B,-effects. Spectra from homonuclear spin coupled nuclei usually become simpler and of first-order type when the &field is increased (except in the case of very strong coupling). It should be apparent from the discussion in the previous Subsection that this reduces attenuation of the echo which is caused by complex J-modulation effects, and there is an increase in the signal to noise ratio in the spectra. The inherent improvement in frequency dispersion and the increase in signal to noise ratios make it an advantage to carry out FT-PGSE measurements at higher static fields. We have recently started FT-PGSE work on a Varian XL-308 (7 Tesla field strength); only preliminary tests have been made, but the results so far have exceeded our initial expectations and appear very promising (c.f. Figs 1, 17 and 18). Some concern has been expressed (by spectrometer manufacturers, in particular) about the possibility of quenching a superconducting magnet when applying field gradient pulses. There is still very little known about the actual risks, which will anyway vary between different magnets and coil geometries. The danger should be reduced if the field gradient coils are located as close to the sample as possible. Large gradient (1 T m-l) PGSE experiments on supercon magnets have evidently been made successfully in the past, and the reviewer has heard about only one quench which was caused by application of a pulsed field gradient (in a spectrometer manufacturer’s research laboratory). 6.5.9.8. Notes on the use of absolute-value frequency-domain spectra. One might argue that acquisition-related phase errors and instabilities as well as many of the problems discussed in the previous Subsections could be circumvented by the use of absolute value spectra (also named magnitude or power spectra). This may be correct when the spectra are of first-order spin systems. In the case of intermediate coupling this is not correct and the spectral appearance is not restored to absorption-made type by this proceedure, and the inherent amplitude distortions of spin-echo spectra are also not removed. More importantly, individual bandshapes of power spectra are not

Molecular diffusion studies

27

additive in overlapping regions and any attempt to neutralize phase errors of absorption-made

spectra by going to power mode calculation will certainly introduce serious errors in the majority of attempts to study multicomponent self-diffusion quantitatively by FT-PGSE.

6.5.9.9. Notes on FT-PGSE pulse programming on commercial instruments. Virtually any commercial spectrometer system can be modified so as to generate the pulse sequences needed for FT-PGSE; on most systems it is a straightforward software matter. The simpler pulse programming systems are designed to start acquisition directly after the ‘observation pulse’. The first problem that has to be solved is to program the acquisition to start exactly at the echo peak, typically 100 ms later. Instabilities or systematic errors with regard to the actual starting point are not acceptable, since they will lead to frequency-dependent phase errors in the absorption-mode spectra. Firstgeneration, software-based, pulse programmers may have an unacceptable timing jitter and may be unsuitable for this demanding application. For automated measurements of any kind, one must also find a way to increment 6 while keeping A and T constant. This is usually straightforward on presentgeneration instrumentation but may be more difficult with primitive pulse sequence generators. For manual measurements on older (pre-1975) systems with hardware pulse timing or userinaccessible pulse sequence generation, one can easily add a hardware or external software-driven timer, triggered twice by the .two PGSE rf-pulses, so as to generate gradient pulse durations of a suitable length. Timer circuits of various kinds have been described in the literature/“* “* lo9* ilo) most of which are more than adequate for the present purpose. The acquisition-based phase error problem just mentioned makes it essential, however, that the pulse timing be crystal-controlled rather than be derived from analog electronics. The reviewer has built a simple 2 MHz crystaldriven, 5 chip, &pulse triggered one-shot timer that can be thumbwheel set in millisecond steps between 2 and 100 msec for less than 15 dollars. It has performed well for FT-PGSE measurements on a JEOL FX-60 machine (used by the Lindman group in Lund), in conjunction with the optional JEOL FX-60 T,-pulse/acquisition program (see Ref. 85). The simplest approach to external field-gradient timing today is to buy a software-driven function generation card (with host-independent internal timing and logic) with external trigger capability for use on a personal computer of Apple II or IBM PC type. Several such cards are advertised in computer journals. At a reasonable cost one can quickly acquire a convenient, partly automated system with excellent timing stability. Pulse durations can easily be incremented through userwritten software to various levels of sophistication. In nearly all our own studies, the JEOL FX-100 system has been used. This has the full hardware and software capability for FT-PGSE in its standard configuration; a suitable probe with integral field gradient coils, a data system based on a Texas 980 computer, a pulse-sequence adressable multichannel CPU interface and a floppy-disc or moving-head disc system. The pulse programming facility is rather primitive by today’s standards and cannot increment 6 directly while keeping A and T constant. This problem can be overcome by preparing a disk file (MENU) for each combination of the pulse timing parameters with one ‘dummy’ DATA file, then execute these as a series of RUN’s and afterwards analyze them with the standard software as if they were a single RUN. A similar approach can be taken on older Bruker, Nicolet and Varian machines (provided there is some kind of spare output port on the computer/pulse programmer that can logically drive a field gradient pulse generator circuit); write a number of ‘dummy’ spectra into disc files (current timing parameters will then be stored as well), then make a looped read (timing parameters are then read back into the pulse programmer together with a dummy spectrum) zero aquisition memory-execute-storeincrement file identification number. Finally analyze the spectra collected in this way by a procedure analogous to that of a Ti-measurement. 6.5.9.10. FT-PGSE based on the stimulated echo sequence. As discussed in Section 6.5.2, the stimulated echo sequence is traditionally preferred over the ordinary echo if T2 is significantly shorter than T,, since the major time interval for spin relaxation is in the longitudinal direction rather than the transverse. This also applies in FT-PGSE, and one can use this alternative technique

28

P.

STlLBS

FIG. 17. Normal, stimulated-echo FT-PGSE (rl =40 msec) and normal echo FT-PGSE (A= r= 140 msec) proton spectra on a 20% n-propanol/D,O solution. Note the absence of the OH peak in the latter series, this peak is broadened (ca 8Hz) by chemical exchange between alcohol and water sites and therefore has a relatively short value of T2.The real water diffusion rate can be determined through a knowledge of the propanol diffusion rate (from the three other peaks) and the known composition of the solution (cf. Ref. 158, for example).

to monitor signals that are anomalously broadened by intermediately rapid exchange phenomena. An example is given in Fig. 17. In IT-FUSE, the properties of the stimulated echo can be used to advantage in a more subtle way than simply to increase the signal strength of broadened peaks. As discussed in Section 6.5.9.5, Tz in a spin coupled system can be very short compared to Tl and this can lead to ‘amplitude distorsions’ in Fourier transformed spectra. For this reason stimulated-echo IT-PGSE, may therefore have relative signal intensities that are very different from those of normal FT-PGSE. An example is given in Fig. 18. Here the main alkyl -CH,- band is more than five times larger in the stimulated-

II!.! I

6=65ms -4,

FIG. 18. Spectra

analogous to those of octylbenzenesulphonate.

Fig. 17, but on an aqueous The butanol

is micellarly

micellar sample of D,O/n-butanol/sodium solubihzed to about 50 “/,

Molecular diffusion studies

29

echo variant and several other intensity differences are evident upon closer inspection. J-modulation effects are, in general, considerably reduced in the stimulated-echo variant and the spectra look more ‘normal’. When investigating multicomponent self-diffusion in complex mixtures, a combination of stimulated-echo and normal FT-PGSE measurements can be applied to enhance the relative intensity of certain bands in cases of weak intensities, broadened peaks or signal overlap. 6.5.9.11. Ecaluation procedures. The primitive way to evaluate PGSE experiments (at a fixed rvalue) is to plot the logarithm of the signal intensity against S’(A--S/3)G2 [c.f. eqn. (13)]. For isotropic unrestricted diffusion, the slope of this linear plot then gives the diffusion coefficient. Deviations from linearity (that are not caused by instrumental artifacts) can be separated into diffusion contributions from different species or interpreted in terms of restricted diffusion and requiring more complex relations than those proposed by Stejskal and Tanner. In the case of nonlinearity, the echo amplitude must generally be monitored over a large range of amplitude and s2(A4/3)G2 values. Modern computing techniques make it easy to apply non-linear least-squares fitting procedures. It is a trivial matter to fit amplitudes to equations of the StejskalLTanner type directly, through routines that are available on virtually any mainframe computing facility. Non-linear fitting procedures do not suffer from problems of a different weighting of the data points (in the linearized form just mentioned, the data points should be weighted in an exponentially decreasing fashion with increasing &*(A-S/3)G2). Also, it is easy to change and extend the form of the fitting function, for example to use multi-exponential echo decay functions in the case of overlapping peaks in FTPGSE. The problems in separating sums of exponential decays from a composite signal in the general case are well-known; it is difficult to make a separation unless the data have very good precision and the individual time-constants differ by a factor of three or more. In the frequency-resolved FTPGSE proton NMR-experiment on a multicomponent mixture the situation is much brighter; it is almost always possible to find an isolated proton NMR band from some part of a particular molecule. This makes it possible to evaluate that particular diffusion coefficient, and thus the ‘decay timeconstant’ for that particular contribution to any other frequency band in the spectrum is known. It is important to note that the ‘time-constant for that particular molecule’ is the same throughout the whole absorption-mode spectrum and can be entered as a known constant in a subsequent simulation. The actual amplitude of that contribution at some other frequency location is not known, however, even with regard to its sign (see Sections 6.5.9.5. and 6.5.9.8) and it must enter as a freely optimizable fitting parameter. An example of such a data evaluation is given in Fig. 19. In this particular case, the constituent diffusion coefficients are quite different. It is actually safe to make a similar analysis on overlapping peaks from molecules that have diffusion coefficients that only differ by a factor of two; provided that the signal to noise is good, that the signal amplitude is monitored over a large dynamic range and that the two signals have similar amplitudes at a particular frequency channel. Note that this evaluation procedure will nor work for absolute-value or power spectra (Section 6.5.9.8). An important factor in the analysis of experiments of the present kind is the number of significant digits in the representation of spectral intensities. This may vary quite a lot between different instrumental designs. Manual registration typically provides two significant digits and by manually changing the spectrometer or display gain, one can achieve the same accuracy at any amplitude. ITPGSE measurements will almost always be based on semi-automatic or automatic data reduction (like in a common automated T,-experiment). Some spectrometer systems (the JEOL FX series, for example) make the data reduction with four significant digits, others only achieve two significant digits (all referred to the ‘strongest’ peak in a measurement series). This makes an enormous difference in the evaluation of high dynamic-range experiments (i.e. when studying a small signal in the presence of a large one or in a case where one wishes to follow the intensity decay over a long 6*(A-6/3)G*-interval). Hopefully, the latter deficiencies will be rectified in future versions of spectrometer software. It is a common practice in traditional PGSE experiments to follow the echo decay over at least one, possibly two or three decades of amplitude. This, of course, is necessary if one wishes to study

30

P. %lLBS

r

;c,2ES~octone

100

6=95ms

Field gradlent duratlod

s

FIG. 19. Experimental FT-PGSE spectra on HDO/n-octane/C, ,Es(C, 2 Ha, (OCH, CH2hOH) a nonionic microemulsion system that forms inverted micellar structures at that particular composition. The water and CIz E, diffusion coefficients can be evaluated separately from the two leftmost peaks (note that water is the most slowly diffusing species in this system). The analysis of the two combined peaks (predominantly n-octane) to the right is then made through fitting to a ‘double Stejskal-Tanner exponential’ with freely varying relative amplitude factors corresponding to the two components, C,, E, and n-octane. The Cl2 ES diffusion coefficient determined from the single peak is entered as a known constant in the simulation. The diagram to the right shows the agreement between theory and experiment. The echo amplitudes were recorded with a precision of four significant digits. The dashed lines indicate the best fits to a single exponential (which result in a systematic underestimation of about 10 % in the octane D-value). The full analysis leads to D-values of 0.078,0.199 and 2.05 10mgm2sm1 for the three components HDO, C,, E5 and n-octane, respectively. (The slightly oscillating phase of the signals is caused by slight field/frequency oscillations of the spectrometer system during data acquisition. The oscillations originate from spectrometer lock perturbations by the magnetic field gradient pulses.)

restricted diffusion in heterogeneous systems or in the case of a multi-exponential decay which is caused by signal overlap. Provided that diffusion is isotropic and that there is no signal overlap, this procedure is neither necessary, nor recommendable. The data of Fig. 1, for example, lead to D-values

that have statistical 90% confidence intervals that are of the order of 1%. With adequate signal amplitude representation in the data acquisition system and a good signal to noise ratio for the experiment, diffusion coefficients can still be determined to an accuracy of a few percent even in a case where the diffusion al effect on the echo decay is only lo-20 y0 (as based on say 10 data points in the interval studied). This is an important positive aspect when designing low-gradient FT-PGSE experiments in cases of slow diffusion or when monitoring the diffusion of nuclei with low magnetogyric ratio. 659.12. Interpretation of PGSE experiments on polydisperse or heterogeneous systems. With regard to polydispetsity, one should first consider the time scale of the experiments; PGSE experiments monitor time-averaged transport during a selected period of typically l-1000 ms (see Section 6.56). Self-aggregated systems such as micelles have a shorter lifetime than that and they are consequently not polydisperse in the present context. Synthetic polymers in solution, on the other hand, are polydisperse to more or less extent on any timescale. The procedures for evaluating PGSE experiments in terms of different polymer polydispersity distributions have been presented by von Meerwall et al.” I1.1lL) and by Callaghan and Pinder. (13) Fleischer also discusses the subject in a recent communication.(1’4) The monitoring of diffusion in polydisperse systems or restricted diffusion in heterogeneous systems is more demanding with regard to data handling procedures than normal diffusion studies. In essence, one has to interpret deviations from the simple Stejskal-Tanner relation in terms of

Molecular diffusion studies

31

different echo attenuation models for the system in question. Naturally, one then should monitor the echo decay over a long S’(A-6/3)G2-interval and collect many more data points than in the case of normal, isotropic diffusion. The question of data analysis in the case of restricted diffusion, restricted diffusion in polydisperse cavities and diffusion within systems of semipermeable carriers has been discussed by Packer and Rees!“‘) von Meerwall et al.J116*‘5) and by Callaghan et al. (I1 ‘I) A complete computer program for the analysis of several transport models of the type mentioned is available.‘116) 6.5.9.13. Survey of diffhrent nuclei. The majority of NMR diffusion measurements to date have been made on protons; the most abundant ‘NMR nucleus’ and also the most facile nucleus for the purpose. Two alternative, ‘favourable nuclei are “F and ‘Li (as Li+-ions) and most spin-l/2 nuclei are usually accessible for FT-PGSE studies. Quadrupolar nuclei have more or less unfavourable spin-echo characteristics. Deuterium, for example, has relatively rapid spin relaxation and a low magnetogyric ratio and studies are possible only on isotopically enriched samples of low molecular weight. Multi-component deuterium self-diffusion investigations are impossible since the shift dispersion in deuterium frequency-domain spectra is too small, even at high magnetic fields. It is typically only 1 y0 of that of 13C on the same compound, when accounting for the differences in spin relaxation rates. 13C has low natural abundance and a low gyromagnetic ratio and isotope enrichment procedures often require difficult and expensive synthetic work. Natural-abundance “C FT-PGSE measurements on concentrated systems are actually quite easy, however, and very complex systems then become accesible for study as a benefit of the large “C chemical shift dispersion. (83.118.119) In the context of experimental design, one should remember’ “) that ’ 3C T,-relaxation under proton broadband decoupling is dependent on the efficiency of the proton decoupling and that the (artificial) 13C T,-values are usually much less than the corresponding T’-values, at least for low-molecular weight compounds. A few quadrupolar nuclei are accessible for FT-PGSE, where the quadrupole moment is small and there is a symmetric electronic environment, as found for hydrated ions in aqueous solution. Examples are ,35C10;, 6Li+, ‘Li+, 13’Cs+ and ‘Be’+. Spin-lattice relaxation rates for the latter ions in D,O solution are several se&nds, and they are seldom below one second, even in aqueous solutions containing macromolecules. Most other quadrupolar nuclei are more or less inaccessible

TABLE 2.

Nucleus ‘H ZH ‘Li 9Be “C 19F Z’Na “Mg 3’P WI ““Ag ‘13Cd 13%s

NMR parameters of relevance for PGSE for some nuclei

Spin quantum number (1)

Gyromagnetic ratio (lo-’ rad/r s)

(Yr&J”)2

l/2

27.752 4.107 10.396 3.759 6.726 25.167 7.076 1.638 10.829 2.621 - 1.245 5.933 3.509

1.000 0.022 0.140 0.018 0.059 0.822 0.065 0.003 0.152 0.009 0.002 0.046 0.016

3;2 312 l/2 l/2 312 512 112 312 l/2 :;

*Instrumentally determined and dependent on proton decoupling conditions; see text. tAs aqueous ions in solution. SPerchlorate; the figure for chloride ions is below 25 ms

Typical T, range in soln /s 0.05-30 0.01-3 0.3-15t 0.1-W 0.05-l * 0.1-3 below 0.067 below 0.22t 0.1-10 below 0.3tj very long 0.1-20 0.5-lot

32

P.

STILBS

because of a combination of low gyromagnetic ratios and rapid spin relaxation rates. For example studies on 23Na+ and 35C1- ions in water are impossible with the low magnetic field gradients of F-T-PGSE. Callaghan et al. have pointed out that the original analyses of the SE and PGSE experiments were made for spin-l/2 nuclei, and that it is not self-evident that they should apply for quadrupolar nuclei.““) They show that the original results are valid for spin-l nuclei in isotropic environments, but not in the anisotropic case. Table 2 summarizes the typical NMR parameters of relevance for potentially interesting PGSE nuclei.

659.14. Comments on the accessible range of diffusion coefficients in FT-PGSE NMR. If competing echo attenuation effects from 7’_‘_-relaxationdid not exist, the range of application of PGSE techniques would be unlimited. With present day technology, high-resolution FT-PGSE will operate with pulsed field-gradients of say 10-30 mT m- ‘. Considering the typical span of diffusion coefficients (see for example, Table l), i.e. 1O-g-1O-12 mz s- ‘, the situation is quite bright for FTPGSE-based studies on solutions. Here spin relaxation rates are relatively long, typically between 0.3 and 30 s for most spin-l/2 nuclei in small to medium-size molecules. Figures 20 and 21 illustrate the diffusional echo attenuation as a function of 6 for various combinations of the parameters in the Stejskal-Tanner relation, such as A equal to 0.1 or 0.3 s and G equal to 10 or 30 mT m-l. Figure 20 shows that proton-based isotropic and unrestricted diffusion rates down to lo-” m2 s-l are accessible even with such weak gradients as 10 mT m-r, provided, of course, that the Tz value is

1 G cm-'

gradlent

duration/

s

3 G

cm-’ gradlent

rluratlon/

5

1 G cm-l

gradlent

duration/

s

3 G

cm-’ gradlent

duration/

5

FIG.20. Simulations diffusion

coefficients.

of the echo attenuation for various combinations of gradient strength and duration and selfThe rf and intergradient pulse intervals (T and A) are both set equal to the maximum gradient duration (6).

Molecular diffusion studies

33

not very much below 0.3 s. For a shorter T2 the lower limit of accessible diffusion coefficients is considerably higher, since one is then forced to compress the experiment by reducing r and A to get an echo of measurable intensity. When studying restricted diffusion or deviations from the simple Stejskal-Tanner relation, the situation is much worse for low-gradient high-resolution PGSE. Firstly, most systems which exhibit some form of restricted diffusion have rather short relaxation times. One must therefore use short rf and field gradient intervals in PGSE experiments. Secondly, one must follow the decay over a much larger dynamic range; at least a factor of five or sometimes a factor of a hundred for adequate experimental determination of the quantities sought. This is rarely possible with low-gradient PGSE. A number of genera1 rules for experimental design on isotropic systems can be. stated. (a) An increase in the field gradient strength has essentially the same effect on the echo attenuation as a correspondingly large increase in A with a proportional increase in the &value interval. (b) A ten times smaller value of D can be studied by increasing G or A by a factor of three to achieve the same echo attenuation. Other nuclei than the proton are also accessible for PGSE experiments, if T2 permits. Figure 21 demonstrates, for example, that ‘%-based FT-PGSE experiments on isotropic systems are possible when D is below 10-r’ mz s-l, provided that T2 is not too far below 300 ms. the curves for ‘Li, 13C and *H may serve as a good first reference for proper pulse parameter settings for nuclei with similar gyromagnetic ratio, such as 31P, “‘Cd and (9Be, lJ3Cs).

1.0

B 3

2%

0.6

c” : 0.6 l-z ,u’

0.4

-2 cs’

0.2

,I

o.“o.oo

0.02

0.04

3 G cm-l

FIG 21. Simulations

0.06

0.10

0.00

gradient

duration/

s

gradient

duration/

e

of the echo attenuation proportions

1

0.0 0.00

0.10 1 G cm-’

0.20

0.30

gradient

duration/

s

gradient

duration/’

e

for four representative nuclei whose magnetogyric 1 :0.4:0.25 and 0.15. respectively.

ratios are in the

P. SllLBS

34

7. APPLICATIONS

OF FT-PGSE

7.1. Isotropic Systems 7.1.1. Miscellaneous Applications of FT-PGSE

Spktroscopy

7.1.1.1. Analytical utilization; ‘Size-resolved NMR’. As evident from the discussion in Section 659.5, J-modulation and other T,-related effects combine into a single term, which depends only on the 90”-180” rf pulse interval. This is true for each part of the bandshape in absorption-mode FTPGSE since individual absorption bandshapes are additive at each point. Furthermore, each atom in a given molecule has the same self-diffusion coefficient. It is therefore possible to separate and assign signals from different components in a complex mixture by FT-PGSE spectroscopy, based on their common diffusion-related echo attenuation.‘“@

7.1.1.2. Solvent signal suppression. It is evident that signal contributions from low-molecular weight solvents can be completely suppressed in FT-PGSE experiments, by extending the diffusion period to a suitable length; for example a 99 % H,O proton signal (D=2.3 x lo-’ m2 s-l at 25°C) is completely nulled with a gradient of 10 mT m-‘, ~=A=140 msec and 6=70 msec (c.f. Fig. 16). Under the same conditions, most of the signal of components with a value of five times lower ‘survive’. In biological applications, where the T2 values are often quite short, one may have to utilize stronger gradients and shorter rf pulse intervals than those given in the example. If one is content with spin-echo spectra on non-spinning samples this technique outperforms by far all other approaches to solvent signal suppression. The stimulated-echo variant of FT-PGSE has less amplitude distortion effects (cf. Fig. 18) and is probably a better’technique in this particular application. J-modulation effects essentially disappear in the limit of zero interval between the 180” pulses of the CPMG spin echo experiment. The successful application of the narrow pulse interval CPMG technique for water signal suppression in the particular case of exchangebroadened (short T2) water signals was described recently by Rabenstein et al.(I”) The ultimate choice would probably be an FT-PGSE variant of that approach; the solvent is usually the most rapidly diffusing species and thus has the most easily suppressed signal in FT-PGSE experiments. 7.1.2. Difusion in Simple Liquid Mixtures. Diffusion in simple liquids is a matter of much fundamental theoretical interest. SE investigations on binary solutions (with one component deuteriated) were reported as early as 1967 by McCall and Douglass.“21’ FT-PGSE techniques can provide the same information, without the need for deuteriation or component masking, even on 12component solutions (c.f. Fig. (1)). A more detailed approach to the problem was taken in an investigation of 1,1,2,2-tetrabromoethane and solvent diffusion in solutions in alkylbenzenes (chain lengths 0-14),‘122’ a study that had been essentially impossible with any other method than FTPGSE. The original papers by James and McDonald (diffusion in DMSO/water)(32) and Kida and Ueidara (diffusion in methanol/water) (so) should also be mentioned in the present Subsection. An FT-PGSE investigation of carbohydrate diffusion in water is also noteworthy.“23) 7.1.3. Diffusion in Solvent-Polymer Systems: SolutejSolvenr Interactions. The subject of diffusion in polymer solutions has been ,reviewed recently by von Meerwall,“s) and also by Callaghan.” ‘) Lowgradient FT-PGSE techniques almost always fail for the investigation of polymer diffusion (because of short T2 values, and slow diffusion). An exception is the polymer poly(ethyleneoxide), (PEO), which is unusually flexible in solution and has long proton spin relaxation times (TI as well as T,; there are no disturbing spin-echo effects from homonuclear spin coupling in the system) (c.f. Fig. 9). We have successfully investigated non-aqueous and aqueous PEO solutions with the purpose of testing the existence of the suggested anomalous structure of PEO in aqueous solution.i124’ FTPGSE techniques also cope easily with the otherwise difficult experimental problem of investigating

Molecular diffusion studies

35

diffusion of polymer ‘A’ in a solution of polymer ‘Br.(125) The good precision of the measurements allowed combined studies to be made on self-diffusion (FT-PGSE), mutual diffusion (quasielastic light scattering, QELS) and sedimentation (ultracentrifugation), for the purpose of studying experimentally the theoretically dij’erent frictional coefficients [c.f. eqn. (3)] for the three processes. (I*w Dextran diffusion (a linear polysaccaride) in aqueous solution could also be studied in a similar way, except that elevated temperatures had to be. used.“*‘) A combined QELS and FTPGSE study made possible definite statements on the origin of the so-called ‘slow-mode’ tails in the observed autocorrelation function in QELS on concentrated solutions.“*s) The experimental investigation of solvent diffusion in polymer solutions by low-gradient FTReference 129 should be PGSE techniques is, on the other htid, trivially simple. (129*130~131,132) noted as being a ‘“C m-PGSE study.

the

7.1.4. Solute Diffusion in Gels. The self-diffusion of small- to medium-sized molecules in serum albumin solutions and gels,““) and in cellulose gels~134~13s*136~ has been investigated by FT-PGSE NMR. Such experiments have the advantage that permeant diffusion rates can be determined selectively to quite low concentrations even in the presence of a huge water peak (c.f. Section 7.1.1.2). In addition, the signal from the gel framework is completely abSent with the relatively long rf pulse intervals used (ca. 100 ms). 7.1.5. Counterion Diflusion in Polyelectrolyte Systems in Solution 7.1.5.1. Polymeric polyelectrolytes: counterion binding. Ion binding and counterion transport processes in polyelectrolyte solutions is a field of much current interest and has attracted much experimental, theoretical and computational effort. Counterion self-diffusion rates are easily related to concepts such as ion binding and also to the numerical results of computer simulation experiments on polyelectrolyte solutions. Much interest has also been focused on the relations between counterion size and type and the ion binding. The low-gradient FT-PGSE technique is a good method for studying the relatively rapid counterion diffusion found experimentally (counterion diffusion rates are seldom less than 25 % of those of corresponding simple salt solutions), and is easily applicable to all organic counterions (monitored through proton-based FT-PGSE), and several inorganic counterions with ‘favourable’ PGSE characteristics (19F-, 3sC104-, ‘Li+, IJ3Cs+, ‘13Cd2+, and ‘Be*+ in particular). In many cases, the lower concentration limit (with overnight time-averaging) is in the submillimolar range, so that total concentration range of four decades may then be accessible for study, which is a highly relevant factor in investigations of the present type. The applications and physico-chemical aspects are discussed in greater depth in Refs 137-141. 7.1.5.2. Ionic surfactant aggregates: counterion binding. The problems and experimental conditions for these systems are very much the same as in the previous Subsection. Since these are selfaggregated systems, one can conveniently monitor the whole aggregation process through multicomponent self-diffusion studies, thus providing a detailed picture of the aggregation conditions of each constituent (i) down to millimolar or submillimolar concentrations in water.‘142’ The analysis of the aggregation process is based on the two-site micelle/free model, such that

which is known as Lindman’s first law; 0

36

P. STILBS

b HDO 2d

-

5~10”’1 50

100

1 150

L 200

ctotp

.

Molecular

diffusion

studies

37

CtotlM FIG. 22. (a) An FT-PGSE experiment on a micellar decylammonium dichloroacetate 40°C. (b) The diffusion coefficients of the constituents as a function of concentration. in terms of a two-site micellarly bound/free model; B denotes the degree of ion

solution in heavy water at (c) An analysis of the data

binding (from Ref. 143).

The problem of ion transport (rather than aggregation and binding) in concentrated micellar solutions is of current theoretical interest. Experimentally (and theoretically), traces of divalent counterions (‘Be2+) and monovalent counterions (‘Li+ of lithium dodecylsulphate) in a monovalent surfactant solution exhibit quite different transport conditions, as monitored by FT-PGSE.“4s) 7.1.6. Diffusion in Binary SurfactantJWater Systems: Surfactant Aggregation. The micellization process, and subsequent changes in micelle shape can be studied by measuring self-diffusion coefficient by the FT-PGSE method. Multicomponent self-diffusion data, (in particular when combined with laser light ‘scattering and sedimentation measurements(146)) provide a detailed picture of the aggregation processes of nonionic surfactant systemso4’) 7.1.7. Diffusion in Ternary SurJactantfWater Systems: Solubilization and Mixed Micelle Formation. Solubilixation (the incorporation of a highly or partly hydrophobic third component into the micellar pseudophase of a surfactant/water solution) is a process of considerable technical and theoretical interest. Multi-component self-diffusion data on the micellar solution provide a new and direct method(14*~14g) for the quantification of the partitioning of solubilizate(s) between the aqueous and micellar pseudophases, according to the simple relation:

where p,,,icoa.r is the degree of solubilization (O148-154) Th e same technique is easily adaptable for investigations of solubilization into vesicular membranes.‘1s5) Mixed micellization, which is the partial partitioning of two or more micellizable surfactants into the same micelle, can also be studied by FT-PGSE along the same lines.“56*157) 7.1.8. Diffusion in Microemulsions. Microemulsion Structure. The structure of ‘microemulsions (isotropic surfactant systems, simultaneously containing large amounts of water and hydrocarbon, and often also co-surfactant; commonly a medium-chain alcohol) has been a subject of considerable controversy during the last decades. Are they at all ‘emulsion-like’ or are they just normal liquids with ‘partly aggregated’ constituents?

38

P. Smas

The idea is to utilize the combined information of the microemulsion.

on the self-diffusion of the individual constituents:

(1) In a micellar (or ‘oil-in-water’) system, water diffusion is rapid, while that of surfactant and solubilized hydrocarbon is slow. (2) In an inverted micellar system, the hydrocarbon diffusion rate is high, while those of water and surfactant remain slow. (3) In a ‘structureless’ system, all constituents diffuse rapidly. (4) In a bi-continuous system, the constituents confined to the structural framework diffuse slowly, while all other constituents diffuse rapidly. Investigations along these lines were started with cumbersome radioactive tracer and traditional PGSE techniques in the late seventies by Lindman et al. The FT-PGSE technique, of course, is the ideal method in this case; there are usually no problems whatsoever to monitor simultaneously the self-diffusion of say water, hydrocarbon, co-surfactant alcohol and surfactant in a matter of minutes per sample (c.f. Fig 10 and 19). We,t1ig*‘5*-163i and later also other groups,“64-170) have made several studies along these lines. The work has been summarized in overviews and review papers. (17’-174*24) According to our findings, microemulsion ‘structure’ can span all extremes, depending on composition and constituent chain length. 7.1.9. Diflusion-Based Monitoring of Miscellaneous Aggregation Processes in Aqueous Solution. We have studied the binding of various substrates to cyclodextrins in aqueous solution through FTPGSE-techniques,“75i with essentially the same approach as that in the solubilization studies just mentioned. Mononucleotide aggregation” 76*177)and ion binding to nucleotide aggregateso78) have also been monitored successfully. All these studies are analogous to those on surfactant systems, but the approach is not quite as .powerful here; the binding had only a small effect on the time-averaged diffusion coefficients since the cyclodextrin and nucleotide aggregate diffusion is not as slow as that of micelles.

1.2. Restricted and Anisotropic Diffusion in Heterogeneous Systems 7.2.1. Solute Diflusion in Liquid Crystals and Emulsions-Phase Structure. Diffusion

rates of small molecules in heterogeneous systems provide information on the phase structure and may have significant interest in their own right. Some thermotropic and lyotropic liquid crystals align in a magnetic field, or at least keep their macroscopic alignment long enough to allow NMR PGSE measurements to be made. Depending on the situation, one can measure the anisotropy of diffusion in the mesophase by merely rotating the sample tube through any chosen angle, or to utilize two different field gradient coils (for example one in the x-direction and one in the z-direction in an iron magnet geometry) on a static sample. It is also possible to rotate a quadrupole gradient coil in the main field in ‘order to change the gradient directiono7g*‘80) (c.f. Fig. 12). Moseley et al.‘99J*1*1*2) have utilized the two-coil approach in their elegant FT-PGSE studies on permeants such as methane and chloroform in various mesophases (Fig. 23). C’allaghan and SGderman(i83i and later Blum et a1.(184)investigated the echo decay from water in (macroscopically randomly oriented) lamellar and smectic lyotropic mesophases in terms of different models for phase structure and domain size. Callaghan et al. have presented an interesting FT-PGSE study of fat and water diffusion in cheese (Fig. 24.) They discuss the results in terms of different models for water and fat confinement in polydisperse emulsion domains.~“‘) NMR spectra of quadrupolar nuclei in anisotropic environments exhibit quadrupolar splittings. Callaghan et al. analyzed echo decay relations for the pulsed-gradient modification of the ‘solid echo sequence’.(which differ from the normal Stejskal-Tanner relation) and applied the technique for investigations on the anisotropy of water diffusion in single-crystal samples of potassium palmitate in D,O.” OS)

39

Molecular diffusion studies

e

55mm

FIG. 23. Diffusion experiments on methane in the smectic B phase of p-butylbenzylidene-n-hexylaniline (40.6) at 26.8”C. The methane diffusional echo attenuation is shown as a function of field gradient duration and direction (z and x magnetic axes). The intergradient pulse interval was 90.9 ms and the gradient strength was of the order of 8 x IO-“ T cm-‘. The Ff spectra are based on four accumulated spectra and 5 Hz artificial line broadening. The coil design is outlined in the lower part of the Figure (from Ref. 99).

C’/T2m’2 FIG. 24. A medium-resolution proton NMR spectrum of Swiss cheese (‘water’ and ‘fat’. respectively) and the results of PGSE experiments on the fat emulsion droplets at two different PGSE pulse parameter settings. The solid curves are fits to a model of restricted diffusion in spherical cavities (about 5000 nanometres in diameter). (from Ref. 117).

40

P. STILBS

1.2.2. Monitoring of Transport Through Cell Membranes. A very powerful and selective technique for

the monitoring of influx of various substrates into living cells was described some years ago by Brown et aL(185-‘88’ It is basically a geld gradient SE technique as applied to a case of restricted diffusion. The field gradient in question is one that arises near the outer cell surface as a result of different magnetic susceptibilities inside and outside the cells (Fig. 25). In the case of spherical geometries there will be no field gradients inside the cells, unless there are internal susceptibility differences. By artificial means, such as the addition of paramagnetic ions to the extracellular solution, the internal/external susceptibility difference can be enhanced. The field gradient magnitude will be proportional to Be in the case of naturally occurring susceptibility-related gradients and thus it is highly advantageous to use larger magnetic fields in these particular experiments; there is also a gain in sensitivity and spectral dispersion. With appropriate field gradient and SE pulse parameters, the signal from the extracellular solution will be completely absent in the SE spectra. The influx into the cells is then monitored simply by recording spectra at

(a)

(d)

FIG. 25. Illustration of the effects of differences in magnetic susceptibility on the lines of magnetic flux in different geometries: (a) x,~
Molecular

diffusion

studies

41

Alanine

Lactate

2 min

v

4 min

--_q$

8 min

14min

35min

L

10

8

1

1

I

6

.

1

4

I

1

2

*

0

6ip.p.m) RG. 26. Proton

spin-echo

spectra

(r=60 ms, 270 MHz) obtained during human erythrocytes (from Ref. 185).

an alanine

influx

experiment

into

different time intervals and measuring the increasing amplitude of the substrate peak(s) (Fig. 26). This experiment is adaptable to any supercon NMR spectrometer that can achieve the pulse programming for a normal Hahn echo with data collection starting at the echo peak, followed by normal Fourier transformation. For practical reasons, the influx kinetics must be reasonably slow (with a half-life of at least a few minutes). The major problems with these experiments appear to be biology-related rather than NMR-related. The cells die unless the solutions are oxygenated; the periodic oxygenation (which also suppresses sedimentation of the cells to the bottom of the tube) causes secondary problems because of bubbles in the sample, flow effect on SE signals, and so on. Andrasko(‘s9) has studied the influx of Li+ ions into erythrocytes with a ‘Li PGSE experiment in a similar way (i.e. with external pulsed field gradients of the normal kind). With appropriate PGSE pulse parameters it is possible to suppress the signal from the extracellular solution completely (see Section 6.5.7.). One then proceeds along the lines just mentioned. We have successfully tested Andrasko’s approach on organic substrates using low-gradient ‘H-based FT-PGSE techniques at 100 and 300 MHz”~” 8. CONCLUSIONS It is evident that Fourier transform PGSE techniques provide a new and very powerful tool for the investigation of a large spectrum of physico-chemical problems. When properly applied, the techniques have good accuracy and they are uniquely selective when applied to multicomponent systems. There is still a great potential for methodological refinements of this technique and for extending the application into many new areas of chemistry. It must be said that NMR self-diffusion measurements seem to be almost completely overlooked by people who use classic radioactive tracer diffusion techniques. As an example, NMR techniques did not receive a single word of mention in an otherwise excellent book on multicomponent diffusion, published in 1984.‘35) Acknowledgements-It is a pleasure to acknowledge the stimulating collaboration in the field of IT-PGSE during the past years with several people mentioned in the list of references. In particular, the reviewer wishes to express sincere thanks to M. E. Moseley and B. Lindman, and to S. For&n who kindly drew the reviewer’s attention to Ref. 32 and to the possible potential of the techniques discussed in this paper.

42

P. STILBS

This work has been supported by the Swedish Natural Sciences Research Council. The Knut and Alice Wallenberg foundation i$ heartily thanked for a grant that made possible the purchase of an XL-300 spcct rometer. -The reviewer would also like to express thanks to J. W. Emsley for many linguistic corrections and improvements of the original manuscript and to all the authors and publishers for permission to use their Figures in the present review.

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43

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44

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Molecular diffusion studies

45

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NOTE ADDED IN PROOF After the submission of the present manuscript in December 1985 some additional papers applying IT-PGSE techniques appeared by May 1986. Two review papers require special mention: StepiSnik has reviewed methods for measuring and imaging of flo~“~~) and von Meerwell has reviewed NMR diffusion measurements in polymer/solvent systems. (1g2’An excellent monograph on diffusion that also contains an introduction to timedomain SE and PGSE techniques”93) was overlooked when preparing Sections 3 and 5 of the present review. 191. J. STEPISNIK, Prog. Nucl. Magn. Reson. Spectrosc. 17, 187 (1985). 192. E. D. VONMEERWALL, Rubber Chem. 58,527 (1985). 193. H. J. TYRELLand K. R. HARRIS,Djffusion in Liquids, Butterworths, London (1984).