Copyright is owned by the Author of the thesis.  Permission is given for 
a copy to be downloaded by an individual for the purpose of research and 
private study only.  The thesis may not be reproduced elsewhere without 
the permission of the Author. 
 
q-SPACE, RESTRICTED DIFFUSION AND 
PULSED GRADIENT SPIN ECHO 
NUCLEAR MAGNETIC RESONANCE 
A thesis presented in partial fulfilment of the requirements 
for the degree of Doctor of Philosophy in Physics 
at Massey University 
"ift1r'y""'' liff1f1ii ' 'liiiry 
1061960757 
Andrew Coy 
1995 
p. ii, line 5 
p. ii, line 7 
p. 25, bottom of page 
p. 28, eqn (3.28) 
Errata 
p. 29, lines 5 and 9, section 3.3.2 
p. 36, eqn (4.6) 
p. 40, line 4 
p. 40, line 7 
p. 42, line 10, section 4.2.3 
p. 42, line 3, section 4.3 
p. 48, line 8 
p. 78, line 13, section 6.6 
p. 81, line 9, section 7.1 
p. 93, bottom line 
p. 94, line 10, section 7.4.3 
p. 118, line 17, section 8.3.3 
p. 126, line 23 
'pervides' should read 'provides' 
'give' should read 'given' 
't' should read ''t' 
full stop is in the wrong place 
r' should read r' 
2 _ Ps(Z,oo)should read Z Ps(Z,oo) 
'lograthmic' should read 'logarithmic' 
'ration' should read 'ratio' 
'planer' should read 'planar' 
'criteria' should read 'criterion' 
'on the order of should read 'of the order of 
'move' should read 'more' 
'hoping' should read 'hopping' 
'However' should read 'however' 
'm 1' should read 'm-1' 
'losses' should read 'loses' 
'repatative' should read 'reptative' 
FOR 
Reference Or'":?? 
NOT TO BE REMOVED FROH r::; '? ??-- ',.,. 
MASSEY UNIVERSITY 
lllll lllllll llll llll llt;. 
1095030826 
Abstract 
The theory and technique of Pulsed Gradient Spin Echo (PGSE) Nuclear Magnetic 
Resonance (NMR) are presented. Particular attention is paid to the Fourier relation?
ship between the average propagator of motion and the echo attenuation function. 
Using the q-space formalism, existing PGSE theory for diffusing molecules trapped 
between parallel barriers is extended to include the effects of relaxation at the walls. 
Computer simulations have been performed to test this extension to the theory and 
also to investigate the effect of finite gradient pulses in such an experiment . 
PGSE experiments were performed on pentane inside rectangular microslides of 
1 00 pm width. Diffraction-like effects predicted by theory for such experiments were 
observed where the PGSE data has a minimum when the gradient wavevector q is 
equal to the reciprocal width of the microslides. Through the use of non-linear least 
squares fitting techniques the PGSE data is fitted to theories for perfectly reflecting 
walls, partially reflecting walls and wall with variable spacings. 
NMR microimaging experiments were performed on the microslide capillaries. 
The images revealed edge enhancement effects which can be explained through the 
signal attenuation expressions used in PGSE experiments. A brief theoretical dis?
cussion shows that the effect is due to the restricted diffusion of the molecules at 
the boundaries compared with the center of the sample. 
A pore hopping technique is presented which allows analytic expressions to be 
found for diffusion in porous media. PGSE experiments are performed on water 
diffusing in the interconnecting voids formed by close packed, monodisperse, micron 
sized polystyrene spheres. Diffraction-like interference effects predicted by theory 
are obsevered where the PGSE data has a maximum when q is equal to the recip?
rocal lattice spacing of the porous network. Using non-linear least squares fitting 
techniques the PGSE data is fitted to the pore hopping theory for a pore glass with 
some variation in pore spacing. The use of an appropriate structure function for 
the pore shape is analysed by modelling the true pore shape and comparing it to 
the structure function for a sphere. The parameters revealed by fitting t heory to 
data are consistent with the known dimensions and show that important structural 
information can be revealed by this technique. 
Electron Spin Resonance (ESR) experiments are performed on the quasi-one?
dimensional organic conductor (FA)2PF6 . PGSE experiments on the conduction 
electrons show restricted diffusion effects. The PGSE data is analysed using both 
an impermeable relaxing wall model and a permeable pore hopping model . Fitting 
the data to these models show that a hopping model is  more consistent with the 
data. 
PGSE experiments are performed on semi-dilute solutions of high molecular 
weight polystyrene dissolved in CC14? The reptation model of diffusion is  reviewed 
and features of this model relevant to PGSE experiments are detailed. PGSE exper-
11 ABSTRA CT 
iments are performed and the mean square displacement of the entangled polymers 
is obtained as a function of diffusion time. Transitions from t to t112 scaling of 
the mean square displacement are found, and a region exhibiting t114 scaling is also 
observed, this region often being considered the signature for reptation. 
The PGSE-MASSEY technique, which pervides a method to correct for gradient 
pulse mismatch, is described. The details of the hardware and software implemen?
tation of this technique are also give. PGSE-MASSEY experiments are performed 
on the semi-dilute polymer solutions and enable structure functions to be acquired . 
These structure functions are compared to the primitive chain structure function 
enabling an estimate of the Doi-Edwards tube diameter to be made. 
Acknowledgements 
It is a pleasure to thank the following people who have helped me with this work: 
Prof. Paul T. Callaghan, my supervisor, who has provided me with endless 
inspiration and has been a teacher, a colleague and a friend throughout my time at 
Massey University; 
Assoc. Prof. Bob D. O 'Driscoll, my second supervisor, for many helpful discus-
siOns; 
Assoc. Prof. Rod K. Lambert , for support and encouragement ; 
Dr. Craig Eccles, for valuable discussions and helpful advice on programming; 
Professors Peter Stilbs, Olle Soderman, Rainer Kimmich, Noam Kaplan and 
Janez Stepisnik ,  sabbatical visitors , who have helped with their discussions, insight 
and assistance with experiments. 
Dr. Yang Xia, Craig Rofe, Lucy Forde, Bertram Manz, John Van Noort and 
everyone else in the NMR Lab, who have been my workmates and friends; 
Sarah Codd, fellow lecturer in 1993, for friendship ;  
Lynley Drummond, Karen Owens, additional inhabitants of the NMR Lab who 
have helped restore sanity to acceptable levels; 
Di Reay, Rose, Deanna and the staff of the Physics Department , who have not 
only put up with me, but have also provided engouragement and support throughout 
my time at Massey University; 
Peter Saunders, fellow PhD student , who has been helpful both as a physicist 
and a friend ; 
Robin Dykstra and the staff of the Electronics Workshop, for help with the many 
electronic problems; 
The staff of the Mechanical Workshop, for help with probes and for putting a 
box on the water cooler for the FX-60 so I still have some hearing left ; 
Pauline, my mum, for proof reading and everything else; 
My family and friends, for being there for me in the good and not quite so good 
times; 
And finally to Theo Behn, my son, for being. 
111 
IV ACKNOWLED GEMENTS 
Contents 
ABSTRACT 
ACKNOWLEDGEMENTS 
TABLE OF CONTENTS 
LIST OF FIGURES 
LIST OF TABLES 
1 INTRODUCTION 
1 . 1  Introduction . . . 
1 .  2 Organisation of thesis . 
2 NMR Theory 
2. 1 Quantum mechanical description 
2 . 1 . 1  Nuclear magnetic moment 
2 . 1 . 2  T he Zeeman interaction 
2 . 1 . 3  The ensemble average 
2.2 The semi-classical description 
2 .2 . 1 The rotating frame . 
2 .2 .2  
2 .2 .3  
Resonant excitation . 
Relaxation . . . . . . 
2 .2 .4 Bloch equations . . . 
2 .2 .5  Signal-to-noise ratio 
2.3 Nuclear interactions . . . . .  
2 .3 . 1  Magnetic field inhomogeneity 
2 .3 .2 Chemical shift . . 
2.4 Pulse sequences . . . . . 
2 .4.1 Signal averaging . 
2 .4 .2  Phase cycling . . 
2 .4 .3 Spin echo . . . . 
2 .4.4 Stimulated echo . 
3 PGSE Theory 
3 . 1  Magnetic field gradients . . . . . . . . . . . . .  . 
3 . 1 . 1  Effect of gradients . . . . . . . . . . . .  . 
3 . 1 . 2  k-space imaging and the Fourier transform 
3 .2 Gradients and spin motion . . . 
3 .2 . 1 Gradients and diffusion . .  
V 
1 
lll 
V 
lX 
Xl 
1 
1 
2 
5 
5 
5 
6 
8 
9 
9 
10  
12  
1 3  
1 3  
1 5  
16  
16  
16  
16  
1 7  
1 7  
1 8  
21 
21 
2 1  
22 
24 
24 
VI 
3.2 .2 Pulsed field gradients . 
3 .2 .3 Stimulated echo . . . .  
CONTENTS 
26 
27 
3.3 PGSE, scattering and q-space . 27 
3.3 . 1  The conditional probability function and the average propagator 28 
3.3.2 The narrow-pulse approximation 29 
3.4 Hardware . . . . 3 1  
3 .4 . 1  FX-60 . . 3 1  
3.4 .2 AMX-300 32 
3. 4 .3 GX-270 . 32 
4 PGSE and restricted diffusion 33 
33 
3 4  
36 
37 
37 
37 
38 
38 
4 0  
42 
42 
42 
4 3  
45 
4 . 1 Long t ime limit case for average propagator 
4. 1 . 1  Parallel plane pore 
4. 1 .2 Cylinders . . . . . . . . . . . . . .  . 
4. 1 .3 Spheres . . . . . . . . . . . . . . . . 
4. 1 .4 Comparison of planes, cylinders and spheres 
4 . 1 .5 Reciprocal q-space . . .  
4 .2 Finite t ime expressions for E (q) 
4.2 . 1  Parallel planes . 
4.2.2 Cylinders . . . . . . . . 
4 .2 .3 Spheres . . . . . . . . . 
4 . 3 The effect of finite gradient pulses 
4.3 . 1  Gaussian phase approximation . 
4 .3 .2 Computer simulations 
4.3.3 Harmonic potential 
5 PGSE-MASSEY 
5 . 1  Theory . . . . . . . . . . . . . . . . .  . 
5 . 1 . 1  The cause of Echo instabilities . 
5 . 1 .2 Effect of a read gradient 
5 .2 Hardware and Software . . .  
5 .2 . 1  Combining gradients 
5 .2 .2 Signal acquisit ion 
5 .2 .3 Software . . . . . . . 
6 PGSE experiments on parallel barrier samples 
6 . 1  Introduction . . . . . . . . . . . . . . . . . . .  . 
6 .2 Theory of enclosing boundaries with edge relaxation 
6.2. 1 Boundary conditions 
6.2 .2 Parallel plane pore 
6.3 Computer simulations . 
6 .3 . 1  Hopping method . 
6 .3 .2 Gaussian method . 
6 .3 .3 Effect of wall relaxation 
6.4 Experiments . . . . . . . . .  . 
6 .4 . 1  S ample and geometries 
6.4.2 100 fLm microslides . .  
6 .4 .3 Fitting theory to data 
6 .5 Edge enhancement effects . .  
47 
47 
47 
48 
52 
52 
53 
53 
55 
55 
55 
57  
57 
60 
60 
6 1  
62 
64 
64 
68 
69 
72 
CONTENTS 
6.5. 1 Theory . . . . 
6 .5 .2 Experiments . 
6 .6 Summary . . . . . . 
7 Porous Media 
7 . 1  Introduction . . . . . . . . . . .  . 
7 .2 Porous theory . . . . . . . . . . . 
7 .2. 1 Gaussian envelope model . 
7 .2 .2 Pore hopping model 
7 .2 .3 Regular lattice 
7.2.4 Pore glass . . . 
7.3 Computer simulations 
7.3. 1 Method . . . .  
7 .3 .2 Results for regular lattice 
7 .4 Experiments on polystyrene spheres . 
7 .4. 1 Theoretical considerations 
7 .4.2 Samples . 
7 .4.3 Results . . . . . . . . . . . 
7 . 5  ESR experiments . . . . . . . .  . 
7 .5 . 1  Apparatus and experiments 
7 .5 .2 PGSE theory 
7 .5 .3 Results . 
7 .6 Summary . . . . .  . 
8 Dynamics of Semi-Dilute Polymer solutions 
8 . 1  Introduction . . . . . . . . . . . .  . 
8 . 2  Polymers . . . . . . . . . . . . . . . 
8 .3 Polymer reptative diffusion theory . 
8 .3 . 1  The primitive chain . . . . .  
8 .3.2 Internal dynamics: Rouse motion and the primitive chain . 
8 .3 .3 Time scale regimes . . . . . . .  
8 .4  Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 
8 .4. 1 Self-diffusion dependence on M .. . . . . . . .  . 
8 .4 .2 Mean squared displacement dependence on time . 
8 .4 .3 Structure factor 
8 .5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 
9 Conclusion 
9 . 1  Summary 
9.2 Future work 
BIBLIOGRAPHY 
A Publications 
B S oftware 
B . 1  PGSEPLOT . 
B .2  PGSE-JEOL . 
B . 3  PGSE-FX 60 
B .4 Simulations . 
Vll 
72  
75 
78 
81 
8 1  
8 1  
82 
83 
84 
87 
89 
89 
89  
90 
90 
93 
94 
99 
99 
99 
. 1 0 1  
. 1 07 
109 
. 109 
. 109 
. 1 1 1  
. 1 1 1  
. 1 1 6  
. 1 18 
. 1 18 
. 1 1 9  
. 1 1 9  
. 122 
. 1 24 
125 
. 125 
. 1 25 
127 
133 
135 
. 1 35 
. 1 35 
. 1 35 
. 1 36 
V Ill 
B.5 Theory . . . . . . 
B.6 BrukerTranslate . 
CONTENTS 
. 1 38 
. 1 38 
List of Figures 
201 Energy levels for an I = 1/2 spin 0 0 0 0 0 0 7 
202 Counter rotating components of B1 0 0 0 0 0 1 0  
203 Laboratory and rotating frames of reference 1 1  
2 .4 A free induction decay (FID) 14 
205 Fourier transform of an FID 0 0 0 0 14 
206 The spin-echo pulse sequence 0 0 0 1 8  
207 The stimulated -echo pulse sequence 19  
301 NMR imaging 0 0 0 0 0 0 0 0 0 0 0 0 23 
302 The pulsed gradient spin echo (PGSE) pulse sequence 0 26 
303 The pulse gradient stimulated echo (PGSTE) pulse sequence 28 
401 Wiener-Khintchine theorem for a parallel plane pore 0 0 0 0 0 35 
402 Long time limit E(q) for parallel planes, cylinders and spheres 38 
403 E( q) theoretical curves for parallel planes 0 0 0 0 0 0 0 0 0 0 0 0 39 
4.4 Surface-to-vol ume ratio effect for parallel planes 0 0 0 0 0 0 0 0 41  
405 Computer simulations of E( q) for parallel planes using finite gradient 
pulses 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 43 
406 Dependance of the diffractive minimum on gradient pulse duration 0 44 
501 The PGSE-MASSEY p ulse sequence 0 0 0 0 0 0 0 0 0 0 0 0 0 49 
502 Processing PGSE-MASSEY data 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1  
503 Diagram of hardware used for PGSE-MASSEY experiments 52 
601 Parallel barrier p( z, t) predicted by computer simulation using hop -
ping method 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1  
602 Parallel barrier p(z, t )  predicted by computer simulation using the 
Gaussian jump method o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 63 
603 Parallel barrier Ps (Z, .6.) predicted by computer simulation using the 
Gaussian jump method 0 o 0 0 0 o 0 0 0 o 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 65 
6.4 Parallel barrier E( q) predicted by computer simulation using the 
Gaussian jump method 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 66 
605 The effect of relaxation on E( q) for parallel barriers o 0 0 0 0 0 0 0 0 0 67 
606 NMR image of the 100 f.Lm microslide stack 0 0 0 0 0 o 0 0 o 0 0 0 0 0 0 68 
607 E(q) data from microslide stack with c urves using known parameters 70 
608 [E(q) data from microslide stack with curves using fitted parameters 0 71  
609 The spin-warp imaging sequence 0 0 0 0 0 o o o 0 o 0 0 0 74 
601 0  Theoretical curves for diffusive edge-enhancement effect 76 
601 1  Images showing the diffusive edge-enhancement effect 77 
701 C ( Z, T) for a regular and irregular lattice 0 85 
X LIST OF FIG URES 
7 .2  E(q) for a regular and irregular lattice . . . . 86 
7 .3  C (Z, T ) for a regular and irregular pore glass . 87 
7.4 E( q) for a regular and irregular pore glass . . 88 
7.5 The radial density function of the unit void . . 92 
7 .6 The spherically averaged radial distribution function of the unit void 92 
7 .  7 E( q) for the 9.870 1-lm polystyrene sphere system 94 
7.8 E( q) for the 14.6 1-lm polystyrene sphere system 95 
7 .9 E( q) for the 15 .8 1-lm polystyrene sphere system 97 
7 . 1 0  ' Image' of the pore glass lattice . .  ? .  . . . . 97 
7 . 1 1  E( q) for all three polystyrene sphere system . . 98 
7 . 1 2  Stejskal-Tanner plot for electron E(q) data . . . . 1 0 1  
7 . 1 3  E(q) for diffusing electrons with relaxation model fits . 1 02 
7 . 14 E(q) with all ? values for diffusing electrons with relaxation model fits 1 03 
7 . 1 5  E(q) for diffusing electrons with pore hopping model fits . . . . . . .  1 04 
7 . 1 6  E(q) with all ? values for diffusing electrons with pore hopping model 
fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 04 
7 . 1 7  Fourier inversion of electron E(q) data . . . . . . . . . 1 06 
7 . 1 8  Fit of a two population model to electron E( q) data . . 1 07 
8 . 1  
8 .2  
8 .3 
8 .4 
8 .5 
8 .6 
8 .7 
Idealised freely-jointed polymer chain . . . . .  
The Doi-Edwards tube and the primitive path 
The process of tube disengagement . . . . . . 
Mean squared displacement , <l>n ( t ) ,  of a chain segment plotted against 
time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . 
Dependence of self-diffusion coefficient ,  Ds, on molecular mass M 
(z(t)2) vs t data for polystyrene . . . . . .  . 
E( q) vs q2 for 15  x 1 06 daltons polystyrene . 
B . 1  Sample output from PGSEPLOT program . 
B .2  Sample output from PGSE-FX60 program . 
B . 3  Sample output from the RECTSIM program 
. 1 1 1  
. 1 12 
. 1 15 
. 1 19 
. 1 20 
. 1 2 1  
. 1 23 
. 1 36 
. 137  
. 1 39 
List of Tables 
6. 1 
6 .2 
7 . 1  
7.2 
7 .3 
7 .4 
7 .5 
7.6 
8 . 1  
8 .2  
8 . 3  
Parameters used in parallel barrier sirnulations . 
Parameters a and D obtained by fitting to E(q) 
The characteristics of the monodisperse polystyrene spheres 
Parameters obtained by fitting to 9.870 f-liD E( q) data . 
Parameters obtained by fitting to 14.6 f-tm E( q)  data 
Parameters obtained by fits to all three sphere systems 
? ? ?  0 
. . . 
Parameters of fits to electron E( q) data using relaxing wall model 
Parameters of fits to electron E(q) data using pore hopping model 
The monodisperse high molecular weight polystyrene samples 
The characteristics of the timescale regions in polymer reptation 
Parameters for the polymer systems . . . . . . . . . . . . . . . . 
V1 
62 
72 
93 
95 
96 
98 
. 103 
1 05 
1 1 0  
1 18 
. 1 22 
LIST OF TABLES