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The lipid-water interaction in lyotropic measophases : an NMR study Abdolall, Khaled 1978

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THE LIPID-WATER INTERACTION IN LYOTROPIC MEASOPHASES AN NMR STUDY by KHALED ABDOLALL B. Sc., U n i v e r s i t y of Waterloo, 1972 M. Sc. U n i v e r s i t y of B r i t i s h Columbia, 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS  We accept t h i s t h e s i s as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  In p r e s e n t i n g t h i s  thesis  in p a r t i a l  fulfilment of  the requirements f o r  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, the I  L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e  f u r t h e r agree t h a t p e r m i s s i o n  for  that  r e f e r e n c e and study.  f o r e x t e n s i v e copying o f t h i s  thesis  s c h o l a r l y purposes may be granted by the Head of my Department or  by h i s of  for  I agree  this  representatives. thesis  It  i s understood that copying o r p u b l i c a t i o n  f o r f i n a n c i a l gain s h a l l  written permission.  Department of The  University of B r i t i s h  2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  Date  V 57  t 6  Columbia  not be allowed without my  Abstract A nuclear magnetic resonance study of the l i p i d - w a t e r i n t e r a c t i o n has been c a r r i e d out i n the l a m e l l a r mesophase of the sodium laurate-water system.  Deuterium quadrupole s p l i t t i n g s and spin l a t t i c e r e l a x a t i o n time  measurements of perdeuterated f a t t y a c i d chains and the quadrupole s p l i t t i n g s of water (D2O)  and the sodium counter ion are used to study the  e f f e c t s of t h i s i n t e r a c t i o n . interaction  The  r e s u l t s i n d i c a t e that the  lipid-water  has a strong i n f l u e n c e on the conformations and motions of  the. l i p i d chains, p a r t i c u l a r l y those chain segments near the l i p i d water interface.  The  d e t a i l s of t h i s i n t e r a c t i o n are not included i n the  theories which attempt to e x p l a i n hydrocarbon chain ordering i n b i l a y e r membranes i n terms of chain-chain (or l i p i d - l i p i d ) i n t e r a c t i o n s  only.  A thermodynamic a n a l y s i s of the r e s u l t s i n d i c a t e s that a d e s c r i p t i o n  of  the ordering of the hydrocarbon chains e n t i r e l y i n terms of chain-chain i n t e r a c t i o n s i s not complete, and that a complete theory should include the l i p i d - w a t e r i n t e r a c t i o n e x p l i c i t l y . The  f i r s t experimental evidence f o r a s p i n l a t t i c e r e l a x a t i o n  mechanism between water protons and l i p i d protons i n a l i p i d water system i s also reported.  The e f f e c t of i s o t o p i c m o d i f i c a t i o n  of the methylene  hydrogen n u c l e i on the proton s p i n l a t t i c e r e l a x a t i o n rate i n l^O  and  also of the e f f e c t of the isotope m o d i f i c a t i o n of the water on the r e l a x a t i o n rates of the l i p i d protons i s i n v e s t i g a t e d .  Although the measurements  show that protons deep i n the b i l a y e r make a s u b s t a n t i a l  contribution  to the s p i n - l a t t i c e r e l a x a t i o n rate of the water protons, a d e t a i l e d theoret i c a l a n a l y s i s demonstrates that the experimental r e s u l t s can be accounted for without invoking deep penetration  of the water i n the b i l a y e r .  Table of Contents Page Abstract L i s t of Figures  i  L i s t of Tables  1 1 1  Acknowledgements  iv  Chapter 1  Introduction  2  Theory 2.1 2.1.1 2.1.2 2.2 2.2.1 2.2.2 2.2.3  3  2 3  1  Quadrupolar I n t e r a c t i o n s Deuterium Magnetic Resonance a) Chain deuterons b) Deuterium i n D2O N a NMR  13 15 17 13  S p i n - L a t t i c e Relaxation Deuterium S p i n - L a t t i c e Relaxation Chain Protons. ( L i p i d ^ O mixtures) Chain Protons and H2O Protons (Lipid/H 0 mixtures) 2  Experimental 3.1 Fatty acids 3.2 D0 2  19 23 24 26 27 27 27  • 3.3 3.4 3.5  Deuteration of the f a t t y acids Proton l a b e l l i n g of deuterated f a t t y acids Samples  27 27 29  3.6  NMRApparatus.  30  A) The Spectrometer B) Probehead and V a r i a b l e Temperature Oven  30 30  NMR Measurements A) Spectroscopy B) Relaxation Measurements  32 32 33  3.8  4  Results 4.1 4.2  Quadrupole S p l i t t i n g s S p i n - L a t t i c e Relaxation Times A) Deuterium Results B) a-Protons C) H 0 Results Sources of E r r o r  35 36 36 38 38 38  2  4.3 5  The Lipid-Water I n t e r a c t i o n  . . .  -.  A Microscopic I n t e r p r e t a t i o n 5.1  Quadrupole S p l i t t i n g s  52  5.2  S p i n - L a t t i c e Relaxation (Perdeuterated Chains)  56  5.3  A Model f o r Molecular Motions Mediated by the Lipid-Water I n t e r a c t i o n Isotope E f f e c t s  58 59  5.4 6  52  The Lipid-Water I n t e r a c t i o n A n y l y s i s In terms of , Macroscopic V a r i a b l e s '• '  62  ?  i 6.1 6.2 7  Quadrupole S p l i t t i n g s . Discussion i n r e l a t i o n to e x i s t i n g theories S p i n - L a t t i c e Relaxation  S p i n - L a t t i c e Relaxation between Water protons and L i p i d Protons  84  7.1  A n a l y s i s and Discussion of the H 0 The Model  7.2  a-CH Results  Appendix A.  Appendix B. Appendix C.  2  results  84 85 92  2  The temperature Dependence of Water and Counter Ion order i n Soap-Water Mesophases. A deuterium and Sodium NMR study . 9  6  Determination of The equation of State f o r the Sodium Laurate-Water system Using low angle X-ray S c a t t e r i n g  104  Water s e l f D i f f u s i o n and s p i n - s p i n r e l a x a t i o n i n Sodium Laurate/H 0  113  2  References  62 77 78  117  i  L i s t of Figures Figure 1 2 3  Page A Schematic representation o f the lamellar l i q u i d c r y s t a l phase of a l i p i d - w a t e r system.  3  A schematic representation of the geometry i n a l i p i d bilayer.  14  Experimental arrangement f o r the deuteration o f the f a t t y acids  28  4  V a r i a b l e temperature oven and sample holder.  31  5  Representative deuterium p a r t i a l l y relaxed spectra  34  6  Representative deuterium, proton and ^3^a J J ^ R spectra .  40  7  v  n  versus T  41  8  V  n  versus n  42  9  v  versus C  43  10  Quadrupole s p l i t t i n g s versus temperature f o r D2O ,  n  23fla and the f i r s t few p o s i t i o n s on the hydrocarbon chain  44  11  1/Ti as a f u n c t i o n . o f p o s i t i o n on the hydrocarbon chain  45  12  1/T  46  13 14 15  ln  Versus 103/T  versus n  47  1/Ti versus C 1/Ti versus 10 /T f o r the ct-CD i n d__C._-Na/6H 0 and d C -Na/6D 0 ^ '1/Ti versus 10 /T f o r the C1-CH2 protons i n d„ C -Na/6H 0 and d C -Na/6D 0 "  48  3  2  2  49  1  16  23  12  2  3  12  2  50  Z X  2]  17  2  1/Ti versus 103/T f o r H 0 i n d C -Na/6H 0 , d C -Na/6H 0 2  0  12  2  21  12  2  and d C -Na/6H 0  51  l / v Ov /8C)  55  23  18  12  n  12  n  2  T  versus n  ii  Figure  Page  19  1 / l n versus  61  T  20(a,b)  n  (a) Order parameter curves obtained f o r two d i f f e r e n t l a m e l l a r phases o f dC^K-I^O » having the same A value, (b) Order parameter curves v a r i a t i o n s f o r d i f f e r e n t a values i n the l a m e l l a r phase of dC^2 H20 . 65 K-  20C 21  Quadrupole s p l i t t i n g s curves f o r two samples o f dC^2~Na/H20 having the same A value.  66  The f r a c t i o n a l v a r i a t i o n of the quadrupole s p l i t t i n g w i t h temperature keeping the area per p o l a r head constant and keeping the water concentration constant.  71  22  The f r a c t i o n a l v a r i a t i o n of the quadrupole s p l i t t i n g s w i t h temperature keeping the area per p o l a r head constant.  23  The f r a c t i o n a l v a r i a t i o n of the quadrupole s p l i t t i n g s w i t h water concentration keeping the area per p o l a r head constant. ,  73  24  The f r a c t i o n a l v a r i a t i o n of the quadrupole s p l i t t i n g w i t h water concentration keeping the area per p o l a r head constant f o r d i f f e r e n t C values a t a f i x e d temperature.  25  The r a t i o n of the change i n V w i t h T keeping A f i x e d to the change i n v w i t h T keeping C f i x e d .  75  26  A semilog p l o t o f roo-r .  76  27  R a t i o o f the a c t i v a t i o n energies f o r the chain deuterons at constant surface area per p o l a r head t o that at constant water concentration.  81  n  n  n  28  E f f e c t of systematic e r r o r s due to the X-ray : measurements  29  Schematic representation of the d i f f e r e n t s p a t i a l regions, i n which water protous and l i p i d protons move  30  Log A versus Log C f o r the sodium laurate water system.  110  31  The dependence of A Q on temperature  111  32  The dependence of A on temperature f o r the sodium laurate-water system.  33  Log, (S/S ) + 2T/T versus Q  2  T  3  83  86  112  116  ) III  L i s t of Tables Table  Page Ratio of the change i n the s p i n - l a t t i c e r e l a x a t i o n rate of protons due to the -CH2 protons to the change of the r e l a x a t i o n rate of the -CH2 protons due t o the water protons i n dj-jC^-Na/SI^O .  94  Temperature dependence of the l a m e l l a r repeat distance, thickness of the b i l a y e r and the mean area per p o l a r head f o r the sodium l a u r a t e water system.  109  Dependence on the water concentration of the l a m e l l a r repeat d i s t a n c e , the b i l a y e r thickness and the area per p o l a r head.  109  Ratio D/D of the s e l f d i f f u s i o n c o e f f i c i e n t of water i n the sodium laurate-water system to that of pure water at 100 °C f o r s e v e r a l G values.  115  Quadrupole s p l i t t i n g s of chain deuterons i n the sodium laurate water system; dependence on temperature and water concentration.  1-20  S p i n - l a t t i c e r e l a x a t i o n times of chain deuterons i n the sodium laurate water system; dependence on temperature and water concentration.  121  0  iv  Acknowledgements I am very g r a t e f u l to my research supervisor  Professor Myer Bloom  who provided the stimulus and continuing help through-out the course of t h i s work.  I have benefited g r e a t l y from h i s i n s t r u c t i o n and the  many discussions we had. I thank my wife Khadija f o r typing the t h e s i s and f o r her patience and continuing encouragement and moral support. I express my sincere gratitude t o the f o l l o w i n g people who through i n d i v i d u a l help have contributed  t o t h i s work:  Dr. E l l i o t B u r n e l l of the Chemistry Department f o r many discussions and u s e f u l suggestions  and f o r p r o v i d i n g the help and equipment i n f  sample preperation. Dr. J . Charvolin of the Laboratoire de Physique de S o l i d e s , Orsay f o r the u s e f u l discussions we had and f o r suggesting the work on the sodium laurate-water system. Dr. Alex Mackay, Dr J . Davis and Dr M. I . V a l i c f o r the t e c h n i c a l consultations and help i n the experiments and f o r the many discussions and u s e f u l suggestions.  I n t h i s regard the continued support o f  Alex i n many other respects i s g r e a t l y appreciated. Dr. T. P.Higgs f o r h i s assistance i n preparing the s p e c i f i c a l l y protonated  samples.  Dr. W. N. Hardy f o r the u s e f u l suggestions he gave f o r e x p l a i n i n g the dependence of the deuterium r e l a x a t i o n rates on water concentration. Dr. K. J e f f r e y of the U n i v e r s i t y of Guelph f o r p r o v i d i n g the assistance and f a c i l i t i e s t o do the X-ray measurements.  1  Chapter 1 Introduction Water i s a major component of c e l l s and t i s s u e s of a l l l i v i n g organisms.  The importance o f i t s r o l e i n l i f e processes on the c e l l u -  l a r l e v e l i s w e l l recognised  but has not yet been f u l l y understood.  Many of the p r o p e r t i e s of the c e l l membrane involve water d i r e c t l y o r indirectly.  For example most of the transport mechanisms i n the c e l l  membrane are mediated by water.  Another example i s the membrane a c t i o n  of some anesthetics which i s b e l i e v e d by some t o involve membrane associated water i n a p h y s i c a l way rather than causing chemical changes i n the c e l l membrane.  From such examples i t i s , apparent that an under-  standing o f how the water i n t e r a c t s w i t h the " b u i l d i n g b l o c k s " of the c e l l membrane i s very important t o the understanding of membrane processes. The complexity of r e a l b i o l o g i c a l membranes has motivated s c i e n t i s t s i n the f i e l d to look f o r simpler systems that could be used as "models". Since l i p i d s form a large f r a c t i o n o f the " b u i l d i n g b l o c k s " of the c e l l membrane, l i p i d b i l a y e r s have been used as such models because of t h e i r s i m i l a r i t y i n s t r u c t u r e t o r e a l b i o l o g i c a l membranes.  The purpose o f t h i s  work i s to contribute t o the understanding o f the mechanism of the l i p i d water i n t e r a c t i o n i n a l i p i d b i l a y e r membrane. In the presence of water, l i p i d molecules form a v a r i e t y of l y o t r o p i c mesophases characterized by the existence of long range order and short range disorder.  These phases have been i d e n t i f i e d by X-ray studies ( 1 ) ,  2  nuclear magnetic resonance (2,3) and other techniques  (4).  Of p a r t i c -  u l a r i n t e r e s t i s the l a m e l l a r l i q u i d c r y s t a l ( L ) phase where the l i p i d Q  molecules form b i l a y e r s of i n d e f i n i t e extent and a l t e r n a t e i n a r e g u l a r l a t t i c e with l a y e r s of water and counter ions as shown i n f i g . 1 . Each b i l a y e r can be thought of as a two dimensional f l u i d w i t h the l i p i d chains p r e f e r e n t i a l l y o r i e n t e d along the normal to the b i l a y e r surface. Within the b i l a y e r the hydrocarbon chains of the molecules are f l e x i b l e (melted) and the molecules undergo r a p i d l a t e r a l d i f f u s i o n (5) and the r o t a t i o n about t h e i r long a x i s .  D i f f e r e n t parts of hydrocarbon chain  can a l s o undergo small and r a p i d angular excursions (such as bending, t w i s t i n g and flopping) perpendicular to the molecular a x i s (the long a x i s of the molecule).  The ordering of the hydrocarbon chains w i t h i n  the b i l a y e r i s then described i n terms of averages over the f a s t motions.  molecular  A measure of t h i s ordering w i l l thus give information on the  p h y s i c a l s t a t e and f l u i d i t y of the b i l a y e r . There have been numerous studies on model and b i o l o g i c a l membranes using X-rays (1) nuclear magnetic resonance (2,3), and other (4).  techniques  With few exceptions (6,Appendix A) most of these s t u d i e s are e i t h e r  concerned w i t h the s t r u c t u r e and p r o p e r t i e s of water i n these systems per se (7-11), or w i t h the dynamics and s t r u c t u r e o f the a m p h i l i c region. The i n f l u e n c e of the i n t e r a c t i o n between the water and the l i p i d on the s t r u c t u r e and dynamics of the two regions has not yet been developed appreciably. The a b i l i t y of water to form hydrogen bonds which i s simply understood as an e l e c t r o s t a t i c a t t r a c t i o n between the e l e c t r o p o s i t i v e hydrogen of one water molecule and the electronegative oxygen of another water or  Figure 1.  The l a m e l l a r l i q u i d c r y s t a l phase o f a l i p i d - w a t e r system. The c i r c l e s represent the p o l a r p o r t i o n s ( p o l a r heads) o f the molecules, and the zig-zag l i n e s represent the hydrocarbon chains.  4  l i p i d molecule, can have profound i n f l u e n c e on the o r d e r i n g and dynamics of the hydrocarbon chains w i t h i n the b i l a y e r .  Thus l i p i d and water  mutually a f f e c t each other v i a hydrogen bonding (Appendix A).  The  d e t a i l s of such an i n t e r a c t i o n have not been i n c l u d e d i n many of the theories which attempt to e x p l a i n hydrocarbon chain d i s o r d e r i n b i l a y e r membranes (12, 13, 14). The c e n t r a l questions that w i l l be h i g h l i g h t e d i n t h i s work are: how does the water between the b i l a y e r s influence the ordering of the hydrocarbon chains and how deeply does i t penetrate i n t o the b i l a y e r ?  To answer such questions i t i s necessary t o use l o c a l  probes that are s e n s i t i v e t o t h e i r environment as w e l l as t o the dynamics of the system.  An NMR study has been c a r r i e d out on the l a m e l l a r l i q u i d  c r y s t a l (L^) phase o f the sodium laurate/water system.  There a r e two  reasons f o r choosing t h i s system: ( i ) a very d e t a i l e d phase diagram i s a v a i l a b l e (15), and ( i i ) one can obtain complementary  information on the  sodium counter i o n . I n t e r a c t i o n s between the nuclear quadrupole moments o f deuterium 23 i n D^O or  Na and the e l e c t r i c f i e l d gradients at the n u c l e a r s i t e s w i l l  provide information on the charge d i s t r i b u t i o n i n the v i c i n i t y of the sodium and/or the s t r u c t u r e and ordering o f water a t the l i p i d - w a t e r interface.  The conformations and. motion o f the l i p i d r m o l e c u l e s can be  studied by measuring the quadrupole s p l i t t i n g s i n the deuterium magnetic resonance spectrum f o r deuterons along the hydrocarbon chain o f a perdeuterated molecule (16-18), o r the proton d i p o l a r s p l i t t i n g s i n a a-CH^ group (19) i n an otherwise deuterated hydrocarbon chain. deuterium s p l i t t i n g s are r e l a t e d to order parameters S  r n  The  which give a  5  measure of the o r i e n t a t i o n a l order of the C-D bond d i r e c t i o n i n a methylene group while the proton d i p o l a r s p l i t t i n g s are r e l a t e d to order parameters S  which give a measure of the o r i e n t a t i o n a l order of the  proton-proton vector i n a 01-CH2 group.  Thus a knowledge of  and  w i l l completely s p e c i f y the o r i e n t a t i o n a l order of the methylene chain segment.  Information on the l i p i d water i n t e r a c t i o n can a l s o be obtained  by r e l a x a t i o n measurements which can y i e l d information regarding the nature and the strength o f the i n t e r a c t i o n of the n u c l e a r s p i n system w i t h i t s molecular environment.  Parameters such as c o r r e l a t i o n times (x )  associated w i t h the molecular motions can be determined from r e l a x a t i o n measurements.  Complementary information on the l i p i d water i n t e r a c t i o n  can a l s o be obtained from c o r r e l a t i o n s i n the quadrupole s p l i t t i n g s of 23 Na, water (D2O), and chain (Appendix A) deuterons. Deuterium magnetic resonance has been used by s e v e r a l workers i n the study o f the conformation and motion of the l i p i d molecules i n b i l a y e r membranes (16-18, 20-26).  The most prominent o f these studies are those  by Charvolin et a l (17) and Mely et a l (18) on the potassium laurate-water system, S e e l i g and Niederberger (23) on sodium deconate-water, S e e l i g and S e e l i g (23, 26) on p h o s p h o l i p i d b i l a y e r s , and Davis and J e f f r e y (19) on the potassium palmitate-water system. I n these s t u d i e s the order parameter p r o f i l e f o r the hydrocarbon chains i n the l i q u i d c r y s t a l phase was determined by measuring the quadrupole s p l i t t i n g s f o r deuterons on s p e c i f i c a l l y deuterated (24) or perdeuterated chains (16, 17, 18). A c h a r a c t e r i s t i c feature common to a l l o f these r e s u l t s i s the appearance of a "plateau" region where about h a l f of the chain methylenes a f t e r the a p o s i t i o n have v i r t u a l l y the same order parameter.  This plateau  6  disappears a t higher temperature and water concentration w i t h a decrease i n the quadrupole s p l i t t i n g s .  The plateau was b e l i e v e d t o have some  o r i g i n s i n the s t e r i c repulsions between neighbouring chains (27, 28). Some models have been proposed where the s t e r i c i n t e r a c t i o n s are taken i n t o account i n terms of the c r o s s s e c t i o n a l area o f a given chain conformation (13). This c r o s s s e c t i o n a l area i s c h a r a c t e r i z e d by the l a t e r a l space occupied by a l i p i d molecule, o r the mean area (A) per p o l a r head at the l i p i d water i n t e r f a c e .  Mely et a l (17) have studied the  order o f the l i p i d chains i n potassium laurate-water mesophases as a f u n c t i o n o f the temperature i n samples having constant water concentration  and as a function o f water concentration a t constant temperature.  They have found that the deuterium quadrupole s p l i t t i n g s decreased w i t h i n c r e a s i n g temperature o r water concentration.  The same authors have a l s o  measured the quadrupole s p l i t t i n g s f o r samples having the same area p o l a r head but a t d i f f e r e n t water concentration and temperature. for such samples though not identical,were s i m i l a r . }  The s p l i t t i n g s  On the b a s i s of  these r e s u l t s the authors concluded that A i s a "good parameter t o represent the average microscopic order".  They b e l i e v e that the v a r i a t -  ion  o f the quadrupole s p l i t t i n g s w i t h temperature and/or water concentrat-  ion  depends p r i m a r i l y on the v a r i a t i o n of A w i t h these parameters Davis  and J e f f r e y (18) studied the hydrocarbon chain d i s o r d e r i n  the  potassium palmitate-water system.  I n the l i q u i d c r y s t a l l i n e phase  the  C-D order parameters o f the f i r s t methylene chain segments were found  to increase w i t h i n c r e a s i n g temperature t o a maximum o f 100°C and.then decrease at higher temperature.  I n contrast the C-D order parameters f o r  the rest of the methylene chain segments decreased w i t h i n c r e a s i n g temperature.  In the same system Higgs and Mackay (19) have determined  the complete ordering tensor f o r the a-methylene group by measuring the CI-CH2 d i p o l a r s p l i t t i n g s i n an otherwise perdeuterated chain and the 0.-CD2 s p l i t t i n g s i n s p e c i f i c a l l y deuterated chains. dependence splittings.  The temperature  o f the a-CI^ s p l i t t i n g s was s i m i l a r to that of the However the order parameters  a-CD^  and S^, d i f f f e r e d by LU  0-20%  tin  i n the temperature range studied (40-100°C), i n d i c a t i n g that the motion of t h i s p a r t i c u l a r segment i s not t r u l y a x i a l l y symmetric around the normal t o the b i l a y e r .  From these s t u d i e s the behaviour o f the methylene  chain segment near the l i p i d water i n t e r f a c e was b e l i e v e d to be due to a l i p i d water i n t e r a c t i o n .  A b d o l a l l , B u r n e l l and V a l i c (Appendix A) 23  studied the hydrocarbon chain, D 0 2  and  Na counter i o n order i n the  potassium palmitate/D20 and sodium palmitate/D20 systems.  There was a  s t r i k i n g c o r r e l a t i o n between the ordering o f the f i r s t few methylenes of 23 the hydrocarbon chain, deuterium i n D 0 2  and the  Na counter i o n . T h i s  c o r r e l a t i o n was ascribed to the s t r u c t u r i n g e f f e c t o f the water v i a hydrogen bonding w i t h the p o l a r heads.  A model consistent w i t h the  experimental r e s u l t s was proposed (Appendix A ) .  In terms of t h i s model  the l i p i d water s t r u c t u r e a t low temperatures imposes a d i r e c t i o n f o r which a l l the order parameters are smaller than f o r the higher temperature s t r u c t u r e f o r purely geometric reasons.  At higher temperatures the  s t r u c t u r i n g e f f e c t of water decreases and there i s an "apparent" increase i n order u n t i l the i n t r i n s i c decrease i n order parameters r e s u l t i n g from the thermal e x c i t a t i o n s dominates at s t i l l higher temperatures.  There have been many t h e o r e t i c a l studies on the chain ordering i n l i q u i d c r y s t a l s and b i l a y e r membranes (12-14, 29-32). One of the most successful  o f these i s Marceljas (13) molecular f i e l d c a l c u l a t i o n .  In the mean f i e l d approximation the i n t e r a c t i o n energy o f a s i n g l e chain i n the molecular f i e l d i s given by the sum o f the i n t e r n a l energy o f the s i n g l e c h a i n , the Vander Waals i n t e r a c t i o n s of the chain w i t h i t s neighbors v i a the molecular f i e l d and a l a t e r a l pressure term which i s p r o p o r t i o n a l to the cross s e c t i o n a l area per p o l a r head.  The l a t e r a l  pressure term takes i n t o account the s t e r i c repulsions between the hard cores o f the atoms.  S t a t i s t i c a l averages are then c a l c u l a t e d by summing  over a l l conformations o f a s i n g l e chain i n the molecular f i e l d o f i t s neighbours.  This model has been s u c c e s s f u l l y u ed to i n t e r p r e t the  deuterium NMR r e s u l t s f o r the sodium-deconate-deconol-water (13) and other NMR data (33). However the r e s u l t s do depend on the assumed o r i e n t a t i o n o f the i n i t i a l chain segment, which implies that a d e s c r i p t i o n of the ordering of the hydrocarbon chains i n a b i l a y e r membrane e n t i r e l y i n terms o f chain-chain i n t e r a c t i o n s i s incomplete. What has been ignored action.  i n these t h e o r e t i c a l c a l c u l a t i o n s i s the l i p i d - w a t e r i n t e r -  E m p i r i c a l evidence that such a l i p i d water i n t e r a c t i o n has a  strong i n f l u e n c e on the o r i e n t a t i o n a l order o f the hydrocarbon chains i n a l i p i d b i l a y e r , e s p e c i a l l y those chain segments near the p o l a r head i s , as already mentioned, demonstrated by the s t r i k i n g s i m i l a r i t y i n the temperature dependence of the quadrupole s p l i t t i n g s of the f i r s t few 23 methylene chain deuterons,  and  Na counter i o n i n the palmitates-  water systems (Appendix A) and i n the sodium laurate-water system studied i n t h i s t h e s i s .  9  In order to i n v e s t i g a t e the i n t e r a c t i o n s that are most important i n determining the o r i e n t a t i o n a l o r d e r i n g of the hydrocarbon chains i n the l i p i d bilayer i n a quantitative  way we have made a systematic study on the  sodium laurate-water system i n the l i q u i d c r y s t a l (L^) phase.  The  quadrupole s p l i t t i n g s of the methylene chain deuterons were measured as a function of temperature and water concentration.  An equation of  s t a t e r e l a t i n g the area per p o l a r head, temperature and water concentration was determined by X-rays f o r t h i s system. thermodynamics  Using elementary  the r e s u l t s were analysed by examining the s e n s i t i v i t y  of the quadrupole s p l i t t i n g s to v a r i a t i o n s i n the d i f f e r e n t variables. the  The r e s u l t s of the a n a l y s i s i n d i c a t e that a d e s c r i p t i o n o f  disorder of the hydrocarbon chain e n t i r e l y >'n terms of chain-chain  i n t e r a c t i o n s i s indeed not complete. the  thermodynamic  A complete theory should include  l i p i d water i n t e r a c t i o n e x p l i c i t l y . S p i n - l a t t i c e r e l a x a t i o n measurements i n l i p i d b i l a y e r membranes  using deuterium NMR i s f a i r l y recent.  Using a s e r i e s of f a t t y a c i d  probes of d i f f e r e n t lengths and l a b e l l e d a t s e v e r a l p o s i t i o n s Stockton et a l (34) showed thac the molecular motions w i t h i n the phosphotldylc h o l i n e b i l a y e r s increase r a p i d l y w i t h distance from the l i p i d water interface.  More r e c e n t l y Davis, Bloom and J e f f r e y (35) measured the  s p i n - l a t t i c e r e l a x a t i o n times as a function of temperature and p o s i t i o n on the hydrocarbon chain f o r the methylene deuterons i n perdeuterated chains of the potassium palmitate-water system.  In the l i q u i d c r y s t a l -  l i n e (k^) phase, t h e i r r e s u l t s i n d i c a t e the presence of complex molecular motions of the d i f f e r e n t methylene chain segments.  A simple model  10  proposing two d i f f e r e n t types of motions (a f a s t and a slow motion) w i t h d i f f e r e n t c o r r e l a t i o n times was considered. In terms of t h i s model they were able to e x p l a i n i n a q u a l i t a t i v e way the change i n r e l a x a t i o n times p r o f i l e with i n c r e a s i n g temperature.  In the present t h e s i s the  s p i n - l a t t i c e r e l a x a t i o n rates f o r methylene chain deuterons o f perdeuterate d chains i n the l i q u i d c r y s t a l (L^) phase of the sodium laurate-water system were measured as a f u n c t i o n of temperature, water concentration and p o s i t i o n on the hydrocarbon chain.  The r e s u l t s obtained  suggest that complex molecular motions mediated by a l i p i d - w a t e r i n t e r a c t i o n must be taken i n t o account.  A model i s proposed to e x p l a i n the  i n f l u e n c e of the l i p i d water i n t e r a c t i o n on the r e l a x a t i o n rates o f the methylene chain deuterons. While deuterium quadrupole s p l i t t i n g s and s p i n l a t t i c e r e l a x a t i o n time measurements can be used to study the e f f e c t of the l i p i d - w a t e r i n t e r a c t i o n on the o r i e n t a t i o n a l ordering and m o b i l i t y of the hydro carbon chains, i t i s not p o s s i b l e to obtain from such measurements information on the extent o f water penetration i n t o the l i p i d b i l a y e r . In order to obtain such information i t i s necessary to measure q u a n t i t i e s that depend on the s p a t i a l l o c a t i o n of the l o c a l probe used and on the strength of the i n t e r a c t i o n between the probe and i t s molecular environment. The spin l a t t i c e r e l a x a t i o n rate f o r a proton due to d i p o l a r couplings (which i n most cases are the main r e l a x a t i o n mechanisms f o r protons) w i t h another n u c l e a r s p i n depends on the inverse of the 6  t n  power of the  separation between the spins and on the product o f the square of t h e i r magnetogyric r a t i o s .  Thus a study o f the e f f e c t of isotope m o d i f i c a t i o n  of the l i p i d region on the s p i n l a t t i c e r e l a x a t i o n rates of the water protons should i n p r i n c i p l e give u s e f u l information on the extent of water penetration i n t o the b i l a y e r .  Complementary information can  also be obtained by studing the e f f e c t o f isotope m o d i f i c a t i o n of the water on the s p i n l a t t i c e r e l a x a t i o n rates o f proton s p i n l a b e l s i n otherwise perdeuterated  chains.  sodium laurate-water system.  We have performed such s t u d i e s on the  The r e s u l t s obtained suggest a t f i r s t  glance that water penetrates much f u r t h e r i n t o the b i l a y e r than the a-position.  However, a d e t a i l e d t h e o r e t i c a l a n a l y s i s , shows that  the experimental r e s u l t s can be accounted f o r without  invoking  deep penetration of the water i n the b i l a y e r . Information on the extent of water penetration i n the b i l a y e r can a l s o be obtained from neutron d i f f r a c t i o n measurements.  Just  r e c e n t l y Buldt e t a l (37) reported some r e s u l t s on p h o s p h o l i p i d bilayers.  The authors conclude that water penetrates i n t o the  b i l a y e r up to the g l y c e r o l back bone of the l i p i d molecules. studies by Schoenborn (36) on a s i m i l a r system d i d n o t have s u f f i c i e n t r e s o l u t i o n t o detect such water p e n e t r a t i o n .  Earlier  12  Thesis O u t l i n e This thesis w i l l c o n s i s t of 6 more chapters. w i l l i n c l u d e the relevant NMR  The second chapter  theory i n l y o t r o p i c mesophases .  Experimental d e t a i l s and r e s u l t s are given i n chapters 3 and 4.  In  chapter 5 an i n t e r p r e t a t i o n of the r e s u l t s w i l l be discussed i n terms of a microscopic model f o r the l i p i d - w a t e r i n t e r a c t i o n .  In chapter 6  a thermodynamic a n a l y s i s of the r e s u l t s i s used to i n v e s t i g a t e the i n t e r a c t i o n s that are important i n determining the s t a t e of the l i p i d chains i n a l i p i d water system.  The l a s t chapter deals with the  mechanism of s p i n l a t t i c e r e l a x a t i o n between water protons and methylene protons i n a l i p i d water system and the problem of water p e n e t r a t i o n i n t o the b i l a y e r .  13  Chapter 2 Theory 2.1 Quadrupolar I n t e r a c t i o n s The t o t a l Hamiltonian f o r a nucleus w i t h s p i n  i n an a p p l i e d  magnetic f i e l d H i s given by ( i g n o r i n g chemical s h i f t terms e t c . ) H where H  z  =  H  z  +  H  [l]  Q  i s the Zeeman Hamiltonian and H Q i s the quadrupolar  Hamiltonian due to the i n t e r a c t i o n o f the quadrupole moment eQ a s s o c i a t e d w i t h the s p i n I and the e l e c t r i c f i e l d gradients (efg) e x i s t i n g a t the s i t e o f the nucleus. I f H Q <(<( H ion  z  , i t can be shown ( 3 8 ) that the f i r s t order perturbat-  to the Zeeman energy l e v e l s due t o H Q are given by ( i n frequency u n i t s )  where the angles G , $ s p e c i f y the magnetic f i e l d d i r e c t i o n r e l a t i v e to the p r i n c i p a l coordinate system of the e f g as shown i n f i g . 2b ,m i s the magnetic quantum number i n the represention where I  z  i s diagonal,  e^qQ i s the quadrupole coupling constant and n i s an asymmetry parameter h , defined such that 0<n<l and i s a measure o f the d e v i a t i o n of the e f g f  from a x i a l symmetry.  I f r|=0 then the energy l e v e l s of the t o t a l  Hamiltonian i n frequency u n i t s are E m  .  E<°> • m T  E <» m  14  Figure 2.  (a) Schematic representation of the geometry i n a l i p i d b i l a y e r . n i s the normal to the bilayer. i s the angle between the magnetic f i e l d H* and n, 0 i s tlje angle between the C-D bon§ d i r e c t i o n and H and 9 i s the angle between the C-D bond d i r e c t i o n and n. (b) O r i e n t a t i o n of the magnetic f i e l d d i r e c t i o n r e l a t i v e to the p r i n c i p a l coordinate system of the e l e c t r i c f i e l d gradient.  15  - - v + F( 4^)(3" -i« i)) i£2  w  2  +  where v. i s the Larmor frequency YH and v 2n L  _ 3e^qQ h2I(2I-l)  n  4  2.1.1 Deuterium Magnetic Resonance. a)  chain deuterons. The deuterium  nucleus has a n u c l e a r spin 1=1.  we concentrate on one deuteron on the n  If  p o s i t i o n of the hydrocarbon  chain and assume f o r the moment that the chain i s not moving ( r i g i d l a t t i c e ) then the energy l e v e l s as given by eqn.  n  E  - v =  =  are  (3cos 0-lA 2  6  \  2  )  I 3cos 0-l 3 V 2 2  0  E.  + VQ_  T  1  [5]  -V  1  +  T  L  VQ 6  /3cos 0-l 1 2 2  The corresponding resonance frequencies are  E -E 0  x  = V _ L  Vp. ^3cos20-l ^  and the NMR spectrum w i l l c o n s i s t o f two sharp peaks separated by. AV,  =  V  ^3cos e-l^  [4]  2  Q  The e f g at the s i t e o f a deuterium nucleus on the hydrocarbon c h a i n i s along the C-D bond d i r e c t i o n .  I f the molecules are undergoing f a s t  a n i s o t r o p i c motion at frequencies much greater than those of the quadrup o l a r s p l i t t i n g s , then the r e o r i e n t a t i o n of the C-D bond w i l l  modulate  16  2 0 and a time average o f 3cos Q-l has t o be considered. For the 2 molecules i n a l i p i d b i l a y e r , the symmetry a x i s of the motion i s the normal n t o the b i l a y e r .  Thus f o r a deuteron on the n  t n  position of  the hydrocarbon chain eqn. [4] w i l l give f o r the quadrupole s p l i t t i n g s  U SS Qn  / / 33ccoossf2tj-Ml \ — J  £ ]  2  5  [  where as shown i n f i g . 2a on page \t\ Q i s the angle between the C-D n  bond d i r e c t i o n and the normal n to the b i l a y e r  i s the angle between  n and the magnetic f i e l d d i r e c t i o n and the quantity 0 ]  z 3cos Vl\ /3cos-Q -l y 2  g  3  =  n  i s defined as the order parameter, l i t e r a t u r e by  SQJJ  normally denoted i n the * .  •  A l l the hydrocarbon chain -CD2 deuterons are chemically e q u i v a l e n t and therefore w i l l have the same V Q . Since the b i l a y e r s are randomly o r i e n t e d , the d i f f e r e n t values o f cosft are e q u a l l y probable and the s u p e r p o s i t i o n of the l i n e s a r i s i n g from the d i f f e r e n t o r i e n t a t i o n s gives r i s e t o a broad absorption curve c h a r a c t e r i s t i c of a powder pattern of the form (39) g(v)  =  /  d cosft  w i t h two intense peaks separated by  [7]  17  = |—|v S ) Q  [8]  n  For a perdeuterated hydrocarbon chain, the spectrum w i l l c o n s i s t of a number of overlapping powder patterns a r i s i n g from the various deuterons s i t u a t e d along the hydrocarbon chain. spectrum i s shown i n fig.6a  A representative  on page 40 •  Since V Q i s o f i n t r a m o l e c u l a r o r i g i n (C-D bond) i t i s not expected to be temperature dependent.  Consequently the order parameters f o r the  methylene chain deuterons can be obtained d i r e c t l y from the measured quadrupole s p l i t t i n g s using equation [8] . b) Deuterium i n D 0 2  For water deuterons although the main c o n t r i b u t i o n to the efg i s from the i n t r a m o l e c u l a r 0-D bond, there can be other c o n t r i b u t i o n s of i n t e r m o l e c u l a r o r i g i n such as the charge d i s t r i b u t i o n near the p o l a r heads and other i n t e r a c t i o n s (Appendix A ).  Moreover, hydrogen bonding  can reduce the c o n t r i b u t i o n of the 0-D bond from 312 KHz (as measured i n the gas phase) to 213 KHz (40). different for different sites.  For such reasons V Q could be  A l s o chemical exchange can take place  between n u c l e i i n d i f f e r e n t environments.  I f the exchange rate i s much  f a s t e r than the s p l i t t i n g d i f f e r e n c e , the observed s p l i t t i n g ted  i s a weigh-  average over the d i f f e r e n t s i t e s and i s given by. V  ,o 2' -  D;  i  l i Qi p  v  s  il  M  where Pj_ i s the f r a c t i o n of n u c l e i i n s i t e i w i t h associated quadrupole coupling constant | V Q  ±  and order parameter S  ±  defined by  18  S  .  ±  3cos 9 2  1  -1  [10]  2 where  i s the angle between ri and the e f g p r i n c i p a l a x i s and the b a r  denotes a time average.  As discussed i n Appendix A the separation  of the various terms i n eqn. [9] i s not p o s s i b l e without making c e r t a i n The l a r g e s t c o n t r i b u t i o n to V Q ^ i s assumed t o come from the  assumptions.  i n t r a m o l e c u l a r 0 - D bond.  This could mean that V Q ^ remains roughly the  same f o r the d i f f e r e n t s i t e s .  I f i n a d d i t i o n t o V q being independent of i  i i t i s f u r t h e r assumed that  does not vary s i g n i f i c a n t l y w i t h tempers  ture then the measured s p l i t t i n g s are roughly p r p o r t i o n a l to an average order parameter S=£ P^S^ which can provide u s e f u l i  information i n a  q u a l i t a t i v e way. 2.1.2 N a NMR 2 3  The  Na nucleus has a s p i n 1= ^ • From eqn. £3] the resonance  frequencies are given by E  E  -3/2  ~ -l/2  "  -l/2  " l/2  =  E  E  *L  V  +  V  Q  (3cos2e-l^  L  and the corresponding NMR spectrum w i l l c o n s i s t o f 3 peaks separated by  A Na v  Since r a p i d exchange can take place between the sodium ions i n  19  d i f f e r e n t s i t e s i n the aqueous region then as o u t l i n e d i n the previous s e c t i o n the observed s p l i t t i n g i s a weighted average given by  ^  i  The e f g a t the s i t e of a sodium nucleus i s o f i n t e r m o l e c u l a r o r i g i n and i s l a r g e l y due to the charge d i s t r i b u t i o n near the p o l a r head groups and the asymmetric d i s t r i b u t i o n of waters of hydration (see appendix A page 77)•  Thus i t i s expected that there w i l l be a d i s t r i b u t i o n o f  Vqi  , p^ and S^ that can be q u i t e temperature dependent making the 23 i n t e r p r e t a t i o n o f the Na NMR s p l i t t i n g s i n terms o f an order parameter rather d i f f i c u l t . However i f c o r r e l a t i o n s e x i s t i n the temperature 23 dependence of the methylene chain deuterons  1 . and  , changes i n  23 the  Na s p l i t t i n g s w i l l give a measure o f the order of the surrounding  charge groups. 2.2 Spin - L a t t i c e Relaxation The general problem o f s p i n - l a t t i c e r e l a x a t i o n i n l y o t r o p i c mesophases i s complicated and not completely  understood.  Only the general aspects o f the relevant theory w i l l be discussed here. The t o t a l Hamiltonian f o r a nuclear s p i n system i s , i n most cases, given by H where H  z  =  H  z  +  H (t) ±  [12]  i s the Zeeman Hamiltonian and H^(t) i s a time dependent  Hamiltonian corresponding to quadrupole o r d i p o l a r couplings ( f o r some 19 n u c l e i , such as F , l a r g e chemical s h i f t terms must a l s o be included)  20  These couplings are modulated by the l a t t i c e : The quadrupolar couplings depend on the e l e c t r i c f i e l d gradients at the s i t e o f a nucleus and d i p o l a r couplings depend on the r e l a t i v e p o s i t i o n s of the s p i n s . H^(t) can always be decomposed i n t o an average  and a f l u c t u a t i o n  about the average as follows (41) Hi(t)  <H >  =  ±  =  ( H ^ t ) - ^ ^ )  +  < i>  +  H  HiCt)  For an a n i s o t r o p i c environment  [13]  i s nonzero  and causes  s p l i t t i n g s i n the NMR spectrum and H^(t) i s a s p i n l a t t i c e  coupling  responsible f o r the r e l a x a t i o n o f the spin system toward thermal e q u i l i b r i u m w i t h the l a t t i c e . H (t) ±  =  ^F  ( m )  H ^ t ) can be w r i t t e n (38) (t)A  [14]  ( m )  m=-2 m (m) where the F ( t ) are random functions of time and the A are operators a c t i n g on the s p i n v a r i a b l e s .  F^ ^ m  under r o t a t i o n s as the s p h e r i c a l harmonics s p i n - l a t t i c e r e l a x a t i o n rate T  and A ^  transform  o f order two. The  which describes the rate o f energy  l  t r a n s f e r from the nuclear s p i n system to the l a t t i c e may be expressed i n the form  i  =<o*i\ y z  2  <%  j(mo)) 0  [15]  m=0 where ^ l ^ i ^ ^ P ^ I coupling, Jn/^o)  i s  st n e  1116311  squared value o f the s p i n - l a t t i c e  are numerical f a c t o r s , OJQ i s the Larmor frequency and t n e  F o u r i e r transform o f the reduced c o r r e l a t i o n f u n c t i o n  gjjjCx) defined by gjx)  G (T) _ < W  =  M  G  F  m(0) -  <F  ( t )  ( m )  >  [ ]  2  (t)>  1 6  2  G (x) i s c a l l e d the c o r r e l a t i o n f u n c t i o n o f F ^ ( t ) defined by m  G^x)  <F  =  ( m )  (t) F  ( m )  (t+T)>  [17]  and describes how F ^ ^ ( t ) a t time t i s c o r r e l a t e d t o i t s value at m  some l a t e r time t+T . The time v a r i a t i o n o f F ^ \ t ) i s due t o some m  p h y s i c a l motions i n the system. c r i t i c a l time T  C  For times x much shorter than some  ( c a l l e d the c o r r e l a t i o n t i m e ) , the motions are  n e g l i g i b l e and F ^ (t) = F ^ ( t + x ) .  For times ' uch greater than T  C  there i s no c o r r e l a t i o n between F ^ (t) and F ^ (t+x) and G (x) = ^ F ^ ( t ) y . 2  m  Therefore the reduced c o r r e l a t i o n function  g ( x ) has a maximum value o f u n i t y at x=0 and f a l l s o f f to zero f o r m  T »T  C  . The c o r r e l a t i o n time x  time scale f o r the motions.  c  may then be used as a measure o r  Thus i f there i s a model f o r the  molecular motions, the c o r r e l a t i o n function can be c a l c u l a t e d , which i n t u r n would allow the c a l c u l a t i o n o f the r e l a x a t i o n r a t e s .  On the  other hand, measurements o f s p i n l a t t i c e r e l a x a t i o n rates allow the determination o f the reduced s p e c t r a l density functions which can be used to t e s t models f o r the molecular motions. calculation of g^x) i s n o n t r i v i a l .  I n most cases the  A crude assumption that i s often  made i s to assume that g ( x ) decays e x p o n e n t i a l l y , i . e. m  g (T) M  -  e -  T / T  *  [18]  22  then, the reduced s p e c t r a l density j ( u ) i s oo m  -OO  2?c  [19] c 2  For very short c o r r e l a t i o n times, a) xf  1 and a l l the s p e c t r a l  density functions j (mu) ) are independent o f frequency and equal., t o m  JiCo)  0  = J ( o ) = j (o) = 2  In t h i s case the expression f o r ^ T  2T  [20]  C  as given by eqn. 15 reduces to a  l  product o f an i n t e n s i t y f a c t o r and a c o r r e l a t i o n time T  c  • For example  the s p i n - l a t t i c e r e l a x a t i o n rate f o r a p a i r o f d i p o l a r coupled protons undergoing i s o t r o p i c motion i s given by (38) i  -  aL±y? ,  [21]  T  Another example i s the quadrupole r e l a x a t i o n through i s o t r o p i c molecular r e o r i e n t a t i o n .  For a n u c l e a r s p i n w i t h 1=1, the r e l a x a t i o n  rate jj- i s ( f o r an a x i a l l y symmetric e l e c t r i c f i e l d gradient) T  l  T  i  1°\~ir-j ( is&Y "T  [22]  c  In equations 21 and 22 , the q u a n t i t i e s - -^-5- and e qQ are the d i p o l e r h 2  dipole and the quadrupolar coupling constants r e s p e c t i v e l y . coupling constants are known the  I f these  measurements w i l l provide T . c  I f the f l u c t u a t i o n s i n the i n t e r a c t i o n H^(t) a r i s e from molecular motion that v a r i e s w i t h temperature, r e l a x a t i o n measurements can be used to study the temperature v a r i a t i o n o f T . Often there i s a  23  " b a r r i e r " to motion and an a c t i v a t i o n energy E T  c  .  x  w  e V  k  a  such that [23]  T  where Too i s the c o r r e l a t i o n time at i n f i n i t e temperature. temperature v a r i a t i o n of ^  Thus the  should give a measure o f the a c t i v a t i o n  energy. 2.2.1 Deuterium S p i n - L a t t i c e Relaxation. For the deterium nucleus, the c o u p l i n g Hq(t) o f the n u c l e a r quadrupole moment w i t h the f l u c t u a t i n g e l e c t r i c f i e l d gradients at the nuclear s i t e i s almost always the main r e l a x a t i o n mechanism.  Since the  e f f e c t i v e n e s s of a r e l a x a t i o n mechanism depends on the magnitude ( o r i n t e n s i t y ) of the corresponding s p i n - l a t t i c e c o u p l i n g (see eqn. 15 ) , the magnetic d i p o l a r couplings, which are very e f f e c t i v e f o r protons make a n e g l i g i b l e c o n t r i b u t i o n to the deuterium r e l a x a t i o n r a t e s . This i s due to the f a c t that the d i p o l a r i n t e r a c t i o n between a deuterium neucleus and another nucleus S depends on the product 2 2 " Y-r Y » where Y^ and Y are the magnetogyric r a t i o s of the deuterium I S I S nucleus and the S nucleus r e s p e c t i v e l y (Abragam p.2^5"). Thus the deuteron-proton  d i p o l a r i n t e r a c t i o n s are weaker by  2 _3_(YH/YD^ —  8  than the corresponding proton-proton d i p o l a r i n t e r a c t i o n .  .24  Therefore  considering the large value of the deuterium quadrupole c o u p l i n g constant (^170 KHz) i n the systems studied here d i p o l a r i n t e r a c t i o n s make a t most a minor c o n t r i b u t i o n t o the deuterium r e l a x a t i o n r a t e s . 1/T^ f o r deuterons i s then given by  24  (  <I"QI > 2  T L  where ^  |HQ  H*o)  +  PQ  J (2«O)) 2  | ^> i s the mean square value of Hg(t) and 2  J J ^ Q )  and  ^(^Uq)  are the reduced s p e c t r a l density functions associated w i t h Hq(t). For deuterons on the hydrocarbon chain these s p e c t r a l density functions are expected t o be f a i r l y complicated r e c e i v i n g c o n t r i b u t i o n s from the r e o r i e n t a t i o n o f the C-D bond around a given chain segment, s i n g l e molecule motions r e l a t i v e t o the d i r e c t o r as w e l l as c o l l e c t i v e motions of many molecules which are associated w i t h the motions o f the d i r e c t o r i t s e l f about i t s e q u i l i b r i u m o r i e n t a t i o n . For a s i n g l e type o f motion i t can be shown t h a t , i n the short c o r r e l a t i o n time l i m i t (W T. « 1 » where T„ i s the c o r r e l a t i o n time On " c h a r a c t e r i s t i c of the motion o f the n*"* methylene chain segment)* 2  2  1  t"h  1  the r e l a x a t i o n rates ^ f o r deuterons on the n ln hydrocarbon chain are given by (35)  p o s i t i o n of the  X  2 where S  n  i s the order parameter defined by eqn. 6 . The f a c t o r 1-S  takes i n t o account the anisotropy of the system.  n  For S =0 n  eqn. 25 becomes the usual expression (eqn. 22) f o r deuterium quadrupole r e l a x a t i o n i n i s o t r o p i c f l u i d s (38). 2.2.2 Chain Protons.  ( L i p i d / D 0 mixtures) 2  For protons on the hydrocarbon c h a i n , the problem i s l e s s simple than that f o r deuterons. coupling responsible f o r r e l a x a t i o n i s  In t h i s case the s p i n l a t t i c e  25  HdU>  =  H  d  (t)  -  <H >  [26]  d  Where H<j(t) i s the time dependent d i p o l a r Hamiltonian f o r the s p i n system.  Due t o the large magnetic moment o f the proton, the d i p o l a r  i n t e r a c t i o n s between protons on the neighbouring methylenes of the hydrocarbon chain are quite strong and tBnd t o cause a f a s t e s t a b l i s h ment of a common s p i n temperature f o r a l l the protons on the hydrocarbon chain.  This makes the i n t e r p r e t a t i o n of proton r e l a x a t i o n  measurements rather d i f f i c u l t .  This d i f f i c u l t y  can be made e a s i e r by  having protons only i n one p o s i t i o n i n an otherwise perdeuterated chain. This e s s e n t i a l l y eliminates  the i n t e r .-CH2 d i p o l a r c o n t r i b u t i o n t o  the r e l a x a t i o n rates f o r methylenes on the same chain.  The c o n t r i b u t i o n  of the i n t e r molecular d i p o l a r i n t e r a c t i o n s to the r e l a x a t i o n  rates  of the methylene protons i s expected to be much weaker than that due to the i n t r a molecular d i p o l a r i n t e r a c t i o n s .  This i s because the magnitude  of the d i p o l a r i n t e r a c t i o n s depends on r  (where r i s the proton-proton  distance so that the c o n t r i b u t i o n to ^  f a l l s o f f very r a p i d l y ( r ~ ^ )  with increasing r . Thus i n the short c o r r e l a t i o n time l i m i t , the main r e l a x a t i o n mechanism f o r a p a i r o f methylene protons i n an otherwise perdeuterated chain, i s mainly due t o the r e o r i e n t a t i o n of the H-H vector j o i n i n g the proton p a i r .  The proton r e l a x a t i o n rate i n t h i s case  i s , i n analogy with the deuterium case,  given by  where S . i s an order parameter defined by tin « ut  v 3cos 0-l\ \ 2 / 2  HH  [28]  where 0 i s the angle between the H-H vector and the normal n to the b i l a y e r . 2.2.3 Chain protons and  protons (Lipid/H^O m i x t u r e s ) .  The theory of s p i n - l a t t i c e r e l a x a t i o n f o r water and chain protons i n l i p i d / H 2 0 samples prepared w i t h nondeuterated, perdeuterated and s p e c i f i c a l l y protonated chain w i l l be dealt w i t h i n chapter 7 .  27  Chapter 3 Experimental. 3.1 Fatty acids The f a t t y acids (reagent grade) were purchased from the Eastman Kodak Co. and used without f u r t h e r p u r i f i c a t i o n . 3.2  P20 The  (99.7% enrichment) was purchased from Merck Sharpe and  Dohme (Montreal). 3.3 Deuteration of the f a t t y a c i d s . The procedure of deuterating the f a t t y acids i s the same as that of Hsiao et a l (42).  The f a t t y a c i d  and palladium on charcoal, as a c a t a l y s t , i n the r a t i o of 5:1  by  weight, are placed i n a two neck f l a s k as shown i n f i g . 3 and  heated-  to 180°C, w i t h deuterium gas (obtained by e l e c t r o l y s i s of D^O)  passing  over the s u r f ace c o n t i n u a l l y at the rate of 35 cc. per. minute f o r 1 week. The palladium on charcoal was  removed by d i s s o l v i n g the mixture of f a t t y  a c i d and palladium on charcoal i n chloroform, f i l t e r i n g through a c e l i t e column, and then evaporating the chloroform using a r o t a r y evaporator. Mass s p e c t r a l a n a l y s i s revealed b e t t e r than 99.2% d e u t e r a t i o n . 3.4 Proton l a b e l l i n g o f deuterated f a t t y a c i d s . L a u r i e ($,-W)d2i a c i d having a -CH2  group at the. a p o s i t i o n i n an  otherwise deuterated chain was prepared by exchange w i t h KOH/^O at 230°C.  Equimolar amounts of the f a t t y a c i d and KOH  (reagent grade)  w i t h .25 m o l e s / l i t r e of water excess KOH were d i s s o l v e d i n ^ 0 heated f o r 24 hours at 220°C i n a sealed s t a i n l e s s s t e e l tube.  and The  28  D  2  gas  cold water  ELHYGEN M A R K IV Milton R o y  Figure 3.  Experimental arrangement f o r the deuteration of the f a t t y a c i d s .  29  exhanged f a t t y a c i d s a l t s o l u t i o n was HC1  to p r e c i p i t a t e the f a t t y a c i d .  a c i d i f i e d w i t h concentrated  Separation of the f a t t y a c i d  accomplished by shaking-with d i e t h y l ether i n a separating The  ether l a y e r was  was  then evaporated i n a rotary evaporator.  the f a t t y a c i d was s p e c t r a l and NMR  was  funnel.  dried over anhydrous sodium s u l f a t e and the ether Further p u r i f i c a t i o n of  accomplished by s i l i c a g e l chromotography.  a n a l y s i s of the d 3 2  Mass  and d j acids i n d i c a t e d b e t t e r than 2  99% H a t t h e a p o s i t i o n and 99.2%D at the (3-w)  p o s i t i o n of the l a u r i c - d  2 1  a c i d sample. 3.5  Samples The f a t t y a c i d s a l t s were prepared by d i s s o l v i n g equimolar  amounts of the f a t t y a c i d and the corresponding base (KOH  or NaOH)  i n ethanol and slowly c r y s t a l l i z i n g the f a t t y a c i d s a l t s .  After  f i l t r a t i o n and washing with ethanol, the p r e c i p i t a t e d s a l t was r e c r y s t a l l i z e d , washed with ethanol, and d r i e d under vacuum at 140°C . The samples were made by weighing the corresponding molar amounts of the dry s a l t and H 0 2  or D 0  accomplished by c e n t r i f u g i n g i n the glass tube.  2  and sealed i n a glass tube.  Mixing  was  back and f o r t h through a c o n s t r i c t i o n  The samples were f u r t h e r homogenized by  leaving  them 4 days i n an oven at 120°C. Notation; D i f f e r e n t samples w i l l be r e f e r r e d to by the number of < deuterated p o s i t i o n s on the hydrocarbon c h a i n , the concentration of water and whether.it i s prepared with H 0 2  or D 0. 2  For example d iCi -Na/6H20 2  2  stands f o r a sample prepared to have 6 moles of water per 1 mole of sodium l a u r a t e with one p a i r of protons at the a p o s i t i o n i n an  30  otherwise perdeuterated chain; s i m i l a r l y c^-jC^2 Na/6D20 stands f o r a _  sample w i t h 6 moles of D2O per 1 mole of perdeuterated sodium l a u r a t e . 3.6 NMR Apparatus. A)  The Spectrometer. The NMR measurements were c a r r i e d out on a Bruker  SXP4-100 NMR pulse spectrometer w i t h a N i c o l e t BNC-12 computer.  The  spectrometer i s capable of p u t t i n g out a t r a i n of up to 4 RF p u l s e s o f c o n t r o l l e d amplitude and whose phases and lengths could be'varied independently.  The computer i s equipped w i t h a Diablo Disk Drive  ( s e r i e s 31 s i n g l e density) and was used f o r storage and a n a l y s i s o f the /. data. A programable timer ( N i c o l e t 293 I/O c o n t r o l l e r ) i n t e r f a c e d to the computer was used to automate the NMR measurements.  Thus the  t r i g g e r i n g of the i n d i v i d u a l RF p u l s e s , the spacing between theia, the r e p e t i t i o n rate as w e l l as the changing of the sample temperature was computer c o n t r o l l e d .  Automation of the measurements, e s p e c i a l l y the  measurement of the s p i n l a t t i c e r e l a x a t i o n rates r e s u l t e d i n a tremendous saving of time.  The accumulation of the t h e s i s data would have other-  wise taken another 2VJ years. B)  Probehead and V a r i a b l e Temperature Oven. The probehead and the a i r flow heating system provided w i t h the  spectrometer were found to be u n s a t i s f a c t o r y due to the large temperature gradient across the sample.  To circumvent t h i s problem a v a r i a b l e  temperature oven was b u i l t .  The diagram i s shown i n f i g . ( 4 ) .  The  oven was connected to the temperature c o n t r o l u n i t supplied w i t h the  31  RF connector  RF t o probe arm  Figure A . V a r i a b l e temperature oven and sample holder. (a) Heater arrangement, (b) Oven w i t h sarnie holder. On the i n s i d e of the copper block there i s a s h i e l d (not shown i n the diagram )made up of a f i h e brass screen to prevent r i n g i n g and eddy current e f f e c t s a f t e r the a p p l i c a t i o n of an RF pulse.  32  spectrometer.  Automatic temperature c o n t r o l was a l s o made p o s s i b l e by  i n t e r f a c i n g the temperature c o n t r o l u n i t to the computer.  Temperature  gradients over a sample space of 1 cm diameter and 3 cm i n height were undetectable (less than .2°C at 100°C ) w i t h i n 1°C  over a 24 hour period.  The temperature s t a b i l i t y  The time required f o r the sample t o  reach thermal e q u i l i b r i u m was l e s s than 20 minutes f o r an i n temperature of 10°C  was  increment  .  3.8 NMR Measurements A)  Spectroscopy. The conventional method of o b t a i n i n g NMR  spectra  c o n s i s t s of applying a 90° RF pulse and then F o u r i e r transforming the free i n d u c t i o n decay (FID). r e c e i v e r of the NMR  During the a p p l i c a t i o n of the R.F.. p u l s e , the  spectrometer gets saturated and a c e r t a i n time  ( c a l l e d the recovery time or dead time) has to elapse before i t returns to i t s normal operating c o n d i t i o n . Therefore the e a r l y part of the FID cannot be observed due to the recovery time of the r e c e i v e r . The u s u a l method, delaying data a c q u i s i t i o n u n t i l the r e c e i v e r has  recovered,  r e s u l t s i n the l o s s of the information contained i n the e a r l y part of the FID (which i s very important f o r wide l i n e s ) and i n v a r i a b l y l e a d s to d i s t o r t i o n o f the spectrum.  I t also introduces f i r s t order phase  s h i f t s and a poorly defined base l i n e . the NMR  To circumvent  t h i s problem  spectra were obtained using the s o l i d echo by the simple  method of Davis et a l  (43).  This method c o n s i s t s of applying a 90^ 0  pulse followed by another 90° pulse whose phase i s s h i f t e d by 90 w i t h respect to the f i r s t pulse at a time later.An  ( t y p i c a l l y 100-200V|s)  echo i s formed at 2T due to the r e f o c u s i n g of the nuclear  33  magnetization.  By F o u r i e r transforming the echo s t a r t i n g at t=2x  the f u l l spectrum i s obtained. B)  Relaxation measurements. S p i n - l a t t i c e r e l a x a t i o n times, T j , were  measured using a 180- T - s o l i d echo pulse sequence where the 180°  pulse  was applied on only every second c y c l e and a l t e r n a t e scans were subtracted from the computer memory.  The i n t e n s i t i e s of the i n d i v i d u a l  peaks of the F o u r i e r transformed spectra decay according to  M -M (t) 0  where M  Q  z  =  2M e  _ T / T  Q  ln  i s t h e e q u i l i b r i u m magnetization  • and n i s the p o s i t i o n on  the hydrocarbon chain. Fig.  5  shows p a r t i a l l y relaxed F o u r i e r transformed s p e c t r a f o r  d i f f e r e n t x values.  The r e p e t i t i o n r a t e , or time between pulse .  sequences, was chosen to be at l e a s t 5 times longer than the longest T j i n the spectrum.  The T j ' s were obtained  from  semilog  p l o t s of T versus peak amplitudes. For H 0 2  was  used.  T^ measurement the conventional 180-T-90 pulse sequence  Figure 5.  Representative deuterium p a r t i a l l y relaxed spectra f o r sodium laurate/water at 13.8 MHz and 90 C. R e p e t i t i o n rate = 10 seconds; 90 pulse length = 4.5 us; number of scans = 200. The spectra were obtained using the quadrupolar echo method as described i n the t e x t .  35  Chapter 4 Results 4.1 Quadrupole S p l i t t i n g s . The deuterium quadrupole s p l i t t i n g s of the (-CT>^) groups on perdeuterated hydrocarbon chains i n the l i q u i d C r y s t a l ( L ) phase of a  the sodium laurate-water system, were measured as a f u n c t i o n o f temperature , p o s i t i o n on the hydrocarbon chain and water concentration. Fig.  6a  obtained.  i s a representative spectrum fronr.which the s p l i t t i n g s are I t c o n s i s t s o f 11 overlapping powder patterns a r i s i n g from  deuterons a t the d i f f e r e n t p o s i t i o n s on the hydrocarbon chain.  At high  temperatures and water concentration i t i s p o s s i b l e to resolve 10 o f the 11 peaks i n the spectrum.  The assignment of the peak p o s i t i o n s was distance , t  made assuming that the order decreases w i t h ^ Fig.  7  from the head groups (39)i  i s a representative diagram showing the temperature  dependence of the quadrupole s p l i t t i n g s f o r ' ^ ^ j ^ - ^ / G H 0,  '^  ie  quadrupole s p l i t t i n g s f o r the 01-CD2 and the 3,4 p o s i t i o n s show a temperature dependence that i s q u i t e d i f f e r e n t from the r e s t of methylene chain deuterons:  The s p l i t t i n g s increase w i t h temperature reach a  maximum at /vL25°C and then show a s l i g h t decrease at higher temperatures. In c o n t r a s t , the s p l i t t i n g s f o r the r e s t o f the chain deuterons decrease w i t h i n c r e a s i n g temperature. Fig  8 . shows the quadrupole s p l i t t i n g s as a function o f p o s i t i o n  on the hydrocarbon chain a t two d i f f e r e n t temperatures.  These s p l i t t i n g s  are l a r g e f o r the a, and the f i r s t few methylenes and become p r o g r e s s i v e l y  36  smaller f o r the methylene p a i r s at the t a i l of the hydrocarbon chain. In the same f i g u r e i t i s i n t e r e s t i n g to note the absence o f the "plateau" observed i n other systems ; f o r example (17, 18) . A representative diagram f o r the dependence of the quadrupole s p l i t t i n g s on water concentration i s shown i n f i g .  .9 . At low water  concentrations the s p l i t t i n g s are large f o r the f i r s t few hydrocarbon chain segments and decrease r a p i d l y w i t h i n c r e a s i n g water concentration. Near the t a i l o f the hydrocarbon chain the s p l i t t i n g s are p r o g r e s s i v e l y smaller and seem to have a somewhat weaker dependence on water . concentration. The e f f e c t of water isotope composition on the quadrupole s p l i t t i n g of the chain deuterons was i n v e s t i g a t e d f o r two samples prepared w i t h 1^0 and D 0 r e s p e c t i v e l y but having the same molar r a t i o o f water t o 2  fatty acid s a l t .  There was no observable d i f f e r e n c e i n the measured  quadrupole s p l i t t i n g s o f the two samples t o w i t h i n the experimental e r r o r . The quadrupole s p l i t t i n g s as a f u n c t i o n o f temperature f o r 23  deuterium i n D£0 , shown i n f i g .  Na and the f i r s t 3 p o s i t i o n s on the chain are  10 . They a l l . have a s i m i l a r temperature dependence  suggesting a c o r r e l a t i o n i n the ordering o f water, counter i o n and the hydrocarbon chain segments close to the p o l a r head.  A s i m i l a r behaviour  was observed f o r the potassium palmitate-water system (see appendix A) . A.2 S p i n - L a t t i c e Relaxation Times, A)  Deuterium Results: The s p i n l a t t i c e r e l a x a t i o n time T j , f o r each-CD2  group of a perdeuterated f a t t y a c i d chain i n the sodium laurate-water  37  system were measured at 13.8 MHz as a f u n c t i o n of temperature and water concentration. Fig.  11  shows 1/Tp  The r e l a x a t i o n rates  as a function of chain p o s i t i o n a t 105°C » ) are l a r g e f o r the methylenes close to the  r  ln  p o l a r region and p r o g r e s s i v e l y get smaller towards the centre of the b i l a y e r . In f i g . 12  the temperature dependence of -jT  c l e a r from f i g . 12 energy.  i s shown.  It is  ln  that a l l the T^'s are characterised by an a c t i v a t i o n  Except f o r the methyl (-CD3) group, the a c t i v a t i o n energies  are approximately the same.  F i g . .13  i s a p l o t of the a c t i v a t i o n  energy versus chain p o s i t i o n • The dependence of 1/T-j, on water concentration i s shown i n f i g . 14  f o r the 2 and 10 p o s i t i o n s and f o r the CD^-group.  The r e l a x a t i o n  rates f o r the 3-9" p o s i t i o n s (not shown i n the diagram) also increase w i t h i n c r e a s i n g water concentration.  The r e l a x a t i o n rates f o r the  methylenes near the p o l a r head seem to be more s e n s i t i v e to the water concentration than the methylenes near the centre of the b i l a y e r .  In  f o r the CD3 i s almost independent of water concentration.  fact ^  The e f f e c t o f water isotope composition on the r e l a x a t i o n rates of the methylene chain segments was studied f o r two samples prepared with  H2O and D2O r e s p e c t i v e l y but w i t h same molar r a t i o of H2O or D2O  to the f a t t y a c i d s a l t . 01-CD2  .  The r e s u l t s are shown i n f i g . 15  f o r the  The r e l a x a t i o n rates f o r the chain deuterons were not measurably  a f f e c t e d by changing the isotope composition of the water •  38  B) ct-protons: The spin l a t t i c e r e l a x a t i o n time f o r the cc-C^ i  na n  other-  wise perdeuterated chain were measured as a f u n c t i o n o f temperature f o r two samples prepared w i t h H^O and D^O r e s p e c t i v e l y but w i t h the same molar r a t i o o f water t o f a t t y a c i d s a l t .  For b r e v i t y they w i l l be  refered to as d„ C -Na/6H 0 and d C -Na/6D 0 . T here was a small b u t 21 12 2 21 12 2 measureable d i f f e r e n c e i n the T 's o f the a protons o f the two samples . 1  the  r e s u l t s are shown i n f i g .  16 . To ensure t h a t the s t a t e o f the two  samples was the same, the quadrupole s p l i t t i n g s o f the chain deuterons i n the two samples were measured and were found t o be i d e n t i c a l , c) H^O Results The e f f e c t o f isotope composition on the H^O T^ i n the sodium laurate-water system was measured f o r  i n 3 d i f f e r e n t samples:  H^O w i t h perdeuterated chains, H^O w i t h orprotonated chains and H^O w i t h a l l protonated chains. These samples w i l l be r e f e r e d t o by d,,C, -Ka/6H0, (JB- ) d„,C -Na/6H 0 and d C -Na/6H 0 . Where the 23 12 2 21 12 2 o 12 2 U  s u b s c r i p t on the d stands f o r the number o f deuterated p o s i t i o n s on the hydrocarbon chain.  The r e s u l t s are shown i n f i g .17 -.  For a l l the  temperatures studied the r e l a x a t i o n r a t e (1/Tj) f o r the H 0 protons 2  increase w i t h i n c r e a s i n g number o f protons on the hydrocarbon chain. 4.3 Sources o f e r r o r Quadrupole s p l i t t i n g s were obtained from the peak p o s i t i o n s i n the NMR spectra. the  I n the presence o f d i p o l a r broadening o f  quadrupolar s p e c t r a , the p o s i t i o n s o f maximum i n t e n s i t y are no l o n g e r  coincident  with the p o s i t i o n s o f the 90° edges o f the powder p a t t e r n s .  This introduces a small decrease i n the measured quadrupole s p l i t t i n g s .  39  Ih a d d i t i o n f o r spectra c o n s i s t i n g of a superposition o f quadrupolar powder p a t t e r n s , e.g. of the perdeuterated  chains, the overlap between  neighbouring patterns causes a small apparent increase i n the measured quadrupole s p l i t t i n g s .  Due t o the small magnetic moment o f the deuteron  and i n view of the large quadrupole s p l i t t i n g s considered here such e f f e c t s were assumed to introduce only a n e g l i g i b l e systematic e r r o r i n the measurements. The accuracy of determining the peak p o s i t i o n s of maximum i n t e n s i t y i s e s s e n t i a l l y l i m i t e d by the s p e c t r a l r e s o l u t i o n o f the computer, which i s 12.5 Hz f o r a 50 KHz spectrum.  However, the s i g n a l to noise r a t i o i s  d i f f e r e n t f o r the d i f f e r e n t p o s i t i o n s on the hydrocarbon chain and ranges from about 400:1 f o r the CD3 to 50:1 f o r the a p o s i t i o n . Therefore i t i s more d i f f i c u l t to l o c a t e the p o s i t i o n s o f maximum i n t e n s i t y f o r the 2-4 p o s i t i o n s , thus i n t r o d u c i n g an a d d i t i o n a l e r r o r of about 50 Hz i n the s p l i t t i n g s o f those p o s i t i o n s . The main sources of e r r o r i n the measurements of the deuterium r e l a x a t i o n rates are the reduction i n the s i g n a l to noise r a t i o a t long T values and i n t e r f e r e n c e between overlapping powder patterns.  Since  the deuterium magnetic resonance spectrum i s a s u p e r p o s i t i o n o f powder p a t t e r n s , then the peak i n t e n s i t y of a c e r t a i n p o s i t i o n w i l l contain c o n t r i b u t i o n s from those powder patterns w i t h the l a r g e r s p l i t t i n g s . These c o n t r i b u t i o n s are only s i g n i f i c a n t f o r short xvalues because the r e l a x a t i o n rates increase w i t h i n c r e a s i n g s p l i t t i n g s .  For t h i s reason  the T^'s f o r the overlapping peaks were obtained from the p l o t s o f T versus peak amplitudes  at longer T values.  40  Figure 6.  (a) A deuterium magnetic resonance spectrum f o r 23 12~ ^ 2° 120°C and 13.8 MHz using the quadrupolar echo method. (b) A proton magnetic resonance spectrum of the a-protons i n a d2^-laurate water sample obtained at 90 MHz and 90 C using the s o l i d echo and quadrature detection. The c e n t r a l peak i s due to water and r e s i d u a l protons on the chain. (c£ A N a spectrum of d 3C -Na/6H 0 at 23.8 MHz and 86 C obtained using the echo method. See text f o r abbreviations. d  C  Na  6H  o b t a i n e d  a t  23  2  12  2  80 Figure 7.  100 120 140 TEMPERATURE(°C )  Temperature dependence of the quadrupole s p l i t t i n g s for d 3Cj -Na/6H 0. The numbers beside the curves denote the p o s i t i o n s on the hydrocarbon chain. The s o l i d curves are l e a s t square f i t s to the experimental data (dots and c i r c l e s ) where a three parameter f i t of the form (T - T ) (n) o v ( T ) = a< + a 5 ( - T ) + a n o 1 o 2  2  2  n)  n )  v  was used.  T  42  16.01 —  A  1—  — A  IM X  •  at 105 °C  A  at 135 °C  —  •  A  9  A  6.0 —. 4.0 —  • A  -  •  A  2.0  A  . I 2 (3,4)  . 1 . 1 . 1 6 8 10  n (CARBON  1  1 12  NUMBER)  Figure 8. The quadrupole s p l i t t i n g s as a f u n c t i o n of p o s i t i o n on the hydrocarbon chain f o r d 2 2 C j 2 ~ ^ / ^ 2 ^ * a  43  3 4 5 6 7 C ( MOLES OF H^IMOLE OF dC-Na)  Figure 9. Dependence of the quadrupole s p l i t t i n g s on water concentrations at 120 C. The s o l i d curves are l e a s t square f i t s t o the experimental data ( c i r c l e s ) where a f i t of the form . »2 v (c) = b n o was used.  + b- (c - c ) + b_ 1 o i l  x-, .  44  T E M P E R A T U R E (°C) Figure 10. The quadrupole s p l i t t i n g s as a f u n c t i o n of temperature f o r deuterium i n D^O ( t r i a n g l e s ) , N a (squares), the a (open c i r c l e s ) and the 3 and 4 p o s i t i o n s ( s o l i d c i r c l e s ) . The D 0 s p l i t t i n g s are f o r d23Ci2 Na/6D20 and the r e s t are f o r d 2 3 C i 2 ~ / ^ ^ * 2 3  2  -  Na  H  2  45  2 (3,4)  6  8  10  12  n (CARBON NUMBER) Figure 11.  T  as a f u n c t i o n of p o s i t i o n on the hydrocarbon chain at 13.8 MHz and 105°C f o r d C -Na/6H 0. 23  12  2  2.5  2.6  2.7  I0 /T  (K )  3  Figure 12.  2.8 H  Temperature dependence of the r e l a x a t i o n r a t e s of the chain deuterons i n d23C -Na/6H 0 a t 13.8 MHz. The s o l i d l i n e s are the l e a s t square f i t s to the experimental data where a f i t of the form 12  Log  2  = L  ln  a  n  + b  -J n T  was used. The numbers appearing beside the s o l i d l i n e s indicate the p o s i t i o n n on the hydrocarbon chain.  47  4.0  -< —  .1 ? ro g  3.0  I  •  i  T T  c o LU 2.0 —  T  I  1.0 1 1 1 2 (3,4) (5,6)  1  n (CARBON Figure 13.  1 , 1 , 1 8 10  12  NUMBER)  A c t i v a t i o n energy versus chain p o s i t i o n f o r the deuterium r e l a x a t i o n rates i n d C , -Na/6H 0. OQ  o  o  48  14.0 —  o  2  12.0 l O LU CO  o  IQO 8.0 — 6.0 4.0  •  2.0 - a i  3  •  •  ft  X  i  i  4  5  o  © 10  A  A  1 6  CD  3  I  7  C (MOLES OF H 0/1M0LE OF C -Na) 2  Figure 14.  )2  Dependence of the r e l a x a t i o n rates f o r the chain deuterons on water concentration a t 13.8 MHz and 105°C. P o s i t i o n s 3-9 are not shown i n the diagram (see text) .  49  Figure 15 . Dependence of the r e l a x a t i o n rates on inverse temperature f o r the 01-CD2 i n d23Ci2 / 2° (open c i r c l e s ) and 2 3 1 2 ~ / 2 ° ( t r i a n g l e s ) a t 13.8 MHz . _Na  d  c  Na  6D  6H  50  1°-  (K)  Figure 16 . Dependence of the r e l a x a t i o n rates of the (X-CH2 protons on inverse temperature i n ($-w)d2]C 2~ /6 2° (open c i r c l e s ) and ( B - w ) d 2 i C " / 2 ( s o l i d c i r c l e s ) at 90 MHz . Na  1  N a  1 2  6 l )  0  H  51  Figure 17 . Dependence of the r e l a x a t i o n rates on inverse temperature f o r H 0 i n d Ci2-Na/6H O ( s o l i d c i r c l e s ) , (3-w)d iC - / 2° ( t r i a n g l e s ) and d 3 C i 2 - / 6 H 0 (open c i r c l e s ) at 90 MHz Na  2  0  2  2  Na  2  2  12  6H  52  Chapter 5 The Lipid-Water I n t e r a c t i o n A Microscopic I n t e r p r e t a t i o n . 5.1 Quadrupole  Splittings, I t was pointed out e a r l i e r that the quadrupole  s p l i t t i n g s are p r o p o r t i o n a l to order parameters which can provide informt i o n on the conformations and motion of the l i p i d molecules w i t h i n the b i l a y e r as w e l l as on the ordering o f water and counter ions a t the lipid-water interface.  The r e s u l t s shown i n f i g . 10 on page  indicate  a c o r r e l a t i o n between the quadrupole s p l i t t i n g s of the deuterons on the f i r s t few methylene chain segments, the deuterons i n D 0 and the % a 2  2  counter i o n . The s p l i t t i n g s increase w i t h temperature, reach a maximum o at  125 C and then decrease at higher temperature.  In contrast the  s p l i t t i n g s f o r the r e s t o f the hydrocarbon chain segments shown i n f i g . 7 on page 41 decrease w i t h i n c r e a s i n g temperature.  A similar  behaviour was observed f o r the potassium palmitate-water system where t h i s c o r r e l a t i o n was ascribed to the s t r u c t u r i n g e f f e c t of the water v i a hydrogen bonding.  A model consistent w i t h the experimental r e s u l t s  (see Appendix A f o r d e t a i l s ) proposes two configurations which i n t e r change r a p i d l y compared w i t h NMR s p l i t t i n g s .  At lower temperatures,  the water, v i a some complicated hydrogen bonded s t r u c t u r e s w i t h the oxygens o f the l i p i d carboxyl groups, imposes a c o n s t r a i n t on the f i r s t C-C bond d i r e c t i o n causing i t to be p a r a l l e l to the normal n t o the b i l a y e r , l e a v i n g the t a i l on the average some what t i l t e d .  This, f o r  53  purely geometric reasons, r e s u l t s i n smaller quadrupole s p l i t t i n g s than those at the high temperatures (see table 1 Appendix A ). Since the  l i p i d - w a t e r i n t e r a c t i o n tends t o t i l t the hydrocarbon chain, i t  i s i n competition w i t h chain-chain ( i . e . l i p i d - l i p i d ) i n t e r a c t i o n s whose i n f l u e n c e i s to cause the hydrocarbon chain to be p a r a l l e l to n . When the molecular axis i s p a r a l l e l to n , the quadrupole s p l i t t i n g s are l a r g e r than f o r the s t r u c t u r e imposed by the l i p i d - w a t e r i n t e r a c t i o n . Therefore when the hydrogen bonded s t r u c t u r e s tend to break up a t h i g h e r temperatures and the chain-chain i n t e r a c t i o n s become more dominant, the  quadrupole s p l i t t i n g s f o r deuterons near the head of the chain  increase even though the order of the system as a whole decreases.  The  D 0 and % a r e s u l t s could a l s o be explained i r terms o f t h i s model. 2  2  For  the deuterons i n D2O and the % a counter i o n the average environment 2  i n the low temperature c o n f i g u r a t i o n are such that t h e i r p r i n c i p a l .es w i t h n g i v i n g average order  e l e c t r i c f i e l d gradients (ej  parameters which are smaller than those f o r the h i g h temperature c o n f i g u r a t i o n (see table 1 appendix A'). The dependence o f the quadrupole s p l i t t i n g s on water concentration i s shown i n f i g . 9 on page 43  .  There i s a marked decrease i n the  quadrupole s p l i t t i n g s w i t h i n c r e a s i n g water concentration.  It is  important to note that the f r a c t i o n a l v a r i a t i o n i n the quadrupole s p l i t t i n g s a t high water concentrations shown i n f i g . 18 i s greater f o r the methylene chain segments close to the p o l a r head than those near the centre of the b i l a y e r .  This r e s u l t i s e a s i l y accounted f o r i n terms  of the proposed model.  Near the surface of the b i l a y e r i t i s the  54  l i p i d - w a t e r i n t e r a c t i o n that i s responsible f o r the v a r i a t i o n i n the order parameters.  The more water there i s between the b i l a y e r s the more  degrees of freedom there are f o r forming d i f f e r e n t hydrogen bonded s t r u c t u r e s which would r e s u l t i n an average conformation o f the hydrocarbon chains that would lead to a reduction i n the quadrupole s p l i t t i n g s . On the other hand, near the centre of the b i l a y e r , i t i s the i n t e r a c t i o n between the chains, which i s l e s s dependent on water concentration, that controls the v a r i a t i o n o f the quadrupole s p l i t t i n g s . Samples prepared w i t h H2O and D2O r e s p e c t i v e l y but having the same molar r a t i o o f water to f a t t y a c i d s a l t show no observable d i f f e r e n c e i n the measured quadrupole s p l i t t i n g s of the two samples.  This  i n d i c a t e s that the average o f ( 3 c o s 0 - l ) over n thylene motions i s not 2  2  measurably a f f e c t e d by m o d i f i c a t i o n of the isotope composition o f the hydrogen i n the water.  2 (3,4) 6 n CARBON Figure 18.  8 10 12 NUMBER  F r a c t i o n a l v a r i a t i o n s of the quadrupole s p l i t t i n g s of.the chain deuterons w i t h water concentration a t f i x e d temperature (120 C). P a r t i a l d e r i v a t i v e s are evaluated at C= 6  56  5.2  S p i n - L a t t i c e Relaxation (Perdeuterated Chains) While deuterium quadrupole s p l i t t i n g s give information on the  o r i e n t a t i o n a l ordering and degree o f motion o f the l i p i d chains, s p i n l a t t i c e r e l a x a t i o n measurements allow the determination of a time s c a l e and an i n t e n s i t y f a c t o r f o r the molecular motions. In chapter 2 i t has been already s t a t e d that f o r chain deuterons quadrupolar i n t e r a c t i o n s are the main r e l a x a t i o n mechanism.  The s p i n -  l a t t i c e r e l a x a t i o n rate }• , was expressed i n the form (eqn. 24 chapter 2) l  A  = T l  O N *  > ( l j  ( u  o>  J  +  ( 2 o ) 2  o)  z  where <^|HQ|^  i s the mean square value o f the s p i n l a t t i c e coupling  Hq(t) due to the quadrupolar i n t e r a c t i o n s and J i ( w ) , ,J2(2w ) are 0  0  the reduced s p e c t r a l density functions associated with Hq(t) . Thus  1  /-jj-  measurements allow the determination o f J i ( w ) and J2(2co ) 0  0  which can be used to define a c o r r e l a t i o n time f o r the molecular motions o c c u r r i n g at the Larmor Figure'11 on page 45 hydrocarbon chain.  frequency w  and a t 2u) .  0  shows ^  0  as a f u n c t i o n o f p o s i t i o n on the  The r e l a x a t i o n rates are large f o r the -CD2 groups  near the p o l a r head and become p r o g r e s s i v e l y smaller towards the centre of the b i l a y e r .  This i n d i c a t e s that the motions o f the chain segments  near the l i p i d water i n t e r f a c e are considerably d i f f e r e n t from those of the methylene chain segments o f the t a i l o f the hydrocarbon c h a i n . The temperature dependence of ^ A  i s shown i n f i g . 12 on page 46  ln  f o r the d i f f e r e n t p o s i t i o n s on the hydrocarbon chain.  A l l the r e l a x a t i o n  57  rates are c h a r a c t e r i z e d by an a c t i v a t i o n energy.  Except f o r the methyl  (-CD3) group, a l l the a c t i v a t i o n energies are roughly the same as shown i n f i g . 13 on page 47  .  The decrease i n the r e l a x a t i o n r a t e s  w i t h i n c r e a s i n g temperatures i n d i c a t e s that the short c o r r e l a t i o n time 2  2  l i m i t (w T<s.l ; where T 0  N  i s the c o r r e l a t i o n time c h a r a c t e r i z i n g the  motion of the n*"* methylene chain segment) i s s a t i s f i e d . 1  Further  evidence that t h i s l i m i t i s s a t i s f i e d comes from the d i r e c t  comparison,  as w i l l be discussed below, of the r e l a x a t i o n r a t e s o f the a-CH at 90 • 2  MHz and the a-CD a t 13.8 MHz . 2  The d e t e c t i o n of d i s t i n c t types o f molecular motion as manifested by d i f f e r e n t c o r r e l a t i o n times requires a study o f the dependence o f ^ on the n u c l e a r Larmor frequency w . The l a r g r proton magnetic moment would permit ^ measurements at much higher values of u thus making 0  Q  1  p o s s i b l e the detection o f s h o r t e r c o r r e l a t i o n times.  However, i n the  short c o r r e l a t i o n time l i m i t , the r e l a x a t i o n rates are independent o f frequency (see chapter 2, s e c t i o n 2.2 ) and the expression f o r ^ to a product o f an i n t e n s i t y f a c t o r and a c o r r e l a t i o n time.  reduces  From the  experimental data o f f i g u r e s 12 and 16 , the r e l a x a t i o n rates of the a-CD a t 13.8 MHz and of the a-CH a t 90 MHz were found t o be i n the 2  2  r a t i o of 17.8 as compared w i t h 18.7 f o r the r a t i o of the square of the coupling constants.This i m p l i e s that the c o r r e l a t i o n times associated w i t h the motions o c c u r i n g at these frequncies are s h o r t For a s i n g l e type o f f a s t motion, i t was p r e v i o u s l y shown (eqn. 25 chapter 2 ) that i n the short c o r r e l a t i o n time l i m i t , the r e l a x a t i o n r a t e s are given by  58  where S  n  i s the order parameter f o r the n  t h  C-D bond defined by equation,  2 6 i n chapter 2 and the f a c t o r 1 S -  n  takes i n t o account the anisotropy o f  the system. A l o g a r i t h m i c p l o t of ^ p r o p o r t i o n a l to S  n  versus v  n  ( r e c a l l the V_ i s  ) i s shown i n f i g . 19 f o r 4 d i f f e r e n t temperatures.  Except f o r the f i r s t few p o s i t i o n s , the dependence of ^ on V seems 1 to obey a power law ( = ^ ) where p = 1.1 at 90°C and becomes s m a l l e r n  l n  In  a t higher temperatures. It i s likely that more complex motions must be taken i n t o account. I f the above equation i s v a l i d .however,the f a c t o r 1-S hardly changes since S ~.2 . Therefore changes i n the r e l a x a t i o n n  n  rates ~ are expected to be l a r g e l y ln times x . i  due to changes i n the c o r r e l a t i o n  n  The dependence o f J-  on water concentration i s shown i n f i g . 14  In  on page 48 .  The r e l a x a t i o n rates f o r the methylenes near the p o l a r  haad increase w i t h i n c r e a s i n g amounts of water more than the r e l a x a t i o n rates of the methylenes near the centre of the b i l a y e r . ^  i s almost independent of water concentration.  For the -CD^  In c o n t r a s t , the  quadrupole s p l i t t i n g s a l l decrease w i t h i n c r e a s i n g water concentration. This i s f u r t h e r evidence that complex molecular motion mediated by the l i p i d - w a t e r i n t e r a c t i o n must be taken i n t o account.  A simple model  e x p l a i n i n g the i n f l u e n c e of the water on the r e l a x a t i o n rates of the methylene chain deuterons w i l l be discussed below. 5.3 A Model f o r Molecular Motions Mediated by The l i p i d - W a t e r I n t e r a c t i o n . The increase i n the r e l a x a t i o n r a t e s w i t h i n c r e a s i n g water  59  concentration maybe due to the i n t e r a c t i o n o f the water with the p o l a r heads v i a hydrogen bonding.  Increasing the water concentration  will  r e s u l t i n greater formation o f hydrogen bonded s t r u c t u r e s ( l i p i d molecules and the water engaged i n hydrogen bonding with the p o l a r heads). Therefore,  i t i s the motion o f the combination ( l i p i d +water hydrogen  bonded to the p o l a r heads) that has to be taken i n t o account.  It is  q u i t e p o s s i b l e that i n c r e a s i n g water concentration w i l l increase the e f f e c t i v e mass o f the hydrogen bonded s t r u c t u r e s and r e s u l t i n slowing down o f the r o t a t i o n a l motion as w e l l as the t w i s t i n g , bending and w i g g l i n g motions about the long a x i s of the molecules and thus give r i s e to higher r e l a x a t i o n r a t e s .  In addition  ,  i n c r e a s i n g water concen-  t r a t i o n w i l l increase the area per polar head (£ee appendix B f i g . 29). This w i l l increase the a v a i l a b l e space occupied by the hydrocarbon chains which would allow f o r s i n g l e molecule motions o f l a r g e r amplitude also around the d i r e c t o r r e s u l t i n g i n l a r g e r r e l a x a t i o n r a t e s . v  However, an  increase i n the area per p o l a r head would a l s o give more freedom o f movement f o r the i n d i v i d u a l chain segments causing a reduction  i n the  observed quadrupole s p l i t t i n g s (see f i g . 9 on page 43 ) 5.4. Isotope e f f e c t s . The isotope m o d i f i c a t i o n o f the water may i n f l u e n c e the r e l a x a t i o n rates o f the chain deuterons i n two ways: ( i ) by changing the s p e c t r a l d e n s i t i e s (or c o r r e l a t i o n times) through m o d i f i c a t i o n o f the motions o r ( i i ) by a f f e c t i n g the c o n t r i b u t i o n o f the d i p o l a r i n t e r actions to the r e l a x a t i o n r a t e s . In the previous s e c t i o n i t has been pointed out t h a t , due to the  60  hydrogen bonding of the water with the p o l a r heads, the motion o f the combination ( l i p i d + water hydrogen bonded to the p o l a r heads) should be considered.  Since the deuteron i s heavier than the proton, then  r e p l a c i n g H2O w i t h D 0 would r e s u l t i n an increase i n the mass o f the 2  hydrogen bonded s t r u c t u r e s r e s u l t i n g i n the slowing down o f the motions ( l o c a l bending and t w i s t i n g around the chain segments as w e l l as the o v e r a l l t r a n s l a t i o n a l and r o t a t i o n a l motion o f the hydrogen bonded structure).  This would increase the c o r r e l a t i o n times and therefore  result i n larger relaxation rates.  However, an examination o f the  r e l a x a t i o n data f o r two samples prepared with H 0 and D 0 2  2  respectively  but w i t h the same molar r a t i o of water to f a t t y a c i d s a l t , reveals  that  t h i s e f f e c t i s n e g l i g i b l e , since there was no measurable d i f f e r e n c e ( w i t h i n the experimental error) i n the r e l a x a t i o n r a t e s o f the two samples as shown i n f i g . 15 on page 49  f o r the a-CD  2  . The r e s u l t s  also i n d i c a t e that the c o n t r i b u t i o n t o the r e l a x a t i o n rates due t o dipolar interactions i s also n e g l i g i b l e .  This r e s u l t i s not s u r p r i s i n g  s i n c e , as was discussed e a r l i e r , the quadrupolar i n t e r a c t i o n s o f the chain deuterons are much l a r g e r than deuteron-proton d i p o l a r i n t e r a c t i o n s and hence are much more e f f e c t i v e i n s p i n - l a t t i c e r e l a x a t i o n .  We -  conclude that j u s t as isotope m o d i f i c a t i o n of water does not change the average quadrupolar i n t e r a c t i o n , as discussed e a r l i e r , the f l u c t u a t i o n s of the quadrupolar i n t e r a c t i o n s about the average which are responsible f o r s p i n l a t t i c e - r e l a x a t i o n are not a f f e c t e d  significantly.  1  2  4  6  4v (kHz) n  Figure 19. A l o g - l o g p l o t of the r e l a x a t i o n rates versus quadrupole s p l i t t i n g s f o r the chain deuterons i n d C -Na/6H 0 at 80°C ( s o l i d d o t s ) , 105°C (open c i r c l e s ) , 125°C ( t r i a n g l e s ) , and 135 C (squares). 23  12  2  62  Chapter 6 The Lipid-Water I n t e r a c t i o n A n a l y s i s i n terms of Macroscopic V a r i a b l e s 6.1 Quadrupole S p l i t t i n g s The order parameters of the C-H bonds along a hydrocarbon chain i n a l i p i d b i l a y e r give a q u a n t i t a t i v e measure of the nature of the f l u i d i t y of the b i l a y e r . the  These order parameters are p r o p o r t i o n a l to  deuterium quadrupole s p l i t t i n g s of the -CD2 groups on the d i f f e r e n t  p o s i t i o n s of the hydrocarbon chain.  In t h i s chapter a systematic  i n v e s t i g a t i o n i s made o f the dependence of these l o c a l order parameters on the macroscopic thermodynamic parameters which c h a r a c t e r i s e the l i p i d water system. I t has been suggested by Mely e t a l (17) that A , the mean area per  p o l a r head,  i s a "good parameter t o represent the average  microscopic order" i n the b i l a y e r .  T h e i r conclusion i s based on  measurements on the potassium l a u r a t e -water system which show that where-as the p r o f i l e of the quadrupole s p l i t t i n g s changes appreciably when the temperature T i s v a r i e d w i t h the water concentration C kept constant and a l s o when C i s v a r i e d w i t h T kept f i x e d , the v a r i a t i o n i s much l e s s than when both T and C are v a r i e d i n such a way as t o keep A constant.As may be seen from t h e i r data which are reproduced i n f i g . 20 (a.b)the quadrupole s p l i t t i n g s i n the samples having i d e n t i c a l surface area are not i d e n t i c a l .  Rather the p r o f i l e s of S  n  versus n f o r these  samples resemble each other more c l o s e l y i n a q u a l i t a t i v e sense than do the p r o f i l e s i n which C and T are separately v a r i e d without keeping A constant . As shown i n fig.20C the curves f o r the quadrupole s p l i t t i n g s  63  A*32.8A'  X  30  -X  5=  \  <J 20 \A  at c  "3."  w» 10 u a 9 £X  •o a  a3  . :247.HjO-50*C  s  . :21V.HjO-74*C  -JO 02 2  4 S 3 _l_ Carbon number (from polar head)  JL 1_ 10~  12  2  i  6  8  10  12  Carbon number (from polar head)  Figure 20 . (a) Order parameter curves obtained f o r two d i f f e r e n t l a m e l l a r samples o f dC^K-R^O,having the same A value. (b) Order parameter curves v a r i a t i o n s as the area per p o l a r head i s increased i n the l a m e l l a r phase o f dC K-H 0 (• :21%-31°C;A :24%-50°C;§ :30%-51°C;- - :30%-110 C ) . (The white dots come from computer s i m u l a t i o n o f the unresolved l i n e s . ) 12  2  Reproduced from reference (17) .  r  64  9  16 O  14  —  o  c = 6 j T = 105°C  ©  C = 5 j T=13 5 C :  N  i  12  HIM  -  CM  \ 1 0 8  o  • o  —  8  —  o  Q  S  • o  i  _  0  I  i  I  I  .  I  I  i  i  i  2 (3,4) 6 8 10 12 n(CARBON NUMBER)  Figure 20C . Quadrupole s p l i t t i n g s curves f o r two samples of dC..-Na/R^O having the same A value.  65  for two samples of dC^-Na/l^O having the same A are a l s o s i m i l a r . The hypothesis of Mely e t a l that the quadrupole s p l i t t i n g s v^CC.T) depend on C and T i n such a way that they vary only i f the surface area v a r i e s , i . e . v (C,T) i s a function of A(C,T) makes good p h y s i c a l sense i f n  the  dominant i n t e r a c t i o n s responsible f o r the o r i e n t a t i o n a l order of the  hydrocarbon chain segments are the chain-chain i n t e r a c t i o n s .  These i n t e r -  actions are expected to be s e n s i t i v e to the average s e p a r a t i o n o f the hydrocarbon chains, which would be simply r e l a t e d t o the surface area per p o l a r head.  In view of the f a c t that we have s t r o n g e m p i r i c a l  evidence that the l i p i d water i n t e r a c t i o n has a s t r o n g i n f l u e n c e on the o r i e n t i o n a l order of the hydrocarbon chains, e s p e c i a l l y those chain segments near the p o l a r head, i t i s of great i n t e r e s t to examine the v a l i d i t y of t h e i r hypothesis i n a systematic manner f o r another system. I n order to do t h i s , i t i s not necessary to compare samples having the same values of A at d i f f e r e n t C and T as has been done by Mely e t a l .  Since  A i s r e l a t e d to C and T through an equation o f s t a t e , as w i l l be discussed l a t e r , i t i s a simple procedure, using elementary thermodynamics, examine the dependence of V  n  to  f o r any s p e c i f i e d v a r i a t i o n o f the  thermodynamic v a r i a b l e s so long as data i s a v a i l a b l e over a wide range of any set of independent v a r i a b l e s .  In t h i s s e c t i o n we carry through  such a thermodynamic a n a l y s i s u s i n g the measured values o f V  n  over a  range of C and T which were presented i n chapter 3 and the e x p e r i m e n t a l l y determined e m p i r i c a l equation o f s t a t e presented i n Appendix B. Thermodynamic Formulation. I f the quadrupole s p l i t t i n g  of the -CD2  groups on the n  t Q  66  p o s i t i o n o f a hydrocarbon chain i s a f u n c t i o n o f s e v e r a l independent thermodynamic v a r i a b l e s {Qj} where Qj could be water concentration, temperature, PH, pressure o r i o n i c species. in V  n  A d i f f e r e n t i a l change dv,  i s given by :  \  f3v (... Q ,..) n  t  ±  \  [1]  i  In t h i s work the NMR measurements were c a r r i e d out on samples o f d i f f e r e n t water concentration a t d i f f e r e n t temperatures but always under the c o n s t r a i n t of the e q u i l i b r i u m water vapor pressure because the samples were sealed.  Therefore i t w i l l be assumed that there are only  two independent thermodynamic v a r i a b l e s that have t o be considered i n this analysis:  the water concentration and the temperature.  Eqn. [ l ]  then s i m p l i f i e s t o .  where T and C are the temperature and water concentration r e s p e c t i v e l y . In order t o t e s t f o r the s e n s i t i v i t y of the quadrupole s p l i t t i n g s V  n  t o changes i n the d i f f e r e n t thermodynamic  to examine the p a r t i a l d e r i v a t i v e s of v  n  variables, i t i s sufficient  w i t h T and C keeping A the  surface area per p o l a r head constant. That i s , to determine whether v depends on T and C only through A . I f these v a r i a t i o n s are d i f f e r e n t from zero then A i s not n e c e s s a r i l y "a good parameter" which  n  67  c h a r a c t e r i s e s the state of the b i l a y e r as proposed by Mely e t a l . From [2] the p a r t i a l d e r i v a t i v e s of v  n  w i t h respect to T and C keeping  A, the surface area constant are given by  A l l the p a r t i a l d e r i v a t i v e s on the r i g h t of equations [3]»[4] are obtained from the NMR experiments except f o r O C / 3 T ) a n d O T / 3 C ) A A  have to be determined independently from an equation o f s t a t e .  which  In  appendix B the equation of s t a t e f o r the Cj^-Na/H^O system was found to be A  =  A (T)C  [5]  P  C  where A i s the surface area per p o l a r head, C i s the water c o n c e n t r a t i o n , p i s a constant equal to .24 and A (T) i s the area per p o l a r head i n the Q  l i m i t of zero water concentration and i s o n l y a f u n c t i o n o f temperature. From [ 3 ] , [4] and [5] ..we o b t a i n : / 3Vn \ \3T c  C .1 P A  /^L  \  V3C  )  P . A C  )  N  X  .  T  A l l the p a r t i a l d e r i v a t i v e s [  V aT  dT  b  Q  / 9Vn \ V8C ;  . J _ 7 \ \ dAo h i dT  \  and  j  c  (^n \  ac  [6] T  [7]  / C  j were obtained J  T  from l i n e a r l e a s t square f i t s to the experimental data as i n d i c a t e d i n .  68  the  captions of f i g u r e s 7 and 9 on pages 41  and 43.  Due to the wide v a r i a t i o n i n the quadrupole s p l i t t i n g s of the methylene deuterons on the hydrocarbon chain, i t i s the r e l a t i v e q u a n t i t i e s 1  ( n  \  9v  1 (  ,  Figure  21  3 v  n \  that are meaningful.  v£\Tc)i  311(1  i s a p l o t of ^  1 / 8v \ [ ] n  A  and  i f 3v \ ~ [jT IC  p o s i t i o n number n on the hydrocarbon chain. 1 —  n  V e r s u s  I t i s c l e a r that  ( 9v \ n  I -^"Y" K  does not vanish except f o r the n=7 p o s i t i o n which may' be  just accidental.  In fact keeping A constant seems to be l e s s e f f e c t i v e  i n c o n t r o l l i n g the f r a c t i o n a l v a r i a t i o n o f the quadrupole s p l i t t i n g s o f the f i r s t few -CD2 groups than keeping C constant. I n f i g . 22 a p l o t o f 1 (3v \ n  \T~  \1TT~ JA versus n the p o s i t i o n number on the hydrocarbon chain i s  shown f o r four d i f f e r e n t temperatures. varation of v  n  I t can be seen that the f r a c t i o n a l  w i t h T keeping A constant f o r the f i r s t 6 p o s i t i o n s  depends on temperature but i s independent o f temperature f o r p o s i t i o n s (7-12) i n c l u s i v e . 1 ( 3Vn \ —  ^  JA.  A S  S N O W N  A s i m i l a r observation can be made f o r 1 / 3v \ n  I N  ^ S * 23.Fig 24 shows that  depends on C only f o r the f i r s t 6 p o s i t i o n s .  \ Xc~ /A  This suggests that the  ordering o f the hydrocarbon chain segments close to the p o l a r head i s i n f l u e n c e d by the c o n s t r a i n t s which water imposes on the f i r s t C-C bond v i a hydrogen bonding.  The i n f l u e n c e of the water gets weaker  towards the centre o f the b i l a y e r . c l e a r l y i n f i g . 25 where the r a t i o  This e f f e c t can be seen more  69  (_3Vn_\ V 9T /C i s p l o t t e d aganist the p o s i t i o n number f o r d i f f e r e n t temperatures. Again t h i s r a t i o i s a function of temperature f o r the f i r s t few chain segments but i s independent of temperature f o r the chain segments close to the centre of the b i l a y e r and asymptotically approaches a constant value of .7. the  This i n d i c a t e s that the v a r i a t i o n w i t h temperature i n  ordering of the hydrocarbon chain segments near the l i p i d - w a t e r  i n t e r f a c e keeping A constant i s much greater than i t i s keeping C constant.  However i n the centre of the b i l a y e r A tends to be s l i g h t l y  more e f f e c t i v e than C i n c o n t r o l l i n g the v a r i a t i o n i n the ordering of the  methylene chain segments.  This v a r i a t i o n i s c o n s i s t e n t w i t h the  model mentioned i n the preceding chapter. At lower temperatures hydrogen bonding of the water w i t h the p o l a r head groups i s more favourable than at higher temperature where the hydrogen bonding s t r u c t u r e s tend t o break up. "An e f f e c t i v e range" <1JI> which i s a measure o f the extent of the persistence of the i n f l u e n c e of the water can be obtained from f i g 25. A semilog imit rn ° p l o t of roo-r„« versus n i s shown i n f i g . 26 where roo= ln->oor  corresponds to an i n f i n i t e l y long chain.  I t was f i t t e d by i n s p e c t i o n  to a value of .8 assuming an exponential dependence of r  Q  on n and an  e f f e c t i v e range of 4<v5 chain segments was obtained. The above observations imply that i n s o f a r as the system s t u d i e d i n t h i s t h e s i s i s concerned, A i s not the best parameter to c h a r a c t e r i s e the s t a t e of the b i l a y e r p a r t i c u l a r l y f o r the methylene chain segments that are close to the l i p i d water i n t e r f a c e  . The  70  conformations of these chain segments are d i c t a t e d by the hydrogen bonded s t r u c t u r e s of water at the surface.  However i n the centre of  the b i l a y e r keeping A constant but varying T or C does not i n f l u e n c e the v a r i a t i o n of the order parameters s i g n i f i c a n t l y .  This i s i n agreement  w i t h the i n t u i t i v e p h y s i c a l p i c t u r e where i n the centre of the b i l a y e r i t i s chain chain i n t e r a c t i o n s and the a v a i l a b l e , area per p o l a r head which determines the ordering and conformations of the hydrocarbon chains, whereas  i n the v i c i n i t y of the p o l a r head i t i s the l i p i d water i n t e r -  a c t i o n v i a hydrogen bonding that c o n t r o l s the v a r i a t i o n of the order parameters.  71  e  o  -A  •  A  CVJ I O  A  x  -2  Q =  A  A  Q =  C _  9  —  o  •  A A  >- TO  —  •  4  A  •  • A  e  A  —  1  (3,4)  i  1  6  1  1  8  1  1  10  •  n (CARBON N U M B E R ) Figure 21.  —  A  ;?~6  -8  •  1  12  The f r a c t i o n a l v a r i a t i o n of the quadrupole s p l i t t i n g s w i t h temperature keeping the area per polar head constant (open c i r c l e s ) , and keeping the water concent r a t i o n s constant ( t r i a n g l e s ) . P a r t i a l d e r i v a t i v e s are evaluated at T=105°C and C= 6 .  72  0.6 9  0.4  I o  5  A105°C -o-  o  ° 1 2 5°C A135°C  0.2  CM  I o  8 6 °C  2  0 <r  -0.2 C  o  -0.4  A  9  C  2  l  3,4  1  6  n (CARBON Figure 22.  O  6  =6  -0.6 1  A  1  8  i  10  1  12  NUMBER)  The f r a c t i o n a l v a r i a t i o n of the quadrupole s p l i t t i n g s w i t h temperature keeping the area per p o l a r head constant.  73  0.3 •  8 6 °C  o 105°C A 12 5 °C  0.2  a  • 1 3 5 °C 0.1  A  —  n 8  X  °  • 8  c  s e  @  -0.1 -  -0.2  0 o  o  • • 1 2  I (3,4)  C  I 6  n (CARBON Figure 23.  i 8  =6  i 10  i 12  NUMBER)  The f r a c t i o n a l v a r i a t i o n of the quadrupole s p l i t t i n g s w i t h water concentration keeping the area per polar head constant.  74  c  O  -1^  2 3,4  6  8  n (CARBON Figure 24.  10  12  NUMBER)  The f r a c t i o n a l v a r i a t i o n of the quadrupole s p l i t t i n g s w i t h water concentration keeping the area per p o l a r head constant.  75  -°-  —  6.0  C  o 4.0  —  =  6  - x -  _X  2.0  —  a  A. a  c  a  a  <T5  C  4  2.0  °  — x — m  -4.0  - i -  *  o  1  A  -6.0  -8.0  "T  X *86°C o 105°C A 12 5°C d 135°C  —  i  i (3,4)  .  1 6  n(CARBON  ,  8  I . I . I 10 12  NUMBER)  Figure 25. The r a t i o of the change i n . v w i t h T keeping A f i x e d t o the change i n V w i t h T keeping G f i x e d . r  76  Figure 26.  A semilog p l o t of r abbreviations.  r . n  See t e x t f o r  77  Discussion i n r e l a t i o n t o e x i s t i n g t h e o r i e s . There have been many t h e o r e t i c a l c a l c u l a t i o n s on the chain ordering i n l i q u i d c r y s t a l s and b i l a y e r membranes (12-14, 29-32). successful  The most  o f these i s Marcelja's (13) molecular f i e l d c a l c u l a t i o n .  In the mean f i e l d approximation, the i n t e r a c t i o n energy of a s i n g l e chain i n the molecular f i e l d i s given by. E - E. int  fc  + E  + disp  PA  where E. i s the i n t e r n a l energy of a s i n g l e chain and depends on int the p a r t i c u l a r conformation, E ^  i s due to Vander Waal's i n t e r a c t i o n s  of the chain w i t h i t s neighbors v i a the molecular f i e l d .  The l a s t  term i s due to the l a t e r a l pressure on each chain and stems from the s t e r i c repulsions among the hard coresf'of the atoms.  I t i s proportional  to the pressure P and the average cross s e c t i o n a l area A of the chain. S t a t i s t i c a l averages are then evaluated by summing over a l l of a s i n g l e chain i n the molecular f i e l d o f i t s neighbors.  conformations By a d j u s t i n g  P and assuming an i n i t i a l o r i e n t a t i o n f o r the f i r s t C-C bond t h i s model i s able to p r e d i c t the order parameter p r o f i l e o f some NMR (13, 33)data. of However the r e s u l t s do depend on the assumed o r i e n t a t i o n ^ t h e i n i t i a l chain segment, which i m p l i e s that a d e s c r i p t i o n o f the ordering o f the hydrocarbon chain i n b i l a y e r membranes e n t i r e l y i n terms of chain chain i n t e r a c t i o n s i s incomplete.  Instead of f i x i n g the o r i e n t a t i o n o f the  f i r s t C-C bond i n an ad hoc way a more complete theory should, as could be seen from the thermodynamic a n a l y s i s , i n c l u d e the l i p i d water interaction explicitly.  78  6.2 S p i n - L a t t i c e Relaxation In the previous s e c t i o n i t was found that the l i p i d - w a t e r i n t e r a c t i o n played an important r o l e i n c o n t r o l l i n g the ordering and f l u i d i t y of the  hydrocarbon chains w i t h i n the b i l a y e r .  In t h i s s e c t i o n , the same  thermodynamic approach w i l l be used to see t o what extent the l i p i d water i n t e r a c t i o n a f f e c t s the dynamics of the l i p i d chains. Using the same formalism o f the previous s e c t i o n , the r e l a x a t i o n rate R  n  of the -CH2 group i n the n  t n  p o s i t i o n of the hydrocarbon chain  i s assumed to be a function o f s e v e r a l thermodynamic v a r i a b l e s  Qj  Since a l l the r e l a x a t i o n rates i n t h i s work were found t o have an a c t i v a t i o n energy, i t would bei more convenient t o consider instead of R  n  (since the a c t i v a t i o n energies a*  l o g RQ  p r o p o r t i o n a l to  (31og Rn/31/T) and are independent of temperature) . A d i f f e r e n t i a l change i n l o g R  n  i s then given by  [9]  I f the same assumptions of the previous s e c t i o n are made i . e . the  only independent thermodynamic parameters that have t o be cons-  idered are the temperature and water concentration then  9  reduces t o [10]  where 3 =  -  From (10) we immediately obtain  3  79  A l l the p a r t i a l d e r i v a t i v e s on the r i g h t o f equation (11) are obtained from the NMR r e s u l t s except f o r (3C/33) which i s r e a d i l y obtained A  from the equation o f state and i s given by  \ 38/C/  \ 3B/A  =  T  \dC /3  C I dAo P AodT  2  S u b s t i t u t i n g 12 i n t o  [12]  11 gives  31°g M f ^ i M , 2_C I ^ / 3 logJR^X 33 /A \ 33 /C P AodT \ 3C / 3 =  T  {^\°^ ^  In 13 we i d e n t i f y  a  n  d  k  {^^~~^)c  b y  a c t i v a t i o n  energies (E ). and .(E ) and. 13 becomes. r  ( E  an A }  where E  a n  -  <%m>C  * P  A  Q  dT  \  3C  /g  i s the a c t i v a t i o n energy f o r the r e l a x a t i o n process  deuterons on the n  t n  L * 1  J  fo'r the  p o s i t i o n o f the hydrocarbon chain.  Analysis of the r e s u l t s . In order t o t e s t f o r the s e n s i t i v i t y o f the dynamical s t a t e o f the hydrocarbon chain on the d i f f e r e n t thermodynamic parameters we examine the f o l l o w i n g r a t i o ( ( E  y  an A (Ean)c }  =  2  C . 1 . ^ 0 f 3 l o g *n) * P A dT V 3 C / (^an)c 0  The quantity  ft  r  1 L  5 l J  ^3 l o g ^n\ was obtained from the dependence o f \ 3 C /8 the r e l a x a t i o n rates on water concentration. (E„ )-, was obtained from n  80  semilog p l o t s of  versus - f o r each p o s i t i o n on the chain.  F i g . 27 shows ( E hydrocarbon chain.  a n  ) /(E A  a n  )  c  f o r the d i f f e r e n t p o s i t i o n s on the  This r a t i o i s greater than 1. f o r the f i r s t few  p o s i t i o n s but tends towards u n i t y i n the centre of the b i l a y e r .  ,  This  r e s u l t i s somewhat s i m i l a r to those obtained f o r the order parameters. Near the p o l a r head i t i s the l i p i d - w a t e r i n t e r a c t i o n that c o n t r o l s the dynamics of that part of the chain but i n the centre o f the b i l a y e r i t i s chain chain i n t e r a c t i o n that i s the c o n t r o l l i n g f a c t o r which i n turn i s dependent on the a v a i l a b l e area per p o l a r head.  81  2  (3,4)  (5,6)  n (CARBON Figure 27.  8  10  12  NUMBER)  Ratio of the a c t i v a t i o n energies f o r the chain deuterons at constant surface area per p o l a r head to that a t constant water concentration.  82  Error Analysis In estimating the e r r o r associated with the various .... quantities  . • — [ — ^ V V 3T A u  .  * , I* i — v V9C  n  , ••• e t c ,  the standard  n  formula f o r c a l c u l a t i n g the e r r o r i n a f u n c t i o n f of s e v e r a l v a r i a b l e s Xj was used, 2  2  6  X j  where e i s the e r r o r i n f and 6Xj i s the e r r o r i n Xj . I t should be noted that only the e r r o r s i n the quantites obtained  from the NMR  / 3v \ measurements such as \"g"Y"/ C  included i n  n  a  /3v \ \3~C~ / T n  **'  e t c  »  w e r e  computing the e r r o r i n the derived q u a n t i t i e s where the equation of s t a t e (eqn. 5  ) was used.  This was done because e r r o r s a r i s i n g  from the use of the equation of s t a t e ( i . e . e r r o r i n p, A , dAo ) as dT determined by X-rays are propagated only i n a systematic manner. This i s i l l u s t r a t e d i n f i g . 28  where as an example the value of P  _1_ dAo A dT 0  has been increased by 1 standard e r r o r i n c a l c u l t i n g the quantity 1 / H i \ —  ^ -jpjT J  . As can be seen from the comparison of figures 22  28 such e r r o r s only s h i f t the whole family of curves without changing the q u a l i t a t i v e v a r i a t i o n with n and T.  83  •  A  —  o  A  © 8 6 °C  •  A 105°C  A  o12 5°C A «  °  A135°C  -  A  4 A  —  1 ' '•"»  A  O  A •  A O A •  —  1  (3,4)  Figure 28.  i l l  i  1  1 1  •  I  6 8 10 12 n ( C A R B O N NUMBER )  E f f e c t of systematic e r r o r s due to the X-ray measurements on the f r a c t i o n a l v a r i a t i o n of the quadrupole s p l i t t i n g s w i t h temperature keeping the area per p o l a r head constant. The parameters P, A and dA /dT obtained from the X-ray measurements have been increased by 1 standard e r r o r . A comparison of the data shown above with those of f i g . 22 shows t h a t the e f f e c t of t h i s systematic e r r o r i s to s h i f t the whole family of curves without changing the q u a l i t a t i v e v a r i a t i o n w i t h n and T. b  e  84  Chapter 7 S p i n - L a t t i c e Relaxation between Water Protons and L i p i d Protons In the previous chapters i t was found out that water does play an important  r o l e i n the ordering of the hydrocarbon chains w i t h i n the  b i l a y e r v i a hydrogen bonding with the p o l a r heads.  The remaining  question to be answered i s : does water penetrate i n t o the b i l a y e r and i f so how deeply? In an attempt to answer t h i s question, the e f f e c t of i s o t o p i c m o d i f i c a t i o n o f the methylene hydrogen n u c l e i on the proton s p i n - l a t t i c e r e l a x a t i o n rate i n H2O and a l s o of the e f f e c t c" the isotope m o d i f i c a t i o n of the water on the methylene protons on the hydrocarbon chain was s t u d i e d . 7.1 A n a l y s i s and Discussion of the H2O r e s u l t s . F i g . 17 on page 51  shows the temperature dependence of ^ f o r  the protons i n H 0 i n d C ~Na/6H 0 , (3-w)d C -Na/6H 0 and 2  23  12  2  21  12  2  d^C^-Na^R^O , where the f i r s t sample contains no methylene protons, the second has -CH groups only i n the a - p o s i t i o n and the t h i r d has 2  protons a l l along the hydrocarbon chain.  The s u b s c r i p t on the d  i n d i c a t e s the number o f deuterons on the hydrocarbon chain.  I t can be  seen that f o r a l l the temperatures s t u d i e d , the r e l a x a t i o n rate ^ f o r the H 0 protons increase w i t h i n c r e a s i n g number o f protons. 2  This r e s u l t  i n d i c a t e s that the d i p o l a r i n t e r a c t i o n s between the -CrL, and H 0 protons 2  contribute appreciably to the r e l a x a t i o n rate o f the protons i n the R^O V  t This work was done i n c o l l a b o r a t i o n w i t h Dr. A.L. Mackay and professor M. Bloom.  85  In fact the increase i n the r e l a x a t i o n rate of the H2O A(i  )  protons  i n the d sample due to the -CH protons and the increase i n 2 the r e l a x a t i o n rate of the R~0 protons A ( — — V i n the doi sample r  l  2  _ C H  0  l  due to the a-CH protons were found to be i n the r a t i o 2  R - ^4^CH* " A (  f a-CH  W  6  )  1  2  As discussed i n the t h e o r e t i c a l s e c t i o n , the c o n t r i b u t i o n to the r e l a x a t i o n rate due to d i p o l a r i n t e r a c t i o n s between a p a i r of spins i s p r o p o r t i o n a l to r ^ f o r two spins at a f i x e d separation -  r .  I t would  appear, t h e r f o r e , that i n order to account f o r the large c o n t r i b u t i o n to \  1  of the HoO  protons due the l i p i d protons f a r from the l i p i d water  i n t e r f a c e , appreciable penetration of the b i l a y e r by H 0 2  have to be involved.  molecules would  However, a d e t a i l e d t h e o r e t i c a l a n a l y s i s (M. Bloom  p r i v a t e communication) shows that t h i s experimental r e s u l t can  be  accounted f o r without invoking deep penetration of the water i n the bilayer.  The mathematical d e t a i l s of t h i s a n a l y s i s are rather  and w i l l  not be presented here.  lengthy  Only the r e s u l t s and a d e s c r i p t i o n o f  the model used and the d i f f e r e n t assumptions made w i l l be  discussed.  The Model. F i g . 29  shows a schematic representation of the d i f f e r e n t  s p a t i a l regions of the water and l i p i d separated by a plane at the l i p i d water i n t e r f a c e . The r e l a x a t i o n rates are p r o p o r t i o n a l to the s p e c t r a l density  86  Region 1 (water region)  Region 2 ( l i p i d  Region 2 ( l i p i d  region)  region)  Region 1 (water region)  1 Figure 29 . Schematic representation of the d i f f e r e n t s p a t i a l regions i n which a spin of type l ( a water proton ) and a s p i n of type 2 (a l i p i d proton ) move. The p o s i t i o n of the two spins i s s p e c i f i e d by the space cordinates {z,p ,<f> , £ } i n a coordinate system having the z a x i s normal to the planes separating regions 1 and 2 . L j i s the thickness of the water layer,L2 i s the thickness of the l i p i d b i l a y e r , I i s the thickness of a proton l a y e r i n i n the l i p i d region , d i s the l a m e l l a r repeat spacing and d i s the distance of c l o s e s t approach between spins 1 and 2 . Note that z + £ , p, <J> are the r e l a t i v e p o s i t i o n s of the two spins i n c y l i n d r i c a l c o r d i n a t e s . The cordinate <J> i s not shown i n the diagram. g  87  functions o f the s p a t i a l part of the d i p o l a r i n t e r a c t i o n between the two spins d i f f u s i n g i n the d i f f e r e n t regions.  These s p e c t r a l d e n s i t y  functions are the Fouler transforms of the c o r r e l a t i o n f u n c t i o n s defined by G (T) m  =  /  Y (r ) Y (r(t)) \ 2 m  ^  0  2 m  r (o)  r (t) /  3  [2]  3  where the2m's are the s p h e r i c a l harmonics o f order 2. Y  The f o l l o w i n g assumptions are made i n t h i s model: 1)  A s p i n o f type 2 (a proton on the hydrocarbon  chain) has a uniform  p r o b a b i l i t y of being i n any of the regions of volume 2 and i s given by r  P ( p , <j>, O  pdpdfldg =<(' A£ '  =  2  , d<C<d+& ' LL-d-i<l -d-£<£<L -d 2  Q  2  2  otherwise  A s p i n o f type 1 (a water proton) w i l l have a uniform p r o b a b i l i t y o f of being i n any of the regions of volume 1 p ( )dz 1  z  dz <; L j  =  0<Z<L.  O  2)  otherwise  The c o n d i t i o n a l p r o b a b i l i t y that the two s p i n s are i n the p o s i t i o n  s p e c i f i e d by the space coordinates Z,p,<{>,£ a t time t , knowing that they were i n the p o s i t i o n s p e c i f i e d by Z ,P »<l>o ?o 0  given by.  a t  t i m e  z e r o  i s  0  •'' -  P(p,<j>,C ,z,t;p ,<J> ,C ,z ) = P(C,? ,t)P(p,4.,p ,4>o»t)P(z,z ,t) 0  0  0  e  0  where P(C,C ,t) 0  =  6(C-C ) 0  '  0  0  [3]  88  and i s equivalent to a s s u m i n g that type 2 spins do not move perpendicular to the b i l a y e r ,  i s the s o l u t i o n t o the two dimensional  PCP.^.PQ.^Q'O  d i f f u s i o n equation w i t h a d i f f u s i o n  D  H  =  D  l  +  D  2  coeffiencient  M  | (  given by the sum of the d i f f u s i o n c o e f f i c i e n t s o f spins 1 and 2 p a r a l l e l to the plane o f the b i l a y e r and P ( z , z , t ) i s the s o l u t i o n t o the d i f f u s i o n 0  equation c h a r a c t e r i s e d by a d i f f u s i o n c e f f i c i e n t Dj_ f o r the d i f f u s i v e motion perpendicular to the plane of the b i l a y e r w i t h r e f l e c t i o n s a t the Z=0 and Z=L^ boundaries.  Mathematical  procedures f o r  P(p,(j>,p ,<|> ,t) and P ( z , z , t ) can be found i n ( 4 4 ) . 0  0  Q  We f i r s t consider some l i m i t i n g cases which are never met i n p r a c t i c e but nevertheless w i l l give some i n s i g h t i n t o the nature o f the problem.  I n a l l o f the cases that w i l l be considered the short 2  2  c o r r e l a t i o n time l i m i t (oa T«l) w i l l be assumed. c ri  The experimental  u  r e s u l t t h a t r a t i o o f the r e l a x a t i o n r a t e s o f the 0C-CH2 a t 90 MHz and o f the 01-CD2 a t 13.8 MHz are i n the r a t i o o f the c o u p l i n g constants squared (see chapter 2 ) together with the fact that the r e l a x a t i o n  rates  decrease w i t h i n c r e a s i n g temperature support t h i s assumption. Case i : A l i p i d b i l a y e r o f i n f i n i t e extent i n contact w i t h an i n f i n i t e r e s e r v o i r of water. This corresponds to the case where L j » d . The spins of type 1 (water protons) are assumed t o have a d i f f u s i o n c o e f f i c i e n t greater than the d i f f u s i o n c o e f f i c i e n t o f the l i p i d . ( D j j > and that  much 10 D y 2  D j ^ . Under these assumption the s p e c t r a l d e n s i t y  )  89  functions at zero frequency (short c o r r e l a t i o n time l i m i t ) were found to be 3TTP a  2 m 8 L D  J (o) n  l  Lo g(—j-)  M  L -d-Jl•)! 2  1  where the a 's are n o r m a l i z a t i o n constants given by m  a* and  =  5/16TT  , a\ = 5/24TT , a* =  i s the number o f spins per u n i t volume i n region 2. To c a l c u l a t e the r a t i o R defined by eqn. 1 i t i s s u f f i c i e n t t o  compare the r a t i o o f two J ( o ) values f o r &=.8A° and £=9.2A° respectm  ively.  The two d i f f e r e n t values of Z correspond to the thickness o f  the l a y e r of protons i n the d 2 l and d  Q  samples and were estimated- from  the thickness of the b i l a y e r and the mean area per p o l a r head as determined by X-ray measurements (Appendix B). A value of 5.75 was obtained f o r the r a t i o R, which i s i n very c l o s e agreement w i t h the experimental value o f ^6 . However a rough c a l c u l a t i o n f o r the absolute values of ^ , by p u t t i n g 2A° f o r d ; 1.73xlO~" cm /see f o r 5  2 2  7.08.x 10  ^  2  and  1  /cm = •—- f o r the density o f protons i n the l i p i d containing J  a-CH2,gave r e s u l t s that were an order of magnitude too small (.014 sec*~l as compared w i t h the experimental value of .08 sec "'" f o r 01-CH2). -  The r e s u l t s f o r t h i s case were i n s e n s i t i v e t o the r a t i o of D,,/^ Case i i P a r a l l e l d i f f u s i o n o f water molecules i n a f i n i t e water l a y e r .  90  In t h i s case the water molecules are assumed to be d i f f u s i n g i n a f i n i t e region 0 z L^with a constant d i f f u s i o n c o e f f i c i e n t i n the p a r a l l e l d i r e c t i o n , i . e D,^(z)=D|j= constant and D^c 0 . The expressions f o r the s p e c t r a l density f u n c t i o n are then given by  where now the c o n t r i b u t i o n from the successive b i l a y e r s has been included.  From  6  a r a t i o R=497 was obtained.  Again the c a l c u l a t i o n s  f o r the a b s o l u t e values of ~ gave r e s u l t s that are an order of magnitude *1 shorter than the experimental values (.02 sec -'- as. compared w i t h the -  ('•>  experimental value of .08;sec  ).  Case i i i Slow d i f f u s i o n of a boundary water l a y e r . In view of the f a c t t h a t the model i n the previous two cases could p r e d i c t the c o r r e c t value f o r R but gave r e s u l t s that are an order o f magnitude too small f o r the absolute value of the r e l a x a t i o n rates suggests that other mechanisms that could be e f f e c t i v e i n r e l a x i n g water protons should be considered.  the  One such mechanism could be t h a t a  small f r a c t i o n f of the water binds to the l i p i d at the w a t e r - l i p i d i n t e r f a c e f o r a long time compared with the c o r r e l a t i o n times considered here (<~10~^see) and thus would d i f f u s e more slowly than the bulk water with a diffusion coefficient  .  By i n c o r p o r a t i n g t h i s i n t o the  theory the c o n t r i b u t i o n to the r e l a x a t i o n rates from the boundary l a y e r  91  can be shown t o be p r o p o r t i o n a l t o  m  ! otir(A)-. otTr( ±^-)4cotTr(y± )-cotTr( i!^+i) s S s . d  d  s°b  C  [9]  L  C  d  d  d  where a d e l t a function was used f o r the p r o b a b i l i t y of the bound water l a y e r .  distribution  In order f o r t h i s model to p r e d i c t the  c o r r e c t absolute value f o r the r e l a x a t i o n  rates and a l s o f o r the r a t i o  R , Djj has to be about t h i r t y times l e s s than the measured d i f f u s i o n c o e f f i c i e n t f o r the water (1.73  xlO  cnr/  s e c  ; Appendix C ) . A lower  l i m i t f o r Db i s the d i f f u s i o n c o e f f i c i e n t f o r the l i p i d molecules themselves.  The d i f f u s i o n c o e f f i c i e n t of the l i p i d chains i n the l i q u i d  c r y s t a l phase of potassium laurate ±s^2.^Xl6^ cm /sec(45)-As f a r as we know 2  the d i f f u s i o n c o e f f i c i e n t of the l i p i d molecules i n the sodium l a u r a t e wacer system has not y e t been measured.  Such a measurement, together w i t h  the measurement o f the d i f f u s i o n c o e f f i c i e n t o f H2O as w e l l as the relaxation  rate as a function o f i t s concentration should be u s e f u l i n  formulating a s u c c e s s f u l theory. From the preceeding analysis  i t seems that the question o f whether  water does penetrate i n t o the b i l a y e r o r not remains to be However, the r a t i o R=6 does not imply water p e n e t r a t i o n .  resolved. Further  t h e o r e t i c a l and experimental work i s required t o r e s o l v e t h i s problem. I t should be noted that information on the extent o f water penetration i n t o the b i l a y e r can also be obtained from neutron d i f f r a c t i o n  92  measurements.  Buldt et a l (37) have r e c e n t l y reported some measurement  on a phospholipid  water system where they conclude that water penetrates  i n t o the b i l a y e r up to the g l y c e r o l backbone o f the l i p i d molecules. E a r l i e r measurements by Schoenborn on a s i m i l a r system d i d not have s u f f i c i e n t r e s o l u t i o n t o detect such water penenetration. 7.2  CC-CH2 Results The  temperature dependence o f the r e l a x a t i o n r a t e f o r  d2j-laurate f o r two samples containing  -^CR^ i n the  R^O and D2O r e s p e c t i v e l y i s  shown i n f i g . 16 on page 50 . There i s a measurable d i f f e r e n c e i n the CA-CH2 r e l a x a t i o n rates f o r the two samples, i n d i c a t i n g the i n f l u e n c e o f the d i p o l a r i n t e r a c t i o n between the H2O protons and OC-CH2 protons. The  f a c t that the a-CH2 protons and the H 0 protons s t i l l have 2  d i s t i n c t T^ values when they i n t e r a c t w i t h each other excludes the p o s s i b i l i t y o f long c o r r e l a t i o n times f o r the d i p o l a r i n t e r a c t i o n between the -CH2 protons which would otherwise b r i n g the two s p i n systems t o a common temperature.  Short c o r r e l a t i o n times f o r the d i p o l a r  i n t e r a c t i o n s between H2O and CV-CH2 protons are expected because o f rapid d i f f u s i o n of water and a l s o of the hydrocarbon chains. The model used t o i n t e r p r e t the r e l a x a t i o n o f the H2O protons due  to the d i p o l a r i n t e r a c t i o n s w i t h the l i p i d protons can a l s o be used 1  to c a l c u l a t e the r e l a x a t i o n rate of the l i p i d protons due t o the same interaction.  Denoting  the number of H2O and -CH2 protons by  and  NB r e s p e c t i v e l y , the model assumes each o f the A spins to be equivalent and independent o f each other and the same f o r each o f the B s p i n s . i s then easy t o show that f o r small perturbations  It  t o the s p i n - l a t t i c e  93  r e l a x a t i o n rate of the A system (H 0) , A ( 1 / T ) i s related, to that o f 2  the  1  A  B system (CH ) by 2  / R  M  =  " l ^ ' ~ " ~  N  B %  since the gyromagnetic r a t i o s and spins o f the A and B systems are e x a c t l y the same.  For the d i sample used here B / N N  2  A  = 1/6 .  In a previous p u b l i c a t i o n ( 46 ) we have reported a value f o r R of  .. 6 ± . 3 which was the average value c a l c u l a t e d from the data o f  f i g u r e s 16 and 17 over the temperature range 80-115 °C . I t was i n i t i a l l y argued that t h i s large discrepency could not be a t t r i b u t a b l e to sample d i f f e r e n c e s or systematic e r r o r s i n the experiment.  This argument was  based on the f a c t that the measured deuterium quadrupole s p l i t t i n g s f o r the d  (3-u) p o s i t i o n s of the d„ C -Na/6H 0 , d C -Na/6D 0 and 21 12 2 21 12 2 C  -Na/6H 0 were found to be i d e n t i c a l , implying that the hydrocarbon  chains have the same f l e x i b i l i t y gradient and therefore i t would be u n l i k e l y that the molecular motions which cause s p i n - l a t t i c e r e l a x a t i o n i n each of the samples are appreciably d i f f e r e n t .  Since t h i s paper was  w r i t t e n we have made a c r i t i c a l estimate o f the r e l a x a t i o n data which has l e d us t o the conclusion that the d i f f e r e n c e between the experimental and predicted values of R may not be inconsistent w i t h each other w i t h i n the experimental e r r o r .  From f i g u r e s 16 and 17 the measured values of R*  are  shown i n t a b l e  1  f o r s e v e r a l temperatures.  I n the high temperat-  ure  region i t i s d i f f i c u l t t o make any conclusions about t h i s r a t i o due  Temperature  /  R  (*C ) 120  2.6 + 2.2  115  1.0 + .7  105  .7  + .3  90  .6  + .3  80  •k  ± .1  Table. (1) Ratio o f the change i n the s p i n l a t t i c e r e l a x a t i o n rate rate  of H2O protons due t o a-protons ti < the change of the r e l a x a t i o n ^ o f the a-protons due to the water pr< tons i n d2]C-Na/6H 0 . 2  95  to the l a r g e u n c e r t a i n t i e s involved.  At lower temperatures t h i s r a t i o  i s almost twice the p r e d i c t e d value of J_ w i t h i n the experimental 6 uncertainties. that  A[ -j-  )  These l a r g e u n c e r t a i n t i e s a r i s e mainly from the f a c t and  A( ^  of comparable magnitudes.  )  are obtained by s u b t r a c t i n g large number Moreover systematic  e r r o r s i n the T^ measure-  ments cannot be exluded. In view o f the l a r g e u n c e r t a i n t i e s i n It'we b e l i e v e that f u r t h e r experimental work i s needed.  Such work should i n v o l v e experiments  where the c o n t r i b u t i o n to the s p i n l a t t i c e r e l a x a t i o n rates due to the i n t r a molecular d i p o l a r i n t e r a c t i o n s are reduced.  For the 1^0  molecule  the i n t r a molecular d i p o l a r i n t e r a c t i o n between the two protons makes an important  c o n t r i b u t i o n to s p i n - l a t t i c e r e l a x a t i o n :  HDO  i s much longer than that f o r D 0  in D 0 2  2  .  The proton T j f o r  Therefore the c o n t r i b u t i o n  of the i n t r a m o l e c u l a r d i p o l a r i n t e r a c t i o n to the water r e l a x a t i o n can  be  e l i m i n a t e d by c a r r y i n g out experiments analogous to those described above using samples which contain a small amount of HDO study of R.' f o r higher values of  in D 0 2  .  should a l s o be c a r r i e d out,  A since  t h i s would give a l a r g e r e f f e c t which would be e a s i t r to measure. I t should be noted that the value of B/NA p r e d i c t e d by the model N  c a l c u l a t i o n s f o r R does not take i n t o account the c o r r e l a t e d motions of the two CH  2  protons.  General coupled equations f o r an AB  a v a i l a b l e , f o r example (47,48 ).  The parameters thereby  2  system are introduced  maybe examined by measuring q u a n t i t i e s such as the d i p o l a r r e l a x a t i o n o f the CH  2  s p i n system (49 ) or by the equivalent methods of " s e l e c t i v e  i n v e r s i o n recovery" ( 50 )•  Summary and Conclusions The flow chart shown below summarizes the areas o f i n v e s t i g a t i o n i n the t h e s i s and the conclusions reached.  PROBLEM: To understand the l i p i d - w a t e r i n t e r a c t i o n QUESTIONS How does the l i p i d - w a t e r i n t e r a c t i o n i n f l u e n c e the conformations and motions o f the l i p i d chains ? F l u i d i t y o f the b i l a y e r  How deep does water penetrate i n t o the b i l a y e r 7  i  -{ NMR  NMR r e l a x a t i o n measurements of H.O and l i p i d protons;effeet o f l s o t o p i c m o d i f i c a t i o n on the relaxation rates  2  H 0 with a-protonated chains 2  (b)  a, HO with all protonated chains (c)  Spectroscopy Quadrupole splittings — ^ I n f o r m a t i o n on the o r d e r i n g o f the l i p i d chains,water'(D20) and counter ions  Spin-lattice relaxation — > I n f o r m a t i o n on the dynamics o f the l i p i d chains  I-Water ( %c H 0 with perdeuterated chains  experiments}-  -protonatd chains  H 0 2  (d)  S p i n - l a t t i c e r e l a x a t i o n between water protons and l i p i d protons; Protons deep i n the b i l a y e r make a s u b s t a n t i a l c o n t r i b u t i o n to the s p i n l a t t i c e r e l a x a t i o n rates; T h e o r e t i c a l a n a l y s i s accounts f o r the r e s u l t s without Invoking deep penetration o f water In the b i l a y e r .  with D0  1  (e)  J n  The n n o d y n a m i c analysis  D0  Perdeuterated chains  l/T (C,T)  2  V <C,T) n  The l i p i d - w a t e r I n t e r a c t i o n has a s t r o n g Influence on the conformations and motions of the l i p i d chains  2 3  2  Na  Correlations.empirical evidence f o r a l i p i d - w a t e r Interaction M i c r o s c o p i c model f o r the l l p l d - w a t e r interaction  V  A d e s c r i p t i o n o f the s t a t e o f t h e b i l a y e r e n t i r e l y i n terms o f c h a i n - c h a i n Interactions ,as done i n e x i s t i n g t h e o r i e s , I s not complete. A complete theory should i n c l u d e the l i p i d water I n t e r a c t i o n e x p l i c i t l y .  APPENDIX A  Chemistry and Physics of Lipids 20 (1977) 115-129 ©Elsevier/North-Holland Scientific Publishers, Ltd.  THE TEMPERATURE DEPENDENCE OF WATER AND COUNTER ION ORDER IN SOAP-WATER MESOPHASES. A DEUTERIUM AND SODIUM NMR STUDY* K. ABDOLALL, E.E. BURNELL and M.I. VALIC 3  Department B.C.,  of Physics and Department  Canada,  of Chemistry",  University of British Columbia,  Vancouver,  V6T1W5  Received January 3, 1977  accepted March 23, 1977  Deuterium magnetic resonance ( D M R ) spectra o f the water and hydrocarbon chains in potassium and sodium palmitates and sodium magnetic resonance spectra of sodium in sodium palmitate demonstrate correlations between water, hydrocarbon chain and counter ion order. In the lamellar phase o f potassium palmitate the order parameters inferred from D M R splittings o f D , 0 and the first few methylene chain segments initially increase and then decrease with increasing temperature. This is explained i n terms o f a model where the lipid-water structure at l o w temperatures imposes a direction for which all the order parameters are smaller than for the higher temperature structure for purely geometric reasons. A s temperature increases the structuring effect o f water decreases and there is an "apparent" increase in order until at even higher temperatures there is an intrinsic decrease in order parameter. In addition, for potassium palmitate the D M R splittings of D , 0 and the first few methylene segments indicate a "phase transition" within the liquid crystalline phase.  I. Introduction In order to understand the role played by water near biological membranes, it is essential to understand the mechanism of lipid-water interaction. In the presence of water, lipid molecules form a variety of lyotropic mesophases characterized by the existence of long range order and short range disorder. These phases have been identified by X-ray studies [1 ], nuclear magnetic resonance [2,3] and other techniques [4]. Their description has received considerable attention in the literature [5] and will not be elaborated on here. Of particular interest are the three lamellar phases: liquid crystal (L ), gel (L^) and coagel. In these phases the soap molecules form bilayers which are stacked parellel to one another and separated from each other by water and counter ions. In the L phase the chains are flexible (melted) and the soap molecules undergo rapid lateral diffusion and rotations abcu» their long axis, while in the lower temperaa  a  * Research supported by the National Research Council of Canada and a special Killam-Canada Council Interdisciplinary Grant.  US  A'. Ahdolall el al.. NMR studies in potassium ami solium palmilales  lure 1.0 phase (he chains arc slil'f(frozen), fully cxlcndcd, lightly packed, and interdigilaled. In the cnagcl phase Ihc lamellar structure is relained but waler is squeezed out into pockets of hulk or "lice" waler. To obtain infoiniation about ilic ordering of cither the water at the lipid water interface or the lipid molecules within the bilayer, it is necessary to use microprobes that are sensitive to their local environment as well as to the dynamics of the system. Interactions between nuclear quadrupole moments and the electric field gradients arising from the surrounding charges affect the NMR spectra in a dear-cut way and enable such information to be obtained. From such studies there is considerable evidence that water near model and biological membranes possesses significantly more strucluic than bulk water |(>|. This is paper III of a series of three papers on the NMR of soap water mesophascs. Paper I | 7 | discusses hydrocarbon chain disorder as observed using DMR, and paper II | 8 | discusses the proton magnetic resonance (I'MK) from potassium palmitate (/}Lo)-dj». In this paper, results as a function of temperature of an NMR study of waler,sodium counter ion and hydrocarbon chains in potassium and sodium palmitates are reported. In particular, we demonstrate a striking correlation between the ordering of water and the first few methylenes of the hydrocarbon chain. An explanation for this correlation in terms of a simple model is given in section IV. We also report a new fluid fluid "phase transition" within the lamellar phase of a potassium palmitatc-water system. This ttansition manifests itself in the order parameters of the water and the methylenes near the polar end of the hydrocarbon chain.  K. Abdokll el al., NMR studies in potassium and sodium palmilales 117  The NMR measurements were carried out on a Bruker SXP 4-100 pulsed NMR spectrometer. The quadrupole splittings were obtained using the quadrupolar echo by the method of Davis et al. | 9 | .  111. Results ^ T h e interaction between nuclear quadrupole moments and electric field giadients efg at nuclei gives rise to splittings in deuterium and sodium magnetic resonance spectra. For water deuterons chemical exchange takes place between nuclei in different sites and. if the exchange is much faster than the splitting difference, the observed splitting is a weighted average over the different sites |10] . F o r an oriented sample, where the normal n to the bilayer makes a uniform angle f l with the magnetic field direction, the high field deuterium (or sodium) NMR spectrum consists of two (or three) peaks separated by Av=l£  PiK)iSj (3 cos a- 1) I  (I)  J  where p, is the fraction of nuclei occupying site i with associated quadrupole coupling constant VQ and order parameter Sj (discussed more fully in paper II). For an axially symmetric efg. S is given by t  Si = 3/2cos i J j - 1 / 2 ,  (2)  J  II. Experimental Palmilic(C| )acid was purchased from Hormel Institute or Calbiochem (California), and was used without further purification. Dcuterated compounds were prepared in our laboratory. The fatty acid salt was prepared by dissolving in ethanol stoichiometric amounts of dry fatty acid and sodium or potassium hydroxide, and slowly crystallizing the fatty acid salt. After filtration and washing with ethanol the precipitated soap was dried at I00°C under vacuum for several hours. The soap-water mixtures were made by weighing the corresponding amounts of dry soap and D 0 or HjO into a glass tube and scaling under dry nitrogen gas. For the sodium soap a quartz tube was used. The samples were mixed by ccntrifuging back and forth through a constriction in the tube. Before each cenlrifugation the samples were heated well above the gel-liquid crystalline phase transition temperature. Further homogenizalion was accomplished by leaving the sample in an oven for several days at a temperature above its gel-fluid phase transition. It was found that the variable temperature unit (gas flow system) suppled with the spectrometer was not adequate when dealing with soap-water mixtures. Therefore, a temperature controlled oven (paper 1) with greatly reduced temperature gradients in the sample was constructed and used for the measurements. 6  2  where i>; is the angle between n and the efg principal axis and the bar denotes a time average. Hq. (I) can be applied to (he case of hydrocarbon chain and water deuterons and counter ions. However, there are differences among these three cases. For example, all hydrocarbon chain C D deuterons are chemically equivalent and will therefore have the same I>Q, . Furthermore, since I>Q is of intramolecular origin ( C - D bond) it is not expected to vary with temperature. Consequently, the splittings Ai> yield order parameters directly. If the efg is axially symmetric about the C - D direction, ^ equals S Q J , the order parameter for the bond direction. The case of water and counter ions is more difficult because p I*Q and Sj may all differ from site to site, hence the separation of the various terms in eq. ( I ) is not possible without assumptions. For DjO the largest contribution to I>Q is from the intramolecular 0-1) bond, and its value is close to that for ice. Therefore, all f g . are similar and are not expected to vary significantly with temperature. The measured splittings are thus roughly proportional to an average order parameter S = ?P,S,, enabling a qualitative comparison of the experimental Ac with model calculations. Also, we do not expect p; to vary greatly with temperature in the narrow temperature region of interest, and hence the temperature dependence of the observed splittings reflects mainly changes in order parameter. 2  jr  v©  118  K. Ahdolall ttal., NMR studies In potassium and sodium pahnltates  K. Abdolall tt al., NMR studies In potassium and sodium pclmilalet  119  In contrast, theefg at a counter ion (e.g. sodium) nucleus arises from intermolccular sources (polar head group changes and water dipoles, i.e., via the asymmetric distribution of waters of hydration). Therefore, one expects to have a distribution of f y (as well as pj and S,) which can be quite temperature dependent, thus making the inlcrpretation of Av in terms of definite order parameters rather difficult. However, should the counter ion splittings correlate with those of hydrocarbon chain and water deuterons then llie temperature changes of counter ion Av should give an indication of changes in order parameters. It will be shown that this is the case for the sodium splittings of perdeuterated sodium palmitate dC Na. 16  A. Potassium palmitate Non-oriented samples yield essentially the same information as oriented samples. For an unoriented sample there is a random distribution of all values of and the deuterium magnetic resonance spectrum (see fig. I for representative half spectra) consists of a broad absorption curve with major peaks separated by Avj2 (paper I and Bloom el al. (111). The spectra of fig. I show"the presence of residual quadrupolar splittings indicating that the water molecules are located in an anisotropic environment in which they experience non-zero time averageikelectric field gradients and are said to be partially oriented. The anisotropy is imposed by the presence of the interface and the charge distribution near it, as well as by the hydrogen bonding of the water molecules and hydialed ions with the polar heads. For the liquid crystalline (fig. I a) and the gel (fig. lb) phases the D 0 deuterium spectrum is the usual "power doublet", the 2  TEMPERATURE ("CI  TEMPERATURE l"C)  Fig. 2a.The temperature dependence of the D , 0 and hydrocarbon chain deuteion quadrupole splittings for a perdeuterated, d C „ K sample. In this case the splittings were measured from the centre to the corresponding peaks in the absorption spectrum. It should be noted that the experiments reported in Fig. 2a were performed to study lipid-water correlation effects, and that the signal to noise ratio was not as good as that for the spectra reported in paper 1. Quantitative differences in the CDj order parameters between Fig. 2 and paper I should therefore not be taken seriously. Fig. 2b. Temperature dependence of the quadrupole splittings of D , 0 and a-CD, in the perdeuterated d C K sample. The insert is the temperature dependence of the D , 0 splittings in a protorutcd C,,K sample. l t  lower temperature spectrum of fig. lc contains in addition a central peak due to the isotropic "bulk" (free) water squeezed out from the water layers of the lamellar phase. This isotropic water is indicative of a coagel phase, and, in principle, the number of water molecules associated with different phases can be determined from NMR intensity measurements. It is likely that fig. Ic is a spectrum of a sample containing a mixture of gel and coagel phases. Alternatively, this spectrum could represent different phase separation zones within the gel region. More work, including X-ray studies, i i needed for a complete understanding of soap structures in the solid phase.  Fig. 1. Representative half spectra for D , 0 In C „ K In (•) lamellar phase, (b) gel phase and (c) mixture of ge! and coagel phases (see text).  The quadrupole splittings, Av, as a function of temperature for the water deuterons as measured from spectra similar to those in fig. I are shown in fig. 2 for perdeuterated potassium palmitate dC,„K. The splittings are appreciable in the low temperature coagel and gel phases and the sudden decrease in splitting denotes the coagel or gel to liquid crystalline phase transition. This decrease may be due to lesser intrinsic order of the  120  )  K. Abdolatl tt at., NMR studies In potassium and sodium palmitates 121  K. Abdotatl et at., NMR studies In potassium and sodium palmitate*  water at the lamellar surface, changes in the average direction of water ordering, and reduction of long-range order throughout the water layer. As shown in fig. 2 the splittings for DjO as a function of temperature in the liquid crystalline phase increase with temperature at first, exhibit a maximum between 70— I00°C. and then decrease. The deuterium splittings of the chain deuterons were also measured as a function of temperature. The results, shown in fig. 2a, indicate that the splittings of the first few methylene pairs exhibit a temperature dependence similar to that of D 0 . This suggests a strong correlation between the ordering of the water molecules and the first few links of the hydrocarbon chain. A model explaining this correlation will be discussed below.  fluid-fluid transition occurs at somewhat different temperatures in different samples; these differences may be due to sample preparation techniques and/or to isotope effects. Representative spectra for D 0 and chain deuterons of dC„,K in the region of this fluid-fluid phase transition are shown in figs. 3a and b. These spectra show the water line (fig. 3a) as well as the lines due to all chain deuterons (fig. 3b). Note thai the or-CD and the D 0 lines are each split near 50°C indicating the co-existence of two fluid phases. (See also the results of paper 1 where the a-CD splitting is more clearly shown, and paper II). 2  2  2  2  2  The results in fig. 2a at ~ 50-60°C indicate a discontinuity in the splittings which is ascribed to a liquid crystalline-liquid crystalline "phase transition". The discontinuity occurs only for the water and the first several C D ' s of the hydrocarbon chain and is thus suggestive of changes in the lipid-water structure and possibly a conformational change in the hydrocarbon chain near the lipid-water interface. Since this transition is a new phenomenon, the experiment was repeated and the temperature was -iried in smaller intervals. The results for the o-CDj and DjO are displayed in fig. 2b. The insert in the same figure shows the results for a protonated C , K sample. It is clear that the 2  6  B. Sodium palmitate Sodium has a nuclear spin of 3/2 and the high field NMR spectrum for an unoriented sample is similar to that for deuterium with two major peaks separated by Ac but includes a sharp orientation independent central line (if chemical shift anisotropy is neglected); fig. 4 is a plot of the quadrupole splittings of D 0 , " N a counter ion, and chain deuterons as a function of temperature for dC, Na. As seen in fig. 4 the temperature dependence of the Na, D 0 , and first several C D exhibit the same slope which demonstrates a definite correlation between the water, " N a counter ion and the first few hydrocarbon chain segments. All of these splittings decrease less rapidly with temperature than those from CD 's near the methyl end of the chain; this agrees with predictions of the Marcelja mean field calculations for chain order parameters [12] be2  6  2  2  2  10  i W  i •op  Fig. 3a. Spectra taken at three different temperatures showing the D , 0 splittings near the "fluidfluid" phase transition in the dC„K sample. The two smaller peaks are due to -CD, groups. The shoulders of the D , 0 signal are also visible. Fig. 3b. Representative half spectra at three different temperatures for the perdeuterated dC„K sample showing the water peak and the peaks due to the methylene pairs on the hydrocarbon chain. The right hand part of the spectra is expanded vertically 16-limes. The arrows Indicate the splittings from the a-CD,.  I100 110 120 OO  I I TFMPF.RATURE (*C )  Fig 4 Temperature dependence or the quadrupolar splittings for D , 0 . " N a counter ion and deuterons on the hydrocarbon chain in a dC,.Na sample. (The C,. results were obta.ned from a H , 0 * dC„ Na sample because the -CD, line overlaps with that of D , 0 making the separation or the two very difficult.).  122  cause increased temperature yields increased area per polar head and decreased lateral pressure. In contrast t o C K no fluid-fluid phase transition is detected,and no increase in splittings with temperature is observed in the lamellar phasefor the DjO, Na* or CDj's. Below the coagel-lamcllar phase transition temperature (~ 83°C), we were unable to observe quadrupolar split lines for Na' or D , 0 . contrary to the case for d C K . Only central Na* and D 0 lines were observed. Further work is needed on this phase.  123  l t  )6  3  IV. Discussion Water exists as various hydrogen bounded structures, and also interacts with the polar head groups of fatty acid salts via hydrogen bonding and other electrostatic interactions. These phenomena give rise to structuring effects: i.e. lipid and water structures mutually affect each other (13]. The details of such an interaction have not been included in many of the theories [12, 14] which attempt to explain hydrocarbon chain order parameters. The interaction between lipid and water is demonstrated in a particularly interesting way by the results of this and papers I and II which show that the NMR splittings of DjO, a-Cllj, a-CDj and the first few CD2's of the alkyl chain for the lamellar fluid phase of C | K all increase with temperature to a maximum value, and then slowly decrease (figs. 2, 3 of I and fig. 2 of II). In general, such an increase in splitting might be though of as an increase in order. However, this is not necessarily true; a strong interaction between lipid and water might well impose a constraint where all the observed order parameters (eq. (2)) are small for purely geometric reasons. A decrease in the strength of this interaction at higher temperatures could lead to larger observed values of S even though the system may be inherently less ordered. To demonstrate how such ideas lead to new information about lipid structure and lipid-water interaction, we present here a simple model which explains qualitatively the results of this and the preceding two papers. S  A. A model consistent with the experimental results of papers Our model proposes two rapidly (compared with NMR splittings) interchanging configurations, A and B, for the lipid-water interface [15]. The main idea here is that at lower temperatures (configuration A) the lipid-water interaction imposes ordering effects near the polar head region. In particular, the first C - C bond is more or less normal to the interface leaving the tail on the average somewhat tilted. At higher temperatures (configuration B) the lipid-water structuring effects are reduced and the dominant steric interaction imposes lipid ordering such that the chain is more or less parallel to n, the normal to the lamellar surface, leaving the first C - C bond at an angle to the surface. The two configurations are now described in more detail. 1. Configuration A As an example, suppose that at low temperatures a lipid-water structure similar to  Fig. 5. Model for the lipid-water Interfac* 1A\ B ^ • B) P - d o m t a . , , c o n n ' * ^ > a C T T ^ " " ' confo ' " " " '™' "'•on, with one p u c h * rotation are .hownTn 1,1 l" conform.See text To, further e x p i a t i o n . "" ° " . ' H H '» Perpendicular to n. 0  p u r a t l o n  a  ?'  l  o  w  '°»<™ «*  , h e  C H  m  u  124  K. Ahdotall et al. NMR studies in potassium and sodium palmltates  K. A bdohtt et al., NMR studies In potassium and sodium palmltates  that shown in fig. 5a (where two of the molecular conformations of the first five carbons are shown) is energetically favoured. For this configuration: (i)The first C - C bond vector r . is aligned at some angle to n, such that for the o>CII both the proton-proton vector r | and the carbon-deuteron vector r make an angle somewhat less than 00° with n. (For simplicity the diagram shows the first C - C bond parallel to nr. a tilt out of the plane of the paper must be introduced to explain the Q-CHJ results), (ii) The average environment of the water deuterons (and of the sodium counter ions) are such that their principal elg axes make angles with n giving average order parameters which are smaller than those for configuration B. The diagram is drawn with the water deuteron-deuteron vector r perpcndicrhr to n and the bisector of the angle DOD parallel to n.  Table I Calculated order parameters  2. Configuration B At higher temperatures the extra thermal energy allows fewer hydrogen bonds leading to a decrease in the lipid-water interaction. As an example we suppose (fig. 5b) that the long axis of the hydrocarbon tail for the all-trans conformation lies along n such that for the a-carhon both r p and are perpendicular to n. This configuration should lead to minimum free energy if hydrocarbon intermolecular interactions are dominant. For the purposes of illustrative calculation we allow isotropic rotation of DjO about the hydrogen bonded OD bond shown in fig. 5b. Two lipid molecular conformations are shown.  D,0  006  0.15  efg  - i  -0.18  c f  125  2  }  ( l (  c n  Configuration A  Configuration B  1 "3  1 *2  3 ' F  3V  2 l P  D D  (  t^i^t^,  d  "Calculations are averaged over conformations tafl'M**' W A T S * I'or A g^j and t 0 give equal contributions to order parameters.  d  * W&Uni  Q  b  Assumes that the C - C bond is at an angle 35V,° to n. Motions about C O O - C axis are not considered (i.e. p°f - 0). F o r gauche conformation S • Vi (P, cos (90) + P, 00s (35M)) • 0. 0  c  ''Assumes hydrogen bond parallel to first C - C bond and free rotation about 0. . . . D - 0 axis.  B. Simple calculations based on the model To illustrate the effects of different structures on the observed NMR splittings, we have listed in table I calculated values of the various order parameters (eq. (2)) for each of configurations A and B. The a-Crl^ has not been included in this simplified model; however, similar qualitative results will obtain for all order parameters (including cr-CH)) if the first C - C bond makes some appropriate angle with n. The values in table I are averaged over the various molecular conformations of the hydrocarbon chain. The following assumptions have been made for the calculations: (i) All CCC and.CCD angles are tetrahedral. For water, angle MOM is 105°. and the angle between lone pair electrons is 120° (16). For ease of calculation, conformer rotations of 120° are used rather than the more likely value of 112.5° | I 7 ] ; this does not significantly modify the calculated order parameters, (ii) There arc rapid (at least threefold symmetric) motions of lipids, water and counter ions about n such that the anisotropic interactions are motionally narrowed and the residual projected along n | I 0 , 18]. (iii) The average order parameter is given by eq. 2, with 0 being the angle between n and the vector of interest (t , T(- , r , q ) . All efg tensors are assumed to be axially symmetric along the bond directions. For sodium the efg used is Ihat^ from the oriented water lone pair electrons .with the principal axis q indicated as efg in fig. 5. (iv)pji (pji) is the probability that the bond between carbons i and j is trans (gauche) to the precedingCj-2-Cj-| bond, with pjJ + 2p'J = 1. IM  n  0 0  n  n  C. Discussion 1. . ie lamellar L phase As shown in Fig. 2 (and papers I and II) the experimental order parameters for DjO, or-CHj and the first several CDj's of the hydrocarbon chain in the liquid crystalline phase of CmK increase with temperature at first to a maximum and then gradually decrease. On the basis of ourmodel these results can be explained in the following way. We note that the calculated order parameters (table I) associated with configuration B are all of larger absolute value than those for A. According to the model the observed order parameters are the average over rapid exchange between both configurations, A being favoured at low, and B at higher temperatures. Hence, as temperature is raised we expect the observed order parameters to increase with (he increasing probability of structure B, until such a point that the probability of configuration A becomes small. In addition, with increasing temperature the thermal motions lead to an overall intrinsic decrease of the "molecular" order parameters, as observed for nematic liquid crystals (19). Thus, the model indeed predicts the experimentally observed temperature dependence of order parameters. Incidentally, for dC| Na (fig. 4) no increase in order parameters is observed, suggesting that configuration B already dominates at temperatures just above the coagcl-lamellar phase transition. a  6  126  K. Abdolcll etal.NMR ttudiet in potassium and sodium palmitates  K. Abdolatl etal., NMR studies in potassium and sodium palmitates 127  2. "Plateau" The experimental order parameters for - C D j ' s along the hydrocarbon chains in soap-water systems (paper I [7, 20)) exhibit a constant value for the first several —CDj's at low temperatures in the L phase. This observation, referred to as the "plateau" in the literature, is usually associated with the influence of "kinks" [21 ], but it can also be accounted for with the proposed model if the lipid-water interaction in the liquid crystalline phase affects to a progressively lesser extent those CD] groups further removed from the polar head region. Let us assume that there is minimum free energy for the intermolecular forces between hydrocarbon chains when the long axis of the hydrocarbon tail lies along n(a situation facilitating the rapid rotational motions about n in the lamellar phase (10, 18]).For configuration B where, as discussed earlier, the water has little or no ordering influence, the steric forces are dominant. However, for configuration A where the first C - C bond lies along (or close) to n. the long axis of the hydrocarbon tail for the all-trans conformation does not lie along n. Since the influence of the surface becomes progressively less as we go down the chain, the steric forces will tend to align parallel to n, that part of the chain which is further from the polar head. This could involve molecular conformations which are slightly distorted from normal trans and gauche. Thus, the orientation of the molecular axis gradually changes from that of configuration A to that like configuration B, and according to table 1 this corresponds to an increase in the observed SCD- On the other hand,because of the extra conformations available to those carbons far removed from the polar head, a general decrease in order parameters is expected as we go down the chain. Thus, the increase predicted above and the general decrease act oppositely and cancel each other in the region of the first few - C D links leaving a constant value for the SCD as observed experimentally.  (table 1) that each successive CDj will have lower order parameter. Hence, as temperature increases the experimental ratio Sg/S increases from the low temperature (configuration A) value of 1, where the 6-CD and e-CD peaks overlap completely, to a higher value where separate peaks are observed. This is consistent with configuration B becoming more probable at higher temperature. Further down the chain the ratio differs from 1 for all temperatures; this is consistent with the long molecular axis becoming aligned along n as one leaves the polar region. E  2  0  2  4. The gel phase The experimental order parameters for the Q-deuterons and a-protons in the gel phase are of slightly lower value thart-those for other chain deuterons [papers 1,11). Furthermore, the experimental valuesare such that ISHH ' slightly smaller than ISrjD both being < M. However, in the lamellar phase ISHH I is equal to or slightly greater than ISCD The results for the gel phase can be explained by the model if a gauche rotation near the polar head (between 0- and 7-carbons, say) is invoked. Such a rotation might accomodate more readily the water and counter ion structure and leave the rest of the tail all-trans and roughly parallel to n, hence facilitating a rotation about a symmetry axis m (not necessarily 1 n). As a result, the first C - C bond will be inclined to n in such a way that both TQ) and r for the cr-position make an angle with n f somewhat less than 90" in agreement with the experimental values ISHH I ~ ISCD V4K  i ( H  Assuming that in the gel phase the soap molecules are rigid and consist of two rapidly interconverting g | j to g^g conformations* and that the only other motion is rapid rotation about m then the results of paper I and II allow the calculation of the angles 0, that various vectors r make with m. The values SQ-CHJ - -34 (fig. 5, paper II) gives = 109° for the a-CH ;the value S = -036 (paper I) in the conjunction with SO>CH gives Syy = - 0.40 where y is thehisector of angle HCH. This value of " gives Oy = 105°. Using the values for angles CCC = 112°, CCH = HCH = 109° [17) and the result that the value for Sct>, for most of the chain is -0.44 (paper I) leads to a rotation of either 21° or 76° about the Cg-C7 bond for formation of the gauche conformation. The latter is close to the "most probable" value of 112.5° for the dihedral angle, and distortions from this value are not unexpected [I7J. The above calculations are crude, and a more plausible picture might be a rotation about C^-C-y of somewhat more that 90° followed by rotations of a degree or so in the opposite sense about bonds further down the chain, hence yielding a distribution of order parameters for different CD 's as indicated by the results of paper I. =  2  2  u  a<:D}  2  3. "Odd-even"effect Measured values (figs 2 and 4 of this paper and fig. 3 of paper I) of the order parameters Sen at lower temperatures in the L;, phase show that Sjj = Sy > S * S , etc. This applies all along the chain except for those -CDj's near the - C D end where each successive -CD] has lower order parameter. This low temperature behaviour of SCD' is referred to in the literature as the "odd-even" effect (7, 20]. With increasing temperature, the "odd-even" effect is observed to disappear progressively in the tail-head direction until at sufficiently high temperatures each successive - C D has lower order parameter (fig. 3 of paper I shows that all except (J and y peaks have separated at ~90°C). This effect is readily accounted for by our model. Indeed, i f the high energy conformationsg*g* are neglected [17], thenconfiguration A predicts equal values for Sfl and Sj. However, ISjjl and ISy I are also predicted to be smaller than IS I due to the term 2/3 p^7 (table 1). Similar results would obtain for S j and S (if the confor mation gy5'{ gf j is neglected), and for each pair of carbons down the hydrocarbon chain to a progressively lesser extent (depending on the probabilities of certain conformations for which the order parameters for both members of the pair are different), thus giving rise to the "odd-even" effect. On the other hand configuration B predicts a  y y  e  3  S  2  2  HT  Q  e  The fact that for the cr-position the order parameter SHH is less than SCD in 'he gel phase, while in the lamellar phase S | is larger than or equal to Sr-ry, depends on ho-v close to n the first C - C bond lies and on the particular motions involved in the averaging. In general, a hydrocarbon chain, even in the all-trans conformation, has no three-fold or  :  e  •Cfl-Cy is the closest C-C bond to the o-carbon around which a gauche rotation will leave the molecular axis roughly parallel to m. We neglect other possible fluctuatiunJ or the molecular axis about m in this rigid molecule calculation.  128  K. A bdohtll tt al, NMR studies in potassium and sodium palmltatts  K. Abdotatl tt al., NMR audits in potassium and sodium palmltatts  greater axis of symmetry, and hence there is no reason to expect that SHU SCD (See paper II for further discussion on this point).  D e  equal to  V . Conclusions This paper has demonstrated that the magnetic resonance spectra of deuterium in water and hydrocarbon chains of potassium and sodium palmitates and of sodium in sodium palmitate indicate correlations between water, hydrocarbon chain and counter ion order. For the lamellar phase of potassium palmitate the order parameters of D O and the first few methylene chain segments are shown to increase and then decrease with increasing temperature. This has been explained in terms of a model where the lipid-water structure at low temperatures imposes a direction on the first C—C bond for which all the order parameters are small for purely geometric reasons. At higher temperatures the structuring effect of water is decreased and there is an increase in the splittings. Details of the lipid-water structure at lower temperatures may involve considerations of a-CH - C O O conformation and complicated hydrogen bonded structures such as two lipid molecules being bridged by one water molecule, and hydrogen bond- • ingof more than one water per lipid oxygen. Also, some water superstructure (possibly tetrahcdral) is likely to persist throughout the water layer. a  2  The above qualitative arguments should be checked with a more complete theory, using for example an intermolecular force mean field to account for fluctuations of the molecular axis about n for each conformation. In such calculations the lipid-water interaction should not be neglected and attention should be given to the first C - C bond orientation which might well depend on both conformation and water structure. A fluid-fluid phase transition is detected within the lamellar phase of C , K and involves the water and first several segments of the hydrocarbon chain. This transition may well indicate a change in the water superstructure, or a conformational change fas trans to gauche) near the polar head. The presence of such a gauche conformation in the gel phase is suggested by a rigid molecule calcul:.-ion. Furthermore, these studies demonstrate that NMR is a very sensitive method of detecting different phases and phase transitions in lipid-water systems. 6  Acknowledgements We thank Dr. T.P. Higgs for synthesizing the deuterated fatty acids and for assisting in sample preparation. We are grateful to the authors of papers I and II for careful criticism of our work. We are indebted to Myer Bloom for many fruitful discussions.  References |1|  V. Luzzati, in: Biological Membranes, ed. by D. Chapman. Academic Pre*. N.Y. (1948)  (2)  129  A. Johansson and B. Lindman, in: Lipuid Crystals and Plastic Crystals, Vol. 2, ed. by G.W. Cray and P.A. Winsor, Kliss llorwood, Chichester (1974); G.J.T. Tiddy, in: Specialist Periodical Reports on NMR, Vol. 16 (1977) [ 31 B. Mely and J. Charvolin, Chem. Phys. Lipids (in press) | 4 | B.D. Landbionke and D. Chapman,Chem. Phys. Lipids 3 (1969) 304 | 5 | P.G. de Genncs, in: The Physics of Liquid Crystals, Clarendon Press. Oxford (1974) |6] II.A. Rcsing, A.N. Garroway and K.R. roster, in: Magnetic Resonance in Colloid and Interface Science, ed. by II.A. Rising and C.G. Wade, ACS Symposium, 34 (1976) 516 ( 7 | J . l l . Davis and K.R. Jeffrey, Chem. Phys. Lipids 20 (1977) 87 [8] A . L . MacKay and T.P. Higgs, ChJm. Phys. Lipids 20 (1977) 105 | 9 | J . l l . Davis, K.R. Jeffrey, M. Bloom, M.I. Valic and T.P. Higgs, Chem. Phys. Letters 42 (1976) 390 [10} G. Lindblom. N.O. Persson and B. Arvidson. in: Lyotropic Liquid Crystals and the Structure of Biomembranes, Chapter 9, ed. by S. Friberg, Adv. Chem. Set. 1976; H. Wennerstrdm, G . Lindblom and B. Lindman, Chem. Scripta 6 (1974) 97 ( I I I M. Bloom, E E . Burncll. S.B.W. Roeder and M.I. Valic, J. Chem. Phys. (in press) 112) S. MarCelja, Biochim. Biophys. Acta 367 (1974) 165 [13] F.Y. Fujiwaraand L.W. Reeves, J. Am. Chem. Soc. 98 (1976) 6790 | 1 4 | J.F. Nagle, J. Chem. Phys. 58 (1973) 252; P. Bothorel, J. Belle and B. Lamaire. Chem. Phyt. Lipids 12 (1974 ) 96; H. Schindler and J. Seelig, Biochemistry 14 (1975) 2283: M L . Scott. J. Chem. Phys. 62 (1975) I 347; R E . Jacobs, B. Hudson and II.C. Anderson, Proc. Nad. Acad. Sci. U.S.A., 72 (1975) 3993; J.A. McCammon and J.M. Deutch, J. Am. Chem. Soc. 97 (1975) 6675: M B . Jackson, Biochemistry 15 (1976) 2555 (15) A two site model has been used previously to explain the NMR of acetylene [P. Diehl. SSykora.W. Niedcrbcrger and E.E. Bumell, J. Mag. Res. 14 (1974) 260] and methyl fluoride | E . E . Bumell, J R . Council andS.E. Ulrich, Chem. Phys. Letters, 31 (1975) 3951 dissolved in nematic solvents | 1 6 | D. F.isenberg and W. Kauzmann, in: The Structure and Properties of Water, Oxford University Press (1969) [17] P.J. Flory.in: Statistical Mechanics of Chain Molecules. Interscicnce, N.Y. (1969) [18] M. Bloom, in: Proc. First Specialized "Colloque Ampere ' ed. by J.W. Hennel. Krakow, Poland (1973), p. 80 M. Bloom, E.E. Bumell, M.l. Valic and G. Weeks, Chem. Phys. Lipids 14 (1975) 107 (19| W.Maierand A. Saupe.Z. Naturforsch. A13 (1958)564; A M , (1959) 882; A15 (1960)287 [20] B. Mely, J . Charvolin and P. Keller, Chem. Phys. Lipids 15 (1975) 161; J . Charvolin, P. Manneville and B. Deloche. Chem. Phys. Letters 23 (1973) 345 [ 211 A. Seelig and J . Seelig. Biochemistry 13 (1974) 4839 1  104  APPENDIX B Determination of The Equation of State f o r The sodium Laurate-Water system Using low Angle X-ray S c a t t e r i n g ^ In order to determine the equation of s t a t e r e l a t i n g the d i f f e r e n t thermodynamic parameters (mean area/polar head, temperature and water concentration) f o r the l a m e l l a r l i q u i d c r y s t a l ( L ) of the sodium a  laurate-water system the l a m e l l a r repeat distance f o r t h i s system was measured using low angle X-ray s c a t t e r i n g as a f u n c t i o n o f temperature and water concentration.  Following the procedure of G a l l o t and Skoulious  (51) the area per p o l a r head was derived from the measured l a m e l l a r repeat spacings. Experimental Results and Discussion Due to the random o r i e n t a t i o n s of the l i q u i d c r y s t a l domains of b i l a y e r of the sodium laurate-water system i n the l a m e l l a r l i q u i d c r y s t a l phase i n the samples used the X-ray d i f f r a c t i o n patterns are c o n c e n t r i c rings whose diameters are r e l a t e d to the l a m e l l a r repeat spacing by Bragg's law  where  mA  =  2 dsin0  m =  0, 1, 2,  [l]  ....  d i s the l a m e l l a r repeat d i s t a n c e , X I s the wavelength o f the X-ray r a d i a t i o n used (1.54A ) and 0 i s the s c a t t e r i n g angle. measured diameter of the m  tn  I f d^ i s the  r i n g , f i s distance between the sample  "''The X-ray measurements were c a r r i e d out i n c o l l a b o r a t i o n w i t h L. Wood and K. J e f f r e y a t the U n i v e r s i t y of Guelph Ontario.  105  and the f i l m the 0 i s given by tan 2 0  =  ^  2  '  i  [2l  f  1  J  Due to instirumental l i m i t a t i o n s only the rings corresponding to 1/2 , 1/3 , ... observed.  of the fundamental spacing ( i . e f o r m-2,3,...) were  Using equations 1 and 2 the fundamental spacing was  obtained from the rings corresponding to m=2 and 3 . Table 2 shows the temperature dependence f o r the l a m e l l a r repeat spacing i n the temperature range (85^145*0 ) f o r a sample having 6 moles of water/1 mole o f sodium l a u r a t e .  I t remains roughly the same between  86 and 125 C and then shows a decrease at higher temperatures. Table 3 shows the concentration dependence f o r d at two d i f f e r e n t temperatures.  There i s a s l i g h t increase i n d w i t h i n c r e a s i n g water  concentration. C a l c u l a t i o n of the b i l a y e r thicVjiess da and the area per p o l a r head A In the model used by G a l l o t and Skoulious the s i m p l i f y i n g assumption that water does not penetrate i n t o the b i l a y e r i s made .'. implicitly. If v c  a  and c  e  a  and v = l-c  d which gives  a  e  are the s p e c i f i c volumes of soap and water r e s p e c t i v e l y , are the f r a c t i o n a l concentrations (per u n i t mass) then  c v + (1-c )v -a a a e  -1 w  106  v  has only been measured f o r the potassium Myristate-water system  a  (51).  For the other soaps v was derived  by assuming the a d d i t i v i t y  a  of p a r t i a l molar volumes i n the f o l l o w i n g way. I f a soap molecule C X o f the i o n type X w i t h n carbons on the n  hydrocarbon chain then i t s molar volume V ( X ) i s given by. n  V (X)  =  n  Where VcR$  V  +  C H 3  i s  t n e  (n-2) V  +  C H 2  [5]  p a r t i a l molar volume o f the methyl group,  Is the p a r t i a l molar voulme o f one..methylene group and V ^ Q ^ i s the volume o f the carboxyl group and the X i o n . In terms o f the molar volume f o r 0^-K eqn. 5 V X )  =  V (K)  Where V -V X  +  U  K  (n-14)V  C R 2  can be rewritten., as  +(V -V ) X  [>]  K  i s the d i f f e r e n c e i n p a r t i a l molar volumes o f i o n X  and the potassium i o n . The s e p e c i f i c volume v r e l a t i o n V (X) =M v n  a  a  where M  a  i s then given by the ,\  i s the molecular weight o f the soap.  a  G a l l o t and Skpulious (51) have  v e r i f i e d that eqn. 6 y i e l d s r e s u l t s that  a r e . i n good agreementwith experimental values f o r v . a  For the Cj^-Na eqn. V (Na) 12  „  The value o f VcH2  6 becomes  V ( ) 1 A  a t  - 2V  K  C H z  + (V -V^) Na  [l\  a given temperature was a r r i v e d a t by comparing  the s p e c i f i c volumes o f the normal alkanes as a f u n c t i o n of the number o f the carbon atoms (52). V - V N a  K  was put equal t o the d i f f e r e n c e o f the  molar volumes o f NaCl and KC1 i n water (53).  107  Using the c a l c u l a t e d v  a  values as o u t l i n e d above and the s p e c i f i c  volume o f water i n e q u i l i b i u m w i t h i t s own vapour pressure f o r v  e  , d  a  was c a l c u l a t e d using 4 . The r e s u l t s are shown i n t a b l e 2 To c a l c u l a t e the mean surface are per p o l a r head the f o l l o w i n g equation was used. A  where N i s Avogadro's number.  Fig.30  i s a l o g - l o g p l o t o f the  dependence o f A on water concentration f o r three temperatures.  The  r e s u l t s at 86°C are those o f G a l l o t and Skoulious (51). L i k e a l l the other sodium soaps studied i n (51) the f o l l o w i n g e m p i r i c a l r e l a t i o n O was found t o hold.  Where A ( T ) i s a Q  function of temperature only ,T i s the temperature.  C i s the water concentration i n moles o f water/1 mole of soap and P i s a constant = .24. Fig.31  i s a plot of A  Q  versus T.  In f i g . 3 2 the temperature dependence o f A i s shown f o r 0=6 i t increases l i n e a r l y w i t h temperature i n the range 86-120 °C and then increases a t a f a s t e r rate a t higher temperatures w i t h a break i n the slope at-125 °C. We b e l i e v e that t h i s i s probably an a r t i f a c t due to s l i g h t perturbations i n the experimental arrangement.  Furthermore there were no  anomalies i n the deuterium quadrupole s p l i t t i n g s of the perdeuterated sample of the same composition. The f a c t that d  a  i s almost independent o f temperature and depends  very weakly on concentration can imply two things : ( i ) water p e n e t r a t i o n  108  i n between the chains o r ( i i ) shrinkage of the chains r e s u l t i n g from bending and t w i s t i n g motionsi  I n the model used f o r c a l c u l a t i o n s the  f i r s t p o s s i b i l i t y has been excluded.  I t must be kept i n mind, however,  that the p o s s i b i l i t y of some penetration of the water i n t o the b i l a y e r s cannot be excluded on the b a s i s o f our experiments.  This w i l l i n no  doubt introduce some u n c e r t a i n t y i n the experimental r e s u l t s .  109  o  :rature (C)  d(A)  d (A)  A (A  86 91 96 105 112 117 122 127 132 137 142.5  30.6 30.8 30.4 30.6 30.4 30.5 30.5 29.7 29.1 28.4 29.4  20.4 20.2 20.3 20.4 20.3 20.4 20.4 19.8 19.4 18.9 18.9  36.2 36.6 36.6 36.9 37.3 37.3 37.5 38.8 39.8 41.0 41.2  a  )  Table 2 . Temperature dependence o f the l a m e l l a r repeat distance d, the thickness of the b i l a y e r d and the mean area per p o l a r head A f o r the sodium laurate-water system. The water concentration i s 6 moles of water/1 mole o f sodium l a u r a t e . a  °c  d(A)  d (A)  A(A  (§105 °C  3 4 5 6 7  29.6 29.9 30.3 28.8 28.1  23.7 22.4 21.4 19.2 17.7  31.8 33.5 35.2 36.9 39.4  @135°C  2 3 4 5 6 7  30.0 29.1 29.1 30.3 27.1 28.2  25.8 23.3 21.8 21.4 18.1 17.8  30.0 33.3 35.5 36.2 40.0 44.0  a  Table 3 . Dependence on the water concentration C(moles o f water/ 1 mole of sodium l a u r a t e ) o f the l a m e l l a r repeat spacing d, the b i l a y e r thickness d and the mean are per p o l a r head A a t 105 °C and 135 °C . a  110  0.3 LOG  0.4  0.5  0.6  0.7  0.8  0.9  C (MOLES OF H 0/1-M0LE OF C ^ - N a ) 2  Figure 30 . A l o g - l o g p l o t of the mean area per polar head versus water concentration f o r the sodium laurate-water system a t 86 C (open c i r c l e s , obtained from r e f . 5 1 ) , 105 C ( s o l i d dots) and 135°C ( t r i a n g l e s ) .  Figure 31  The dependence of A  q  on temperature.  112  Figure 32.  The dependence of the mean area p o l a r head on temperature at a f i x e d concentration f o r the sodium laurate-water system.  113  Appendix C Water s e l f D i f f u s i o n and Spin-Spin r e l a x a t i o n In sodium l a u r a t e / H 0 . 2  The s e l f - d i f f u s i o n c o e f f i c i e n t and the s p i n - s p i n r e l a x a t i o n time T of H 0 were measured, i n the sodium l a u r a t e water system. 2  2  The s p i n echo  method w i t h an e x t e r n a l l y applied f i e l d gradient (38) was used. Theory For  an i s o t r o p i c l i q u i d sample the amplitude of a s p i n echo i n an  NMR experiment i s given by S(2T)  S(0) exp (-2T/T ), exp (- I Y V D T ) 3 '  =  [ l ]  3  2  where T i s the spacing between the 90° and 180° p u l s e s , G i s the a p p l i e d magnetic f i e l d gradient and D i s the d i f f u s i o n c o e f f i c i e n t .  The term  i n v o l v i n g D i n equation 1 takes, i n t o account the e x t r a damping to the transverse n u c l e a r magnetization due to the change i n the Larmor frequency as a r e s u l t o f t r a n s l a t i o n a l d i f f u s i o n o f the molecules across an inhomogenous a p p l i e d magnetic f i e l d .  I f G and T  2  are known D can be  obtained from the slope of a semilog p l o t o f l o g (S(2T) + 2T/T ) versus T 2  For water i n an a n i s o t r o p i c enviroment eqn. [ l j has to be modified. In the sodium laurate water system the d i f f u s i o n o f water between the b i l a y e r s i s mainly p a r a l l e l to the b i l a y e r s .  For an o r i e n t e d sample i t  can be shown (54) that the angular dependence i n the l a b frame o f the d i f f u s i o n tensor i s given by D  ll  -  < 1> D  + (< ..> D  -<Dj>)sin 0 2  DO  114  where ^D ^>  and  x  are  the components of the d i f f u s i o n tensor :  i n the frame of the b i l a y e r and 9 i s the angle between the normal ri to gradient the  b i l a y e r and the magnetic f i e l d d i r e c t i o n .  Since the water molecules  are  confined to narrow regions between the b i l a y e r , the d i f f u s i v e motion  perpendicular to the lamellae does not transport the water molecules very f a r and therefore i t w i l l be assumed that <^D^—  0 .  Equation 2  then becomes %  -  where D =  <n  >  D  s  i  n  2  =  Q  d  (  i  _  u  2  c3  )  J  <^D^and y = cos 0 . and the echo amplitude w i l l be given by:  S(2x)  =  S  exp (-2T/T ). exp (- | Y G D ( l - y ) ) 2  Q  2  [4]  2  2  In randomly o r i e n t e d samples a l l values o f y=cos0 are e q u a l l y probable and an average over a l l a r i e n t a t i o n s has t o be considered g i v i n g S(2T) = S exp (-2T/T ) / d y exp (- | y G D ( l - y ) x ) 2  Q  =  S exp(-2x/T ). exp(0  2  1  Y G DT )X  3 where x ( y )  =  2  2  3  2  2  2  3  (\/| Y V D T )  [5]  J  1 y 2 ~ f dx. exp (x ). 2  2 2  3  / — Y G D. T ) i s a c o r r e c t i o n term f o r l a m e l l a r systems  to equation 1 , the expression f o r S(2T) f o r i s o t r o p i c systems. Experimental Results Using a Bruker B-KR 300Z18 f i e l d gradient u n i t , rtie echo Amplitude was measured f o r s e v e r a l T values at d i f f e r e n t values of G f o r the water i n the sodium l a u r a t e water sample. the  same measurements were made on pure water (R^O) •  To c a l i b r a t e G The sample s i z e s  and diameters of the tubes (.5cm) were chosen t o be i d e n t i c a l .  T  2  for  115  rLjO i n the sample was measured using the 90°-T-180 pulse sequency i n the absence o f a f i e l d gradient and was found to be 30. msec. To c a l c u l a t e D equation 5 was f i t t e d to the data by a 2 parameter non l i n e a r l e a s t squares f i t where the i n t e g r a l was evaluated n u m e r i c a l l y . F i g . 33 i s a representative f i t f o r eqn. 5  .  The r a t i o of D to the s e l f  d i f f u s i o n c o e f f i c i e n t i n pure water D / D Q , i s shown i n table 4 f o r d i f f e r e n t G values.  2  G  2  2  | y G D _0  (Gauss/cm)  (msec  19.4  1.55.  27.3  3.08:  34.4 41.0  9  Q  _2 )xl0  2  2  3 Y G D  " .  Q  q —2 (msec )xlO -J  .30  .19  .59  .19  4.90  1.03  .21  6.93  1.50  .22  '  •  D/D  Table 4. Ratio D/D of the s e l f , d i f f u s i o n c o e f f i c i e n t of water i n the sodium l a u r a t e water system to that of pure water at 100°C f o r s e v e r a l G values. D/D was obtained from the r a t i o of 2y G D/3 c a l c u l a t e d by f i t t i n g eqn. 5 to the experimental data (column 3 ) , t o the slope o f l o g S(2x) + 2T/T ) versus x p l o t f o r pure water (column.-2). G was c a l c u l a t e d using the r e s u l t s i n column 1 and the known d i f f u s i o n c o e f f i c i e n t of pure water at 100°C (55 ). The water concentration f o r the sample i s 6 moles of water per 1 mole of sodium laurate. a  2  0  3  2  2  Figure 33 . Log(S/S )+2T/T2 versus T . The s o l i d l i n e represents the f i t of equation 5 to the experimental data (dots)for the water s e l f d i f f u s i o n c o e f f i c i e n t measurement i n the lamellar phase of the sodium l a u r a t e water system a t 100*0. 3  0  117  References (1)  V. L u z z a t i , i n : B i o l o g i c a l Membranes , ed. by D. Chapman, Academic Press, N.Y. (1968)  (2)  A. Johanson and B. Lindman, i n : L i q u i d c r y s t a l s and P l a s t i c C r y s t a l s , V o l . 2, ed. by G. W. Gray and P.A. Winsor, E l i s s Harwood, Chichester (1974) ; G.J.T. Tiddy, i n : S p e c i a l i s t P e r i o d i c a l Reports on NMR, V o l . 16 (1977)  (3)  B. Mely and J . C h a r v o l i n , Chem. Phys. L i p i d s ( i n press)  (4)  B.D. Ladbrooke and D. Chapman, Chem Phys. L i p i d s 3 (1969) 304  (5)  S.B.W. Roeder E.E. 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The P h y s i c a l Review s e r . 2,111 (1958) 1201 .  P. 120  1-3  CD  cf M CD  pj £ ct- fO CO  2 << o 4  d C -Na/2H 0 23  12  2  Vn (kHz ) 2 •yn 100 105 108 110 111 112 115 116 117 120 125 130 135 140 145 150  2 20.45 20.84 20.90 20.85 20.83 20.80 20.78 20.78 20.78 20.64 20.65 20.52 20.40 20.36 20.28 20.19  3,4  5  14.83 15.42 15.41 15.23 15.27 15.19 15.05 15.10 14.98 14.94 14.76 14.65 14.45 14.31 14.17 14.01  16.05 16.44 16.46 16.33 16...33 16.26 16.11 16.17 16.13 16.04 15.99 15.88 15.72 15.66 15.52 15.36 d  23 12 C  v  2 100 102 103 105 110 115 120 125 130 135 140 145 ' 150  19.63 19.54 19.57 19.59 19.45 19.36 19.25 19.17 19.06 18.97 18.85 18.79 18.67  3,4 14.89 14.86 14.86 14.78 14.67 14.53 14.47 14.38 14.23 14.16 14.01 13.93 13.82  5  7  6  8  *..:7  13.02' 13. 90 13.83 13.67 13.66 13. 61 13.42 13.45 13.31 13.26 13. 16 13.10 12.71 12.41 12.85 12.28 12.67 12.11 12.51 12.38 11.99  _ N a / 3 H  10.53 11.36 11.32 11.21 11.16 11.07 10.93 10.95 10.84 10.72 10.57 10.39 10.14 9.99 9.84 9.67  10  11  12  9.12 9.80 9.68 9.59 9.62 9.51 9.36 9.34  5.93 6.43 6.38 6.32 6.29 6.24 6.12 6.12 6.03 5.98 5.83 5.74 5.58 5.46 5.33 5.20  2.06 2.26 2.22 2.17 2.19 2.14 2.09 2.11 2.08 2.05 1.97 1.90 1.84 1.78 1.75 1.67  9.18 9.01 8.87 8.64 8.56 8.36 8.23  CO Hj  o  o  &  to  c+ H" CD 0  &  CD CD 4 0 c+ CD  ^ 2 CD t3 03  P3 pu 3 s; t+  P tret- CD CO l-J CO O O  O M-  O £ o B ps H  11.39 11.38 11.33 11.22 11.11 10.86 10.10 10.62 10.46 10.28 10.14 10.00 9.84  1  o o  CD t i i— <+ fo 4 0  2°  n (kHz ) 2 6 7  13.24 13.16 "".13.15"1 7" 13.10 12.93 12.81 12.63 12.50 12.32 12.16 12.04 11.91 11.77  9  CD I— ct- CD CO B CO M PL H CO c+ • d ctCD H 0 CS ci, rjq CD m ti  1  ct- 03 H- ctO CD  3  8 10,71 10.62 10760 10.51 10.34 10.13 9.97 9.84 9.68 9.51 9.35 9.23 9.08  9  10  11  8.70 8.62 8.59 8.46 8.36 8.15 8.04 7.89 7.74 7.51 7.40 7.25 7.13  7.35 7.30 7.24 7.18 6.99 6.82 6.68 6.52 6.38 6.25 6.10 5.98 5.86  4.82 4.74 4.72 4.64 4.53 4.41 4.27 4.15 4.06 3.96 3.86 3.77 3.69  12 1.61 1.59 1.57 1.56 1.50 1.43 1.37 1.33 1.29 1.22 1.17 1.12 1. 11  d 5C -Na/6H 0 2  n C 70 75 80 85 90 95 100 105' 110 115 120 125 130 135 140 145 150  2 14.49 14.66 14.84 14.99 15.10 15.25 15.38 15.43 15.43 15.52 15.58 15.58 15.59 15.50 15.45 15.42 15.33  3,4 10.58 10.73 10.79 10.89 10.99 11.10 11.08 11C13 11.13 .11.17 11.18 11.18 11.17 11.11 11.01 11.00 10.94  5  12  6  2  ^SL (kHz ) 2 7 8  9  10  11  12  7.91 7.80 7.69 7.52 7.42 7.24 7.18 7.08 6.93 6.82 6.65 6.57 6.45 6.35 6.18 6.09 5.96  6.59 6.43 6.25 6.10 5.96 5.80 5.71 5.37 5.42 5.31 5.15 5.04 4.96 4.87 4.71 4.60 4.49  5.66 5.51 5.37 5.18 5.03 4.87 4.74 4.54 4.49 4.38 4.27 4.16 4.05 3.99 3.83 3.75 3.66  3.91 3.78 3.64 3.52 3.42 3.30 3.22 3.13 3.03 2.92 2.86 2.75 2.70 2.60 2.50 2.44 2.39  1.34 1.29 1.23 1.18 1.15 1.10 1.06 1.03 .98 .95 .92 .87 .85 .78 .74 .73 .70  7  8  9  10  11  12  7.39 7.32 7.30 7.20 7.03 6.95 6.84 6.77 6.70 6.53 6.35 6.19 6.05  7.02 6.95 6.90 6.79 6.52 6.41 6.32 6.19 6.09 5.87 5.66 5.48 5.30  5.90 5.83 5.77 5.62 5.33 5.21 5.10 4.97 4.85 4.63 4.42 4.21 4.04  5.10 5.03 4.97 4.81 4.53 4.41 4.30 4.16 4.06 3.83 3.64 3.43 3.25  3.58 3.52 3.48 3.34 3.13 3.03 2.93 2.82 2.75 2.59 2.43 2.26 2.11  1.20 1.17 1.16 1.11 1.03 .99 .95 .90 .88 .81 .74 .67 .61  8.42 8.29 8.20 8.15 8.06 7.93 7.86 7.71 7.67 7.56 7.45 7.40 7.29 7.20 7.06 6.95 6.84  9 .55 9 .53 9 .54 9 .52 9 .47 9 .39 9.67 9.33 9.67 9.18 9.18 9.57 9.51 9.07 9.55 9.05 9.42 8.94 9.46 8.86 9.38 8.81 9.25 8.67 8.61 9.24 9.13 8.50  d C -Na/7H 0 23  12  2  (kHz ) n C 70 73 75 81 91 95 100 106 110 120 130 141 150  2 12.18 12.21 12.38 12.38 12.51 12.66 12.77 12.83 12.92 13.00 13.06 13.05 13.09  3,4 8.86 8.90 8.98 8.98 9.11 9.14 9.20 9.22 9.28 9.34 9.38 9.36 9.33  5  6  2  8.20 8.20 8. 19 8. 15 8.09 . 8.04 8.01 8.21 7.92 8.15 7.89 8.09 7. 76 8.03 7.67 7.97 7.51 7.91 7.40  P. 17  d C -Na/4H 0 23  12  2  (kHz ) n °c  2  3,4  2 6  5  7  8  9  10  11  12  9.83 9.79 9.70 9.62 9.42 9.33 9.11 8.91 8.87 8.79 8.70 8.61 8.48  9.03 8.96 8.87 8.74 8.59 8,50 8.24 8.08 8.02 7.91 7.73 7.68 7.56  7.18 7.10 6.99 6.88 6.74 6.62 6.40 6.23 6.19 6.05 5.97 5.87 5.80  6.03 5.93 5.85 5.74 5.62 5.52 5.27 5.11 5.08 4.97 4.87 4.76 4.68  3.96 3.93 3.85 3.78 3.66 3.61 3.42 3.31 3.27 3.20 3.14 3.05 2.99  1.34 1.33 1.32 1.26 1.21 1. 12 1.10 1.05 1.03 .98 .94 .90 .88  7  8  9  10  11  12  8.98 8.85 8.72 8.61 8.53 8.44 8.30 8.07 7.73 7.89 7.76 7.69 7.59 7.54  8.20 8.04 7.89 7.76 7.67 7.48 7.37 7.17 6.87 6.96 6.86 6.74 6.66 6.57  5.48 5.33 5.20 5.09 4.97 4.87 4.74 4.51 4.21 4.35 4.22 4.14 4.06 3.98  3. 70 3.58 3.47 3.42 3.32 3.21 3.14 2.98 2.77 2.86 2.77 2.70 2.64 2.59  1.22 1.18 1.14 1.11 1.07 1.04 1.01 .95 .87 .89 .85 .84 .81 .78  >  90 92 95 100 105 110 120 126 130 135 140 145 150 ,  18.49 18.52 18.47 18.51 18.35 18.32 18.24 18.02 17.93 17.81 17.74 17.66 17.64  13.78 13.76 13.71 13.62 13.48 13.46 13.26 13.18 13.07 13.06 12.93 12.87 12.78  11. 73 11. 69 11. 58 11.45 11. 30 11. 16 10.91 11. 23 10. 79 11. 17 10. 73 10.64 11. 11 10.93 10. 49 10. 89 10.41 10. 80 10. 31  d  23 12 C  _ N a / 5 H  2°  ( kHz ) 2  3,4  5  6  2  C 80 85 90 95 100 105 110 120 128 130 135 140 145 150  17.32 17.38 17.33 17.37 17.27 17.30 17.26 17.09 16.86 16.99 16.89 16.76 16.70 16.67  12.72 12.66 12.61 12.64 12.62 12.54 12.50 12.33 12.10 12.16 12. 18 12.05 11.96 11.87  10. 74 " 10.62 11.01 10.47 10.91 10.44 10.85 10.34 10.68 10.22 10.11 10.64 10.47 9.89 10.14 9.56 10.27 9.70 10.18 9.59 10.08 9.59 10.03 9.40 9.94 9.35  6.51 6.34 6.18 6.09 5.97 5.82 5.71 5.49 5.19 5.30 5.19 5.09 5.03 4.93  S p i n - l a t t i c e r e l a x a t i o n times of chain deuterons i n the sodium l a u r a t e water system; dependence on temperature and •Water concentration.  oo <r r*» vo vo vO m  O  >-i  CM vO  o CM  w  CM  cd J25 1  CN  — •1 u co  m  ON  CM  IN  IN  vO  <* co  c  CO TJ  C o  a  t—i  oIN  vo in  IN  i-H CM CO  •  O  1 CM  u I—1  IN  00  ON  vo IN vo CM co  m -o-  m  IN  vo  ON ON  I  I  o  co  o  oo vo es  •  •  •  co <• m  <f  <r  vo  IN  IN IN  o <r  o  r-H CM c o c o <j"  o  r~H m  00  O CM i-H <r m i-H IN r-H fN vO  |N  CM  CM  i-H CM CM CO CO vT  00 o m  i—i vO CM m ON «* i-H i-H CM CM CM CO  IN  rN o r-H m r—1 i-H  IN ON  m  O  o  i-H m i-H i-H CM CM CO  —^ d r-H H  oo  |  00  TJ  O  CM  C o  co oo  rN  CM CM vO  IN  •-H  O  m O fN ID m co <r  CM  i-H i-H CM CM CO CO  m O  CM O O -J" CO CM  r-H CM CM CM CO  EC cu  OJ CO  CO CM <r m in CM o •—i m ON i-H •—i CM CM CM CM  vO  I—1  in  -CO CM TJ  rN  CO CO in o i-H O O i—1 VO i-H r-H CM CM CM CO  CO  |N  CO  '—.  N-/  S I  H  eg CM  ON  r-H CM CO IN  co  cd S3  1  IN  00  O  CO  *—s  H  o CM a  "N.  cu cn  •  m  •  OO  o m O CM o CM vO ON i-H CM CO  CM  •  ffl O Csl 0> H <f •-H CM CM CM CO -sT  co  CO TJ  co  •  ci vo rs r>  N  <r  vo  •  m m IN f"» co o oo CM co ON m co i-H CM co co <f m  m in i^-  CM  vo  o "-H c o -vi-  r-« m m r~- O oo oo co oi H oo CM CM o- m fN IN  co in o IN m — •< <t in N  CO  vo  CM  CO  Table 6.  o  r-H  c  o m m m CM in o m o  r-H VO  vo •-H CM CM CM  m o  CO CM  TJ  CM  ON IN ON  m  tO  m m o in IN o  r-H rH r-H CM CM CO  <r CO  mom  00  vo H CM CO CO  C / /  /  1  m m m U  H  r»  -H -<  CO  o m m m  o vo m co IN in co vo o m oo m  CO vO CM —' CM CM vo *-H r-H CM CM CM CO  C_)  O  00  O  O  m m m m H  N  C  O  CJ> i-H i-H i-H i-H  r-H i-H CM CM CM CO  / CJ  o  m m m m o o o -H CM CO 00 ON rH i—I r-H r-H  P.-  n t * .  d^C -Na/5H 0 12  2  (seconds) n o C 80 90 105 115 125 135  2 . 0822 .116 . 173 .220 .260 .342  3,4 C9096 .135 . 183 .224 .273 .309  5,6  7  8  9  10  11  12  .097 .137 .177 .228 .266 .310  .0977 . 138 .200 .242 .285 .345  .117 . 155 .220 .267 .312 .380  .148 .184 .325 .320 .370 .437  .187 .243 .327 .375 .445 .510  .290 .335 .370 .537 .614 .735  .750 .815 1.01 1.24 1.19 1.42  9  10  11  12  .090 . 123 .172 .230 .315 .405  . 107 .150 .210 .275 .370 .470  .158 .200 .277 .385 .493 .595  .68 .76 .89 1.05 1.22 1.36  9  10  11  12  .076 .112 .147 .187 .250 .280  .100 .140 .210 .245 .317 .368  .163 .235 .329 .353 .458 .555  .600 . 760 .946 1.02 1.15 1.31  d -T -•Na/6H 0 23^12 T. (seconds) in 8 5,6 7 2  >n o C 80 90 105 115 125 135  2  3,4  .057 .066 .097 .118 .168 .204  .067 .077 .110 .140 . 160 .212  .060 .080 . 117 .150 .173 .257 d  . n  2  3,4  C 80 90 105 115 125 135  .043 .052 .076 .098 .113 .144  .048 .062 .085 . 112 . 138 . 156  .067 .088 .126 .167 .212 .295  .070 .095 .137 .178 .230 .310  •Na/7H 0 23 12T, (seconds) In 5,6 7 8 C  .048 .064 .097 . 120 .147 .187  2  .056 .073 .107 .135 .163 .223  .057 .083 .118 .145 . 182 .210  

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