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A theoretical study of dilute aqueous electrolyte solutions Kusalik, Peter Gerard 1984

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A THEORETICAL STUDY OF DILUTE AQUEOUS ELECTROLYTE SOLUTIONS  By PETER GERARD KUSALIK B . S c , The U n i v e r s i t y of L e t h b r i d g e , 1981  A THESIS SUBMITTED  IN PARTIAL FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in  THE FACULTY OF GRADUATE STUDIES Department o f  (Chemistry)  We accept t h i s t h e s i s as  conforming  to the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA  April,  1984  ® P e t e r Gerard K u s a l i k , 1984  In p r e s e n t i n g requirements  this thesis f o r an  B r i t i s h Columbia,  it  freely available  Library  shall  for reference  and  study.  I  understood that for  f o r extensive copying of  h i s or  be  her  g r a n t e d by  f i n a n c i a l gain  shall  not  be  CAemiS"^  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3  Date  >E-6  (3/81)  of  y  Columbia  make  further this  thesis  head o f  this  my  It is thesis  a l l o w e d w i t h o u t my  permission.  Department o f  the  representatives.  copying or p u b l i c a t i o n  the  University  the  s c h o l a r l y p u r p o s e s may by  the  I agree that  agree that permission department o r  f u l f i l m e n t of  advanced degree a t  of  for  in partial  written  - i i -  ABSTRACT  4  In the p a s t , s t u d i e s of e l e c t r o l y t e s o l u t i o n s have g e n e r a l l y t r e a t e d the  s o l v e n t o n l y as a d i e l e c t r i c continuum.  i n the  theory  of Debye and  Huckel and  T h i s was  is still  widely  s i g n i f i c a n t improvement to t h i s approach would be as a t r u e m o l e c u l a r electrolyte  species.  a p o l a r i z a b l e hard-sphere f l u i d  The  hypernetted-chain  theory  infinite dilution limit.  The  f u n c t i o n s of i o n s i z e and  charge.  p r o p e r t i e s of s o l u t i o n are Both dynamical and  compared w i t h experimental In the present  The  with  i o n charge.  The  theory,  f o r small  of s o l u t i o n are low  ion-solvent c o r r e l a t i o n found to s c a l e e x a c t l y  ions s c a l e to a f a i r  For i o n s i n the w a t e r - l i k e  strong  approximation  s o l v e n t the p o t e n t i a l s o f  mean f o r c e are observed to be l e s s s t r u c t u r e d and l i m i t more r a p i d l y than f o r i o n s i n a simple  as  equilibrium  measurements at the  studied  i o n - i o n p o t e n t i a l s of mean f o r c e demonstrate  dependence on i o n s i z e and  with  i s a p p l i e d to these s o l u t i o n s i n the  f u n c t i o n s f o r t h i s model e l e c t r o l y t e s o l u t i o n are w i t h charge.  solvent  water-like  s o l v e n t at 25°C.  c o n t r i b u t i o n s to the apparent d i e l e c t r i c c o n s t a n t  concentrations.  A  model s o l u t i o n s are systems of hard  s p h e r i c a l i o n s immersed i n t h i s w a t e r - l i k e  examined and  to i n c l u d e the  t e t r a h e d r a l quadrupoles w i t h  parameters as i t s s o l v e n t .  linearized  used today.  T h i s t h e o r e t i c a l i n v e s t i g a t i o n of aqueous  solutions considers  embedded p o i n t d i p o l e s and  the approach taken  approach the  dipolar solvent.  e q u i l i b r i u m c o n t r i b u t i o n to the d i e l e c t r i c decrement f o r a l k a l i  continuum The metal  - iii  and  h a l i d e i o n s i s found  ion s i z e .  is  to be n e g a t i v e but not s t r o n g l y dependent upon  The v a l u e s f o r the k i n e t i c d i e l e c t r i c decrement are a l s o  n e g a t i v e and The  -  are i n f a i r agreement w i t h p r e v i o u s t h e o r e t i c a l  results.  t o t a l d i e l e c t r i c decrement i s dominated by the e q u i l i b r i u m  term  r e l a t i v e l y i n s e n s i t i v e to i o n s i z e f o r aqueous a l k a l i h a l i d e s .  limiting  s l o p e s f o r 1:1  and  2:1  e l e c t r o l y t e s at 25°C o b t a i n e d  e x p e r i m e n t a l data at low c o n c e n t r a t i o n s are found agreement w i t h those p r e d i c t e d by the present  from  to be i n f a i r  theory.  and The  - iv -  TABLE OF CONTENTS Page ABSTRACT  i i  TABLE OF CONTENTS  iv  LIST OF TABLES  vi  LIST OF FIGURES  v i i  ACKNOWLEDGEMENTS  CHAPTER I - INTRODUCTION  CHAPTER I I - MODEL POTENTIALS 1. 2. 3.  Introduction W a t e r - l i k e Solvent Model E l e c t r o l y t e S o l u t i o n Model  CHAPTER I I I - STATISTICAL MECHANICAL THEORY 1. 2. 3.  4.  1  . 6 6 9 23  28  Introduction D i s t r i b u t i o n Functions I n t e g r a l E q u a t i o n Methods  28 29 33  (a) (b)  33 35  The O r n s t e i n - Z e r n i k e E q u a t i o n C l o s u r e Approximations  A p p l i c a t i o n of I n t e g r a l E q u a t i o n Methods t o E l e c t r o l y t e Solutions (a) (b) (c) (d)  A p p l i c a t i o n of the O r n s t e i n - Z e r n i k e E q u a t i o n . . The LHNC C l o s u r e Approximation Average I n t e r a c t i o n Energy and P o t e n t i a l of Mean Force Method o f S o l u t i o n  CHAPTER IV - DIELECTRIC THEORY OF ELECTROLYTE SOLUTIONS 1. 2.  ix  38 39 46 51 54  61  Introduction E q u i l i b r i u m Theory o f the D i e l e c t r i c Constant  61 67  (a) (b)  69 75  The D i e l e c t r i c Constant of a S o l u t i o n The S o l u t e Dependent D i e l e c t r i c Decrement  - v -  Page 3.  Dynamical Theory of the D i e l e c t r i c Constant  80  (a) (b)  85 88  The Debye-Falkenhagen E f f e c t The K i n e t i c D i e l e c t r i c Decrement  CHAPTER V - RESULTS AND DISCUSSION 1. 2. 3. 4.  Introduction E n e r g i e s and C o o r d i n a t i o n Numbers C o r r e l a t i o n F u n c t i o n s and P o t e n t i a l s of Mean Force The D i e l e c t r i c Constant of S o l u t i o n  93 93 95 97 129  CHAPTER VI - CONCLUSIONS  156  LIST OF REFERENCES  162  APPENDIX A - R o t a t i o n a l I n v a r i a n t s  168  APPENDIX B - I t e r a t i v e Procedures  170  APPENDIX C - Numerical R e s u l t s  176  - vi -  LIST OF TABLES Tables I II  III  IV  Page Diameters, d^, f o r a l k a l i metal and h a l i d e i o n s . . . . Reduced parameters and d i e l e c t r i c constants f o r the w a t e r - l i k e and d i p o l a r s o l v e n t s at 25°C w i t h d = 2.8A s E n e r g i e s of i n t e r a c t i o n and c o o r d i n a t i o n numbers f o r a l k a l i metal and h a l i d e ions M o l a r i o n conductances at  V VI  [4] at i n f i n i t e  25°C  Numerical r e s u l t s f o r u n i v a l e n t  25  94  95  dilution 147  ions  Numerical r e s u l t s f o r d i v a l e n t i o n s  177 178  - vii-  LIST OF FIGURES Figure 1  2  3  4  5  Page The o r i e n t a t i o n a l c o o r d i n a t e 6 , < j i , \ | ; are the E u l e r angles  system where 10  M o l e c u l a r a x i s system f o r the w a t e r - l i k e model  solvent 13  A charge d i s t r i b u t i o n p o s s e s s i n g (a) a l i n e a r quadrupole and (b) a t e t r a h e d r a l quadrupole..  15  The d i e l e c t r i c constant o f the w a t e r - l i k e as a f u n c t i o n o f temperature  22  Comparing water-like  n  ^ ^ ( r ) for L i  and I  +  solvent  ions i n the  -  s o l v e n t model  98  f\0 0  6  Comparing water-like  7  8  9  10  11  12  13  14  h  i  g  (r) for L i  +  and I  -  ions i n the  s o l v e n t model  A comparison o f Sw^Cr) size  100 f o r ions o f d i f f e r e n t 104  A comparison o f 3w^j(r) f o r d i f f e r e n t p a i r s o f o p p o s i t e l y charged ions  106  Comparing 3 w i j ( r ) f o r p a i r s o f o p p o s i t e l y charged i o n s f o r which the d_^_. v a l u e s are e q u a l . . . .  110  Comparing gw-^j ( r ) / J z ^ z j pairs of o p p o s i t e l y charged u n i v a l e n t and d i v a l e n t i o n s  112  A comparison o f 3w^j(r) f o r p a i r s o f o p p o s i t e l y charged ions i n d i f f e r e n t s o l v e n t s  115  A comparison o f Sw^Cr) i n d i f f e r e n t solvents  117  f  o  r  2  f o r F"/F~ and I ~ / I ~  The f o r c e a c t i n g between two I water-like solvent  -  The f o r c e a c t i n g between two L i water-like solvent  i o n s i n the 123 +  i o n s i n the 125  - viii  -  Figure 15  16  17  18  Page The f o r c e a c t i n g between two d i p o l a r solvent  Li  +  ions i n  the 127  The f o r c e a c t i n g between a L i / F ~ p a i r i n the w a t e r - l i k e s o l v e n t  130  The f o r c e a c t i n g between a C s / I ~ the w a t e r - l i k e s o l v e n t  pair in 132  The the  pair i n  +  +  f o r c e a c t i n g between a C s / I ~ dipolar solvent +  134  * 19  The  terms c o n t r i b u t i n g to  e  Q)  i  f°  r  t n e  water-like  solvent 70  The  as a f u n c t i o n of i o n diameter ^ dependence of ( ^ ) ^ i ° diameter f o r  univalent 21  22  23  24  e  o  n  n  i o n s i n the w a t e r - l i k e  solvent  141  The dependence on i o n diameter of the k i n e t i c d i e l e c t r i c decrement f o r the w a t e r - l i k e s o l v e n t . . . .  145  A comparison of the e q u i l i b r i u m and dynamical c o n t r i b u t i o n s to the t o t a l d i e l e c t r i c decrement....  148  Comparing e x p e r i m e n t a l and decrements f o r aqueous 1:1  150  theoretical dielectric electrolyte solutions...  Comparing e x p e r i m e n t a l and t h e o r e t i c a l d i e l e c t r i c decrements f o r aqueous C u C l s o l u t i o n s  152  A diagrammatic r e p r e s e n t a t i o n of the i t e r a t i v e method used to solve the i n t e g r a l e q u a t i o n s  172  2  25  138  ACKNOWLEDGEMENTS  I would l i k e  to thank my academic a d v i s o r , Dr. G. Patey, f o r h i s  guidance and support,  and the Chemistry department of the U n i v e r s i t y o f  B r i t i s h Columbia and the N a t u r a l  Sciences  and E n g i n e e r i n g  C o u n c i l of Canada (NSERC) f o r t h e i r f i n a n c i a l a s s i s t a n c e . like  Research I would a l s o  to express my s i n c e r e g r a t i t u d e t o my f a m i l y and e s p e c i a l l y to my  f i a n c e e , S h e i l a Davis,  f o r her t i r e l e s s help and moral  support.  -  1 -  CHAPTER I  INTRODUCTION  The  study of e l e c t r o l y t e  a r e a s of p h y s i c a l t h i s century.  solutions  has  been one  c h e m i s t r y , p a r t i c u l a r l y i n the  This interest  has  been due,  of the most  first  i n part,  charged s p e c i e s p l a y i n many chemical r e a c t i o n s and especially  i n aqueous s o l u t i o n s .  electrolyte  solutions  What do we this  I t was  However, he  was  two  an  electrolyte  term " e l e c t r o l y t e  solution."  yield  many times each  mean by  study the  In r e a l l i f e we  Faraday  few  decades  of  to the  central  role  processes,  are  solution?  For  the  coined the  term  electrolyte.  seeking to d e s c r i b e substances, such as water, which  interpretations.  In p h y s i c a l  Today the  c h e m i s t r y an  in  e l e c t r o c h e m i s t or p h y s i c i s t  s p e c i f i c solvents.  d e f i n i t i o n and  to the  To  conducting s o l u t i o n  be  Physically  electrolyte  an  solution  by  a "large"  M a c r o s c o p i c a l l y the electrolyte  solution  We  use  in  the  term  the  former  solution.  i n a polar solvent, a l i q u i d  l i q u i d i s well  is  dissolved  i s a homogeneous l i q u i d c o n s i s t i n g  d i e l e c t r i c constant such as  resulting  has  which, f o r a l l p r a c t i c a l  viewed as being c o m p l e t e l y i o n i z e d  ionic s o l i d , a s a l t , dissolved  characterized  itself.  c o n s i d e r only those e l e c t r o l y t e s  purposes, can an  the  term  electrolyte  i o n i c a l l y c o n d u c t i n g medium when  refers  of  "ionic  a substance which produces an  usually  by  purposes  i s synonymous w i t h  t h e i r c o n s t i t u e n t elements upon e l e c t r o l y s i s .  different  confronted  day.  solution"  [1] who  active  understood  i s o f t e n d e f i n e d as a s o l u t i o n  water. [2],  having a  An "high"  of  - 2 -  conductance  [3],  Many other macroscopic  d e n s i t y , vapour p r e s s u r e , and apparent r e a d i l y measured and  that  s o l u t i o n s are s t i l l  s o l v e n t - s o l v e n t and  p o o r l y understood We  s p e c i e s i n s o l u t i o n but  with  know  the  i o n - s o l v e n t s t r u c t u r e present i n s o l u t i o n i s s t i l l  of g r e a t c o n t r o v e r s y [4,6-9],  understanding  c o n s t a n t can be  q u e s t i o n s and p u z z l e s remaining unanswered.  the s o l u t e e x i s t s as f r e e i o n i c  matter  dielectric  t h e i r dependence on s a l t c o n c e n t r a t i o n determined.  Microscopically, electrolyte many fundamental  p r o p e r t i e s [2,4,5] such as  A more complete  microscopic  i s needed i n order to e x p l a i n what can be observed  measured m a c r o s c o p i c a l l y .  a  and  I t i s to t h i s p o i n t t h a t t h i s t h e s i s i s  addressed. The  study of e l e c t r o l y t e  s o l u t i o n s began i n , and  the e a r l y days of what i s known as e l e c t r o c h e m i s t r y . i n t r i g u e d by the f a c t t h a t matter was conductors whereas t h i s was Clausius that  in fact  I n v e s t i g a t o r s were  transported l n e l e c t r o l y t e  not the case w i t h e l e c t r o n i c  [10] noted t h a t i o n i c  there must be e l e c t r i c a l l y charged  In 1887  van't Hoff  conductors.  s o l u t i o n s obeyed Ohm's law.  He  colligative  from  Planck  those of non-conducting  i n such a  solutions.  properties distinct  [12] i n t e r p r e t e d  r e s u l t s as a p o s s i b l e i n d i c a t i o n of i o n i z a t i o n of the s o l u t e .  credited solution.  [13], who  first  the  [11] p u b l i s h e d e x p e r i m e n t a l r e s u l t s which  showed t h a t c o n d u c t i n g s o l u t i o n s possess  Arrhenlus  concluded  p a r t i c l e s present to c a r r y  c u r r e n t , p o s s i b l y a s m a l l p o r t i o n of the e l e c t r o l y t e was form.  dominated,  p u b l i s h e d h i s theory i n 1887,  these  However,  i s usually  f o r the present theory of e l e c t r o l y t e d i s s o c i a t i o n i n Although  the i d e a of f r e e i o n s i n s o l u t i o n seemed r a d i c a l at  that time, acceptance of the i d e a d i d come s l o w l y with the support of a  - 3 -  w e a l t h of data.  By the t u r n o f t h i s c e n t u r y the b a s i c concept was more  or l e s s u n i v e r s a l l y accepted and the f i e l d was e n t e r i n g period.  In 1907 Lewis  [14] i n t r o d u c e d  activity  c o e f f i c i e n t s as a t o o l to handle the d e v i a t i o n from that  the f i r s t  i n t e r a c t i o n s made i t p o s s i b l e ionic solutions  [15].  and of Onsager  of i d e a l s o l u t i o n s .  to describe e l e c t r o l y t e  The long-range nature of the i o n - i o n to d e r i v e  exact l i m i t i n g laws f o r d i l u t e  These t h e o r e t i c a l r e s u l t s were found to be i n  v e r y good agreement w i t h experiment. [16]  i n behavior of  decades of t h i s century t o  develop e q u i l i b r i u m and l a t e r dynamical t h e o r i e s s o l u t i o n s and t h e i r b e h a v i o r .  growth  the concept of a c t i v i t y and  r e a l s o l u t i o n s at higher concentrations Many attempts were made d u r i n g  a rapid  The t h e o r i e s of Debye and Huckel  [17] stand out today as landmarks.  The work o f  Onsager l e d t o the development of a dynamical theory f o r e l e c t r o l y t e solutions  [8].  The theory o f Debye and Huckel (DH) f o r the e q u i l i b r i u m  structure of e l e c t r o l y t e solutions s t i l l in  p r o v i d e s a common approach used  the d e s c r i p t i o n and d i s c u s s i o n of i o n i c s o l u t i o n s  and  Huckel only  solvent  dielectric  behavior. Electrolyte this  considered  [2,4,18],  i n t e r i o n i c i n t e r a c t i o n s as i n f l u e n c e d  Many t h e o r i e s have b u i l t  on t h e i r g e n e r a l  s o l u t i o n s have been i n the past  d i e l e c t r i c continuum.  approach.  traditionally  studied  using  o n l y as a  The i o n - i o n i n t e r a c t i o n i n the PM i s j u s t the  coulombic r e l a t i o n , q ^ q j / e r .  The a b i l i t y o f the continuum  Q  to s c r e e n i o n i c charges  dielectric  by the  constant and examined t h e i r e f f e c t upon i o n i c  " p r i m i t i v e model" (PM) approach which t r e a t s the s o l v e n t  solvent  Debye  constant, e . Q  [2,15] i s e n t i r e l y r e p r e s e n t e d by i t s  The PM and v a r i a t i o n s of i t a r e s t i l l  commonly used today t o study  [19,20] i o n i c s o l u t i o n s because o f the  - 4 -  attractive simplicity obtained  using  of these models.  Recent attempts solvent  p o i n t out of the PM. the  i n the  the  The  the m o l e c u l a r nature of  treatment of e l e c t r o l y t e s o l u t i o n s have served w i t h i n i o n i c s o l u t i o n s and  presence of enormous l o c a l e l e c t r i c  i o n s might be  solvent.  that the  solvent  the  dielectric  Instead  constant  f i e l d s generated i n the  i s an i n d i r e c t  of t r y i n g to work w i t h i n  as a true m o l e c u l a r s p e c i e s .  s o l u t i o n s , i t i s important that constant The  bulk  probe of  the PM  such  framework, the  aqueous e l e c t r o l y t e  the model s o l v e n t have the  dielectric  of water. search  f o r b e t t e r models has  t h e o r e t i c a l studies  [21-23,25-30].  commonly used i n l i q u i d useful  In m o d e l l i n g  by  neighborhood  have chosen to study a " b e t t e r " model, a model which i n c l u d e s  solvent  to  inadequacies  i n a q u i t e d i f f e r e n t s t a t e from t h a t of the  solvent  solvation effects. we  [7,9,21-24] to i n t r o d u c e  complexities  The  has  [19,20].  f r e e ions i n s o l u t i o n implies  of the  improve the r e s u l t s  these models, improvement of the t h e o r i e s a p p l i e d  been c o n t i n u a l l y sought  the  To  been the aim The  s t a t e theory  been shown to be  C a l c u l a t i o n s of t h i s type  i n p r i n c i p l e y i e l d a l l e q u i l i b r i u m p r o p e r t i e s of the several studies hard-shere f l u i d  [21-23,35] i t has i s a very  [41]  poor model f o r water.  A fluid  should  From  model i s p a r t i c u l a r l y a t t r a c t i v e i n the  of p o l a r i z a b l e  t e t r a h e d r a l quadrupoles  to be a f a r s u p e r i o r model f o r water.  that the d i e l e c t r i c constants  system.  very  become c l e a r t h a t a simple d i p o l a r  hard spheres w i t h embedded p o i n t d i p o l e s and been found  recent  i n t e g r a l e q u a t i o n approach,  [31-33], has  [21-30, 35-41] i n such s t u d i e s .  of a number of  This  has  "water-like"  study of aqueous e l e c t r o l y t e s i n  c a l c u l a t e d f o r such a f l u i d  system show  - 5 -  good agreement w i t h e x p e r i m e n t a l temperature  and  r e s u l t s f o r water over a l a r g e  p r e s s u r e range [34,41],  A recent study  [23]  has  c o n s i d e r e d a s o l u t i o n of hard s p h e r i c a l i o n s immersed i n t h i s w a t e r - l i k e s o l v e n t i n the i n f i n i t e d i l u t i o n l i m i t . u n i v a l e n t i o n s w i t h the same diameters i o n - i o n p o t e n t i a l s of mean f o r c e and p r o p e r t i e s of  T h i s study examined o n l y as the s o l v e n t .  the e q u i l i b r i u m  I t focussed upon  dielectric  solution.  In the present study, p a r t of which has a l r e a d y been p u b l i s h e d [42], we  extend  the work i n i t i a t e d  of i o n s i z e and  charge  i n Ref. 23.  We  i n v e s t i g a t e the i n f l u e n c e  upon the s t r u c t u r a l and  p r o p e r t i e s of s o l u t i o n .  An examination  equilibrium  of dynamical  dielectric  dielectric  p r o p e r t i e s i s c a r r i e d out; d i f f e r e n t  s o l v e n t models and  t h e o r e t i c a l r e s u l t s are compared and  contrasted.  previous  A l s o , the  theoretical  r e s u l t s are compared w i t h e x p e r i m e n t a l data wherever p o s s i b l e . P r e d i c t i o n s and  comparisons w i l l  tend to be r a t h e r q u a l i t a t i v e i n  n a t u r e , w i t h the b a s i c purpose being to demonstrate the u s e f u l n e s s of the present s o l u t i o n model i n examining macroscopic  I I we  p r e s e n t the c u r r e n t s o l u t i o n model i n d e t a i l .  s t a t i s t i c a l mechanical  d i s c u s s e d i n Chapter t h e o r y of s o l u t i o n s . and  their  implications.  In Chapter classical  m i c r o s c o p i c p r o p e r t i e s and  theory a p p l i e d to our s o l u t i o n model i s  I I I , w h i l e i n Chapter The  The  IV we  examine the  dielectric  r e s u l t s o b t a i n e d are d i s c u s s e d i n Chapter  our c o n c l u s i o n s are g i v e n i n Chapter  VI.  V  - 6 -  CHAPTER I I MODEL POTENTIALS  1.  Introduction In the  study of r e a l  systems and  the p h y s i c a l p r o p e r t i e s  that  c h a r a c t e r i z e them, the development of " u s e f u l " models i s e s s e n t i a l . could  choose to model the  many ways; we case,  systems of i n t e r e s t i n t h i s  c o u l d d e s c r i b e our w a t e r - l i k e  by a d i e l e c t r i c  c o u l d attempt  continuum ( i . e . PM)  to s o l v e the f u l l  to be u s e f u l , i t must be meaningful r e s u l t s , but  simple  study i n a  s o l v e n t , i n the  or i n the most extreme case  quantum mechanical problem.  enough to enable us to produce  i t must a l s o have a s u f f i c i e n t degree o f  p r o p e r t i e s of primary i n t e r e s t ) .  There i s , however, a b a s i c  r e a l i s t i c a model might be and The  d e s i r e to work w i t h  yields realistic important In any  r e s u l t s has  step i n s c i e n t i f i c  system (or at l e a s t tradeoff  e a s i l y i t lends  itself  advancement. interactions within  the  Hence, the p o t e n t i a l  the model.  We  can  then d e f i n e  s t a t i s t i c a l mechanical system i n terms of some energy, Ufj, the  In d e v e l o p i n g terms of the  full  to  made the development of such models an  (or r e p r e s e n t s )  of a l l i n t e r a c t i o n s of the  the  s i m p l e s t model p o s s i b l e that  by a model p o t e n t i a l .  mathematically defines our  the  how  s t a t i s t i c a l mechanical study, the  system are d e s c r i b e d  we  For a model  a d e q u a t e l y our  solution.  great  simplest  s o p h i s t i c a t i o n so as to r e p r e s e n t  between how  We  sum  system.  a u s e f u l model, we  need to determine and  r e t a i n those  i n t e r a c t i o n p o t e n t i a l that make important  c o n t r i b u t i o n s to the p r o p e r t i e s of I n t e r e s t .  I t i s sometimes found  that  - 7 -  we can i g n o r e otherwise l a r g e f a c t o r s because on p r o p e r t i e s we wish to i n v e s t i g a t e . As a f i r s t  they have l i t t l e  influence  1  approximation to our p o t e n t i a l , we w i l l  l i m i t our  treatment o f the p o t e n t i a l to s t r i c t l y a c l a s s i c a l one. We w i l l  ignore  many quantum e f f e c t s due to v i b r a t i o n s , e t c . , which a r e assumed to be s m a l l , and a l l i n t r a m o l e c u l a r i n t e r a c t i o n s w i l l a l s o be i g n o r e d . m o l e c u l e s w i l l be t r e a t e d as r i g i d p a r t i c l e s  The  (Born-Oppenheimer  approximation) and average v a l u e s f o r such q u a n t i t i e s as the d i p o l e moment and p o l a r i z a b i l i t y w i l l be used.  Such approximations a r e  convenient f o r our l e v e l o f study. The  total  interaction potential w i l l  thus depend o n l y upon the  p o s i t i o n s and o r i e n t a t i o n s o f the molecules of the system. t o t a l energy  F o r the  of i n t e r a c t i o n o f the system we w r i t e  UJJ = U(X^ , 5^2» 2^3 »• • • >X^)  where  (2.1)  r e p r e s e n t s the c o o r d i n a t e s of molecule i and N i s the number  of molecules i n the system. particularly forliquid  In g e n e r a l , t h i s N-body i n t e r a c t i o n ,  systems,  problem would be v e r y d i f f i c u l t  i s v e r y c o m p l i c a t e d and the N-body to solve.  We a r e a b l e to make a second major s i m p l i f i c a t i o n by assuming t h a t we can express the t o t a l i n t e r a c t i o n energy  of e q u a t i o n ( 2 . 1 ) as the sum  of terms  Sometimes p r i o r knowledge from p r e v i o u s s t u d i e s can be used t o support such approximations or we may be a b l e t o show that they a r e rigorously true. 1  -  U„ =  where the f i r s t  I i<j  u(X., _  1  X.) ~  +  2  8  Z i<j<k  -  u'(X., X., X.) ~ ~* _  term r e p r e s e n t s the sum  1  +  ...  of a l l unique p a i r  the second term r e p r e s e n t s the sum  of a l l unique 3-body  and  that  so on.  I t i s g e n e r a l l y agreed  interaction potential  complicate  modified  interactions,  A l s o , to i n c l u d e  term of the expansion  i n equation  pair anything  (2.2)  would  the problem, i n most c a s e s , to the degree of r e n d e r i n g i t  unsolvable. neglected  interactions,  f o r most l i q u i d s the  i s a dominate term.  more than o n l y the f i r s t  (2.2)  2  As a r e s u l t , i n almost a l l cases h i g h e r order  terms are  (they are assumed to be s m a l l ) or the p a i r p o t e n t i a l i n an attempt  In what i s u s u a l l y  to take  is  i n t o account h i g h e r order terms r e s u l t i n g  c a l l e d an " e f f e c t i v e " p a i r  potential.  Therefore,  we  write  U  =  I  u(X  X )  i<j  where the p a i r expressed  2  interaction  (2.3)  2  potential  between molecules  1 and 2 can  be  as  u(X  l f  X ) 2  = u(12)  = u(r  1 2  ,  ftj,  fi ) 2  (2.4)  2 Here we choose the c e n t r e of mass of p a r t i c l e 1 as the o r i g i n of the c o o r d i n a t e system.  - 9 -  where r_i2 i s the v e c t o r from the c e n t r e o f p a r t i c l e p a r t i c l e 2 and fij, ft r e p r e s e n t the m o l e c u l a r 2  1 t o the c e n t r e o f  orientations.  In our  c o o r d i n a t e system, the o r i e n t a t i o n fi^ i s d e s c r i b e d , as i l l u s t r a t e d i n F i g u r e 1, by the E u l e r a n g l e s , potential  [31,43],  or an e f f e c t i v e p a i r p o t e n t i a l , predominates w i t h i n  c u r r e n t s t a t i s t i c a l mechanical particularly  9, (j), ty. The use of o n l y a p a i r  s t u d i e s , although i t s v a l i d i t y ,  f o r systems of l i q u i d  pair interaction potential  d e n s i t y , has been q u e s t i o n e d .  i s g e n e r a l l y much l o n g e r ranged than  h i g h e r o r d e r many-body p o t e n t i a l s .  Hence, i t would be expected  e r r o r s r e s u l t i n g from the use of o n l y a p a i r p o t e n t i a l e v i d e n t i n short-range  2.  The other that  should be most  s t r u c t u r e i n most c a s e s .  W a t e r - L i k e S o l v e n t Model T h e r e have been many d i f f e r e n t p a i r p o t e n t i a l s used to model  water-like  fluids  [6,35,41,44,45] or w a t e r - l i k e s o l v e n t s [21-23].  many model p o t e n t i a l s , i t i s sometimes hard terms used and t h e i r r e s p e c t i v e v a l u e s . p o t e n t i a l s a r e determined  through  fitting  parameters a r e v a r i e d so as to reproduce of the system. find  to f i n d j u s t i f i c a t i o n f o r  Often parameters used i n model techniques;  the p o t e n t i a l  some r e a l p h y s i c a l p r o p e r t i e s  However, i n t r y i n g to model an e l e c t r o l y t e  t h a t some b a s i c molecular  ( i . e . they have been measured).  For  s o l u t i o n , we  p r o p e r t i e s of i t s components are known The i n c o r p o r a t i o n o f some o f these  "measured" v a l u e s i n t o our model should not o n l y g i v e us a more realistic  p i c t u r e but a l s o should demonstrate to some degree  relative  importance.  their  - 10  -  z  y  F i g u r e 1 - The o r i e n t a t i o n a l c o o r d i n a t e the E u l e r a n g l e s .  system where 9 , <j>,  are  - 11  The X-ray  p a i r d i s t r i b u t i o n f u n c t i o n s f o r water has been measured both  [46] and neutron  [47] d i f f r a c t i o n ,  t o be a more a c c u r a t e method. agreement. found  -  The  first  The  [48], and the p o l a r i z a b i l i t y known to r e a s o n a b l e  though the l a t t e r i s c o n s i d e r e d  Most r e s u l t s are i n g e n e r a l o v e r a l l  peak of the 0-0  to occur near 2.8A.  radial distribution function i s  d i p o l e moment, the quadrupole  We  tensor  t e n s o r [49] f o r the water molecule  are  accuracy.  In d e v e l o p i n g a model f o r the present study we work [23,41],  by  follow previous  assume that the system we wish to study i s dominated 3  by e l e c t r o s t a t i c dielectric  interactions.  We  would expect p r o p e r t i e s such as  the  constant and p o t e n t i a l of mean f o r c e to be dominated by  e l e c t r o s t a t i c i n t e r a c t i o n s and  thus these p r o p e r t i e s should not depend  s t r o n g l y on the short-range p a r t of the model p o t e n t i a l . The  oxygen atom of the water molecule  p r o t o n s , hence the molecule to t r e a t  the molecule  i s l a r g e compared to the  i s roughly s p h e r i c a l .  as a sphere  T h e r e f o r e , we  s i n c e a n o n - s p h e r i c a l model would  i n v o l v e rather complicated short-range i n t e r a c t i o n s . spherically but  symmetric short-range p o t e n t i a l s  There  are many  [31,43] that c o u l d be  s i n c e the c h o i c e i s somewhat a r b i t r a r y , the molecules w i l l  treated  simply  4  as hard spheres.  choose  The  hard sphere i n t e r a c t i o n ,  used,  be HS u g(r), a  i s g i v e n by  We note t h a t e l e c t r o s t a t i c i n t e r a c t i o n s tend to be c o n s i d e r e d long-ranged, p a r t i c u l a r l y charge-charge, c h a r g e - d i p o l e , and dipole-dipole interactions.  4 U n l i k e s o f t - p o t e n t i a l s which u s u a l l y i n v o l v e 2 or more parameters, the hard sphere p o t e n t i a l has o n l y one parameter, the hard sphere diameter.  -  u „ J ( r ) = -< 0, P  d^  ,  + ^  a  =  -  r > d  7  d where  12  d  (2.5a)  a  3  »  (2.5b)  w i t h r being the s e p a r a t i o n between the c e n t r e s of the molecules 3, and  d ,  a hard  sphere  a  dg b e i n g t h e i r diameter  r e s p e c t i v e diameters.  f o r a molecule  The  exact  s  c o n s i s t a n t w i t h the d i f f r a c t i o n data  [47] mentioned above.  10  permanent  esu cm)  c h o i c e of  i s a g a i n somewhat a r b i t r a r y ,  f o r our w a t e r - l i k e model a v a l u e of d  The  a and  " 2.8A  i s an obvious  but  choice  (gas phase) d i p o l e moment, Pp = 1.855D (1 Debye =  where we  d e f i n e the a x i s system as shown i n F i g u r e 2  jip l i e s a l o n g the z - a x i s .  The  quadrupole t e n s o r , Q, has  the  and  general  form  Q =  (2.6a)  yy zz  where  =  J  I  e  i  (  3  r  i a  r  i3 "  r  l  at?  6  (2.6b)  -  13  -  2  -> X  Figure  2 - Molecular  a x i s system f o r the w a t e r - l i k e  solvent  model.  - 14 -  , and r ^ denotes the a-component o f the v e c t o r r± which i t s e l f  gives  a  the p o s i t i o n o f the charge e^ w i t h r e s p e c t mass.  We know t h a t Q must be t r a c e l e s s  to the m o l e c u l a r c e n t r e o f  ( i . e . Cr  -"•  symmetric or l i n e a r quadrupoles, Q  xx  = -Q /2 L.  zz  and Q  T  yy  i s known as a l i n e a r quadrupole moment. [48] Q  = Q zz  = 2.63B, Q = -2.50B, and Q = -0.13B yy zz  esu cm ) .  We note, however, that Q  approximate  [48] Q, the quadruple t e n s o r , ( t o w i t h i n 5%) by  Q =  T  We  shall refer We  0  0  -Q  0  0  u.  where  Lt  Water i s not a x i a l l y  (IB = 10"  Q  = 0) i f  F o r molecules t h a t have a x i a l l y = Q  xx  +Q  yy  TCX  our molecule i s e l e c t r i c a l l y n e u t r a l .  symmetric and one f i n d s  +Q  z z  " 0 , so we  0 T  0  (2.7)  •  0  to Qj, as a t e t r a h e d r a l quadrupole moment [ 5 0 ] .  can more f u l l y understand the p h y s i c a l d i f f e r e n c e between a  l i n e a r and a t e t r a h e d r a l quadrupole by examining the i m p l i c a t i o n s o f t h e i r geometries more c l o s e l y will  (see F i g u r e 3 ) .  A tetrahedral  quadrupole  i n t e r a c t w i t h p o s i t i v e and n e g a t i v e charges i n an e q u i v a l e n t  f a s h i o n , whereas a l i n e a r quadrupole w i l l i n a much d i f f e r e n t manner.  i n t e r a c t w i t h o p p o s i t e charges  Ions w i l l be s o l v a t e d  f a s h i o n by a s o l v e n t w i t h a t e t r a h e d r a l quadrupole. l i n e a r quadrupole w i l l g i v e r i s e  i n a symmetric A solvent with a  to asymmetric s o l v a t i o n w i t h ions o f  one charge b e i n g s o l v a t e d d i f f e r e n t l y than ions of the o p p o s i t e s i g n . I f we now l o o k at a t e t r a h e d r a l charge d i s t r i b u t i o n , i . e . e q u i v a l e n t charges l o c a t e d at the v e r t i c e s o f a t e t r a h e d r o n (see F i g u r e 3 ) , we  find  15 -  as-  a ) Linear Quadrupole  b)  Figure  lehrxxhedral quadrupole  3 - A charge d i s t r i b u t i o n p o s s e s s i n g (a) a l i n e a r and (b) a t e t r a h e d r a l quadrupole.  quadrupole  - 16  t h a t i t p o s s e s s e s a d i p o l e and o r d e r m u l t i p o l a r moments. molecule  i n the l i q u i d  t e t r a h e d r a l angle. p o s i t i v e charges and between the two  We  state,  Hence, we the two  -  t e t r a h e d r a l quadrupole note that 104.5°,  as I t s two  the bond angle o f a water  i s r e l a t i v e l y c l o s e to the  can see a correspondence  between the  protons of a water m o l e c u l e , and  n e g a t i v e charges and  lowest  two  another  the lone p a i r s of the water  molecule. These m u l t i p o l e moments are added to our w a t e r - l i k e model as moments embedded at the c e n t r e of hard spheres. 2.5B.  point  Qp i s taken as  However, the v a l u e f o r the permanent d i p o l e moment of 1.855D i s  not used. Pp due  I n s t e a d , an " e f f e c t i v e " d i p o l e moment, which i s l a r g e r  to the p o l a r i z a t i o n of the model, i s determined  system.  The  polarizablity  and used  i n our  p o l a r i z a b i l i t y of any molecule i s d e f i n e d by i t s t e n s o r , a, but s i n c e the water molecule  i s o t r o p i c a l l y p o l a r i z a b l e , we  i s nearly  simply take i t s p o l a r i z a b i l i t y ,  a, to be  a = ^ Tr a = 1 . 4 4 4 A .  (2.8)  3  The  " e f f e c t i v e " d i p o l e moment, nig, i s determined  s e l f - c o n s i s t e n t mean f i e l d many-body e f f e c t  u s i n g the  (SCMF) t h e o r y of C a r n i e and Patey  of p o l a r i z a t i o n i s reduced  [41].  to p o l a r i z a t i o n i n t o an e f f e c t i v e d i p o l e moment.  t h e o r y f o r a homogenous f l u i d given  the average  The  i n t h i s t h e o r y to an  " e f f e c t i v e " p a i r p o t e n t i a l by a t t e m p t i n g to i n c o r p o r a t e the t o t a l due  than  effect  In the SCMF  t o t a l d i p o l e moment, m',  is  by  N o t e t h a t were we to take the c e n t r e of the oxygen atom as our o r i g i n i n s t e a d of t a k i n g the c e n t r e o f mass as our o r i g i n , Qj would be changed by l e s s than 2%. 5  (2.9a)  where the induced d i p o l e moment  p = a • < E  and  < ET, > i s the average  d i p o l a r and quadrupolar the average molecules C  2 v  local  L  >  (2.9b)  local electric  components.  6  f i e l d which w i l l have both  In p r i n c i p l e , d e t e r m i n a t i o n of  f i e l d w i l l be a many-body problem, depending on a l l  i n the system.  < Ej, > f o r an i s o t r o p i c f l u i d  o f a x i a l or  symmetry w i l l be non-zero o n l y i n the d i r e c t i o n o f jjp.  the average  induced moment, _p_, w i l l be i n the same d i r e c t i o n as yip.  We can then drop will  Hence  the v e c t o r n o t a t i o n , u n d e r s t a n d i n g  l i e a l o n g the z - a x i s .  that a l l vectors  I n d e t e r m i n i n g mg, the e f f e c t i v e d i p o l e  moment, f l u c t u a t i o n s i n the e l e c t r i c  field  and i n m^,  the  i n s t a n t a n e o u s d i p o l e moment o f p a r t i c l e i , are i g n o r e d .  For d  s  = 2.8A  at 25°C, a v a l u e of m' = 2.56D i s o b t a i n e d which agrees w e l l w i t h v a l u e s that have been c a l c u l a t e d  f o r i c e [51] and estimated  f o r water [ 6 ] ,  a l t h o u g h the v a l u e o f m' does vary somewhat w i t h c h o i c e o f diameter. The  determined  v a l u e of the e f f e c t i v e permanent d i p o l e moment,  I t has been found [41] t h a t the quadrupolar a p p r o x i m a t e l y 25% of the t o t a l f i e l d . 6  7  2 By s e t t i n g m^nu = <m > =  2  ,  I  I  = pH^I  =  m  field  '•  constitutes  -  m  e  * m'  , will  be used i n a l l f o l l o w i n g  d i p o l e moment f o r our w a t e r - l i k e The  18 -  pair interaction, u  s s  c a l c u l a t i o n s as the permanent  solvent.  , between two s o l v e n t  molecules i s g i v e n  by  ss  U  (  1  2  )  =  U  ss  (  r  )  +  U  D D  (  1  HS where u ( r ) i s the p r e v i o u s l y ss  >  2  +  U  D Q  defined  (  1  2  )  +  U  Q  Q  (  1  2  )  <  solvent-solvent  p o t e n t i a l , 11^(12) i s the d i p o l e - d i p o l e  hard  2  '  1  >  0  sphere  i n t e r a c t i o n , 11^(12) i s  d i p o l e - q u a d r u p o l e i n t e r a c t i o n , and U Q Q ( 1 2 ) i s the  the  quadrupole-quadrupole i n t e r a c t i o n . We now employ a method f i r s t multipolar defined  potentials  /  (  0  1  2  r  ^1,  i n terms of r o t a t i o n a l i n v a r i a n t s ,  fi^  j-mnJl/m n SL ^m ,„ D ( y'VA»> y p ' V x  , „  )  E  y  where  by Blum [52-54] f o r expanding  f i t  (  _n  N  D  with respect  x D  „ Jl 0 X '  ,_ (  a  i 2  , „ ...  N )  (  2  '  U  )  vA  the o r i e n t a t i o n s  to a l a b o r a t o r y  f i x e d frame o f r e f e r e n c e .  v  i s a p p r o x i m a t e l y 4% l a r g e r m  (™ "  2  =  £ 2 ~ —1  a n c  *  The Wigner a r e d e f i n e d by  than m' f o r the c u r r e n t  = m' + 3a'kT e where a' i s a r e n o r m a l i z e d p o l a r i z a b i l i t y [ 4 1 ] . 2  r ^  o f p a r t i c l e s 1 and 2, r e s p e c t i v e l y ,  m a t r i x elements, D * ( f i ) , and the 3-j symbols,  e  ,„  vv'<V  r e p r e s e n t s the o r i e n t a t i o n of the v e c t o r  ^2 d e s c r i b e  m  $™^(12),  by  mn& -, \ •yv " A  exploited  model,  -  Rotenberg et a l . [55] the  coefficients f  m n  and  following  ^ given  mnZ  e x c e p t i o n of f  these r o t a t i o n a l g e n e r a l , the which w i l l  infinite  of p a r t i c l e s 1 and  A|/(  are  set of  2.  m  be  m  interactions.  We  we  can  n A  and  and  22A  = 8/35/2.  see  invariants  In Appendix  angles d e n o t i n g the how  a unique set of  used to d e s c r i b e the  even and  the  1  2  )  A,  w r i t e the  In  set  orientations rotational  components of the  pair  of  potential  Symmetry imposes c e r t a i n r e s t r i c t i o n s m n  ^ ( r ) are  uv  symmetry, such as water, r e q u i r e  allowed.  non-zero.  that  p and  v must  be  condition  mn£/  u  must be  >  r o t a t i o n a l symmetry  a x i a l symmetry, o n l y c o e f f i c i e n t s where u=v=0 are 2 v  2  form a complete  J  M o l e c u l e s of C  have  (  hence only c e r t a i n c o e f f i c i e n t s , u  ss For  f  }  rotational  i n terms of some set of $™"^(12). (12)  u s u a l c o n v e n t i o n [38], we  expressed i n a more e x p l i c i t manner.  Hence, we  multipolar  upon u  the  space of E u l e r i a n  $ ^ ( 1 2 ) , can  f i n d that  =  = -2/5~  invariants  span the  invariants,  220  -  by  f  w i t h the  19  uv  ( r )  satisfied.  x  mn£v  =  u  -uv  For  ( r )  \ =  mnZ,  u  u-v  ( r )  » =  u  mn.lt  -u-v  the w a t e r - l i k e s o l v e n t  these c o n d i t i o n s must a l s o be  obeyed.  * \  /<«  ( r )  ( 2  model we  In a d d i t i o n ,  the  are  *  io\  1 3 )  considering,  simplification  - 20 -  to  a t e t r a h e d r a l quadrupole r e q u i r e s a f u r t h e r r e s t r i c t i o n i n t h a t  p + v + 2% = 0 (mod 4)  which does not h o l d f o r g e n e r a l C 2  (2.14)  symmetry.  V  In view of these  r e s t r i c t i o n s , i t i s convenient to d e f i n e the f u n c t i o n s *  1 2 £  *  2 U  *  2 2 £  ( 1 2 ) = *J *(12) 2  ( 1 2 ) = *20*  +  ( 1 2 )  +  £ ( 1 2 )  +  $  J-2  ( 1 2 )  *-20  ( 1 2 )  '  (2.15a)  '  (2.15b)  and  We  (12) = *22  can now w r i t e  $  ?22  ( 1 2 )  +  *2-2  ( 1 2 )  +  $  -2-2  ( 1 2 )  *  (2.15c)  [41] the m u l t i p o l a r i n t e r a c t i o n s of e q u a t i o n  (2.10) as  u (12) = u  1 1 2  (r) $  1 1 2  (12),  u (12) = u  1 2 3  (r) *  1 2 3  (12) + u  u (12) = u  2 2 4  (r) *  2 2 4  (12),  D D  D Q  (2.16a)  2 1 3  (r)  $  2 1 3  (12),  (2.16b)  and  Q Q  where  u  1 1 2  (r) = -p /r ,  u  1 2 3  (r) = -u  2  2 1 3  (2.16c)  (2.17a)  3  ( r ) = "~™— r* , /6 r 4  (2.17b)  - 21  -  224. v T (r) = — j 2r V  u  and  (2.17c)  _  have the r e q u i r e d r dependence. the s u b s c r i p t s 00 from $QQ^ otherwise i n d i c a t e d .  The  I t should be noted  rotational invariant  dropped  $^^(12) = 1 for  i n t e r a c t i o n of (2.10) can  Moreover, the p a i r p o t e n t i a l expressed by e q u a t i o n  (2.10) w i l l h o l d f o r molecules o n l y to r e p l a c e 0^ w i t h ST/l in  have  assuming these to be understood when not  i s o t r o p i c p o t e n t i a l s such as the hard sphere a l s o be o m i t t e d .  that we  Q  p o s s e s s i n g l i n e a r quadrupoles.  L  i n (2.17) and  *  m n £  ( 1 2 ) with  One  has  $ *(12) m n  (2.6). T h i s w a t e r - l i k e s o l v e n t model has been found to be v e r y  successful  g i n d e s c r i b i n g the d i e l e c t r i c quadrupole  was  properties  shown to be very important  c h a r a c t e r i s t i c s of the system essential. crucial;  The  [41],  [41]. The  and p o l a r i z a b i l i t y was  2.8  and  2.9A  tetrahedral  i n d e t e r m i n i n g the g e n e r a l  exact c h o i c e of hard sphere  the v a l u e s of 2.7,  similar results  of water  diameter  found to be proved not to be  were examined and a l l gave v e r y  U s i n g a diameter of 2.8A,  the model was  found  to  g i v e d i e l e c t r i c c o n s t a n t s i n good agreement w i t h e x p e r i m e n t a l p o i n t s on the vapour p r e s s u r e curve over the temperature  range  F i g u r e 4) as w e l l as w i t h p o i n t s above the c r i t i c a l The  s t r u c t u r e of the system,  25 - 300°C (see temperature  as expressed by the r a d i a l  [34].  distribution  f u n c t i o n , g^OO(r) , gave o n l y g e n e r a l q u a l i t a t i v e agreement w i t h  P h y s i c a l q u a n t i t i e s , such as the d e n s i t y and d i e l e c t r i c c o n s t a n t , of water are w e l l t a b u l a t e d as f u n c t i o n s of temperature [18,56].  100  T(°C)  200  300  F i g u r e 4 - The d i e l e c t r i c constant of the w a t e r - l i k e s o l v e n t as a f u n c t i o n of temperature [41], The r e s u l t s f o r the w a t e r - l i k e model u s i n g the SCMF theory are i n d i c a t e d by the s o l i d l i n e ; the r e s u l t s from experiment are i n d i c a t e d by the d o t s .  - 23  -  experiment, but t h i s i s probably g r e a t l y i n f l u e n c e d by the r e p u l s i v e hard core p o t e n t i a l .  unrealistic,  However, the a b i l i t y of t h i s model to  d u p l i c a t e the d i e l e c t r i c p r o p e r t i e s of water makes i t p a r t i c u l a r l y a t t r a c t i v e i n the study of aqueous e l e c t r o l y t e s .  3.  E l e c t r o l y t e Solution Model S o l u t i o n s c o n t a i n i n g symmetric i o n s , p r i m a r i l y a l k a l i h a l i d e s , w i l l  be the focus of t h i s study.  Charged hard  e x t e n s i v e l y to model e l e c t r o l y t e s salts.  They are an obvious  The  by a p o i n t charge at i t s c e n t r e .  The  components due  given  represented  i o n s w i l l not be c o n s i d e r e d to be  Their p o l a r i z a b i l i t y i s spherically  field  as molten  ions i n s o l u t i o n  charge on an i o n w i l l be  assume t h a t a s o l v a t e d i o n e x p e r i e n c e s the e l e c t r i c  have been used  both i n s o l u t i o n and  choice f o r modelling  our c h o i c e of s o l v e n t model.  polarizable.  [6,7]  spheres  symmetric and  we  no net average p o l a r i z a t i o n s i n c e  to the surrounding  solvent  molecules  w i l l cancel. Again we ionic radii  are f a c e d w i t h c h o i c e s f o r hard can be d e f i n e d i n both the s o l i d  sphere and  diameters.  Though  gas phases, there i s a t  p r e s e n t no d i r e c t method of e s t i m a t i n g the r a d i i of i o n s i n s o l u t i o n . The  r a d i i of gaseous i o n s are found  radii.  to be much l a r g e r than  T h i s can be e x p l a i n e d simply by n o t i n g t h a t i n an i o n i c  - the l a t t i c e i s h i g h l y compressed by coulombic possess  their  much s m a l l e r apparent  radii.  An  a t t r a c t i o n and  The  crystal,  hence i o n s  i o n i n s o l u t i o n w i l l have a  r a d i u s of i n t e r m e d i a t e v a l u e , somewhere between i t s c r y s t a l and values.  cyrstal  gas  s o l u t i o n v a l u e i s g e n e r a l l y viewed as being very c l o s e to  - 24  the c r y s t a l  r a d i u s and  thus c r y s t a l  -  radii  are o f t e n used f o r aqueous  electrolytes. We  have s e v e r a l c h o i c e s f o r c r y s t a l  being most w i d e l y used. of  ionic  crystals  radii,  radii.  We  have been used to determine i o n i c  have chosen to use  d e s c r i b e d by M o r r i s  of P a u l i n g  [57]  R e c e n t l y , X-ray e l e c t r o n d e n s i t y measurements  measurements seem to g i v e a more p h y s i c a l l y ionic  those  [58],  these  radii.  Such  r e a l i s t i c method of d e f i n i n g  radii,  in particular  those  Table I summarizes the v a l u e s of the  Table  ion  I  Diameters, d^, f o r a l k a l i metal and h a l i d e i o n s . Both those of M o r r i s [58] and P a u l i n g [59] are i n c l u d e d . A l s o g i v e n are the reduced diameters,  d. = d./d ( w i t h d = 2.8A; rounded to the n e a r e s t 1 i s s accommodate a g r i d width of 0 . 0 2 d ) , used i n a l l numerical calculations.  0.04  s  T  ±  ( i n A)  Ion Pauling  Morris  d.  0.60  0.93  0.68  0.95  1.17  0.84  1.33  1.49  1.08  1.48  1.64  1.16  1.69  1.83  1.28  F~  1.36  1.16  0.84  Cl~  1.81  1.64  1.16  Br"  1.95  1.80  1.28  I"  2.16  2.04  1.44  Li Na  +  +  K+ Rb Cs  +  +  to  - 25 -  d i a m e t e r s f o r the a l k a l i h a l i d e s comparison. larger,  We f i n d  that  and P a u l i n g r a d i i  f o r cations  are also  included f o r  the v a l u e s used are somewhat  and f o r anions they a r e somewhat s m a l l e r than the P a u l i n g  We d e f i n e the p a i r i n t e r a c t i o n p o t e n t i a l , U J J ( 1 2 ) ,  radii.  between two  xons as , . 000. <. HS, v , i j u j(12) = u (r) = u j ( r ) + — ^ q  , . (2.18)  q  1 0  ±  where  0 1 Q  ±  and q^ a r e the i o n i c charges and u ^ ( r )  hard sphere i n t e r a c t i o n .  The i o n - i o n  i s j u s t the i o n - i o n  pair potential  i s spherically  symmetric w i t h $000(12) = 1 a g a i n being understood. In d e f i n i n g simplifications  the i o n - s o l v e n t that  r e s u l t from the i n f i n i t e  the  infinite dilution limit,  the  e f f e c t i v e dipole  permanent d i p o l e  p o t e n t i a l , we a r e a b l e t o make some  the b u l k s o l v e n t  moment o f the w a t e r - l i k e  thus a l s o  an e f f e c t i v e p a i r p o t e n t i a l  account f o r the p o l a r i z a t i o n molecules.  Hence  i s used as the  interaction  i n which we a r e a t t e m p t i n g to  of a s o l v e n t  molecule by a l l o t h e r  solvent  molecules by an i o n i s not  However, we need o n l y worry about one such i o n  because we a r e c o n s i d e r i n g  the i n f i n i t e  o n l y generate a s p h e r i c a l l y  solvent.  solvent  The i o n - s o l v e n t  The p o l a r i z a t i o n of the s o l v e n t  taken i n t o account.  will  remains unchanged.  In  moment, as opposed to u s i n g the gas phase v a l u e i n  d e t e r m i n i n g the c h a r g e - d i p o l e i n t e r a c t i o n . is  d i l u t i o n assumption.  dilution limit.  symmetric e l e c t r i c  field  F o r the model we c o n s i d e r h e r e , t h i s can o n l y  A single ion i n the contribute  - 26 -  spherically  symmetric  terms to the i o n - s o l v e n t i n t e r a c t i o n and we  to i g n o r e a l l such terms except those that might be i n c l u d e d sphere  i s  ( 1 2 ) , i s given  , v  u (12) 10  i g  = u  000, . .  O i l , v 011  i g  i g  (r) + u  u  U  (r) $ A  000. . is  is  =  1 ( r ) =  022, .  v  ,  022 - 0 2 2 , .  (12) + u  / 1 o  ± g  $  (12) 10  ,  .  (2.19a) 0  1 Q  HS, . is^»  ,„ .... (2.19b)  'V ^'  (2.19c)  U  7  ^  ( 2  '  1 9 d  >  0 2 2  ( 1 2 ) = $° (12) + *[J"(12). 2 2  (  2  '  2  0  )  second term of (2.19a) g i v e s the c h a r g e - d i p o l e i n t e r a c t i o n and the  third this  interaction,  define  *  The  pair  [23] by  where  and we  i n the hard  potential.  Thus, f o r the present model the i o n - s o l v e n t u  choose  term g i v e s the charge-quadrupole i n t e r a c t i o n . study w i l l  needed  be a d i p o l a r  hard sphere s o l v e n t .  f o r t h i s s o l v e n t can be o b t a i n e d  s o l v e n t by s e t t i n g Qp = 0.  Also referred A l l pair  to i n  potentials  from those of the w a t e r - l i k e  - 27  Having now  -  d e f i n e d our model i n terms of p a i r  p o t e n t i a l s , we w i l l  interaction  i n Chapter V study t h i s system u s i n g the  statistical  mechanical t h e o r y developed i n Chapter I I I and the d i e l e c t r i c t h e o r y of Chapter  IV.  - 28 -  CHAPTER I I I STATISTICAL MECHANICAL THEORY  1.  Introduction S t a t i s t i c a l mechanics  to study f l u i d s  p r o v i d e s s e v e r a l d i f f e r e n t approaches  [31,59,60].  by which  D i s t r i b u t i o n f u n c t i o n language i s  f r e q u e n t l y used and i s p a r t i c u l a r l y u s e f u l i n s t u d i e s of f l u i d s a t the molecular l e v e l .  D i s t r i b u t i o n f u n c t i o n s a l l o w a complete but compact  d e s c r i p t i o n o f the m i c r o s c o p i c s t r u c t u r e o f f l u i d s  [32,61].  Knowledge  of even the lower o r d e r d i s t r i b u t i o n f u n c t i o n s i s s u f f i c i e n t , i n g e n e r a l , t o determine most e q u i l i b r i u m  (macroscopic or thermodynamic)  p r o p e r t i e s o f a system d e f i n e d u s i n g the p a i r p o t e n t i a l  assumption.  A model p o t e n t i a l u n i q u e l y determines some d i s t r i b u t i o n We can d e p i c t  u(12) model  function.  t h i s s c h e m a t i c a l l y a f t e r Rasaiah [ 2 4 ] :  m e c h a n i c a l ^ ^ S(12) ^, theory  ) thermodynamic p r o p e r t i e s  where g(12) i s the p a i r d i s t r i b u t i o n f u n c t i o n as d e f i n e d below. r e q u i r e d sum over c o n f i g u r a t i o n s  ( i n t e g r a l over phase space) i m p l i c i t i n  going from u(12) t o g(12) i s u s u a l l y performed s t a t i s t i c a l mechanics,  temperature.  using c l a s s i c a l  s i n c e a t o r d i n a r y temperatures most m o l e c u l a r  f l u i d s can be t r e a t e d i n a c l a s s i c a l manner. of course not v a l i d  The  This approximation i s  i n cases such as l i q u i d helium or hydrogen  a t low  The sampling o f phase space can sometimes be performed  d i r e c t l y u s i n g Monte C a r l o or M o l e c u l a r Dynamics t e c h n i q u e s which g i v e  - 29 -  essentially  "exact" r e s u l t s .  Approximate  methods, i n t e g r a l e q u a t i o n  methods b e i n g one such g e n e r a l approach, can be used f o r some simple models.  The P e r c u s - Y e v i c k (PY), Mean S p h e r i c a l Approximation  Hypernetted-Chain  (HNC)  methods are among the most f r e q u e n t l y  i n t e g r a l equation theories.  (MSA), and used  D i s t r i b u t i o n f u n c t i o n s p l a y primary  roles  i n the f o r m u l a t i o n of these t h e o r i e s .  2,  D i s t r i b u t i o n Functions The p a i r d i s t r i b u t i o n f u n c t i o n , g ( 1 2 ) , can be d e f i n e d  [32] by  first  (N) c o n s t r u c t i n g a n o r m a l i z e d c a n o n i c a l p r o b a b i l i t y , P^  , f o r a homogenous  system of N n o n - s p h e r i c a l molecules of a volume V and at a T.  The p r o b a b i l i t y of s i m u l t a n e o u s l y f i n d i n g molecule 1 i n dXi at X_i,  molecule 2 i n dXj> at 2^2 >  p  N ) ( n  temperature  -1»-2 * *»2N ,  ,  )  =  \  e t c  »  I  s  g i v e n by  exp[-$U (X ,X ,...,X )]dX N  1  2  N  1  d^.-.d)^  (3.1)  N  where 3 = 1/kT,  k i s the Boltzmann  configurational integral  ZJJ =  /.../  c o n s t a n t , and  i s the  (the n o r m a l i z a t i o n f a c t o r ) d e f i n e d  exp [ - 3U  N  (X^ ,5^2 , • • •, XJJ ) ] dX_i  by  dX^.t-dX^.  (3.2)  (n) The n-body p r o b a b i l i t y d e n s i t y , P^ is  i n dX]  , the p r o b a b i l i t y that any  at X j , • • • » and any molecule i s i n dX^  molecule  at X Q J i s  o b t a i n e d by i n t e g r a t i n g over the c o o r d i n a t e s of the remaining  N-n  - 30 -  molecules:  It  can be shown [32] that  f o r a s i n g l e molecule  P^V^ where p i s the number d e n s i t y  - N/V = p  (3.4)  of the system.  We now examine the l i m i t o f pf. ^ as the mutual d i s t a n c e s between N 11  the  n p a r t i c l e s become l a r g e .  As these s e p a r a t i o n s i n c r e a s e ,  c o r r e l a t i o n between the p a r t i c l e p o s i t i o n s Hence, i n the l i m i t  can be expected t o d e c r e a s e .  the n-body p r o b a b i l i t y d e n s i t y  i n t o the product o f n s i n g l e - b o d y p r o b a b i l i t y  4 The n - p a r t i c l e  »V "  P  N  1)(  V  P  N  d i s t r i b u t i o n function,  t  P  1 ) (  V  =  (3  '  5)  g^ ^, i s then d e f i n e d by 1  (^1 > • • • »X )  \  4 ( ,...,X ) = N — i —n  can be f a c t o r i z e d  densities:  V"- N  1 ) (  the  =2-  n)  Xl  n n  (3.6)  ,. x p; (Xi) i ;  i=l  which expresses the p r o b a b i l i t y d e n s i t y  of o b s e r v i n g  different  - 31  configurations  -  f o r a set of n molecules i n a system c o n t a i n i n g N  molecules i n t o t a l . I f a model f o r a system i s d e f i n e d u s i n g only p a i r p o t e n t i a l s , i t (2) can be shown t h a t  a pair d i s t r i b u t i o n function,  c o m p l e t e l y d e s c r i b e the e q u i l i b r i u m  g^  ( X i , X2^»  thermodynamics of that  will  system.  We  write  g(12) =  iWtll P  Z  /.../  e X  p[-3U ]dX ...dX N  3  (3.7)  N  N  where g(12) i s a s i m p l i f i e d n o t a t i o n f o r the p a i r d i s t r i b u t i o n function. density  The  radial d i s t r i b u t i o n function,  m o l e c u l e s 1 and  the p a i r d i s t r i b u t i o n f u n c t i o n  over a l l o r i e n t a t i o n s  of  2:  g(r)  = (8TT  where d^ = s i n 9d9d<j)d^.  1  For  probability  of f i n d i n g a p a i r of molecules a d i s t a n c e r a p a r t , i s o b t a i n e d  by i n t e g r a t i n g  function  g ( r ) , the  r  // g(12) dfi.dH, 1  3  8  1  T h i s angle-averaged p a i r d i s t r i b u t i o n  can be o b t a i n e d from X-ray and neutron s c a t t e r i n g  g ( r ) , and l i k e w i s e  ( - )  experiments.  f o r a l l n-body d i s t r i b u t i o n f u n c t i o n s ,  the  ^ h e p r o b a b i l i t y , p ( r ) , of f i n d i n g a molecule a d i s t a n c e r from o t h e r molecule i s g i v e n by p ( r )  = ^  g(D dr .  any  - 32 -  f o l l o w i n g must  hold:  g ( r ) •»• 1 as r -»•  0 0  Explicit  .  formulae f o r the thermodynamic  (3.9)  p r o p e r t i e s o f f l u i d s can  always be w r i t t e n as f u n c t i o n s of u(12) and g ( 1 2 ) .  The average  j  c o n f i g u r a t i o n a l energy of a (NVT) system, UTJQT' which i s important i n many s t u d i e s , i s d e f i n e d as  1 1 = Y~ /.../ e x p [ - 8 U ] [ | . ^ 2 u ( i j ) ] d X . . . d X N i * j N  U  T 0 T  N  which can be s i m p l i f i e d  / u(12) g(12) dXj d X . 2  0  (S* )  T  (3.10a)  2  P  2  For a multicomponent total  N  [32] to g i v e  U, T  1  (3.10b)  2  system, such as e l e c t r o l y t e  s o l u t i o n , we w r i t e the  i n t e r n a l energy per m o l e c u l e as  U/N  P . 2(8TT ) 2  I „X a3 X  2  a  f t  / g„ (12) u R  3  a  e  a  B  (12) dr dfi.dfi, 1 2  (3.11a)  where P = S P a and  X  a  a  = P /P> a  (3.11b)  (3.11c)  - 33  p  a  and  b e i n g the  of component  of the d i e l e c t r i c p r o p e r t i e s  how  be  they may  of e l e c t r o l y t e  solutions  o b t a i n e d from p a i r d i s t r i b u t i o n f u n c t i o n s  i s given  IV.  I n t e g r a l E q u a t i o n Methods  (a)  The  In the  Ornstein—Zernike Equation  above d i s c u s s i o n ,  been d e f i n e d .  In the  the  pair d i s t r i b u t i o n function,  defined  i s c l e a r that  the  function  d i s t r i b u t i o n function [62]  h(12),  h(12)  (3.12)  d e s c r i b e s the  d e p a r t u r e of  from i t s l i m i t i n g v a l u e of  1.  Ornstein  developed a r e l a t i o n s h i p i n which h(12), the  2 o n l y , and  correlations the  pair correlation function,  = g(12)-l.  c o r r e l a t i o n , i s expressed as a sum 1 and  has  by  h(12)  Zernike  g(12),  development of i n t e g r a l e q u a t i o n methods, i t  becomes e s s e n t i a l to i n t r o d u c e the  It  a.  A discussion  i n Chapter  3.  number of d e n s i t y  -  an  h(12)  and  total pair  involving particles  i n d i r e c t p a r t which takes i n t o account a l l  involving  Ornstein-Zernike  of a d i r e c t p a r t  the  other p a r t i c l e s .  (OZ)  = c(12)  e q u a t i o n and  + -Py 8TT  This can  / c(13)  relationship  be w r i t t e n  h(32)  dX  3  i s known as  i n the  form  (3.13)  - 34  where the called  integration  the  i s over a l l c o o r d i n a t e s of p a r t i c l e 3.  direct correlation  function  assumed ( i n c o r r e c t l y as i t t u r n s out) the  p a i r i n t e r a c t i o n , u(12).  more f u l l y understood i f we  h(12)  = c(12)  + -Hy 8ir  dX  +  dX^  3  see  involves  correlations  increasing The  OZ  The  by  the  that  c(12)  dX  +  3  the  C-^) // 8ir 2  t h e o r y of f l u i d s .  i n d i r e c t part  of the  between p a r t i c l e s 1 and  (3.13) i s a b a s i c  c(12)  I t i s an  exact r e l a t i o n s h i p  of  interpretation.  -»• -3u(12).  quickly  2 but  relationship  be viewed as being d e f i n e d by  simple p h y s i c a l  c o n v o l u t i o n of  which has  The  the  the  proves to be v e r y u s e f u l .  OZ  the  i n the  equilibrium be  derived  [32]  correlation,  e q u a t i o n and  has  no  separations  r,  approaches i t s asymptote more  s i m i l a r o s c i l l a t o r y b e h a v i o r to  second term of the  As  etc.  direct  OZ  pose many problems i n t r y i n g to s o l v e f o r h ( 1 2 ) .  with a spherically  c(42)  correlation  which can  the more f a m i l i a r r a d i a l d i s t r i b u t i o n f u n c t i o n , The  c(34)  o n l y those through  In g e n e r a l , f o r l a r g e  In most c a s e s , c(12)  than does h(12)  be  expansion  c(13)  total pair  through c l u s t e r diagram expansion methods. c ( 1 2 ) , can  (3.13) can  (3.14)  number of i n t e r m e d i a t e p a r t i c l e s 3,4,  equation  had  would depend o n l y upon  second term of e q u a t i o n  c(32)  was  o r i g i n a l authors who  i t e r a t e (3.13) to o b t a i n the  / c(13)  c(12)  ...  Thus, we  an  that  -  g(r).  e q u a t i o n (3.13) Fourier  s i m p l e s t case, i f one  symmetric p o t e n t i a l , the  OZ  that  can  transform  i s dealing  e q u a t i o n can  be  only  Fourier  - 35  transformed  immediately  to g i v e the k-space e q u a t i o n  h(k) = c ( k ) +  The  -  ph(k)  c(k).  (3.15)  a l g e b r a i c e x p r e s s i o n t h a t r e s u l t s upon rearrangement of e q u a t i o n  (3.15) expresses h(k) as a simple f u n c t i o n of c ( k ) and  (b)  The  Closure  Approximations  OZ e q u a t i o n  (3.13) serves as the f i r s t  e q u a t i o n theory f o r f l u i d s . necessary  p.  A second  e q u a t i o n i n any  e q u a t i o n or r e l a t i o n s h i p i s  i n o r d e r to s o l v e f o r the two  unknowns, h(12)  and  a r e l a t i o n s h i p i s known as a c l o s u r e f o r the OZ e q u a t i o n . second  r e l a t i o n s h i p which attempts  f u n c t i o n , h ( 1 2 ) , and  the d i r e c t  integral  to r e l a t e the p a i r  c(12).  Such  However, t h i s  correlation  c o r r e l a t i o n f u n c t i o n , c ( 1 2 ) , by  i n c l u d i n g the p a i r p o t e n t i a l , u ( 1 2 ) , i s o n l y (at p r e s e n t ) an approximate relationship. which l i m i t s  I t i s the accuracy of the c l o s u r e approximation the a c c u r a c y of an i n t e g r a l e q u a t i o n t h e o r y .  used  Hence an  i n t e g r a l e q u a t i o n theory i s i n v a r i a b l y known by the c l o s u r e r e l a t i o n s h i p it  employs.  The most commonly used  c l o s u r e s , as mentioned above, are  the Mean S p h e r i c a l , P e r c u s - Y e v i c k , and Hypernetted-Chain The MSA  works from the premise  Z e r n i k e ) t h a t the d i r e c t  (originally  approximations.  t h a t of O r n s t e i n and  c o r r e l a t i o n f u n c t i o n , c ( 1 2 ) , w i l l depend o n l y  on the p a i r p o t e n t i a l , u ( 1 2 ) .  The  c l o s u r e i s d e f i n e d by two  equations:  - 36  and  c(12)  = -3u(12)  g(12)  = 0  -  for r > d  (3.16a)  (3.16b)  for r < d  where d i s a hard-sphere diameter.  The  hard core  (3.16b) s t a t e s that molecules never  i n t o a molecule (equation  c l o s u r e always i n s e r t s a  interpenetrate).  An  even though u(12)  may  MSA  c o r r e c t asymptotic form i n that e q u a t i o n  does have the  exact  i n the  limit  implicit  MSA  contain  r •*• °°.  hard  sphere p o t e n t i a l i s always  other  s p h e r i c a l l y symmetric terms.  I t i s not  implied  (3.16a) i s  s u r p r i s i n g then that the MSA  good r e s u l t s f o r monatomic f l u i d s at very  low  densities.  The  At  gives  higher  d e n s i t i e s where s h o r t - r a n g e c o r r e l a t i o n s become much more i m p o r t a n t , the MSA  g e n e r a l l y g i v e s poor r e s u l t s .  sphere systems where the MSA The  PY  and  HNC  i s equivalent  c l o s u r e s can be d e r i v e d  or c l u s t e r s e r i e s expansions f o r c(12) PY  c l o s u r e i s given  HNC  to the PY  closure.  from f u n c t i o n a l T a y l o r  by n e g l e c t i n g  the c o r r e c t l a r g e r b e h a v i o r .  series  c e r t a i n terms.  The  The  (3.17)  PY  theory  gives  r e s u l t s f o r some monatomic  good  fluids.  good r e s u l t s , p a r t i c u l a r l y at h i g h d e n s i t i e s , f o r hard  sphere, Lennard-Jones, and The  the case f o r hard  = g ( 1 2 ) [ l - exp(Bu(12))]  agreement w i t h computer s i m u l a t i o n I t g i v e s very  i s not  by:  c(12)  which a l s o has  This  closure, given  c(12)  square w e l l p o t e n t i a l s  [32].  by  = h(12)  - Jin g(12)  -  3u(12),  (3.18)  - 37 -  is  s u p e r i o r to the PY c l o s u r e f o r most other systems [32], I t s  s u p e r i o r i t y f o r models p o s s e s s i n g coulombic known [32,35], been found  interactions i s well  F o r the r e s t r i c t e d p r i m i t i v e model, the HNC theory has  [63] t o be s u p e r i o r to both the PY and MSA t h e o r i e s through  comparison w i t h Monte C a r l o r e s u l t s .  The HNC c l o s u r e a l s o has the  c o r r e c t l a r g e r b e h a v i o r ; as r -*• , c ( 1 2 ) -> -3u(12). 00  I n t e g r a l e q u a t i o n t h e o r i e s can be s o l v e d a n a l y t i c a l l y i n some cases.  The MSA, or e q u i v a l e n t l y the PY a p p r o x i m a t i o n ,  analytically  f o r hard sphere p o t e n t i a l s  [64-66],  has been s o l v e d  Wertheim  s o l v e d the MSA theory e x a c t l y f o r d i p o l a r hard spheres. u s i n g r o t a t i o n a l i n v a r i a n t expansions, s p h e r i c a l model f o r any hard  sphere  has t r i e d  [67] has  Blum [ 5 4 ] ,  to g e n e r a l i z e the mean  f l u i d with e l e c t r i c a l multipolar  i n t e r a c t i o n s . He o b t a i n e d formal s o l u t i o n s which must be e v a l u a t e d n u m e r i c a l l y i n a l l but the s i m p l e s t c a s e s . However, s o l u t i o n s t o i n t e g r a l e q u a t i o n t h e o r i e s a r e g e n e r a l l y obtained numerically. procedures.  The n u m e r i c a l techniques used  involve i t e r a t i v e  The t r a d i t i o n a l method o f d e t e r m i n i n g h(12) and c(12)  t y p i c a l l y proceeds  as f o l l o w s :  say h ( 1 2 ) , i s used  i n the c l o s u r e r e l a t i o n s h i p t o o b t a i n c ( 1 2 ) ; the OZ  e q u a t i o n i s then used  an e s t i m a t e f o r one o f the f u n c t i o n s ,  to o b t a i n a new estimate f o r h ( 1 2 ) .  This cycle  can be i t e r a t e d u n t i l a d e s i r e d degree o f convergence i s o b t a i n e d .  A  method known as the Newton-Raphson t e c h n i q u e , r e c e n t l y developed by Gillan  [68], has a l s o proved v e r y u s e f u l f o r some simple models.  A  f u r t h e r d i s c u s s i o n o f these methods and t h e i r a p p l i c a t i o n to the present model can be found  i n Appendix B.  - 38 -  4.  A p p l i c a t i o n o f I n t e g r a l E q u a t i o n Methods t o E l e c t r o l y t e S o l u t i o n s  I n t e g r a l e q u a t i o n methods have been used e x t e n s i v e l y s o l v e n t models [22]  [35],  to study  E l e c t r o l y t e s o l u t i o n s , both a t i n f i n i t e  and at f i n i t e c o n c e n t r a t i o n s  have a l s o been s t u d i e d .  [21], u s i n g  only a d i p o l a r  simple  dilution solvent  F o r these systems, the f u l l HNC theory  can not  2 be  s o l v e d a t present  is  required One  and so f u r t h e r a p p r o x i m a t i o n to the HNC c l o s u r e  i n order  to o b t a i n  a tractable  theory.  such c l o s u r e , the l i n e a r i z e d h y p e r n e t t e d c h a i n  (LHNC) [36]  a p p r o x i m a t i o n , has been shown t o given good r e s u l t s f o r m u l t i p o l a r fluids  [35].  The LHNC c l o s u r e i s obtained  from the HNC c l o s u r e by  r e t a i n i n g o n l y t h e l i n e a r term of i t s expanded l o g a r i t h m . c l o s u r e w i l l be e x p l i c i t e l y d e r i v e d quadratic  hypernetted chain  of up to second For  on  r e s u l t s from r e t a i n i n g terms  s o l v e n t model, the LHNC theory  d i e l e c t r i c properties  earlier.  (QHNC) [37],  A s i m i l a r t h e o r y , the  order.  the present  QHNC t h e o r y ,  below.  The LHNC  gives  r e s u l t s f o r the  that agree very w e l l w i t h r e s u l t s from the  and both agree v e r y w e l l w i t h experiment  [41], as was noted  The LHNC and QHNC t h e o r i e s a r e a l s o i n q u a l i t a t i v e agreement  structural properties.  both t h e o r i e s  (although  computer s i m u l a t i o n  Recent work [69] has shown t h a t r e s u l t s from  the QHNC i s somewhat s u p e r i o r ) agree w e l l w i t h  (MD) r e s u l t s f o r s t r u c t u r a l p r o p e r t i e s  s o l v e n t model, a g e n e r a l i z e d  Stockmayer f l u i d  fora similar  c o n s i s t i n g of  Very recent work ( F r i e s and Patey, t o be p u b l i s h e d ) has shown that the f u l l HNC theory can be s o l v e d f o r simple s i n g l e component systems and i t s a p p l i c a t i o n to the present system looks p r o m i s i n g .  - 39 -  Lennard-Jones spheres w i t h d i p o l e and quadrupole moments those o f our w a t e r - l i k e when one c o n s i d e r s  solvent.  s i m i l a r to  These r e s u l t s a r e even more emphatic  t h a t both t h e o r i e s a r e known  [35] to g i v e  better  r e s u l t s f o r systems w i t h hard, as opposed t o s o f t , s p h e r i c a l potentials. it  The QHNC theory  has been r e p e a t e d l y  multipolar  i s a more d i f f i c u l t  theory  t o s o l v e , though  shown to g i v e s u p e r i o r r e s u l t s p a r t i c u l a r l y f o r  systems a t low d e n s i t y  [35,37],  However, u n l i k e the LHNC  c l o s u r e , the QHNC c l o s u r e i s not a s e l f - c o n s i s t a n t a p p r o x i m a t i o n [ 3 7 ] , The  LHNC theory  simplicity,  will  thus be used i n the present  s e l f - c o n s i s t a n c y , and i t s a b i l i t y  the d i e l e c t r i c p r o p e r t i e s  f o r the s o l v e n t  study because of i t s  to g i v e good r e s u l t s f o r  model.  For m i x t u r e s , such as e l e c t r o l y t e s o l u t i o n s , the g e n e r a l the OZ e q u a t i o n becomes much more c o m p l i c a t e d , task i s much s i m p l e r  at i n f i n i t e  however we f i n d  that  this  dilution.  (a)  A p p l i c a t i o n o f the Ornstein-Zernike  The  f o l l o w i n g development c l o s e l y f o l l o w s  work [21,23,41],  solution to  Equation t h a t of p r e v i o u s  We begin w i t h the OZ e q u a t i o n f o r a m i x t u r e , a  g e n e r a l i z a t i o n o f e q u a t i o n (3.13), which can be w r i t t e n as  V  1  2  )  -  C  a3  ( 1 2 )  = 72 8n  1  Y y P  J  V  1  3  where py i s the number d e n s i t y of s p e c i e s all  coordinates  of p a r t i c l e 3 of species  C  )  Y3  ( 3 2 )  ^3  (  3  '  1  9  )  Y, the i n t e g r a t i o n i s over y,  and  h g(12) a  c o r r e l a t i o n f u n c t i o n between molecule 1 o f s p e c i e s  i s the p a i r  a and molecule 2 o f  - 40  s p e c i e s 3.  The  sum  i n equation  -  (3.19) i s over a l l s p e c i e s y of  the  mixture. In the p r e s e n t electrolyte  study we w i l l  solutions.  Hence we  t h r e e component mixture and  s the s o l v e n t .  system, e q u a t i o n equations. cases  our  We  c o n s i d e r only s i n g l e  can always c o n s i d e r a s o l u t i o n to be  where + and  - w i l l designate  In the most g e n e r a l case  (3.19) can be expressed  r e s u l t , f o r the present  completely  (12) - c  decouples;  as a system of 27  c  s g  been s o l v e d  study we  ss  have o n l y to c o n s i d e r 9  t h a t i s to say the s o l v e n t remains  given  The  (3.20a)  completely  pure s o l v e n t  the s o l u t i o n s f o r h ( 1 2 ) s s  and  There are o n l y 5 unique p a i r s of  c g ( 1 2 ) f o r which to s o l v e ; we  a  a  equations.  dX_ —3  J  0  [41] and  As  by  (12) = -Ar / h (13) c (32) 2 ss ss  ( 1 2 ) are r e a d i l y a v a i l a b l e .  h g ( 1 2 ) and  coupled  d i l u t e , t h a t i s p+ = p_ = 0.  unchanged from i t s pure (or b u l k ) s o l v e n t s t a t e . system has  species  f o r a t h r e e component  Furthermore, the s o l v e n t - s o l v e n t e q u a t i o n , g i v e n  ss  the i o n i c  a  have, however, chosen to make the assumption t h a t i n a l l  solution i s i n f i n i t e l y  h  component  a  choose to s o l v e f o r those  by  h , ( 1 2 ) - c._.(12)  (3.20b)  J  h  (12) - c  (12)  - ^ j / h_ 8ir  S  (13) c _(32) S  dX,  (3.20c)  - 41 -  P  V  "  ( 1 2 )  c  +-  ( 1 2 )  =  r  S  ~ T $ + h  ( 1 3 ) s  c s  -  (  3  2  )  s  (  3  2  d  -3'  (3.20d)  8TT  P  h  +  (12) - c  r  S  (12) - -=2 /  s  (  ~  S  h +  1  3  )  c S  )  d  * >  (3.20e)  3  8TT  and  P  h  - S  (12) - c  — S  r  S  (12) = - 5 - / h o  £•  (13) c  S S  (32) dX„. —  (3.20f)  J  8ir  The s p h e r i c a l symmetry  of the i o n - i o n p o t e n t i a l r e q u i r e s  = h_+(12) = h ^ j ( r ) , and s i m i l a r l y f o r c ^ j ( r ) .  An OZ  f o r a$ = - + would thus be redundant s i n c e e q u a t i o n describes  the u n l i k e i o n - i o n c o r r e l a t i o n s .  about the m i c r o s c o p i c Following  Blum  arguments  They would g i v e no f u r t h e r  f o r ag  information  s t r u c t u r e of the system.  [52], we expand h g ( 1 2 ) and c g ( 1 2 ) i n the same  h (12) =  where  (3.20d) completely  the OZ e x p r e s s i o n s  a  a  s e t of r o t a t i o n a l i n v a r i a n t s p r e v i o u s l y d e f i n e d  a g  expression  S i m i l a r symmetry  (as p r e s e n t e d below) can be used to show that = s+, s- a r e l i k e w i s e redundant.  that h+_(12)  I h ^ mn£  y  v  ( r )  to o b t a i n  ^ ( 1 2 )  (3.21a)  - 42 -  /h n„  .(12) $ —5  (r;  ft  m n £  dO-dO,  (12) ~  .  J[*™*(12)r dOj d n  The  denominator o f e q u a t i o n (3.21b) serves  h™^^ (r). v  The i n t e g r a t i o n s i n e q u a t i o n  o r i e n t a t i o n and we note  JdB  The  0  / J  =  f o r the c o e f f i c i e n t s c  In g e n e r a l ,  0  sinedBdW  = STT .  (3.22)  2  ^(r). ctg,uv m  these expansions w i l l  have an i n f i n i t e  number of terms,  i n the MSA and the LHNC t h e o r i e s o n l y a f i n i t e number o f r o t a t i o n a l  i n v a r i a n t s are r e q u i r e d convolution  to form a c l o s e d  s e t under t h e g e n e r a l i z e d  o f the OZ e q u a t i o n (as d e s c r i b e d  equivalently stated  that  below).  I t can be  this p a r t i c u l a r set o f * ™ * ( 1 2 ) w i l l  generate  themselves and one another when the angular i n t e g r a t i o n s o f  e q u a t i o n (3.19) are performed. find  (3.21b) are over a l l a n g u l a r  a  v  only  to n o r m a l i z e the c o e f f i c i e n t s  f u n c t i o n c g ( 1 2 ) i s a l s o expanded i n t h i s manner w i t h s i m i l a r  expressions  but  2  that  ljl  -  (3.21b)  that only  the c l o s u r e  Under the LHNC ( o r the MSA) c l o s u r e , we  r o t a t i o n a l i n v a r i a n t s belonging  expressions.  c o n s t i t u t e s an exact  In the LHNC ( o r MSA) theory  solution.  This  from which, i n p r i n c i p l e , an i n f i n i t e generated.  to t h i s  this  f i n i t e set  i s not the case i n the QHNC c l o s u r e set of r o t a t i o n a l i n v a r i a n t s i s  The f u n c t i o n s h g ( 1 2 ) and c g ( 1 2 ) w i l l a  s e t can appear i n  a  conform to the  - 43 -  same symmetry as t h a t c o n t a i n e d i n the terms o f the expansion of u g(12). a  appear  We know then t h a t a t l e a s t those r o t a t i o n a l i n v a r i a n t s t h a t  In the expansion o f u g ( 1 2 ) w i l l  appear  a  h g ( 1 2 ) and c g ( 1 2 ) . a  A d d i t i o n a l terms may be r e q u i r e d  a  a closed  i n the expansions of so as t o form  s e t , a l t h o u g h t h i s i s not the case f o r the i o n - s o l v e n t and  ion-ion correlation functions. We can then w r i t e  u /-I'M - u h (12) = h l g  0 0 0 ± g  f  the i o n - s o l v e n t  p a i r c o r r e l a t i o n f u n c t i o n as  \ _i_ 1.011/ . * 0 1 1 , . , ,022. . .022.... (r) + h (r) * (12) + h (r) * (12)  ?i si ^ > r  =  3  o2 9 d ) 2 / \ is  u022. . _ l , 0 2 2 , h (r) - 2 - l , 2  .  r  h  l g  0 U  (12)  ( c f . Chapter I I  dfi.dfi, 1 2  }  l  s  0  (  r  )  +  . ,022 , i s , 0 - 2 h  (  - / h. (12) $ (8rr ) 1  Z  f o l l o w from e q u a t i o n (3.21b).  symmetric  *  2  ( 8 7 r  = 5/8  h^j(12),  N  ( 2 . 2 0 ) ) , and where  h  , and  0  i g  022 w i t h i = + or - and * (12) as p r e v i o u s l y d e f i n e d equation  , (3.23)  l o  i g  1  (3.24a)  N 1  r  )  0 2 2  ]  (12) dfl d n  S  1  9  (3.24b)  Z  The i o n - i o n p a i r c o r r e l a t i o n f u n c t i o n ,  i s s p h e r i c a l l y symmetric  as r e q u i r e d  i o n - i o n p a i r p o t e n t i a l and hence,  by the s p h e r i c a l l y  (3.25)  The  solvent-solvent pair c o r r e l a t i o n function w i l l  be the same as t h a t  of the pure s o l v e n t and i t s expansion i s g i v e n by C a r n i e (cf.  equation  and Patey [41]  (2.11) of Ref. 4 1 ) . T h i s expansion does c o n t a i n  t h a t do not appear i n the s o l v e n t - s o l v e n t p a i r p o t e n t i a l . the i o n - s o l v e n t and i o n - i o n d i r e c t in  terms  Expansions o f  c o r r e l a t i o n functions are equivalent  form to t h e i r r e s p e c t i v e p a i r c o r r e l a t i o n f u n c t i o n s . The  expansions f o r c ^ ( 1 2 ) and h g ( 1 2 ) can then be s u b s t i t u t e d a  a  i n t o e q u a t i o n s (3.20) and the r e s u l t i n g e x p r e s s i o n s transformed  [52],  Fourier  A f t e r much s i m p l i f i c a t i o n and rearrangement, we have  h™r (k) - c " f (k) = p cip ap s  I „ „  n  lV2  Z 21 mnn,  l 1(10 c 1 ( k ) as s (J 2  h  (3.26a)  1  with  ™Vl l n  t,M Z = * mnn^ 2  1  where (cf.  f ^mnx.  g fl  n  *2  . m+n+n. * <?* > ("I) { (2n^ + 1) m +  1  L  i s a 6-j symbol  Chapter I I e q u a t i o n  [70] and f  (2.12)).  o f the OZ e q u a t i o n  m  n  n  % I }(* n^ 0 J  2 n  I SL ) (3.26b) 0 0  ^ has been p r e v i o u s l y  The sum over n^,  f i n i t e , l i m i t e d by the non-zero v a l u e s ' convolution  I  1  of Z  . m n n^  defined  %^ i s always  This  generalized  b  i s e s s e n t i a l l y t h a t of Blum [54].  The  - 45 -  coefficients given  ( i n k space) of the transformed c o r r e l a t i o n f u n c t i o n s are  by the Hankel t r a n s f o r m s  h  (k) =  where the t i l d e  J r  j ^(kr) h  ( r ) dr  denotes a Hankel t r a n s f o r m ,  spherical Bessel I t w i l l now  4TT1  f u n c t i o n of order  i = /-l,  (3.27)  and j ^ ( k r ) i s the  I.  be convenient to i n t r o d u c e  the f u n c t i o n  n(12) d e f i n e d  TI(12) = h(12) - c(12)  which w i l l  by  (3.28)  a l s o be expanded i n the same manner as h(12) and c ( 1 2 ) .  f u n c t i o n h(12) i s p r e f e r e n t i a l l y used s i n c e i t i s a smooth,  The  continuous,  and w e l l behaved f u n c t i o n of r , u n l i k e the f u n c t i o n h(12) which i s discontinuous treat  at r = d.  i n numerical  1_ P s  1_ P s  n " ( r ) are thus e a s i e r to  transformations.  Then e v a l u a t i n g expressions  The coe f f i c i e n t s  the Z  mnn  i n equation  (3.26a) we are l e f t  w i t h the  1  ( i n k-space)  ~000 c"000 .  ti.  is  000 •• is  sj  1 roll c. "Oil +,- B8- n "022 ~022 . c. ,  — - s - n.  ~000 c , ss  3  is  js  5  is  js  (3.29a)  (3.29b)  - 46 -  1 — p P  ~011 1 ~011 ."110 . ~112. , 4 ~022 .~211 , , ~213 n. =-s-h. (c + 2 c )+--=- h. (c + 6 c ), is 3 is ss ss 5 is ss ss  1 — p  "022 rOH (^ c"121 +. c~123. . 4 -022 .-222 ~220 . , ~224. , . r\ = h. ) + -g- h. (c - c + 4c ) (3.29d) is is 6 ss ss 5 is ss ss ss  0  N  „ . (3.29c) Q  and  A  s  0 O  Q  J  , . , "101 " O i l . "202 where we have a p p l i e d the symmetry requirements c ^ = c ^ and c ^ = -  g  ~Q22 c , and s i m i l a r l y 1 s (and  s  g  f o r h ( 1 2 ) . C e r t a i n p r o j e c t i o n s have been s e t to zero  hence do not appear) i f that p r o j e c t i o n does not occur i n a  particular correlation function.  For example, c ^ ^ ( k ) = 0 and ss  n ^ * ( k ) = 0 s i n c e the s o l v e n t has no charge and the ions have no g  dipole.  The i o n - s o l v e n t  correlations  c o r r e l a t i o n s d i r e c t l y , and the i o n - i o n  i n d i r e c t l y , depend on the s o l v e n t - s o l v e n t  i n d i c a t e d by (3.29). shown by the f a c t  The s o l v e n t - s o l v e n t  c o r r e l a t i o n s as  c o r r e l a t i o n f u n c t i o n s , as  that e q u a t i o n (3.20a) c o m p l e t e l y decouples, can be  thought of as constants  depending only upon the pure s o l v e n t  system  [41].  (b)  The  The LHNC C l o s u r e  Approximation  LHNC c l o s u r e was chosen i n the present  e v i d e n c e t h a t the QHNC theory c o r r e l a t i o n functions it  study.  There has been no  g i v e s b e t t e r r e s u l t s f o r the i o n - i o n  [21,22], or f o r d i e l e c t r i c p r o p e r t i e s  [41,69] and  i s these p r o p e r t i e s i n which we a r e most i n t e r e s t e d . The LHNC c l o s u r e  may be d e r i v e d  i n s e v e r a l ways and may appear i n d i f f e r e n t forms.  We  - 47 -  note t h a t the LHNC approximation superchain  (SSC) approximation  i s e s s e n t i a l l y e q u i v a l e n t t o the s i n g l e  of Wertheim [ 7 1 ] .  We s t a r t w i t h the HNC e q u a t i o n h(12) g i v e n by e q u a t i o n spherically  (3.18).  U s i n g the d e f i n i t i o n of  (3.12) and the s i m i l a r r e l a t i o n s h i p  f o r the  symmetric p r o j e c t i o n  ,000, . 000, , ctg cxg " » N  h  we r e a r r a n g e  c  a e  (  r  )  =  g  (  r  )  (3.30)  1  (3.18) t o o b t a i n  ( 1 2 ) = h ( 1 2 ) - In g°°°(r) - i n [ l + X ( 1 2 ) ] - 3 u ( 1 2 ) a p  a p  (3.31a)  ag  where  h X  3  F o r rani * 000,  ag< > 1 2  -  raR  B  ( 1 2 ) - h°° °(r) R  000, "  h™*(r) - g^*(r)  *  by d e f i n i t i o n .  (  3  '  3  1  b  )  - 48 -  We now expand £n[l + X ( 1 2 ) ] i n (3.31a) as a T a y l o r s e r i e s . a R  In the LHNC theory the  only  the l i n e a r  term i n X 3 ( 1 2 ) i s r e t a i n e d ( f o r a  QHNC c l o s u r e , terms to the order  <XB = V  C  (12)  1 2 )  - * CB N  We now r e c a l l the d e f i n i t i o n  .  of X  ( R )  2  are retained)  -W  ( c f . equation  1 2 )  -  0 u  to give  aB<  1 2 )  ( 3  -  3 2 )  (3.21b)) f o r the c o e f f i c i e n t s  J c (12) $ *(12) d n . d f l m n  Q  - ~a ,,,,\1-/  n  'aB,uv  /[$^(12)]  0  o 2  (3.33)  •  d^dfi  2  The  LHNC c l o s u r e e x p r e s s i o n s  f o r c . ( r ) a r e obtained  the  l i n e a r i z e d HNC e q u a t i o n (3.32) i n t o (3.33), expanding both h g ( 1 2 )  and  u 3 ( 1 2 ) , and p e r f o r m i n g the r e q u i r e d  Olp  by s u b s t i t u t i n g  a  a  integrations.  i n t e g r a t i o n s a r e g r e a t l y s i m p l i f i e d by u s i n g  The  the o r t h o g o n a l i t y  o f the  I t should be noted that such an expansion w i l l r i g o r o u s l y h o l d o n l y f o r -1 < X g ( 1 2 ) j< 1. I t i s not c l e a r then why the expansion should work. For any systems p o s s e s s i n g a s p h e r i c a l l y symmetric term i n the p o t e n t i a l , t h i s expansion w i l l always be v a l i d as the a n i s o t r o p i c ( e . g . m u l t i p o l a r ) terms o f the p o t e n t i a l become s m a l l . The expansion w i l l a l s o , i n most systems, be v a l i d a t l a r g e s e p a r a t i o n s , r , s i n c e i n a  general  g^g^C ) * 1 as r ->1  0 0  and the d i f f e r e n c e , h  aft  (12) - h ^ ^ ( r ) , w i l l  v e r y q u i c k l y go to zero. Near c o n t a c t ( i . e . r " d) however, we can i n most cases expect |X > 1. T h i s f a c t can be t o l e r a t e d simply because even f o r l a r g e a n i s o t r o p i c p o t e n t i a l s , the t h e o r i e s that r e s u l t from t r u n c a t i o n of the expansion s t i l l g i v e r e s u l t s that agree w e l l [35,69] w i t h exact s o l u t i o n s (computer s i m u l a t i o n s ) . A l s o as mentioned e a r l i e r , the LHNC a p p r o x i m a t i o n can be obtained by other methods which do not i n v o l v e t r u n c a t i o n o f the l o g a r i t h m i c expansion.  - 49 -  rotational invariants.  c  ,  and  000/ >. a3 ( r ) =  c  maZ, a  3  h  .  We  ,000, a3 ( r  then have  )  ,mn^/  (r) = h  x  „  "  Z n  N  g  000, \ a3 ( r ) "  mn£,  a  3 u  q  r  s  **\/ \  (3.34a)  )  (3.34b)  K  N  Q  g  for a l l other p r o j e c t i o n s .  (  ,mn£, . a3 '  (r) - 3uaj3 (r) -  a g  000, a3  Q  0  a3  ( r )  It i s interesting  to note  that  (3.34a), the c l o s u r e r e l a t i o n s h i p f o r the s p h e r i c a l l y e x a c t l y the same as t h a t g i v e n by the f u l l HNC  equation  symmetric term, i s  equation.  We  rearrange  these f u r t h e r to o b t a i n  000/  c  ,  and  cx3  c  -,  ( r )  rr,0()0, . =  e x p t n  mn£, \ a  g  a3  000,  ( r )  "  000, *,  3u  a3  " c*3  ( r ) ]  n  000, .  x. „ m n £(,r ) - 3„umn&,( r )», ] -  (r) = g ^ ( r ) [ n r  N  ae  a 0  where these c l o s u r e s are expressed  "  ( r )  ,  1  ( 3  „mnJl,  ( ), N  r  as f u n c t i o n s of n(12) i n s t e a d  *  0  o  c  .  3 5 a )  ,(3.35b) _v 0 ocl  h(12).  The c l o s u r e r e l a t i o n s h i p s g i v e n by (3.35) a r e , i n g e n e r a l , o n l y approximate.  However, i n the present model f o r r < d $ a  have a s p e c i a l case because the hard sphere Rearranging  (3.35) we  can show that  0  a  r  find  we  potential i s i n f i n i t e .  f o r hard sphere  c° 3°(r) + T r ™ ( r ) 4- 1 = gJ00( ) = exp[  we  models  ( r ) - 0u° ° (r)] - 0 O  a  for r <  d  a e  (3.36a)  - 50 -  and  mnJi/_\ :  a3  (  r  ) +  rjamHf  , ^mn-fc/x _ _mn£/_\ n  a3  (  r  )  g  a3  (  r  ) =  [ n  a3  (  r  )  • >  " a3 0 u  (  0  r  )  ] g  mn£/ \i  a3  (  r  ) =  000/ 0  for  3  ,~  n  N  < -  36b  ^r,\  >  r < d „.  a3  We have from (3.36) t h a t g(12) = 0 f o r r < d 3 »  5  a  result  f o r hard  sphere  c  and  systems.  000/ a3 (  r  ,  ) =  "  ( 1  We then can w r i t e the c l o s u r e s  /, , 000/ a3 n  (  r  )  „, . (3.37a) / 0  )  c^ (r) = " ^ ( r )  (3.37b)  £  for  r < d g . The e x p r e s s i o n s a  (3.37) can a l s o be o b t a i n e d  the c o n d i t i o n g(12) = 0 f o r r < d g d i r e c t l y a  of  which i s an exact  n(12) and r e a r r a n g i n g . Together,  the e q u a t i o n s  to the d e f i n i t i o n  (3.28)  The c l o s u r e s g i v e n by (3.37) a r e thus  exact.  o f (3.35) and (3.37) c o n s t i t u t e the f u l l  LHNC c l o s u r e f o r the present model. will  be a d i s c o n t i n u o u s f u n c t i o n .  zero  for r < d  a B  by a p p l y i n g  We f i n d  that c ( 1 2 ) , l i k e  h(12),  However, u n l i k e h(12) which  , c(12) (and hence n(12)) w i l l  equals  have non-zero v a l u e s  5  Note t h a t t h i s i s the same e x p r e s s i o n as e q u a t i o n (3.16b) of t h e MSA c l o s u r e e x p r e s s i o n s . The MSA c l o s u r e (3.16) can thus be o b t a i n e d from the LHNC c l o s u r e by s e t t i n g for  large r ) .  =  1 f ° r > d (which r  i s exact  - 51 -  for r < d 3. a  (c)  Average I n t e r a c t i o n  Energy and P o t e n t i a l o f Mean F o r c e  We a r e very r e s t r i c t e d i n the number o f thermodynamic p r o p e r t i e s o f an e l e c t r o l y t e s o l u t i o n we a r e a b l e to i n v e s t i g a t e chosen to examine o n l y the i n f i n i t e d i l u t i o n l i m i t .  because we have Any v a l u e s we a r e  a b l e to c a l c u l a t e a r e always the l i m i t i n g v a l u e s a t i n f i n i t e d i l u t i o n . By d e f i n i t i o n , the a c t i v i t y c o e f f i c i e n t , y, w i l l be u n i t y . p r e s s u r e w i l l be z e r o . the  pure  The s o l v e n t  will  The osmotic  r e t a i n a l l the p r o p e r t i e s o f  solvent.  We can, however, c a l c u l a t e  the average i o n - s o l v e n t  interaction  energy f o r a s i n g l e i o n  U  IS  / N  i  f CD  =  u  +  U  CQ^  / N  i  (3.38a)  where Urjjj, UQQ a r e the average c h a r g e - d i p o l e and average charge-quadrupole e n e r g i e s , r e s p e c t i v e l y , of species i .  and  i s the number of i o n s  Ufjp and UQQ a r e determined by s u b s t i t u t i n g the  expansions f o r u ( 1 2 ) and g ( 1 2 ) i n t o e q u a t i o n (3.11a) and i s  the  limit  +.O.  i g  Simplifying,  taking  a g a i n u s i n g the o r t h o g o n a l i t y  o f the  r o t a t i o n a l i n v a r i a n t s , we have  ir i  r  p  q s  i  y  V  h  i s  is  (  r  )  d  r  (  3  -  3  8  b  )  - 52  -  and  CQ NT i  16TT —  U  =  The  _ » i °T V f  p  s  q  h  is  terms of the average i o n - s o l v e n t  i n themselves and moments, but we  The  ( r )  d r  '  i n t e r a c t i o n are not  that the e x p r e s s i o n  theory  for  to be d e s c r i b e d  average i o n - s o l v e n t  will  i n Chapter  i n t e r a c t i o n energies  a l s o be used i n c a l c u l a t i n g s o l v a t i o n e n e r g i e s commonly used Born theory,  the  i n f i n i t e d i l u t i o n i s given by  a d i e l e c t r i c constant).  The  We  6  (completely  Born theory,  solvent-solvent of an  i o n has  solvent  long-ranged  solvent-solvent  correlations.  solvent-solvent  term was  Q  C  important  for a single ion In  described  can  the  s t r u c t u r e due [72]  an  ion  by s p e c i f y i n g  however, makes no  d i l u t i o n by  correlation functions.  a very  _ 8  a l s o be needed  for ions.  have r e c e n t l y developed a theory  i o n at i n f i n i t e  3  IV.  the change i n energy of a d i p o l a r hard sphere s o l v e n t presence of an  z'  multipole  the energy gained by c h a r g i n g  take i n t o account changes i n the ion.  only  3  s o l v a t i o n energy of a s i n g l e i o n a t  immersed i n a continuum s o l v e n t  of the  (  i n g i v i n g a r e l a t i v e importance of the  find  i n the d i e l e c t r i c  v j  ,022, is  attempt  to the  to  presence  f o r determining due  to  the  c a l c u l a t i n g the changes i n  I t was  ( i . e . a 1/r  found t h a t the  presence  dependence) e f f e c t on  For the d i p o l a r hard sphere system,  found to be o p p o s i t e  i n s i g n and  roughly  the half  E q u i v a l e n t l y , the s o l v a t i o n energy i s g i v e n by the energy r e l e a s e d i n t r a n s f e r r i n g an i o n from a near vacuum (the i d e a l gas phase) to being c o m p l e t e l y immersed i n the s o l v e n t .  - 53 -  the magnitude o f the i o n - s o l v e n t  interaction.  The r e s u l t i n g s o l v a t i o n  energies  f o r i o n s i n a d i p o l a r hard sphere s o l v e n t  constant  equal to that o f water ( u s i n g c r y s t a l  d i a m e t e r s ) are w e l l w i t h i n the c o r r e c t order experimental values. water-like  solvent.  T h i s theory  having a d i e l e c t r i c  radii  o f magnitude o f  i s c u r r e n t l y being  I n i t i a l r e s u l t s look  I t i s a l s o convenient f o r us to i n t r o d u c e i  g i j  ( 1 2 ) = exp [ - e  W i j  a p p l i e d to the  p r o m i s i n g , and a r e  s u r p r i s i n g l y s i m i l a r to those f o r the simple d i p o l a r  mean f o r c e a t i n f i n i t e d i l u t i o n , w j ( 1 2 ) .  to determine i o n  solvent.  an i o n - i o n p o t e n t i a l o f  w j ( 1 2 ) i s d e f i n e d by i  (12)l  (3.39)  where g ^ j ( 1 2 ) i s the i o n - i o n p a i r d i s t r i b u t i o n f u n c t i o n . the  potential associated  at i n f i n i t e d i l u t i o n . i  w i t h the average f o r c e between two i o n s i and j  ( I t i s the p o t e n t i a l o f the f o r c e a c t i n g between  and j i n the pure s o l v e n t . )  I t includes a l l solvent e f f e c t s that  i n f l u e n c e the i o n - i o n c o r r e l a t i o n s . the  w^j(12) i s  Rewriting  e q u a t i o n (3.39) and u s i n g  f u l l HNC a p p r o x i m a t i o n (3.18), we have  3 w ( 1 2 ) = -in g ±j  i;j  ( 1 2 ) - 011^(12) - 11^(12)  .  N o t i n g t h a t any i o n - i o n p o t e n t i a l must be s p h e r i c a l l y symmetric present  model), we o b t a i n  8w  (3.40a)  ( f o r the  f o r r > d^j  (r)= ^  1  - nj°°(r)  (3.40b)  - 54 -  where - T l ^ j ^ ( r ) ^  s  t  h  e  c o n t r i b u t i o n t o the i o n - i o n p o t e n t i a l of mean  f o r c e which depends upon the s o l v e n t . e a s i l y o b t a i n e d once r|000( ) ^as r  It  D  e  e  determined  n  ( c f . equation  (3.29)).  can be shown [21,25,25] t h a t as r •*  0 0  q  where e is  The p o t e n t i a l of mean f o r c e i s  i  q  i  i s the pure s o l v e n t d i e l e c t r i c  Q  the c o r r e c t  (d)  constant.  Expression  (3.41)  continuum l i m i t .  Method o f S o l u t i o n  S o l u t i o n s f o r the present equations (3.37).  system a r e o b t a i n e d by s o l v i n g the  g i v e n by (3.29) s u b j e c t to the c l o s u r e s g i v e n by (3.35) and Upon c l o s e r i n s p e c t i o n of e q u a t i o n  n^_.(12) = n ^ j ^ ( r ) depends d i r e c t l y functions.  (3.29), we f i n d  that  o n l y upon i o n - s o l v e n t c o r r e l a t i o n  Hence, once s o l u t i o n s have been o b t a i n e d  f o r the i o n - s o l v e n t  c o r r e l a t i o n s , the v a l u e o f the i o n - i o n c o r r e l a t i o n f u n c t i o n s a r e f i x e d by (3.29a).  We f i n d a l s o  that i n the LHNC t h e o r y n?°°(r) depends o n l y is  on o t h e r s p h e r i c a l l y decouples  symmetric c o r r e l a t i o n s  ( c f . equation  It  from the e x p r e s s i o n s f o r the o t h e r p r o j e c t i o n s of r i i ( 1 2 ) , s  Oil 022 namely n. ( r ) and n. ( r ) , g i v e n by the two remaining IS  (3.29).  (3.29b)).  Xs  equations of  The major task thus becomes to o b t a i n c o n s i s t a n t s o l u t i o n s t o  - 55 -  the coupled equations  (3.29c) and (3.29d).  For computational purposes,  i t i s convenient  to express a l l  e q u a t i o n s as f u n c t i o n s of two unknowns; we have chosen The  closure relationships  i n t h i s form.  and  (3.35) and (3.37) have a l r e a d y been expressed  We have, however, s t i l l  from the reduced  h(12) and c ( 1 2 ) .  OZ equations  rearranging, equations  to remove the dependence on h(12)  (3.29).  R e p l a c i n g h ™ ^ by  C"!™^ + c^g^) 1  (3.29c) and (3.29d) become  .- ~011 . - ~ 0 2 2 - . . - ,~ -022 . ~ - O i l . l is 2 is s " 4> 2 4 is 3 is ; = = — W l /  -Oil h, (k) = ^  (  c  c  +  c  c  )  (  1  ( l / p  /  P  c  +  c  - c )(l/P  a  ( c  +  - c ) -  s  4  c  x  C  c  c c  3  2  )  (3.42a)  and  f  ~022 TI. (k) = i  ( c  - ~022 . ~ -OIK,., 4 is 3 is s " = c  +  c  C  )  (  1  (1/P  s  where  /  S  =  c  2  C  l  - . . ~ ,~ ~011 . " -022. 3 l is 2 is = —  }  +  c )(l/p  -  4  c  +  e  (  -  8  1 -110 3" <  l  c  P  J  C  c  )  l  +  -  c  c  ss  }  4 ,~211 , , ~213. ss ss ( c  +  C  2  c  c  }  (3.42b)  3  "112 2  s s  =  c  6  C  }  (3.42c)  ,„ (  3  *  4  2  d  /  O  J  .  )  When e v a l u a t i n g these e x p r e s s i o n s n u m e r i c a l l y , we must take i n t o account the i = ~1 that r e s u l t s from and product o f any two p r o j e c t i o n s 2  n ( k ) and ( r ) where both V and £" are odd. F o r odd V these k-space f u n c t i o n s are p u r e l y imaginary. I B a A ,  and i"  - 56 -  ~ _ ,1 "121 , "123, c„ = (T- c + c ) 3 6 ss ss  4 ,~222 ~220 . , ~224 c, = - F - (c - c + 4 c ), 4 5 ss ss ss  and  We  (3.42e)  (3.42f)  N  also  rewrite  e q u a t i o n (3.29b) as  „ -000 "is  (10 -  rooo rooo  P h . s is  h  ss  (3.43a)  1 + p h s ss  where we have used the r e l a t i o n s h i p  "000 "000 "000 n = p h c ss s ss ss  which comes from s o l v i n g  (3.43b)  (3.20a), the pure s o l v e n t  OZ e q u a t i o n .  Using  (3.43a), we can show that  ,000 n. is  , 000 "000 is js h  then i n s e r t t h i s r e s u l t i n t o  000 n..  "000 is  (3.44a)  c  1 + p s  We  000 h. js h ss 0  0  0  (3.29a) which becomes  "000 js , 1,~011 , "011 000 3 is is 1 + P h s ss  "011 js  . 8 "022 5 is  . "022 is  "022  (3.44b)  - 57 -  The  r e l a t i o n s h i p s g i v e n by (3.35) and (3.37) c o n s t i t u t e complete  c l o s u r e s and c o u l d be s o l v e d as they some i n a c c u r a c y would be i n t r o d u c e d sphere p o t e n t i a l . technique  first  are.  However, at h i g h d e n s i t i e s  from the HNC treatment o f the hard  To reduce t h i s e r r o r , we apply a p e r t u r b a t i o n  suggested by Lado [73].  charge-multipolar  T h i s technique  ensures t h a t f o r  000 HS sphere systems g ( r ) •*• g g ( ) *  hard  r  a  R  a  n  the l i m i t  HS of v a n i s h i n g charge and m u l t i p o l e s , where g ( r ) i s the "exact" ctp sphere r e s u l t .  hard  0  The  a p p l i c a t i o n o f t h i s method i s v e r y s t r a i g h t f o r w a r d i n the LHNC  theory.  In the LHNC c l o s u r e e x p r e s s i o n  for^g^( )> c  o n l y the s p h e r i c a l l y  r  Hence c^^P ( r )  symmetric p r o j e c t i o n s o f n ( 1 2 ) and u ( 1 2 ) appear. Q  Q  Otp  dp  Ctp  can o n l y depend upon the i s o t r o p i c p a r t o f the p a i r p o t e n t i a l ( c f . equation  (3.34a)).  I f ^ g ^ ( ) c o n t a i n s o n l y the hard u  000 i n t e r a c t i o n , we have o n l y t o r e p l a c e g r e l a t i o n s h i p s f o r c™g^ ( r ) . hard  For the present  [74] f i t to Monte C a r l o d a t a .  011 (3.42) f o r n is  and  a f t  HS ( r ) by  ( r ) i n a l l closure  study,  g^g ( r ) , the "exact"  sphere r a d i a l d i s t r i b u t i o n f u n c t i o n , i s taken to be the  Verlet-Weis  c  sphere  r  is  (  r  )  g  i s  We can then s o l v e  equations  022 ( r ) and n ( r ) , s u b j e c t to the c l o s u r e r e l a t i o n s Is  (  r  )  [  c. Is  n  i s  (  r  )  "  3 u  is  ( r ) = -n. is  (  r  )  1  " is  (r),  n  ( r )  >  r  > is d  r < d. is  (  3  '  4  5  a  )  (3.45b)  - 58  for  (mnJO = ( O i l ) and  -  (022).  In o b t a i n i n g s o l u t i o n s f o r n ( 1 2 ) , we a g  Hankel transforms (3.29) has  (as given by  d i r e c t l y evaluate numerically. i n t e g r a l transforms  transforms,  and  equation  the c l o s u r e e x p r e s s i o n  Hankel transforms  f u n c t i o n s of second-order or h i g h e r  introduce  to perform  (3.27)) s i n c e the reduced OZ  been s o l v e d i n k-space and  (3.37) are i n r e a l space.  must be prepared  (3.35) and  using s p h e r i c a l Bessel  (i.e. I  2) are v e r y d i f f i c u l t  to  For t h i s purpose i t i s u s e f u l to as d i s c u s s e d by Blum [54],  t h e i r i n v e r s e s , needed i n the present  The  study  integral  are  defined  by  c  m n 0  (r) = c  m n 0  (r),  (3.46a)  c  m n l  (r) = c  m n l  (r),  (3.46b)  "mn2, . mn2, , _ r<*> c c (r) = c (r) - 3 J  c  mn2,  v  ~mn2,  (r) = c  (r) s  3  r  where the hat  (~)  The  J  o  i  2  ~mn2  s  c  •  %  / o  used i n o b t a i n i n g c  ,  mn  Hankel ( F o u r i e r ) t r a n s f o r m  f i r s t - o r d e r Hankel transform  of c  i f I i s odd.  m n  We  ^  the  i f % i s even, or  write  We  ^ ( r ) depends o n l y  c a l c u l a t i o n of c - ^ ( k ) i s then reduced to t a k i n g  zeroth-order  t t i r * \  (3.46d)  function. m n  / ^  (3.46c)  ( s ) ds  i n d i c a t e s the i n t e g r a l transformed  note that the i n t e g r a l t r a n s f o r m on I.  rr  J  ( s ) , —^ ds,  m n 2  the  \  - 59 -  c^Oc)  c  mn£  = 4IT J " r  ( r )  =  _1_  2  j ( k r ) c \r)  dr,  mn  Q  j» 2  ~mn*  k  (k)  d  (3.47a)  k  (  3  >  4  ?  b  )  2ir  when % i s even and  c  c  m n J l  ( k ) = 4rri  mn*  ( r )  r  2  j (kr) c  j» Z  =  m n £  x  k  r  )  ( r ) dr,  rmn£  ( k )  (3.48a)  d f c  (  3  >  4  g  b  )  2ir  when % i s odd. h™ -^. 1  are  S i m i l a r expressions  The i n t e g r a l  the z e r o t h  can a l s o be w r i t t e n f o r n  m n  ^ and  transforms are e a s i l y performed n u m e r i c a l l y , as  ( F o u r i e r ) and f i r s t - o r d e r Hankel t r a n s f o r m s ,  done by Fast F o u r i e r transform  which can be  techniques (see Appendix B ) .  C a l c u l a t i n g the Hankel t r a n s f o r m s i n t h i s manner has a second advantage i n t h a t i t allows be  treated exactly.  c  long-range p a r t s of c e r t a i n  £ s ^ ) w i l l have a long-range term (due t o r  u ^ ^ ( r ) ) w i t h a 1 / r dependence. Xs 2  I t must be t r e a t e d w i t h great  s i n c e i t becomes a 1/r term i n the Hankel t r a n s f o r m result,  projections to  t h i s term i s always s u b t r a c t e d  transformed a n a l y t i c a l l y .  integral.  from ^ g ^ ( ) and F o u r i e r c  r  What remains of c ^ ^ ( r ) 1 s  can then be  care As a  - 60 -  n u m e r i c a l l y transformed.  S p e c i a l a t t e n t i o n must be p a i d i n i n v e r t i n g  n ^ j ^ ( k ) , as g i v e n by e q u a t i o n  (3.44b).  to have the c o r r e c t 1/r b e h a v i o r  ~ n  0 0 0  / , >. (k)  We can show t h a t i f 3 w ^ ( r ) i s  f o r l a r g e r r , then as k ->• 0  / 4TT q ^  o ~ —  £  o  J  Thus upon n u m e r i c a l  1  1  /, x (3.49)  2 .  /  t  k  t r a n s f o r m a t i o n o f T\j*?^(k), the k = 0 c o n t r i b u t i o n  to the i n t e g r a l must be taken  i n t o account  analytically.  Oil 022 The d i s c o n t i n u i t i e s i n c. ( r ) and c. ( r ) must a l s o be handled is is care.  f  with  022 c. ( r ) must be t r e a t e d as being dual v a l u e d a t r = d. when is is  performing  i t s integral  t e c h n i q u e s , we f i n d  transform.  i t convenient  In u s i n g Fast F o u r i e r t r a n s f o r m to perform  the i n t e g r a t i o n s i n (3.47)  Q  and  (3.48) u s i n g the t r a p e z o i d a l r u l e ,  and hence we have o n l y t o  average the d u a l valued p o i n t s at the d i s c o n t i n u i t i e s . S o l u t i o n s f o r the p r e s e n t model a r e o b t a i n e d procedure  d e s c r i b e d i n Appendix B.  represented  by the i t e r a t i v e  In a l l c a l c u l a t i o n s  f u n c t i o n s were  n u m e r i c a l l y u s i n g 512 p o i n t s with a g r i d width  The r e s u l t s o b t a i n e d w i l l  be presented  Ar = 0.02d  and d i s c u s s e d i n Chapter  V.  s  A l l i n t e g r a t i o n s are performed u s i n g the t r a p e z o i d a l r u l e . As a check, Simpson's r u l e i s used to repeat the i n t e g r a t i o n s i n the d e t e r m i n a t i o n o f i n t e r a c t i o n e n e r g i e s o f the system. Good agreement was always o b t a i n e d .  - 61  -  CHAPTER IV  DIELECTRIC THEORY OF ELECTROLYTE SOLUTIONS  1.  Introduction  E l e c t r o d y n a n i i c and  e l e c t r o s t a t i c p r o p e r t i e s are among the most  d i s t i n c t i v e used i n r e c o g n i z i n g  an e l e c t r o l y t e s o l u t i o n [4-6,8,75],  e l e c t r o l y t e s o l u t i o n i s c h a r a c t e r i z e d by a l a r g e e l e c t r i c The  solvent  possesses a l a r g e d i e l e c t r i c constant  ionic concentration.  The  conductivity.  which decreases w i t h  d i e l e c t r i c p r o p e r t i e s of any  solution  substance) depend on the e l e c t r i c moments, both permanent and of the molecules which compose i t . The o c c u r s because of l a r g e e l e c t r o s t a t i c the by  solvent molecules. the  charge-dipole  s o l v a t i o n and  (any  induced,  s o l v a t i o n of an i o n by a  i n t e r a c t i o n s between the  These i n t e r a c t i o n s w i l l g e n e r a l l y be  potential.  An  solvent  ion  and  dominated  Hence i n e l e c t r o l y t e s o l u t i o n s , i o n  the d i e l e c t r i c p r o p e r t i e s of s o l u t i o n have t h e i r b a s i s i n  r e l a t e d p r o c e s s e s at the m i c r o s c o p i c  level.  A more complete  u n d e r s t a n d i n g of t h e i r d i e l e c t r i c p r o p e r t i e s would improve the  general  u n d e r s t a n d i n g of e l e c t r o l y t e s o l u t i o n s . The its  d i e l e c t r i c response of an e l e c t r o l y t e s o l u t i o n i s r e p r e s e n t e d  "apparent" d i e l e c t r i c c o n s t a n t .  apparent d i e l e c t r i c constant  To  of a s o l u t i o n , we  n o t i o n of a s t a t i c d i e l e c t r i c c o n s t a n t . the  understand what i s meant by  s t a t i c d i e l e c t r i c constant  must f i r s t  the  understand  For a non-conducting  i s well defined.  by  material  I t i s r e l a t e d to  p o l a r i z a t i o n , P (the mean d i p o l e moment per u n i t volume), of  the  an  the  - 62  i n s u l a t i n g material placed  -  i n a homogeneous e l e c t r i c  field  , E_, by  the  equation  ( e - l ) E = 4TTP  where e i s the exists within  static  dielectric  constant.  The  [76]  The  net  electric  field  that  the m a t e r i a l , which takes i n t o account the p o l a r i z a t i o n  ( i . e . the o r i e n t a t i o n of permanent and related  (4.1)  to i t s  induced d i p o l e s ) , can  be  capacitance.  capacitance,  C,  for a p a r a l l e l  p l a t e c a p a c i t o r can be  expressed  as  where A i s the area  of the p l a t e s of the  between the p l a t e s , and The  dielectric  £/5o» where 5o principle, field  constant, i s the  £ i s the e l e c t r i c e, i s d e f i n e d  electric  e w i l l depend on  limit.  c a p a c i t o r , d i s the p e r m i t t i v i t y of the  strength,  Hence, the d i e l e c t r i c  constant  but we  In  consider  of a m a t e r i a l  dielectric  a  [78]  constant,  i . e . between two capacitor.  to now  introduce  o n l y the can  determined by i t s response c h a r a c t e r i s t i c s i n an e l e c t r i c a l It i s useful  material.  as the r e l a t i v e p e r m i t t i v i t y ,  p e r m i t t i v i t y of f r e e space.  field  distance  low  be  circuit.  a frequency dependent complex  e*(w), which f o r both c o n d u c t i n g and  o p p o s i t e l y charged i n f i n i t e p a r a l l e l  non-conducting  p l a t e s as  in  - 63  systems i s g i v e n  -  by  e*(a)) = e'(u)) -  The  real  above.  part,  e'(u),  r e p r e s e n t s the  I t approaches the  frequency since  the  for reorientation  part,  i s known as  orientation  by  dielectric  s t a t i c v a l u e as  (4.3)  constant  w ->• 0.  the  of m o l e c u l a r d i p o l e s ) .  dielectric  loss.  charge; e i t h e r  intermolecular  This  from the  forces  discussed  e' depends  p o l a r i z a t i o n process requires  (i.e.  motion of e l e c t r i c a l  ie"(uj).  on  a f i n i t e time  e"(to), the  period  imaginary  l o s s term r e s u l t s from impedence of  or from a c t u a l  the  dipole  charge  transport  (conduction). For  a conducting s o l u t i o n ,  a c c o r d i n g to the  relationship  E*(UJ) w i l l d i v e r g e at low [8]  E*(U)) •*• 4Tro/io) as  where a i s the  conductance of the  apparent d i e l e c t r i c  constant,  e  where cr i s taken as r e p r e s e n t s the  SOL  £  c m  OJ ->- 0,  solution. ,  limit io 0  (4.4a)  Hence, one  defines  of a c o n d u c t i n g s o l u t i o n  [e*(u) -  4TTQ-  ioj  a) •*• 0.  the  by  ]  the measured z e r o - f r e q u e n c y c o n d u c t i v i t y .  measured d i e l e c t r i c  remain f i n i t e as  frequency  (4.4b)  e  goL  c o n s t a n t of a s o l u t i o n which w i l l  - 64 -  The c o n d u c t i v i t y of an e l e c t r o l y t e  s o l u t i o n has made the  d e t e r m i n a t i o n of i t s d i e l e c t r i c constant e x p e r i m e n t a l l y d i f f i c u l t . measurement  of the pure s o l v e n t d i e l e c t r i c constant  i s relatively  The easy  to perform ( a t low f r e q u e n c i e s ) due to the small or n e g l i g i b l e c o n d u c t i v i t i e s of pure l i q u i d s .  As i o n i c c o n c e n t r a t i o n s i n c r e a s e ,  h i g h e r f r e q u e n c i e s are r e q u i r e d to o f f s e t c o n d u c t i v i t y and other i o n i c e f f e c t s . "apparent" d i e l e c t r i c  the i n f l u e n c e of i n c r e a s e d  E a r l y measurements of the  c o n s t a n t s of e l e c t r o l y t e  s o l u t i o n s were a l l  c a r r i e d out at low c o n c e n t r a t i o n s and low f r e q u e n c i e s ( i . e . < 10 MHz)[6,15,75].  Though many d i f f e r e n t  trends  (some c l e a r l y  i n error)  appear i n the e a r l y l i t e r a t u r e , some c a r e f u l measurements[15,79] gave s t r o n g evidence  of the Debye-Falkenhagen (DF) e f f e c t  at low c o n c e n t r a t i o n s .  ( d i s c u s s e d below)  L a t e r measurements have g e n e r a l l y a l l been at  h i g h e r c o n c e n t r a t i o n s and at h i g h e r f r e q u e n c i e s ( i . e . > 100 [6,75].  MHz)  C o - a x i a l t r a n s m i s s i o n l i n e s , waveguides, and microwave i n f e r o -  meter techniques  have been a p p l i e d i n h i g h frequency measurements.  these h i g h frequency constructed imaginary  techniques, Cole-Cole  showing the frequency  In  [95] diagrams are u s u a l l y  dependence of e  1  and e", the r e a l and  p a r t s , r e s p e c t i v e l y , of the complex d i e l e c t r i c  However, d e t e r m i n a t i o n of the zero frequency  constant.  value u s u a l l y r e q u i r e s a  l o n g e x t r a p o l a t i o n from at best s e v e r a l p o i n t s and at l e a s t one assumed value.  U s u a l l y i t i s the v a l u e of the h i g h frequency  dielectric  constant  has  to o b t a i n .  e<x»  the  t h a t a r i s e s o n l y from i n t r a m o l e c u l a r p o l a r i z a t i o n  e f f e c t s , t h a t i s assumed. been hard  limit,  Hence even moderate ( i . e . 1-2%) a c c u r a c y has  At moderate to h i g h c o n c e n t r a t i o n s t h i s  accuracy  proved to be more than adequate to f o l l o w the l a r g e trends t h a t  - 65  exist.  High frequency methods have g e n e r a l l y  concentrations [80,81] has  where trends w i l l be  been used at low  l e s s pronounced.  i o n i c concentrations.  f o r comparison w i t h the present may  not  Only  also define  Coulomb's law.  The  [76]  constants  These r e s u l t s w i l l be  useful  study.  the d i e l e c t r i c constant of a system  f o r c e , F ^ j ( r ) , between two  medium w i t h a d i e l e c t r i c  ionic  recently  an attempt a g a i n been made at measuring d i e l e c t r i c  of s o l u t i o n s at low  We  -  constant  £g  (the  charges i n a continuum  s u b s c r i p t E i s used here  to denote the e q u i l i b r i u m c o n t r i b u t i o n as d i s c u s s e d  F ,(r) ±  using  below) i s given  =  by  2  (4.5a) e r E  and  hence the p o t e n t i a l , u ^ j ( r ) , i s  i i u..(r) = - i - i q  As we  will  describe  below, t h i s e x p r e s s i o n  q  .  (4.5b)  (4.5b) a l s o d e f i n e s  the  large  r asymptotic form of an e f f e c t i v e i o n - i o n p a i r p o t e n t i a l f o r a s o l u t i o n containing  a f i n i t e concentration  the d i e l e c t r i c  of i o n i c s o l u t e p a r t i c l e s .  constant f o r a s o l u t i o n which may  the pure s o l v e n t .  Here the  not  be equal to that  d i e l e c t r i c constant r e p r e s e n t s  long-range s h i e l d i n g of e l e c t r i c  charges by the  is  solvent.  of  the Therefore,  £g  One must always be m i n d f u l of which system of u n i t s one i s working i n and whether i t i s r a t i o n a l i z e d or u n r a t i o n a l i z e d [77]. We have chosen to use the u n r a t i o n a l i z e d system where the f a c t o r 4TT w i l l appear i n Maxwell's e q u a t i o n s and not i n Coulomb's law.  - 66 -  will  depend on the e q u i l i b r i u m  s t r u c t u r e can be d e s c r i b e d  structure  i n the s o l u t i o n and t h i s  by m o l e c u l a r c o r r e l a t i o n f u n c t i o n s .  Thus f o r  a s o l u t i o n , £„ i s an exact analogue ( d e s c r i b i n g the same p r o c e s s e s o r responses) to the s t a t i c d i e l e c t r i c pure  apparent d i e l e c t r i c  i s not a true e q u i l i b r i u m authors SOL  e a r l i e r of a  solvent. The  e  constant d e f i n e d  a  n  d  constant of a c o n d u c t i n g s o l u t i o n ,  quantity  and w i l l not equal e^,.  [8,15,82,83] have i d e n t i f i e d ^  n  §eneral S 0 L e  =  +  €  ~D'  d e f i n i t i o n of  EgQ^*  first  en, t o  These dynamical  a r i s e because the c o n d u c t i v i t y , e q u a t i o n (4.4),  a( co), t h a t appears i n the  i s a complex f u n c t i o n of  The dynamical c o n t r i b u t i o n s  imaginary c o n d u c t i v i t y  goL»  Several  dynamical c o n t r i b u t i o n s ,  contributions  frequency.  e  to  e  gQ  L  a r i s e then r e a l l y as  terms which have not been taken i n t o acount.  o f these terms, which i n c r e a s e s  e  g  n  T  j  »  w  a  s  i d e n t i f i e d and  e s t i m a t e d many years ago i n the theory of Debye and Falkenhagen second term, which d e c r e a s e s Hubbard and Onsager decrement  [84],  EgQj/  This  w  a  s  The  [15].  A  r e c e n t l y i d e n t i f i e d by  term, known as the k i n e t i c d i e l e c t r i c  (KDD), has been i n v e s t i g a t e d  f u r t h e r from a m i c r o s c o p i c  point  of view by Hubbard, Colonomos, and Wolynes [ 8 2 ] , In the present study we w i l l dynamic terms o f out  e  g Q T /  examine both the e q u i l i b r i u m and  Although the present study w i l l be c a r r i e d  a t i n f i n i t e d i l u t i o n , we a r e a b l e  different  contributions  to determine l i m i t i n g  to the apparent d i e l e c t r i c  s o l u t i o n a t low c o n c e n t r a t i o n .  laws f o r the  constant of a  - 67 -  2.  Equllbrium Theory o f the D i e l e c t r i c Constant  Empirical  and s e m i - e m p i r i c a l  s u c c e s s f u l i n accounting  e q u i l i b r i u m t h e o r i e s have proven  f o r much of the v a r i a t i o n  [4-6,75] i n the d i e l e c t r i c c o n s t a n t concentration  i s increased.  observed  o f a s o l u t i o n as the i o n i c  The d i e l e c t r i c constant  decreases  l i n e a r l y w i t h c o n c e n t r a t i o n a t low t o moderate c o n c e n t r a t i o n s f o r most  strong e l e c t r o l y t e s ) .  At v e r y low c o n c e n t r a t i o n s  i ti s relatively  d i e l e c t r i c constant Equilibrium behavior  increases.  t h e o r i e s have p l a c e d much emphasis on e x p l a i n i n g the l i n e a r  o f the d i e l e c t r i c decrement and arguments have been put forward at l e a s t  Many s t r o n g e l e c t r o l y t e s permanent d i p o l e moments.  qualitative  explanations.  ( e g . the a l k a l i  h a l i d e s ) possess no  Hence, the " s t a t i c " d i e l e c t r i c  s o l u t i o n i s lowered by a simple  decrement  ( i n most cases  constant  of a  volume e f f e c t due to the displacement  s o l v e n t m o l e c u l e s by non-polar s o l u t e p a r t i c l e s .  of  A further  the dominant term) r e s u l t s from what i s  sometimes known as d i e l e c t r i c results  can be  The r a t e o f decrease o f the  becomes s m a l l e r as the c o n c e n t r a t i o n  [4,6,75] which supply  polar  small.  ( i . e . < 1M  the DF  e f f e c t , which has a square r o o t dependence upon c o n c e n t r a t i o n , observed although  roughly  saturation.  from the i n t e n s e e l e c t r i c  field  This local high-field  (=10  neighbor s e p a r a t i o n ) which surrounds an i o n .  effect  V/cm at the n e a r e s t This f i e l d  w i l l greatly  T h i s assumes a s m a l l d e n s i t y change r e l a t i v e to the drop i n d i e l e c t r i c c o n s t a n t , which i s the case.  - 68 -  reduce the a b i l i t y of s o l v e n t molecules themselves w i t h an a p p l i e d f i e l d . m o l e c u l e s immediately surrounding  c l o s e to the i o n to o r i e n t  For a s m a l l i o n , the s o l v e n t i t ( f o r a monovalent i o n , i t s n e a r e s t  neighbor s h e l l ) a r e s a i d to be i r r o t a t i o n a l l y unable to o r i e n t w i t h an a p p l i e d  bound and, t h e r e f o r e ,  field.  C l e a r l y , the volume e x c l u s i o n of the s o l v e n t w i l l  result  in a linear  c o n c e n t r a t i o n dependence of the d i e l e c t r i c decrement at low i o n i c concentrations.  Though i t has been long assumed  by many  authors,[4,6,75] the l i n e a r i t y o f the d i e l e c t r i c decrement due to dielectric  s a t u r a t i o n has o n l y been r e c e n t l y proven  non-polarizable  systems u s i n g m i c r o s c o p i c  theory.  express both e f f e c t s i n terms of the m o l e c u l a r the s o l u t i o n .  T h i s i m p l i e s t h a t one know how  [72,86] f o r One should  c o r r e l a t i o n f u n c t i o n s of these  f u n c t i o n s depend  upon i o n i c c o n c e n t r a t i o n .  R e l a t i o n s h i p s f o r determining  constant  [35,75,85] or of a s o l u t i o n of  of a pure s o l v e n t  non-polarizable  particles  [21], from s o l v e n t - s o l v e n t  f u n c t i o n s have been o b t a i n e d . dielectric  constant  be a b l e to  the d i e l e c t r i c  correlation  The c o n c e n t r a t i o n dependence of the  c o u l d be examined by r e p e a t i n g a  finite  These molecules should not be confused w i t h those t h a t make up what i s known as the h y d r a t i o n s h e l l , the proposed s h e l l of s o l v e n t molecules t h a t surrounds and moves w i t h an i o n i n s o l u t i o n . The molecules of the h y d r a t i o n s h e l l , i f i t e x i s t s , must be a subset of these m o l e c u l e s . Although the n o t i o n of a h y d r a t i o n s h e l l seems to be supported by thermodynamic d a t a , t h e r e has been no c l e a r m i c r o s c o p i c evidence (except f o r i o n s t h a t c o v a l e n t l y bond s o l v e n t molecules or p o s s i b l y f o r v e r y s m a l l i o n s , eg. L i ) of i t s a c t u a l e x i s t e n c e or the time s c a l e on which i t may e x i s t . I t may be a " s t r u c t u r e " which e x p e r i e n c e s very r a p i d exchange of s o l v e n t m o l e c u l e s . +  P o l a r i z a b i l i t y may potential. 5  be i n c l u d e d i f expressed  as p a r t of an e f f e c t i v e  - 69  -  c o n c e n t r a t i o n c a l c u l a t i o n at a number of d i f f e r e n t However, at i n f i n i t e d i l u t i o n , as i n the present  concentrations.  study,  the  dielectric  constant  of the s o l u t i o n w i l l always be that of the pure s o l v e n t .  Friedman  [86] has  r e c e n t l y obtained  solvent-solvent direct  constant  at i n f i n i t e  (some c o r r e c t i o n being necessary  c a l c u l a t e the l i m i t i n g low  (as a f u n c t i o n of  c o r r e l a t i o n ) f o r the l i m i t i n g  dependence of the d i e l e c t r i c expression  an e x p r e s s i o n  the  concentration  dilution.  [ 2 3 ] ) , we  Using  this  are a b l e to  slope which determines the decrement i n e^, at  concentration.  (a) The  The  D i e l e c t r i c Constant  determination  of a S o l u t i o n  of the d i e l e c t r i c c o n s t a n t ,  u s i n g e q u i l i b r i u m theory i s best understood s o l v e n t case. constant  We  develop  c o n s i d e r an i n f i n i t e molecules equation  and  0  examining the pure dielectric  proposed by Kirkwood  examine a s p h e r i c a l sample c a v i t y i n our  (4.1), i t i s e a s i l y  [85],  system.  shown [6,77] that the e l e c t r i c  We  Using  field  inside  by  —  represent  the  solution  s o l v e n t composed of n o n - p o l a r i z a b l e d i p o l a r  the sample, _E, i s g i v e n  where  by f i r s t  the e x p r e s s i o n f o r e ,  of a pure s o l v e n t system, f i r s t  e^, of a  =  7 T T o  ^o  i s the homogeneous e x t e r n a l a p p l i e d f i e l d .  ( 4  We  the average t o t a l d i p o l e moment of the system.  c l a s s i c a l s t a t i s t i c a l mechanics, we  let M Using  determine <Uj_*'e>, the average  '  6 )  - 70 -  moment of the i  t  applied f i e l d .  Assuming M«E_ /kT «  and  n  molecule i n the d i r e c t i o n , e^ (a u n i t v e c t o r ) , o f the 1, a v e r a g i n g over a l l d i r e c t i o n s  0  s i m p l i f y i n g , we  obtain  <y.-e> = <V '^[>  (4.7a)  ±  where M* i s the average moment o f the system when o n l y the i m o l e c u l e i s f i x e d and <y_j/ _ > i s the z e r o - f i e l d M  l  average.  We  t  Q  define  the d i p o l e moment y_* to be that o f the s p h e r i c a l sample when a c e n t r a l (i  t n  ) molecule i s h e l d  fixed.  so that the system o u t s i d e  Our sphere must be s u f f i c i e n t l y  can be regarded as b e i n g  homogeneous from the p o i n t of view of the i  t  n  large  electrically  molecule,  _y* w i l l  depend on the alignments o f a l l the molecules w i t h i n our s p h e r i c a l sample.  We can show [77] t h a t  9e 2  Thus i t f o l l o w s  ( e + 2 ) ( 2 e + l ) ±*' o o  =  ( 4  7 b )  that  9e  E  <V^> TTWuTT+TY Ji'Ji* ife ' =  o  and  '  o  from e q u a t i o n s (4.1) and (4.6) we have  (4  *  7c)  - 71 -  4 ^ E  o  4  VE  *  (  £  o  +  2  )  M  4  * (e +2)H <P .e> o  3VE o  ±  3VE o  *  {  8 )  where V I s the volume of the system and N i s the number o f m o l e c u l e s . S u b s t i t u t i n g the v a l u e f o r <ui«e> rearranging  from (4.7c) i n t o (4.8) and  we o b t a i n  (e - l ) ( 2 e +1) o o v  . 4 up  o  , 2 y = ^ L £ ,  where  (4.9b)  p i s the m o l e c u l a r d i p o l e moment and g i s known as the Kirkwood c o r r e l a t i o n parameter o r g - f a c t o r . In o r d e r to determine a more e x p l i c i t  e x p r e s s i o n f o r g, we f o l l o w a  s i m i l a r d e r i v a t i o n f o r the same system proposed by F r o h l i c h [87], I t can be shown [87] that the average moment of the system i n the d i r e c t i o n of the f i e l d  i s g i v e n by  <M'e> =  <M > 2  (4.10)  2 where <M > i s the mean squared moment i n the absence o f an a p p l i e d field.  I t i s then easy t o show that  - 72 -  g = <*!| = 1 + ^ ± < y - M > . Ny y 1  (4.11a)  2  N o t i n g t h a t we can i n t e g r a t e over the a p p r o p r i a t e c o r r e l a t i o n f u n c t i o n (see Appendix A) i n o r d e r  to o b t a i n  c o r r e l a t i o n between d i p o l e o r i e n t a t i o n s , we have g = 1 + 4TT P /" h (r) J d ss ss  [88] f o r the s t a t i c  obtained  from (4.9) by s e t t i n g g = 1.  general,  underestimate e  Q  the average  [35,36] r  1 1 0  The Onsager e x p r e s s i o n  p r o j e c t i o n of the p a i r  2  dr  dielectric  (4.11b)  constant  can be  The Onsager formula w i l l , i n  s i n c e g > 1 f o r systems where d i p o l a r  c o r r e l a t i o n s s t r o n g l y i n f l u e n c e the o r i e n t a t i o n a l s t r u c t u r e . shown [97] that the d i e l e c t r i c  constant  can be e q u i v a l e n t l y  I t can be obtained  through the l i m i t  11? n ( r ) •»• ss , -5 4ire pyr o (  h  £  _  1  )  as r •> ».  (4.12)  ~112 This expression note t h a t give  i s derived  by examining the k-K) l i m i t of h  i n i n t e g r a l equation theories equations  (k).  g g  We  (4.9) and (4.12) must  the same r e s u l t . In the LHNC and QHNC t h e o r i e s , the above r e l a t i o n s h i p s f o r e  been shown [35] to g i v e good agreement w i t h the v a l u e s determined by computer s i m u l a t i o n t h e o r i e s become l e s s a c c u r a t e , relative  to Q.  We  for e  have  Q  f o r d i p o l e - q u a d r u p o l e systems.  overestimating  Q  The  e , as y becomes l a r g e Q  can use (4.9) to c a l c u l a t e the d i e l e c t r i c  constant  - 73 -  f o r the c u r r e n t  s o l v e n t model even though we have allowed the s o l v e n t  m o l e c u l e s to be p o l a r i z a b l e .  T h i s i s because i n the SCMF theory [41]  e f f e c t s due to p o l a r i z a t i o n a r e a l l absorded i n t o the e f f e c t i v e d i p o l e moment, nig, and e  i s then determined f o r the e f f e c t i v e system.  Q  We now c o n s i d e r  a solvent  f i n i t e concentration non-polarizable.  system i n t o which there  has been added a  of s o l u t e p a r t i c l e s which are assumed to be  It i s possible  component of the g e n e r a l  [25,26] t o w r i t e  OZ e q u a t i o n s d e s c r i b i n g  the s o l u t e - s o l u t e solute-solute  c o r r e l a t i o n f u n c t i o n s as e f f e c t i v e OZ e q u a t i o n s of the form,  h  k) - c^ (k) - E p h f f  i j (  a  l a  where the sum i s o n l y over s o l u t e s p e c i e s . [21,35] e  E  by c o n s i d e r i n g  ( k ) ~cf ( k ) ,  (4.13)  We can then determine  the e f f e c t i v e s o l u t e - s o l u t e d i r e c t c o r r e l a t i o n  ef f f u n c t i o n , c ^ ( r ) (which i n c l u d e s a l l s o l v e n t  e f f e c t s ) , as d e f i n e d by  ef f ef f can(be e l a t e d [21,25] to the e f f e c t i v e s o l u t e - s o l u t e p(4.13). air potec n t? ir ( a lr,) u^. r ) ,rby  c .(r) e  As mentioned e a r l i e r  + -Su..  ( c f . equation  (r) * — - i - l  (4.5b)),  as r + ~.  (4.14) can serve as  (4.14)  - 74  d e f i n i t i o n of E g .  -  ~ef f (4.13) can be s o l v e d f o r c ^ and  Equation  the k  0  ~ef f limit  of c ^  ( k ) i s determined by examining the k  0 behavior  c o r r e l a t i o n f u n c t i o n s c o n t r i b u t i n g to i t . From (4.14) we 'eff „ , c , (k) ( I F  ±  ~ ^  -»-  ±  -A-  1  ^ f-  [21]  that i n the i n f i n i t e d i l u t i o n l i m i t  Kirkwood e x p r e s s i o n , £g = e .  (4.9)  and  Kirkwood r e l a t i o n s h i p .  At  e^, and  g.  (p^ = p.. = 0 )  (4.11), i s a g a i n o b t a i n e d  This constitutes a microscopic  Q  have that  (4.15)  K  which i s then used to o b t a i n an e x p r e s s i o n r e l a t i n g found  the  as k •»• 0,  -*  E  of  d e r i v a t i o n of  f i n i t e concentrations  It i s the  with  the  (p^ > 0 ) , however,  the Kirkwood r e l a t i o n s h i p does not hold because of the Debye s c r e e n i n g of  the long-range d i p o l e - d i p o l e c o r r e l a t i o n s .  In t h i s case, one  obtains  [21]  -3  where h *  1  0  ^  yg = y (1 + j  i s the k •>• 0 l i m i t .  i n the i n f i n i t e d i l u t i o n  limit.  P h s  g s  Thus (4.12) and £g i s a l s o r e l a t e d  )  (4.16)  (4.9) w i l l o n l y h o l d [21]  to the  i o n - i o n p a i r c o r r e l a t i o n f u n c t i o n by  £  E  =  4 7 r 3  ^ . i j i j i j P  p  q  q  h  }  (  4  a  7  )  - 75 -  where ~(2)  ij  h  e„ should  1  2  r°°  V  =  -  -  r )r dr  (4  18)  be g i v e n c o n s i s t e n t l y by both (4.16) and (4.17) i n the  Ci  MSA, LHNC, and QHNC t h e o r i e s .  (b)  The S o l u t e Dependent D i e l e c t r i c Decrement  At low c o n c e n t r a t i o n s ,  e„  = e  can be expressed  p  [64] by the  expansion e  where p - Ep^ and  p  =  e  o I (l)i i +  E  p  +  (4  i s c o n c e n t r a t i o n of i o n i c  species i .  *  19)  Friedman  [86]  has proven that the l e a d i n g term of t h i s expansion Is l i n e a r i n  p^.  It i s this linear  term which w i l l dominate a t low ef f  concentrations. to  By examining and f o r m u l a t i n g a theory  t h a t used i n d e t e r m i n i n g  with  the OZ e q u a t i o n  equation  forc ^  (similar  (4.17) f o r e ; i . e . working E  a t the McMillan-Mayer l e v e l ) , Friedman  [86] has  f o r m a l l y shown t h a t as p •> 0  ( £  where  (l)i  V  ~iio  1 ) 2  = - 9 y —  P  s  6  i ss c  (  4  '  2  0  a  )  - 76 -  - "110 3 "110,. o.c = [-5— c (k)]. „ i ss 3p^ ss k=0,  and  p  s  i s the number d e n s i t y  of the s o l v e n t .  expansion methods, he then p r o v e d  * 110 Vss  F^^(r)  6  limit k - 0  =  Applying  t h a t i n the HNC  F  cluster  theory  -110,, x ' ( k )  = / h (12)h s s  i s  (32)h  where p a r t i c l e s 1 and 2 are s o l v e n t s We note that F ( 1 2 ) ,  s i  (  (13) dr  4  ,, * 2  x  0 1 1  a  )  (4.21b)  3  and p a r t i c l e 3 i s an i o n .  and hence £(i)i» depends on the s o l v e n t -  c o r r e l a t i o n convoluted  w i t h s o l v e n t - i o n and  c o r r e l a t i o n s over a l l p o s i t i o n s of the i o n . only  n  i s the (110) p r o j e c t i o n of the f u n c t i o n  F(12)  solvent  _.. (4.20b)  _ (^,=0  r  on the s o l v e n t - s o l v e n t  ion-solvent  F(12) w i l l  thus depend not  p a i r c o r r e l a t i o n s , but w i l l a l s o depend on  the c o r r e l a t i o n between two s o l v e n t molecules t h a t e x i s t s through ( i . e . depends on) the i o n present are  the a p p r o p r i a t e  should  then be a b l e  in solution.  Since  h  s s  , h ^ , and h ^  c o r r e l a t i o n f u n c t i o n s at i n f i n i t e to e v a l u a t e  the c o e f f i c i e n t s ,  determine the l i m i t i n g b e h a v i o r of e  p  s  dilution,  ^(i)±t  i n the present  s  we  and hence  study.  The r e s u l t s of Friedman are c o r r e c t only to t h i s p o i n t subsequent r e l a t i o n s h i p s are given i n Ref. 23.  [23] and  - 77 -  U s i n g the d e f i n i t i o n  F °(r) =  ^  U  (8rr  we can r e w r i t e  [23] ( s i m i l a r t o (3.21b) and (3.33))  / F(12) $  1 1 0  (12)  dn.dn ,  (4.21a) as  S. = - ^ - / h (12)h. (32)h ,(13) $ l ss y^g^Z^J ss is si 1 1 0  (4.22)  9  r  3  c  1 1 0  T  where V i s the volume o f the system.  ( 1 2 ) dX.dX dX_ —1 —l —3 0  (4.23)  F o l l o w i n g Patey and C a r n i e [ 2 3 ] ,  we can prove that  V«  =  -—•* / %  o -x  i  (2TT)  / dndaiQ  $  1  1  0  h (n n k)  (i2)  c  (8TI)  J  x  ss  lf  i  z  -  h (n ,k) h (fl ,k), ±8  2  sl  (4.24)  1  where we have taken the i o n , which i s l a b e l l e d as p a r t i c l e 3, as our origin.  of h  a  g  I n e q u a t i o n (4.24), h g ( & i , f l 2 , k ) i s j u s t a  the F o u r i e r  transform  (fti,&2>r_) and we note that h ^ and h ^ do not depend on the g  g  o r i e n t a t i o n o f the i o n . As b e f o r e , we now expand h  , h. , and h ,  SS  i n terms o f r o t a t i o n a l i n v a r i a n t s .  Then by s u b s t i t u t i n g  IS  S X  these  expansions i n (4.24) and p e r f o r m i n g the angular i n t e g r a t i o n s we o b t a i n  - 78 ,-  , "110=  .  Vss  „  110_Jc.0£.0&' £'  f  MMT  f  MNL U'mm'  x  [ Z  f  where we have used  MNL , 101.,0 V Vw 1XM W 1 £ ' N . ( 0 0 0 ) ( 0 0 0 0m-m 0m'-m' )(  Ux0 X-X0 -X0X X0-X )(  X  f  )(  )(  )]I  mm'  )(  )  '  ( 4  '  2 5 )  standard p r o p e r t i e s [73] o f the 3-j symbols and  Wigner m a t r i x elements and use the d e f i n i t i o n  I  The  = — r r Jn k _ 20  , mm'  h . (k) h. , (k) h . ~ ( k ) dk. ss.-m-m' is.Om' si.mO  (4.26)  n  J  constant f a c t o r s , f™"'-, o f e q u a t i o n  (4.25), which r e s u l t  1  from our  mnl d e f i n i t i o n of  , a r e g i v e n by e q u a t i o n  (2.12).  In g e n e r a l , the c a l c u l a t i o n of 6. c ^ ^ would appear t o be a very X ss  difficult  task.  are p o s s i b l e . limits  However, f o r the present model, many s i m p l i f i c a t i o n s The o r t h o g o n a l i t y o f the r o t a t i o n a l i n v a r i a n t s  the p o s s i b l e non-zero terms o f (4.25).  d i p o l e - t e t r a h e d r a l quadrupole of terms.  F o r the  s o l v e n t , there i s a l s o some c a n c e l l a t i o n  We o b t a i n * 110 _ 00110 . 1 11000 Vss ^0 T 0 0 T  =  which i s a l s o the e x p r e s s i o n found equation  greatly  (4.26), we have  T  +  I  [23] f o r a d i p o l a r s o l v e n t .  0 7 ( 4  '  x  2 7 )  Using  79  T  x  00110  oo  1  =  r°° , 2 , 000,, r  7 T 'o  k  [ h  N 1  ( k ) 1  -  2  ,110,, >, ,,  h  i s  ( k )  ,,  d k  ( 4  s s  '  O  .  Q  2 8 a )  2TT  and  11000 -Inn UO  T  IQQ^^  -1 r , 2 ,011,, 2 ,000,, Jo l 4 „ „ I '0 i s ^ ) j „ ss( ) 2TT m  =  — y  r  k  N 1  k  h  can be viewed as e s s e n t i a l l y  depends on h ^ l ( k ) , which i t s e l f w i l l g  interaction  between the i o n and  viewed as b e i n g predominately ion-solvent  t h e o r y , i t can be shown [21] b e h a v i o r of c^**)  h  d k  O  «  term  term  u  N  since i t  charge-dipole  Similarly,  IQQ**^ can  s i n c e i t depends of  be  the  depend on the packing w i t h i n  i n the d e t e r m i n a t i o n of I Q Q ^ ^  remain f i n i t e  Q  (4.28b)  an e l e c t r o s t a t i c  the s o l v e n t .  Some c a r e must be used  N  depend on the  a packing  the i n t e g r a n d of (4.28a) w i l l  as k -*• 0.  In the  (by examining, i n p a r t , the  since  present  limiting  that  e -1 ( k ) > 4Tr[3q.u ( ° _ ) is I y Q  i  0 1 1  3  by  k  r a d i a l d i s t r i b u t i o n , which w i l l  the system.  where i  h  E  as k * 0  (4.29)  k  Hence, the s m a l l k b e h a v i o r of the i n t e g r a n d i s g i v e n  - 80 -  . 2 " O i l , , . . 2 "000,. . ,. . ( i s s s k ) * <47reV} k  We note  [  h  (  k  )  ]  h  .2 , _fb~* 2 J 0 0 0 , . <3^> ss '  ,.  n  h  (  0  )  that h^^^(k) i s w e l l behaved i n the l i m i t k-H). ss  (  4  '  3  0  o  n  .  )  This analytic  term f o r the v a l u e o f the i n t e g r a n d i n the l i m i t k-K) has been found [23] f o r the p r e s e n t model to be l a r g e and n e g a t i v e i n v a l u e . s m a l l k b e h a v i o r tends  to dominate the i n t e g r a l .  o b t a i n e d , u s i n g (4.27) and (4.28), f o r a d i p o l a r found  [23] to agree v e r y w e l l w i t h the l i m i t i n g  finite  3.  e . or  T  The v a l u e of s o l v e n t system was s l o p e of £g from  concentration r e s u l t s [21],  Dynamical T h e o r y o f t h e D i e l e c t r i c  The  In f a c t t h e  apparent  dielectric  Constant  constant o f an e l e c t r o l y t e  = € „ + £ , as measured by experiments,  has been  to be the sum o f both e q u i l i b r i u m and dynamical  solution,  found  terms.  The analogue o f  the s t a t i c d i e l e c t r i c c o n s t a n t , e , d e f i n e d by m i c r o s c o p i c E  can not a t present be r e a d i l y measured e x p e r i m e n t l y . c o n t r i b u t i o n s to the apparent  directly  electric  (4.18)).  fields.  In  i f one c o u l d measure the  i o n - i o n s t r u c t u r e f a c t o r s u s i n g neutron o r X-ray ( c f . equation  The dynamical  d i e l e c t r i c constant a r i s e because  measurements must be made u s i n g o s c i l l a t i n g p r i n c i p l e one c o u l d determine  theories,  scattering  experiments  U n f o r t u n a t e l y these e x p e r i m e n t a l probes  [6] t e c h n i c a l l y d i f f i c u l t  to c a r r y out.  A knowledge o f e  n  remain  would not  - 81  only f a c i l i t a t e e  -  the d e t e r m i n a t i o n of  from the measured q u a n t i t y  , but would a l s o improve the understanding of  dynamical  OULJ  p r o c e s s e s t a k i n g p l a c e In The  dynamical  solution.  c o n t r i b u t i o n to the d i e l e c t r i c  constant of a  £ Q , a r i s e s from the d e f i n i t i o n of EgQL (4.4) i n which the  solution,  imaginary  term of the complex c o n d u c t i v i t y i s assumed to be zero i n the z e r o frequency l i m i t .  At any  between a (GO) and  the a p p l i e d e l e c t r i c  finite  finite  frequency, t h e r e w i l l be a phase l a g field  even i n the z e r o - f r e q u e n c y l i m i t .  and  this lag  I t i s t h i s non-zero  the imaginary term o f o (u) as u + 0 which appears o  f  e  remains v a l u e of  as a dynamical  SOL' The  Debye-Falkenhagen theory [15] was  the f i r s t  theory to d e a l w i t h  the frequency dependent p r o p e r t i e s of an e l e c t r o l y t e s o l u t i o n . t h e o r y was  o r i g i n a l l y developed  to e x p l a i n the frequency  solutions.  I t proved v e r y s u c c e s s f u l  concentrations.  frequency  low  At low c o n c e n t r a t i o n , the z e r o - f r e q u e n c y l i m i t  molar c o n d u c t i v i t y , A , 0  was  DF  electrolyte  [15] i n e x p l a i n i n g the  c o n c e n t r a t i o n dependence of c o n d u c t i v i t y data at  The  and  c o n c e n t r a t i o n dependence of the molar c o n d u c t i v i t y , A, of  and  term  of the  shown to obey  A  O  = A  oo  + Ac  (4.31)  1 / 2  where Aoo i s the molar c o n d u c t i v i t y at i n f i n i t e d i l u t i o n , c i s the s o l u t e c o n c e n t r a t i o n , and A i s a constant dependent on the nature of the electrolyte.  The DF  theory a l s o p r e d i c t e d a d i e l e c t r i c  increment  now  - 82  called  the DF  effect.  The  -  i n c r e a s e i n the d i e l e c t r i c constant  c o n c e n t r a t i o n i s p r e d i c t e d to have a /c concentration  DF  theory was  low  Measurements at  [15,79,80] have shown both the c o n c e n t r a t i o n and  dependence p r e d i c t e d by the DF The  dependence.  at  low  frequency  theory.  developed  [3] by examining how  the  complex 7  c o n d u c t i v i t y i s a f f e c t e d by the r e l a x a t i o n of the i o n i c surrounding  a central ion.  ion-ion interactions. d i l u t i o n l i m i t was and DF  In the  theory would h o l d up  The  DF  effect w i l l  f o r m u l a t i o n of the DF  The  original  effect  authors  to c o n c e n t r a t i o n s  c o n d u c t i v i t y measurements [15] DF  thus depend, i n p a r t , theory,  the  i s most e v i d e n t at low  had  have supported  equivalence this  concentrations.  dependence w i l l  dominate a l l l i n e a r l y dependent e f f e c t s at  concentration.  At h i g h e r  concentrations  the DF  ion  predicted that  of about 10  effect  on  infinite  assumed i n d e f i n i n g the i n t e r a c t i o n between an  i t s i o n i c atmosphere.  mole and  The  atmosphere  the per  claim.  I t s -/c low  i s usually  Q  ignored  s i n c e the DF  concentration valid  has  low  i t i s assumed that i t s /c" behavior  a second dynamical c o n t r i b u t i o n to the d i e l e c t r i c  (KDD).  is  concentrations.  been p r e d i c t e d  decrement  i s o n l y understood i n the  l i m i t , although  at f i n i t e  Recently  effect  [82,84] The  KDD  I t i s known as the k i n e t i c  dielectric  i s i n e f f e c t an analogue to the DF  The d i s t r i b u t i o n of i o n s , equal and o p p o s i t e c e n t r a l i o n , t h a t surrounds a c e n t r a l i o n .  constant  effect  i n charge to  that  the  Q  A l s o , at the h i g h e r f r e q u e n c i e s g e n e r a l l y used [6,80] i n measuring S0L hiS c o n c e n t r a t i o n s , we assume the DF e f f e c t w i l l not be evident. The r e l a x a t i o n of the i o n i c atmosphere i s a r e l a t i v e l y slow process and w i l l not have time to o c c u r . e  a t  n e r  - 83 -  a r i s e s from s o l v e n t - i o n i n t e r a c t i o n s . d e f i c i e n c y by the o r i g i n a l authors the p r i m i t i v e model framework. a l s o been developed  Called  the k i n e t i c  [ 8 4 ] , i t was f i r s t  and examined  [82] f o r i o n s i n water.  decrease  ( c f . 4.31), a = cAa>«  i n the apparent  which, f o r d i l u t e  Theories f o r  i n the d i e l e c t r i c constant as a  f u n c t i o n o f the c o n d u c t i v i t y o f the s o l u t i o n . approximation  examined w i t h i n  A m i c r o s c o p i c theory f o r the KDD has  the KDD p r e d i c t a l i n e a r decrease  a first  polarization  At low c o n c e n t r a t i o n , t o  Thus the KDD p r e d i c t s a  d i e l e c t r i c constant o f an e l e c t r o l y t e  solution  s o l u t i o n s , w i l l have a l i n e a r dependence on i o n i c  concentration. An attempt has been made behavior at  [80,83] to account  o f the d i e l e c t r i c c o n s t a n t s o f aqueous a l k a l i h a l i d e s o l u t i o n s  low c o n c e n t r a t i o n s e n t i r e l y i n terms o f the two dynamical  mentioned above. Although  However,  obtained,  s c a t t e r i n the data p o i n t s , i n d i c a t i n g t h a t the  a c c u r a c y o f the measurements  was not s u f f i c i e n t  s m a l l e f f e c t s a t low c o n c e n t r a t i o n . a c c u r a t e l y reproduce  the frequency  effects  the r e s u l t s a r e f a r from c o n c l u s i v e .  the measured r e s u l t s appear to have been c a r e f u l l y  there i s s u b s t a n t i a l  to  f o r the observed  to f o l l o w the r e l a t i v e l y  A l s o , van Beek  [80] seemed  unable  (when compared w i t h t h e r e s u l t s of Wien [79])  dependence o f the DF e f f e c t .  The i n c r e a s e s i n the  measured d i e l e c t r i c c o n s t a n t s a t very low c o n c e n t r a t i o n s were found t o be l a r g e r than t h a t p r e d i c t e d by the DF t h e o r y other known e f f e c t s t h a t should concentrations).  The observed  still  ( u n l e s s one i g n o r e d a l l  be present at these  decrease  [80] i n the measured  c o n s t a n t , a f t e r removal o f the DF term, was expressed f u n c t i o n o f cr, a l t h o u g h  dielectric  as a l i n e a r  the s c a t t e r i n the data p o i n t s l e a v e s much  - 84  ambiguity theory  i n this operation.  [83]  f o r the KDD  -  Moreover, the slope p r e d i c t e d by  i s o n l y about o n e - t h i r d t h a t of the  original  slope obtained  from the e x p e r i m e n t a l  r e s u l t s of van  Beek [80] w i l l be d i s c u s s e d i n more d e t a i l i n Chapter  The  data  continuum  i s a l s o a l i n e a r f u n c t i o n of T  KDD  f o r aqueous s o l u t i o n s .  d  the d i e l e c t r i c  r e l a x a t i o n time [6,75], i n continuum and m i c r o s c o p i c would expect  then t h a t the KDD  The V.  (Debye)  theories.  One  would be more e v i d e n t , or even a dominant Q  f a c t o r , f o r p o l a r s o l v e n t s w i t h long r e l a x a t i o n times.  This  c o n c l u s i o n seems to be c o n s i s t e n t w i t h r e c e n t r e s u l t s  for solutions  of i o n s i n methanol  The r e l a x a t i o n  times  f o r the KDD  found  [89]  at low  [89] and  for sulfuric acid  [90].  of d i l u t e aqueous and methanol s o l u t i o n s have been  to be the same o r d e r of magnitude as  (approximately  c o n c e n t r a t i o n s ) the s o l v e n t d i e l e c t r i c r e l a x a t i o n times.  p r o x i m i t y of the r e l a x a t i o n times would f u r t h e r complicate observed  i n Cole-Cole  diagrams  e x p e r i m e n t a l l y observed is  the c o n c e n t r a t i o n and  frequency  through theory and  dependence of the  - T  d  behavior  There  experiment, about  KDD.  -  lt  The  b e h a v i o r which remains p o o r l y understood.  S o l v e n t s c o n s i s t i n g of l a r g e p o l a r molecules would d i e l e c t r i c r e l a x a t i o n times. T h e i r s i z e would r e s t r i c t o r i e n t a t i o n freedom and hence they would take l o n g e r to an a p p l i e d f i e l d ; e.g. Water - 9ps, methanol n ~ 2  the  to  and would help to e x p l a i n such  1 0  s t i l l much to be l e a r n e d , both  H S0  equal  T  have l o n g e r their orientate with 48ps,  = 400ps (Ref. 89,90).  P l o t s [95] of e"(oj) a g a i n s t e'(co) are f r e q u e n t l y used by e x p e r i m e n t a l i s t s [6,75] to determine d i e l e c t r i c c o n s t a n t s and d i e l e c t r i c r e l a x a t i o n times. 1 0  - 85  (a) The  The  -  Debye-Falkenhagen E f f e c t  theory  of Debye and  work of Debye and  Huckel  attempted to p r o v i d e  Falkenhagen [91,92] b u i l t on the  [93]  and  of Onsager  to a s t a t i c e l e c t r i c  i o n i n s o l u t i o n w i l l be caused to move by  motion w i l l not s o l v e n t , but an i o n and The  first  force.  o n l y be  a l s o by  r e s i s t e d by the  two  f a c t o r s due  such a s t a t i c  frictional  to the  the long-range e l e c t r o s t a t i c  f o r c e due  The  the  f o r c e s t h a t e x i s t between  to t h a t of the i o n .  frictional ionic  ordinary  second e f f e c t r e s u l t s from the motion of  the  At e q u i l i b r i u m , the charge d i s t r i b u t i o n around a c e n t r a l i o n i s  isotropic.  However, an Ion moving w i t h a constant  v e l o c i t y must always  b u i l d up an i o n i c atmosphere i n f r o n t of i t , w h i l e behind i t , the d i s t r i b u t i o n must r e l a x .  The  assymmetry of the i o n i c atmosphere  increase with increasing ion v e l o c i t y . r e t a r d i n g f o r c e , sometimes c a l l e d are p r o p o r t i o n a l Debye and  to the  Falkenhagen  relaxation force.  frequencies  the assymmetry of the  a  concentration.  i n an a l t e r n a t i n g e l e c t r i c  i n s t a n t a n e o u s v e l o c i t y of the  will  Both e f f e c t s  [91,92] extended t h i s approach to  s o l u t i o n placed  applied f i e l d .  the  charge  T h i s assymmetry g i v e s r i s e to  square root of i o n i c  electrolyte  the  ions.  T h i s motion  s o l v e n t along w i t h i t , thus i n c r e a s i n g  r e s i s t a n c e to motion.  to  Its  i o n i c atmosphere s u r r o u n d i n g  I t r e s u l t s from the motion of the o p p o s i t e l y charged  tends to drag the  field.  field.  e f f e c t , known as e l e c t r o p h o r e s i s , i s an a d d i t i o n a l  atmosphere i n a d i r e c t i o n o p p o s i t e  ion.  T h i s e a r l y work  a d e s c r i p t i o n of an e l e c t r o l y t e s o l u t i o n ( u s i n g  o n l y a continuum model approach) s u b j e c t e d An  [94].  initial  fied.  an At  i o n i c atmosphere w i l l depend on  i o n as i t moves to and  low the  f r o i n response  to  I f however, the p e r i o d of o s c i l l a t i o n i s comparable  - 86 -  to  (or smaller  atmosphere can It  11  than) the c h a r a c t e r i s t i c r e l a x a t i o n time f o r the i o n i c  then p r a c t i c a l l y  be s e t up.  no assymmetry  i n the charge  distribution  Thus the r e l a x a t i o n f o r c e d i s a p p e a r s almost  i s t h i s frequency dependent r e l a x a t i o n f o r c e which g i v e s  completely. rise  to a  d i s p e r s i o n i n the c o n d u c t i v i t y and p e r m i t t i v i t y o f an e l e c t r o l y t e s o l u t i o n , the Debye-Falkenhagen e f f e c t . An e x p r e s s i o n that  f o r the mean v e l o c i t y was found  under s t a t i o n a r y c o n d i t i o n s ,  the four  above) a c t i n g on an i o n i n s o l u t i o n must  [91,92] by r e q u i r i n g  forces  (those  sum to z e r o .  considered  In o b t a i n i n g  expression  s e v e r a l approximations were n e c e s s a r y , many assuming  conditions  present  Falkenhagen the  only  i n the i n f i n i t e  [91,92] showed  that  dilution limit.  the t o t a l c u r r e n t  sum of the c o n d u c t i o n and the displacement  J T  where E i s the e l e c t r i c  strength.  o . „ itot . j _ 2 , l 2 J = — icoEe + {En.q.co + ~—:-=• 4TT .11 3ekT J o q  J  where n  Q  Debye and  d e n s i t y , J , must  q  J  i s the s o l v e n t v i s c o s i t y ,  They  (4.32a) 0  0  N  [15] expand to o b t a i n  2 _ 2 _ j j i _ icot En.q.tox - £ 7 } Ee .11 6TTTI J J o n  K  equal  current,  l ^ t , ico ,, s„ i w t = aEe + e(io)Ee ,  field  this  q  J |  J  K i s the r e c i p r o c a l Debye  N  (4.32b)  radius of  A l t h o u g h c o n c e n t r a t i o n d e p e n d e n t , the r e l a x a t i o n time f o r the i o n i c atmosphere (" 20 ns f o r a 10" M aqueous s o l u t i o n ) i s s e v e r a l o r d e r s o f magnitude l o n g e r than the s o l v e n t d i e l e c t r i c r e l a x a t i o n time.  x i s a f u n c t i o n which  the i o n i c atmosphere ( p r o p o r t i o n a l to / c ) , and r e l a t e s i o n v e l o c i t y to the  field  frequency, OJ.  The  f u n c t i o n x i s given  by  X(b,oj0 )  / b { ( l + i(Q0 1  where  9  )  /  2  - l//b|  i s the r e l a x a t i o n time of the i o n i c atmosphere ( g i v e n  equation  (5.35) of Ref.  15)  and  equation  (5.36) of Ref.  15)  depending on the charge and  b i s a dimensionless  (4.32b) can be  rearranged  to o b t a i n an e x p r e s s i o n  dependent s p e c i f i c c o n d u c t i v i t y , CT( OJ) , w h i l e represents  the  by  parameter ( c f .  c o n d u c t i v i t i e s at i n f i n i t e d i l u t i o n of a l l ions p r e s e n t . of  (4.33)  - l//b  iO)0  +  1  equivalent The  f o r the  real  ionic part  frequency  imaginary p a r t  an e x t r a c o n t r i b u t i o n to the d i e l e c t r i c constant  of  the  solution. G e n e r a l formulae f o r the c o n d u c t i v i t y and dielectric  constant  the d i m e n s i o n l e s s  have been d e r i v e d  I t i s convenient to  define  quantities  _ 1 R = /2  and  The  [15].  the apparent i n c r e a s e i n  Q = — /2  [(1 +  a> 0 ) 2  2  1 / 2  [(1 + c o © ) / 2  2  1  2  by Debye and  (4.34a)  1]  -  l] ' .  apparent i n c r e a s e i n the d i e l e c t r i c constant  s o l u t i o n , as d e s c r i b e d  1/2  +  Falkenhagen  1  (4.34b)  2  f o r a very  dilute  [15], i s given  by  - 88 -  " o £  e /b{Q(l ~ 1/b) - io9 (R -  l 2 JTYT  4 7 T q  e ( a , )  =  q  O)0{(1  1/b)  2  2  dilution.  It  constant  i s the i n c r e a s e  zero-frequency limit  i s greatest  limit  constant  g i v e n by (4.35) i n the  t h a t i s u s u a l l y known as the DF e f f e c t .  I n the  co + 0 we can show t h a t Q -> co0/2 and R ->- 1, and hence  •**  -  .  •  Thus f o r a g i v e n dilute  o^-'J  4 i r  -  %  l 2  0 /b  q  e l e c t r o l y t e at constant  2  (  1  +  u  r  b  )  2  H  - > 36  temperature T, we can w r i t e f o r  solutions  where A i s a constant  of A f o r d i f f e r e n t  D p  = A/c  (4.37)  depending upon the temperature and the n a t u r e s o f  s o l v e n t and e l e c t r o l y t e .  (b)  q  - -^r-  Ae  the  The i n c r e a s e  f o r co = 0.  i n the d i e l e c t r i c  limit; , . i  very  This  ( c o n c e n t r a t i o n dependence coming  through K) and i t s frequency dependence has been s t u d i e d . in dielectric  (4.35)  + co © }  2  where O c i s the s p e c i f i c c o n d u c t i v i t y a t i n f i n i t e i n c r e a s e w i l l be p r o p o r t i o n a l t o /c  l//b)}  However, a t o r d i n a r y  temperatures  e l e c t r o l y t e s are a v a i l a b l e i n t a b l e s  values  [2,15],  The K i n e t i c D i e l e c t r i c Decrement  Hubbard and Onsager  [84], i n d e v e l o p i n g  t h e o r i e s to study the  motions of ions i n a continuum s o l v e n t , i d e n t i f i e d  specific  dynamical  - 89  -  mechanisms which r e s u l t i n an apparent r e d u c t i o n constant  of a s o l u t i o n .  dynamically  d i s t i n c t , but through the it  i t s b a s i s i n how  ion-solvent  a f f e c t the p o l a r i z a t i o n of the  c l o s e l y r e l a t e d , e f f e c t s were found.  s o l u t i o n under the  d i r e c t i o n to that  dielectric  As  field  by  to the  local  favoured by the a p p l i e d  the d i e l e c t r i c  field,  r e l a x a t i o n time.  r e t a r d a t i o n f o r c e experienced by an  the  solvent.  field  an  apparently  ion migrates field,  to r o t a t e i n the field.  Although  solvent p o l a r i z a t i o n  t h i s r e l a x a t i o n process l a g s behind  the  As  Two  i n f l u e n c e of an a p p l i e d e x t e r n a l  r e l a x a t i o n tends to r e s t o r e the  appropriate  interactions  solvent.  tends to cause s o l v e n t molecules i n i t s v i c i n i t y  opposite  dielectric  Known i n i t i a l l y as the k i n e t i c p o l a r i z a t i o n  d e f i c i e n c y , t h i s e f f e c t has will  i n the  The  second e f f e c t r e s u l t s from  i o n due  to the p o l a r i z a t i o n of  s o l v e n t molecules r o t a t e i n response to the  applied  over a p e r i o d equal to the d i e l e c t r i c r e l a x a t i o n time, an i o n i n  solution will  be  retarded  i n i t s motion.  Both of these dynamical  e f f e c t s are a r e s u l t of the response of an e l e c t r o l y t e s o l u t i o n to applied alternating e l e c t r i c been o b t a i n e d we  as  The  oo  KDD  ->-  field.  Approximations f o r the KDD  only f o r d i l u t e s o l u t i o n s .  would expect the KDD  (i.e.  the  has  been shown [82-84,96] to be p r o p o r t i o n a l  frequency c o n d u c t i v i t y , a, of the  between the  frequencies  D  D  has  high  effect,  1/T ).  of the d i e l e c t r i c r e l a x a t i o n time, x ,  the KDD  have  However, u n l i k e the DF  to d i s a p p e a r o n l y at very  an  been expressed i o n and  [82]  solvent)  by  of the  solution.  s o l v e n t and For  to the the  a continuum  (assuming s l i p boundary  product  low solvent,  conditions  - 90 -  e  -8ir , o  Ae, KDD  which d i f f e r s [84],  slightly  Felderhof  [96],  a f u r t h e r refinement  —  — —  (  (4.38)  V  a l s o u s i n g the continuum model, has  to t h i s r e s u l t , a r g u i n g hold.  f o r Ae^-p  ^  (which r e q u i r e s the s o l u t i o n of two  KDD -  where the sum  i s over  to s p e c i e s i , N  g  coupled  -8TT  f i Di „  J  —  h7~  i  a  ST  T  }  i  s  r  a l l ion species, i ,  molecular  Langevin  equations).  obtained  .—  i <  * — .  2~~ >  ( 4  r  r  S  T  denote an e q u i l i b r i u m average.  Within  from an i o n to the s o l v e n t molecule.  the  vector We  can  write  = y cos6 = u $  and  and  the e q u i l i b r i u m  average, j£ r e p r e s e n t s a s o l v e n t d i p o l e v e c t o r and _r the u n i t  then  3 9 )  represents  assuming s l i p boundary c o n d i t i o n s [82],  i n d i c a t i n g the d i r e c t i o n  '  i s the c o n d u c t i v i t y due  i s the number of s o l v e n t molecules,  Stokes r a d i i o b t a i n e d angular b r a c k e t s  v  only  f o r the decrement.  In t h e i r approximate theory, Hubbard e t . a l . [82]  .  r e c e n t l y made  His expression d i f f e r s  a l s o been d e r i v e d u s i n g  ias  authors  that not a l l the  from (4.38) and must g i v e s m a l l e r v a l u e s  An e x p r e s s i o n theory  N  }  from the e x p r e s s i o n of the o r i g i n a l  assumptions of the o r i g i n a l authors slightly  — e °°  by  definition  0 1 1  (12).  (4.40)  - 91  > =  where V i s the  -  ///  2 2  sample volume.  g  l s  (  1  2  dr  )  dfljd^  Then expanding g £ ( 1 2 ) , u s i n g S  relationships  (3.23) and  (4.40), and  orthogonality  of r o t a t i o n a l i n v a r i a n t s ) we  (4.41a)  the  s i m p l i f y i n g (again u t i l i z i n g  the  have  y • r < ^—f  >  1  r  and  i t i s not  ^-2—  ///  hj (r)[$° (12)] 1  V  (8TT )  difficult  ' -  4TT y »  N  (4.41b)  r e c a l l t h a t the v a l u e  to the average c h a r g e - d i p o l e  Oil, .  f  ~3  V •'d  r  immediately f o l l o w s  dr d^dO,  r  2~~  now  2  to show t h a t t h i s f u r t h e r s i m p l i f i e s to  ^  We  1 1  8  is ^  (4.41c)  is  of the  i n t e g r a l of  (4.41c) i s p r o p o r t i o n a l  energy ( c f . e q u a t i o n ( 3 . 3 8 b ) ) .  It  that  <—  r  > =I T q T V TI s i  I n s e r t i n g t h i s r e s u l t i n t o (4.39), we  obtain  •  ( 4  '  4 1 d )  - 92 -  8TT 3  KDD  Values  (4.42)  f o r the KDD are r e a d i l y o b t a i n e d  we can view e q u a t i o n  i n the p r e s e n t  calculation  since  (4.42) as depending o n l y upon the average  c h a r g e - d i p o l e energy, and o t h e r v a l u e s can be t r e a t e d as system parameters.  I t i s important  to note t h a t two d i f f e r e n t  used i n the c a l c u l a t i o n o f A e ^ ^ . expression  through  the dynamical  t h e o r y of Hubbard e t . a l . [82] and the  r a d i u s e n t e r s i n the c a l c u l a t i o n o f UQQ/N^.  effects  the b e h a v i o r o f AG^QQ as a f u n c t i o n o f i o n s i z e  limit  dielectric by e q u a t i o n  f o r small  l a r g e i o n s , we can use the long-range  form of h?** ( c f . Ref. 35) i n o r d e r is analytically  This greatly  the r a d i i a r e r o u g h l y equal f o r l a r g e r i o n s .  of i n f i n i t e l y  ( c f . equation  constant (4.38).  r a d i i are  The Stokes r a d i u s e n t e r s the  crystal  ions, although  ionic  (3.38b)).  In the  asymptotic  to determine U.^/N., CD i I f we then  take the h i g h  frequency  t o be one, we o b t a i n the continuum r e s u l t as g i v e n  - 93 -  CHAPTER V  RESULTS AND DISCUSSION  1.  Introduction  In  Chapter I I we have d e f i n e d a model p o t e n t i a l f o r an e l e c t r o l y t e  solution.  I n Chapter I I I we have d e r i v e d the LHNC a p p r o x i m a t i o n f o r the  system c h a r a c t e r i z e d by t h i s p o t e n t i a l a t i n f i n i t e  ion dilution.  Then  u s i n g the s t a t i s t i c a l mechanical theory i n t r o d u c e d i n Chapter I I I , we have d e s c r i b e d , i n Chapter IV, dynamical and e q u i l i b r i u m to  £gOL f °  ra  n  infinitely  dilute electrolye solution.  contributions  In the p r e s e n t  c h a p t e r we w i l l d i s c u s s the r e s u l t s o b t a i n e d from s o l v i n g the LHNC t h e o r y f o r the p r e s e n t model s o l u t i o n .  Both s t r u c t u r a l and  dielectric  p r o p e r t i e s and t h e i r dependence on i o n s i z e and charge w i l l be examined.  In the present study temperature  investigated.  The temperature  dependence was not  dependence o f A e  p  and $w^j(r) have  been p r e v i o u s l y examined f o r equal diameter 1:1 e l e c t r o l y t e  s o l u t i o n s by  Patey and C a r n i e [ 2 3 ] , We w i l l , however, make comparisons, as d i d Patey and C a r n i e , [23] between w a t e r - l i k e and d i p o l a r discussion w i l l  focus m a i n l y on a l k a l i  solvents.  halide salt  Our  solutions although  r e s u l t s were o b t a i n e d f o r a wide range o f diameters f o r both u n i v a l e n t and d i v a l e n t In  ions.  our c a l c u l a t i o n s and i n our d i s c u s s i o n i t i s convenient t o  express a l l parameters infinite  i n reduced u n i t s .  d i l u t i o n can be t o t a l l y  The present s o l u t i o n model a t  c h a r a c t e r i z e d by the reduced  parameters  - 94 -  P* - P d , d* = 3  s  g  q? = ( 3 q / d l i s  )  2  1  d l  /  2  /d ,  (V/dJ) , 172  y* =  s  where 3 = 1/kT.  Q* =  (3Q /dJ) 2  A s o l v e n t diameter  and  of d = 2.8A and => c  a temperature of 25°C a r e used throughout  this  of Qr and  I I . F o r the w a t e r - l i k e  y = m  e  as d e s c r i b e d i n Chapter  s o l v e n t these parameters g i v e e experimental  value, e  Q  = 0  study.  1/2  We use the v a l u e s  77.47 which i s c l o s e to the  = 78.5, f o r water at 25°C.  F o r the d i p o l a r  s o l v e n t examined, the d i p o l e moment and d e n s i t y have been a d j u s t e d t h a t the t h e o r e t i c a l d i e l e c t r i c at 25°C. and  Table  constant  I I c o n t a i n s the reduced  dipolar solvents.  The diameters  i s very c l o s e to t h a t o f water parameters f o r the w a t e r - l i k e  assigned  h a l i d e i o n s were d i s c u s s e d i n Chapter  such  to the a l k a l i metal and  I I and are summarized i n Table I .  * The  reduced  charge,  = 14.1527(z^) where z^ i s the v a l e n c y a s s o c i a t e d  w i t h the i o n .  Table  I I . Reduced parameters and d i e l e c t r i c c o n s t a n t s f o r the w a t e r - l i k e and d i p o l a r s o l v e n t s a t 25°C w i t h d = 2.8A. s  Solvent  p  * s  *  Q  P  Water-like  0.732  2.751  Dipolar  0.800  1.5  For the p r e s e n t model the r e s u l t s but  *  e o  T  0.94  77.47  0  78.5  o b t a i n e d f o r i o n s o f the same s i z e  o p p o s i t e charge are i d e n t i c a l i n a l l cases except  f o r the  - 95 -  ion-solvent  c o r r e l a t i o n functions h ^  w i t h i o n charge.  and  g  which change i n s i g n  D e t a i l e d n u m e r i c a l r e s u l t s f o r a l l ions  studied are  p r e s e n t e d i n Appendix C.  2.  Energies  and C o o r d i n a t i o n  Numbers  T a b l e I I I summarizes the reduced e n e r g i e s by  an i o n i n the w a t e r - l i k e  at i n f i n i t e d i l u t i o n there  model.  We note that  s i n c e the s o l u t i o n i s  i s no i o n - i o n I n t e r a c t i o n energy and the  ion-solvent  i n t e r a c t i o n energies  ion-solvent  energy and i t s c h a r g e - d i p o l e  Table I I I .  of i n t e r a c t i o n e x p e r i e n c e d  are f o r a s i n g l e i o n .  The average  and charge-quadrupole  E n e r g i e s of i n t e r a c t i o n and c o o r d i n a t i o n numbers f o r a l k a l i metal and h a l i d e i o n s .  -3U  C D  /  N i  -BU  C Q  /  -3U /N.  N i  IS  C.N.  358.7  73.24  431.9  8.97  316.2  54.09  370.3  10.38  267.9  35.92  303.8  12.33  Rb , C l "  254.9  31.71  286.6  13.15  Cs ,  237.7  26.52  264.2  14.40  218.0  21.19  239.2  16.25  Li  +  Na , +  F  -  K+ +  +  I  Br~  -  components were c a l c u l a t e d u s i n g ion size increases  equations  (3.38).  We f i n d t h a t as the  the average charge-quadrupole i n t e r a c t i o n d e c r e a s e s  much more r a p i d l y than the c h a r g e - d i p o l e  interaction.  This behavior i s  - 96  due  simply  -  to the d i f f e r e n t r dependence of the  charge-dipole proportional  p o t e n t i a l s (U Q C  to 1/r  ).  i s proportional  These trends  prove u s e f u l i n d i s c u s s i n g  charge-quadrupole  to 1 / r  3  i n the e n e r g i e s  whereas u  and is  C D  of i n t e r a c t i o n w i l l  structural features. 2  It  was  accuracy we is  a very  t h a t the ion.  a l s o found that  have c a r r i e d our  calculations) regardless  included  i n T a b l e I I I are  XT  C.N.  where R i s the the LHNC theory  first  / fR = 4TTP J d^  the  i n t e r a c t i o n s do not  r  of i o n s i z e .  This  ion-solvent  i t s s i z e and  coordination  numbers  Thus the not  on  For  number of  study s i n c e  our  hard-sphere  and  the  ion-solvent i n t e r a c t i o n radial  distribution  same time cause i t s c o n t a c t  These e f f e c t s tend to c a n c e l  an  However, t h i s  have a l a r g e r i n f l u e n c e on  minimum of the  at the  In  Ion-solvent  a high density  I n c l u s i o n of the  first  i n , and  i t s charge.  drawback to the present  system an a t t r a c t i v e p o t e n t i a l w i l l not  e f f e c t s would cause the  coordination  i n the LHNC t h e o r y .  system i s at r e l a t i v e l y h i g h d e n s i t y .  o v e r a l l packing s t r u c t u r e .  /c i x (5.1)  radial distribution function.  influence g ? ^ ( r ) 1 s  seen as a s i g n i f i c a n t  f u n c t i o n to s h i f t  below.  000, v , g. (r) dr is  2  minimum i n the  000 HS g^T ( r ) = g ? g ( r ) .  depends o n l y on  increase.  the  using  n  not  (to  i n t e r e s t i n g r e s u l t s i n c e through e q u a t i o n s (3.38) i t i n d i c a t e s Oil 022 i n t e g r a l s over h. and h. must s c a l e w i t h the charge of the is is  (C.N.) c a l c u l a t e d  is  s c a l e w i t h z^  A f u r t h e r d i s c u s s i o n of t h i s s c a l i n g b e h a v i o r i s g i v e n Also  ion  the average e n e r g i e s  so the v a l u e  of  value the  to  - 97  coordination the  C.N.  -  number remains r e l a t i v e l y unchanged.  were found to i n c r e a s e w i t h i o n s i z e .  i n s p e c t i o n of v a l u e s  reveals  t h a t the C.N.  As we  would e x p e c t ,  However, c l o s e r  do not  f o l l o w a smooth  curve as a f u n c t i o n of i o n diameter.  3.  C o r r e l a t i o n Functions  and P o t e n t i a l s o f Mean F o r c e  Correlation functions describe present  study the  solvent-solvent  The  correlations.  theory u  g  g  part, h ^ ^ , 1 s not  f i x e d by the  solvent-solvent  s i n c e the  A  g  In  i o n - s o l v e r i t and  infinite  pair correlation function, h^ (12),  defined  g  _ As d i s c u s s e d  i s , i n the LHNC t h e o r y ,  _ earlier,  the  c o r r e l a t i o n s are  s o l u t i o n i s at  by e q u a t i o n (3.23), c o n t a i n s  0 0 0 .011 . 022 h^ , h^ , and h ^ . u  The  [41]  Ion-solvent  i n the present  s t r u c t u r e of a system.  i o n - i o n c o r r e l a t i o n s are  those of the pure s o l v e n t dilution.  the  only  three  terms,  , „ the s p h e r i c a l l y symmetric J  j u s t the hard-sphere r e s u l t and  so  of g r e a t i n t e r e s t . The  p r o j e c t i o n h^ ^ g  i n t e r a c t i o n and correlation. I .  has  the  same symmetry as the  so e s s e n t i a l l y d e s c r i b e s  In F i g u r e 5 we  t h a t p a r t of the  have p l o t t e d h ^ ^ g  . . As mentioned above, f o r the present  charge-dipole  for L i  . , .011 model h + g  +  ion-solvent  and  ~h^ * g  ,011 = "h_ g  for  - ,022 and h = + g  022 -h_  g  .  I t i s c l e a r from the  s t r u c t u r a l features ability h^ ^ g  to order  t h a t the v e r y feature.  i s greater  solvent  is relatively  f i g u r e t h a t a l t h o u g h the magnitude of for smaller  p a r t i c l e s , the  greater  p a t t e r n of the o s c i l l a t i o n s  independent of i o n s i z e .  r a p i d r i s e of the  ions,indicating their  A l s o , we  f u n c t i o n s near c o n t a c t  see  the  in Figure  in 5  i s a prominent  - 98  -  Figure 5 Comparing h ? * ( r ) f o r L i  +  g  The  solid  0 1 1  (r)  I  -  i o n s i n the w a t e r - l i k e s o l v e n t model.  the dashed curve i s - h (r) is is f o r I . The i o n diameters are those g i v e n i n Table I. Note that two s c a l e s are used i n the p l o t . -  curve i s h  and  for L i  +  and  0 1 1  - 100  -  Figure 6 Comparing h ?  2 2  is  ( r ) for L i  +  and I  -  i o n s i n the w a t e r - l i k e s o l v e n t model.  022 022 curve i s -h, ( r ) f o r Li" " and the dashed curve i s h. (r) for is is I . The i o n diameters are those g i v e n i n T a b l e I. Note that two s c a l e s are used i n the p l o t .  The -  solid  1  -  022 projection h 1s  The  102 -  e s s e n t i a l l y describes  the charge-quadrupole 022  of  the i o n - s o l v e n t  and  h  correlation.  022 (r) for I . -s  general  We f i n d  o s c i l l a t o r y patterns  s i z e i s a l s o very  Figure  6 compares  -  h  +  g  part  + (r) for Li  022 that not o n l y does h. ( r ) have the same is as h ^ ^ ( r ) , but i t s dependence upon i o n  s i m i l a r to that  of h ^ ^ ( r ) .  The c o n t a c t  values of  022 h^  are again  g  For  observed to be the most prominent  d i v a l e n t ions  investigated  i t was found that  f o r the diameter  the a n i s o t r o p i c i o n - s o l v e n t  the n u m e r i c a l a c c u r a c y o b t a i n e d ,  features.  simply  range  c o r r e l a t i o n functions  are,  to  twice those o f the u n i v a l e n t i o n  of the same d i a m e t e r . T h i s s c a l i n g w i t h charge i s i n i t i a l l y somewhat s u r p r i s i n g s i n c e p h y s i c a l l y i t means that f o r the present model and theory  the o r i e n t a t i o n a l s t r u c t u r e i s e f f e c t i v e l y  univalent  ions.  As a d i r e c t  r e s u l t we see that  saturated f o r  the i n t e g r a l s over h?** is  022 and  h  so w i l l The  must a l s o s c a l e w i t h i o n charge, and hence from e q u a t i o n the average i o n - s o l v e n t  energies  of i n t e r a c t i o n .  s c a l i n g b e h a v i o r of the i o n - s o l v e n t  unexpected f o r the present 011 know t h a t both u ^  g  (3.38)  model and theory  c o r r e l a t i o n functions at i n f i n i t e  dilution.  i s not We  022 ( r ) and u ^  g  (r)will  s c a l e w i t h charge and  000 HS g^ = g ^ i s always independent of q. Then from e q u a t i o n s (3.45), , mn£ , , , , mn£ , , , mn£, if n s c a l e s w i t h charge, c. must a l s o s c a l e . We note that c. ( r ) is ° is is g  g  r  N  will  always s c a l e f o r l a r g e r r .  T i ? ^ ( k ) and n9g (k) are simply 2  g  From e q u a t i o n s linear  (3.42) we see t h a t  combinations of j j g ^ ( k ) and c  - 103  022. ' c. (k). xs  Oil j 022 Thus i f both c, and c, xs xs  -  . . . . Oil s c a l e w i t h charge, then n. xs  and  022 h^  l i k e w i s e must s c a l e .  there  should  f o r the  e x i s t a s o l u t i o n which w i l l  T h i s cannot be t h e o r i e s , but  Therefore,  the case at f i n i t e i s true only  at i n f i n i t e d i l u t i o n .  present  s c a l e w i t h the  concentrations  f o r the  model and ionic  However, a l t h o u g h the h  m n  charge.  or i n higher  LHNC treatment of the ^(r)  theory  order  present  w i l l not  model  scale for  XS  small r i n higher large r.  order  Therefore,  we  t h e o r i e s , they must always s c a l e w i t h charge f o r would expect the d i e l e c t r i c p r o p e r t i e s  i o n - i o n s t r u c t u r e p r e d i c t e d by from that p r e d i c t e d shown [22]  that  by the  LHNC theory  LHNC theory  QHNC or HNC  theory.  not  to d i f f e r  greatly  Furthermore, i t has  the p o t e n t i a l of mean f o r c e c a l c u l a t e d i n  i s i n good agreement w i t h Monte C a r l o  P o t e n t i a l s of mean f o r c e as d e f i n e d  by  computations.  (3.39) d e s c r i b e  the  ion-ion  p a i r c o r r e l a t i o n s i n s o l u t i o n . For the present model f o r i o n s of same s i z e , 3w++(r) = gw ( r ) . T h e r e f o r e , i t i s o n l y n e c e s s a r y to know whether the p o t e n t i a l of mean f o r c e i s between a l i k e or an ion p a i r .  gwn  8,  respectively.  and  gw^  it  will  u n l i k e i o n p a i r s are  I t i s c l e a r l y evident  e x h i b i t the most d i s t i n c t i v e  p a i r s of r e l a t i v e l y s m a l l s i n c e we  the  unlike  P o t e n t i a l s of mean f o r c e spanning the a l k a l i metal and  i o n diameter range f o r l i k e and and  been  f o r an i n f i n i t e l y d i l u t e s o l u t i o n of hard s p h e r i c a l i o n s  immersed i n a d i p o l a r s o l v e n t the  the  and  have a l r e a d y  ions.  i n t e r a c t with solvent  s t r u c t u r a l features  smaller  particles.  shown i n F i g u r e s  7  i n both f i g u r e s t h a t both  T h i s perhaps i s not  seen that the  halide  the  for  a surprising result  i o n , the more s t r o n g l y  Thus s m a l l e r  ions w i l l  have  - 104  -  Figure 7 A comparison of 3w£^(r) f o r i o n s of d i f f e r e n t s i z e . The s o l i d , dashed, and dash-dot curves are f o r L i / L i , K / K , and T~/I~ pairs, r e s p e c t i v e l y , i n the w a t e r - l i k e s o l v e n t . The i o n diameters are those given i n Table I. +  +  +  +  - 105 -  - 106  -  Figure 8 A comparison o f 3w^j(r) f o r d i f f e r e n t p a i r s of o p p o s i t e l y charged ions. The s o l i d , dashed, and dash-dot curves r e p r e s e n t L i / F , L i / I , and C s / I p a i r s , r e s p e c t i v e l y , i n the w a t e r - l i k e s o l v e n t . The i o n d i a m e t e r s a r e those g i v e n i n T a b l e I . +  +  -  -  +  -  - 107 -  - 108  -  l a r g e r o r d e r i n g e f f e c t s upon the s o l v e n t - i o n s t r u c t u r e . must i n f l u e n c e how In F i g u r e 7 we  an i o n p a i r w i l l  interact.  observe that f o r l i k e p a i r s of r e l a t i v e l y  the p o t e n t i a l s are s t r o n g l y r e p u l s i v e at c o n t a c t . q u i c k l y drop to a f i r s t  minimum at ( r - d^£)/d  s  s m a l l e r ions such as L i , i s a c t u a l l y a t t r a c t i v e +  sign).  This, i n turn,  This i s a s u r p r i s i n g result  We  a l s o see  « 0.25  which, f o r  ( i . e . negative  i n that i t gives a r e l a t i v e l y  the f i r s t  I f the p o t e n t i a l s of mean f o r c e of s t i l l  examined, the  first  smaller  l a r g e r ions  minimum i s a c t u a l l y found to move r i g h t up  the c o n t a c t value becomes a t t r a c t i v e at  from e q u a t i o n  high  small  minimum s h i f t s to  separations.  and  in  that as i o n s i z e i s i n c r e a s e d the r e p u l s i o n at  c o n t a c t very r a p i d l y decreases and  contact  ions  They then very  p r o b a b i l i t y of f i n d i n g p a i r s of s m a l l l i k e i o n s at q u i t e separations.  small  a  (3.40a) t h a t the p o t e n t i a l must be the sum  1,8. of  are  to We  have  an  HS electrostatic  term and  e l e c t r o s t a t i c term w i l l and  the hard  a hard  sphere packing  term, -Jin g ^ ( r ) .  decrease i n magnitude w i t h  sphere term w i l l  The  increasing ion size  i n c r e a s e i n magnitude.  For l i k e i o n p a i r s  * w i t h d^  > 1.8,  3 w ^ ( r ) at As  the n e g a t i v e  hard  of  contact.  shown i n F i g u r e 8,  3w j(r) f o r i o n p a i r s of o p p o s i t e i;  s t r o n g l y a t t r a c t i v e at c o n t a c t at  sphere term dominates the v a l u e  (r - d ^ j ) / d  g  » 0.35.  We  and  charge i s  then q u i c k l y r i s e s to a f i r s t maximum  observe that as i o n s i z e i n c r e a s e s  magnitude of 3w^j(r) at c o n t a c t  decreases.  A l t h o u g h the f i r s t  the maximum  decreases i n v a l u e w i t h i n c r e a s i n g i o n s i z e , i t s p o s i t i o n remains relatively  unchanged.  It i s interesting  to note t h a t f o r a p a i r of  very  -  s m a l l u n l i k e i o n s such as L i / F , +  -  109 -  the f i r s t  mean f o r c e a c t u a l l y becomes r e p u l s i v e .  The b e h a v i o r e x h i b i t e d by the  second minimum o f Bwj_j(r) w i l l be d i s c u s s e d The Figure  p o t e n t i a l s o f mean f o r c e f o r L i / I +  9.  maximum o f the p o t e n t i a l of  -  below. and C s / F +  -  a r e compared i n  These two p a i r s are such that the sum o f the i o n diameters i s  equal i n each c a s e .  The p o t e n t i a l s a r e found to be almost  identical.  T h i s i n d i c a t e s t h a t a t l e a s t f o r the a l k a l i metal and h a l i d e  ions,  Bw^j(r) depends s t r o n g l y on d ^ j but i s r e l a t i v e l y independent o f the individual  i o n diameters.  We would expect t h i s t o break down, however,  f o r a p a i r c o n s i s t i n g o f a very In F i g u r e univalent  small and a v e r y  10 we have p l o t t e d  3 W ^ J / | Z ^ Z J |  large i o n .  f o r a p a i r of  and a p a i r o f d i v a l e n t i o n s where a l l the i o n s have the same  diameter as the s o l v e n t .  We f i n d t h a t  to a f a i r approximation the  p o t e n t i a l o f mean f o r c e s c a l e s w i t h product of the charge r a t i o s of both ions.  T h i s approximate s c a l i n g has been found to h o l d  u n l i k e i o n p a i r s where d^ _< d . g  not  F o r l a r g e r i o n s the p o t e n t i a l s do  s c a l e so w e l l , p a r t i c u l a r l y near c o n t a c t .  explained  T h i s behavior i s  by r e c a l l i n g t h a t we can express the p o t e n t i a l o f mean f o r c e  as a sum of an e l e c t r o s t a t i c  term and a hard sphere packing term.  hard sphere term w i l l be u n a f f e c t e d We have found f o r the present ions the i o n - s o l v e n t  by i o n i c charge i n the LHNC  model and theory  The  theory.  t h a t even f o r u n i v a l e n t  structure i s o r i e n t a t i o n a l l y saturated  would expect the e l e c t r o s t a t i c small  f o r both l i k e and  and so we  term to s c a l e w i t h i o n i c charge.  For  i o n s the hard sphere term i s small and hence the p o t e n t i a l o f mean  f o r c e appears t o a p p r o x i m a t e l y s c a l e w i t h charge.  However, f o r l a r g e r  - 110  -  Figure  9  Comparing gw^j(r) f o r p a i r s of o p p o s i t e l y charged i o n s f o r which the d^A v a l u e s are e q u a l . The s o l i d and dashed curves r e p r e s e n t L i / I ~ and Cs /F"~ p a i r s , r e s p e c t i v e l y , i n our w a t e r - l i k e s o l v e n t . The i o n diameters are those given i n Table I . +  +  - Ill -  CO  o  -  112  Figure  -  10  Comparing 3 w ^ ^ ( r ) / I f ° p a i r s of o p p o s i t e l y charged u n i v a l e n t and d i v a l e n t i o n s . The s o l i d and dashed curves r e p r e s e n t u n i v a l e n t and d i v a l e n t i o n p a i r s , r e s p e c t i v e l y , i n our w a t e r - l i k e s o l v e n t . The i o n diameters are a l l equal to the s o l v e n t diameter. r  - 113 -  -  ions  114 -  the hard sphere term becomes important near c o n t a c t  and so s c a l i n g  cannot o c c u r . The  p o t e n t i a l o f mean f o r c e o f i o n s i n a p r i m i t i v e model s o l v e n t , a  dipolar  solvent  and the present  Figures  11 and 12.  water-like  solvent  are compared i n  The p o t e n t i a l s o f mean f o r c e f o r i o n s i n a d i p o l a r  s o l v e n t w i t h parameters as given  i n T a b l e I I are determined i n the same  manner as those f o r ions i n the w a t e r - l i k e  solvent.  The p o t e n t i a l of  mean f o r c e between two i o n s , a and 3, i n a PM s o l v e n t continuum l i m i t , constant.  i s j u s t the  PM 3w ( r ) = 3q q J e r , where e i s the s o l v e n t a3 or 3 o o  dielectric  Q  This expression  i s exact  f o r l a r g e r and t h e r e f o r e  3w g(r) a  PM, for  any other  s o l v e n t must approach  obvious from both f i g u r e s that solvent  Q\ ~) r  a  a  s  the w a t e r - l i k e  than the d i p o l a r system.  F o r a given  r  +  °°*  I t becomes  f l u i d i s a "better" i o n p a i r , l i k e or u n l i k e ,  the magnitude of the p o t e n t i a l of mean f o r c e a t contact water-like Also and  solvent  i s always much l e s s  3w g f o r the p r e s e n t a  water-like  f o r the  than f o r the d i p o l a r  solvent  i s much l e s s  approaches the continuum l i m i t much more r a p i d l y .  solvent.  structured  The d i p o l a r  s o l v e n t would be expected to be h i g h l y s t r u c t u r e d  s i n c e only a s m a l l  range o f o r i e n t a t i o n s o f the s o l v e n t w i t h r e s p e c t  to an i o n are h i g h l y  favoured e n e r g e t i c a l l y .  A d d i t i o n of a t e t r a h e d r a l quadrupole moment  would a l l o w a l a r g e r range o f angles t o be f a v o u r a b l e . of the w a t e r - l i k e  solvent  structure despite  i t s l a r g e d i p o l e moment.  In F i g u r e water-like  The quadrupole  thus must a c t to " s o f t e n " much of i o n - s o l v e n t  11 we have p l o t t e d  3w^j f o r L i / F ~ and C s / I " +  and the d i p o l a r s o l v e n t s .  p o t e n t i a l i s always c h a r a c t e r i z e d  +  i n the  F o r the d i p o l a r s o l v e n t the  by a deep w e l l a t ( r - d..)/d  =1.  - 115  -  Figure  11  A comparison of 3w^j(r) f o r p a i r s o f o p p o s i t e l y charged i o n s i n d i f f e r e n t s o l v e n t s . The s o l i d and short-dashed curves r e p r e s e n t the C s / I ~ p a i r i n the w a t e r - l i k e and d i p o l a r s o l v e n t s , r e s p e c t i v e l y . The dash-dot, long-dashed, and d o t t e d curves r e p r e s e n t the L i / F p a i r i n the w a t e r - l i k e , d i p o l a r , and continuum s o l v e n t s , r e s p e c t i v e l y . +  +  -  - 116 -  - 117  Figure  -  12  A comparison of pV-^Cr) f o r F /F~ and I ' l l " i n d i f f e r e n t s o l v e n t s . The s o l i d and short-dashed curves are f o r the I ~ / I ~ p a i r i n the w a t e r - l i k e and d i p o l a r s o l v e n t s , r e s p e c t i v e l y . The dash-dot, long-dashed, and d o t t e d curves are f o r the F / F ~ p a i r i n the w a t e r - l i k e d i p o l a r , and continuum s o l v e n t s , r e s p e c t i v e l y . -  - 118 -  -  119 -  T h i s must correspond to the e n e r g e t i c a l l y v e r y f a v o u r a b l e two  o p p o s i t e l y charged ions separated  f o r L i / F ~ i n the w a t e r - l i k e s  by a s o l v e n t p a r t i c l e .  3WJ[J  s o l v e n t has i t s second minimum at  +  (r - dj:j)/d  s t r u c t u r e of  = 0.75 a l t h o u g h i t does have a s l i g h t d i p a t 1.0.  This  second minimum appears to be a p u r e l y quadrupolar f e a t u r e s i n c e i t does not  appear i n 3w^j f o r the d i p o l a r s o l v e n t and the i o n - i o n  roughly  separation  corresponds t o the s e p a r a t i o n of i o n s p l a c e a t t e t r a h e d r a l  " c o r n e r s " o f a s o l v e n t molecule.  The t e t r a h e d r a l corners  would  correspond to s i t e s on the s u r f a c e o f the s o l v e n t molecule ( i . e . 109° a p a r t ) o f most f a v o u r a b l e  charge-quadrupole i n t e r a c t i o n .  No d e f i n i t e  s o l v e n t s t r u c t u r e around two i o n s a t such a s e p a r a t i o n can be r e a d i l y assigned.  For C s / I ~  3w-£j i s a g a i n separated  i n the w a t e r - l i k e  +  at ( r - d ^ j ) / d  s  s o l v e n t the second minimum i n  = 1.0 corresponding  to a s o l v e n t  i o n p a i r but t h i s minimum i s r e l a t i v e l y shallow.  The apparent  disappearance of the quadrupolar minimum i n 3w^j must r e s u l t  from the  decreased r e l a t i v e importance o f the charge-quadrupole i n t e r a c t i o n as the i o n s i z e i s i n c r e a s e d . the  Hence, the second minimum f o r C s / I ~ +  same p o s i t i o n f o r both the d i p o l a r and w a t e r - l i k e  i n the l a t t e r the quadrupole i s s t i l l  is in  solvents although  a c t i n g to g r e a t l y dampen the  s t r u c t u r e i n the p o t e n t i a l of mean f o r c e . In F i g u r e water-like  12 we have p l o t t e d pw (r)  f o r F~/F~ and I ~ / I ~ i n the  i±  and d i p o l a r s o l v e n t s .  F o r the d i p o l a r s o l v e n t the p o t e n t i a l  i s c h a r a c t e r i z e d by a peak a t ( r - d  i : L  )/d  s  = 1, which corresponds  t o the e n e r g e t i c a l l y u n f a v o u r a b l e s t r u c t u r e of two l i k e i o n s by a s o l v e n t m o l e c u l e . water-like  solvent.  No peak appears a t t h i s s e p a r a t i o n  separated  f o r the  We observe that f o r F~/F~ i n both s o l v e n t s the  - 120 -  first  minimum o f  i s so deep as to be a t t r a c t i v e , more so f o r the  &w±±  dipolar  solvent.  T h i s c o u l d p o s s i b l y be a d i p o l a r f e a t u r e due to some  solvent  s t r u c t u r e which a c t s to c o u n t e r a c t  between the i o n s .  However, t h e r e  hypothetical structure. having  the coulombic r e p u l s i o n  i s no obvious form f o r such a  T h i s f e a t u r e may simply  to go somewhere, so they choose the l e a s t  separation.  a r i s e from the i o n s unfavourable  I t i s important t o bear i n mind that these are p o t e n t i a l s  o f mean f o r c e a t i n f i n i t e d i l u t i o n where the average s t r u c t u r e remains unchanged.  solvent-solvent  I t would be i n t e r e s t i n g to see i f some o f  the above s t r u c t u r e s p e r s i s t a t f i n i t e i o n c o n c e n t r a t i o n s . For both m o l e c u l a r contact  s o l v e n t s the p o t e n t i a l s o f mean f o r c e near  have been found to be v e r y  structured.  C l e a r l y , the PM i s  unable to r e p r e s e n t  the i o n - i o n c o r r e l a t i o n s f o r s m a l l  The  i s approached v e r y  continuum l i m i t  separations.  r a p i d l y by the p o t e n t i a l s o f mean  f o r c e f o r i o n s i n the w a t e r - l i k e  solvent.  e s s e n t i a l l y reached w i t h i n  three  s o l v e n t diameters f o r most i o n s except  i n the case o f small m u l t i v a l e n t  ions which w i l l more r e a d i l y s t r u c t u r e  the  solvent.  Such r e s u l t s f o r 3 w  g i v e good agreement w i t h  aR  using  solvent cospheres.  cospheres i s a v e r y tried  can perhaps e x p l a i n why the PM can  some experimental  [ 4 , 9 8 ] . p a r t i c u l a r l y i f the e f f e c t i v e  The continuum l i m i t i s  data  f o r aqueous s o l u t i o n s  i o n diameters are i n c r e a s e d by  In the PM framework the use o f s o l v e n t  common approach.  Many r e s e a r c h e r s  [7,18,24,75] have  to improve t h i s approach by d e f i n i n g a s e p a r a t i o n  dielectric vicinity  constant  o f an i o n .  to r e p r e s e n t  dependent  the s o l v e n t w i t h i n the immediate  Many approximations have been put forward,  i n c l u d i n g the assumption that the d i e l e c t r i c  constant  a t the nearest  -  neighbor separation w i l l  approach e<», the h i g h - f r e q u e n c y  However, the d i e l e c t r i c constant and  can be d e f i n e d  large separations, longer the  121 -  i s a macroscopic p r o p e r t y  as g i v e n by (4.5b).  dependent d i e l e c t r i c constant ( see  Figure  At small  constant  s o l v e n t on i o n i c i n t e r a c t i o n s .  +  separations  to describe  to d e s c r i b e  11), we would f i n d  3w^j a t small  where the p o t e n t i a l i s r e p u l s i v e .  that t h i s d i e l e c t r i c  constant  a t the s e p a r a t i o n s dielectric  We see then t h a t the u s e f u l n e s s of  t h i s type o f c o r r e c t i o n to the PM s o l v e n t c l e a r t h a t i n order  to use an r  separations  A l s o , the b e h a v i o r o f t h i s  would change w i t h each i o n .  one can no  the e f f e c t o f  F o r example, i f we t r i e d  must go through i n f i n i t y and then becomes n e g a t i v e  constant  o f a system  by the p o t e n t i a l o f mean f o r c e between two ions a t  e f f e c t i v e l y use a d i e l e c t r i c  for L i / F ~  limit.  i s indeed l i m i t e d .  It i s  to study the s h o r t - r a n g e s o l v e n t e f f e c t s upon i o n i c  c o r r e l a t i o n s , a p o t e n t i a l of mean f o r c e which takes i o n - s o l v e n t and solvent-solvent The  s t r u c t u r e i n t o account must be employed.  ion-ion correlations within  s o l u t i o n may a l s o be i n v e s t i g a t e d by  examining the mean f o r c e a c t i n g between an i o n p a i r . by  the n e g a t i v e  d e r i v a t i v e o f the p o t e n t i a l and i s g e n e r a l l y a more  d i r e c t l y interpretable physical quantity. t h a t i n the p r e s e n t solvent  theory r  From e q u a t i o n (3.40b) we have  f5w g i s the sum of a coulombic term and a a  dependent term, ~ ^ ^ ^ ^ '  F g ( r ) , i s g i v e n by a  The f o r c e i s g i v e n  T  Thus the mean f o r c e between two i o n s ,  -  3 F  where the solvent  first  and  a8  ( r )  -9 - 3F  the  3 q  ( 3 w  The  the  We  13,  the  of two  sum  14,  and  15.  3 F  =  (Ti  s  (  5  '  2  a  )  the  -  (r))  ion-solvent  (5  and  We  2b)  solvent-solvent force i s a t t r a c t i v e .  immediately n o t i c e that the  l a r g e terms which are o p p o s i t e The  interionic  We  much l a r g e r r e l a t i v e v a l u e . i n the w a t e r - l i k e  near c o n t a c t  and  find  s t r u c t u r e except near contact  For  i n s i g n and  The  force i s  so tend  to  and  separations.  Li /Li +  +  ions w i l l  In F i g u r e s  Figures  f o r c e between two  14 and  L i  15 compare 8F^^  +  and  solvent  similar  i o n s has  for  dipolar solvents, respectively.  s o l v e n t , the  13  ions, respectively, i n  f o r c e e x h i b i t s much more s t r u c t u r e f o r the  the w a t e r - l i k e  experience  that the mean f o r c e s have very  where the  the  r e s u l t i n g suras have  segments i n d i c a t i n g that  f o r T~/I~  solvent.  observe t h a t the  total  coulombic term i s always r e p u l s i v e whereas  negative  compare P F ^  the w a t e r - l i k e  solvent.  +  dependent term i s always a t t r a c t i v e .  can  +  =  s il T  f o r c e s which p r e f e r s p e c i f i c  +  q  note that by d e f i n i t i o n a n e g a t i v e  both p o s i t i v e and  Li /Li  a 8  second term,  c a n c e l each o t h e r . solvent  ( r ) )  mean f o r c e a c t i n g between p a i r s of l i k e ions i s presented i n  Figures  14 we  a6  f o r c e dependent on the  structure.  -  term i s the coulombic f o r c e which i s independent of  6F  is  122  We  dipolar  dependent f o r c e term  i s much more a t t r a c t i v e which r e s u l t s i n a l e s s r e p u l s i v e  a  -  123 -  F i g u r e 13 The f o r c e a c t i n g between two I i o n s i n the w a t e r - l i k e s o l v e n t . The dash-dot curve r e p r e s e n t s the mean f o r c e between the two i o n s . The dashed curve r e p r e s e n t s the r e p u l s i v e coulombic component and the s o l i d curve r e p r e s e n t s the n e g a t i v e o f the a t t r a c t i v e s o l v e n t dependent component of the mean f o r c e . -  - 124 -  - 125  Figure  -  14  The f o r c e a c t i n g between two L i ^ i o n s :i n the w a t e r - l i k e s o l v e n t . curves are i d e n t i f i e d as i n F i g u r e 13. +  The  - 126 -  -  127  Figure  -  15  The f o r c e a c t i n g between two L i i o n s i n the d i p o l a r s o l v e n t . curves are i d e n t i f i e d as i n F i g u r e 13. +  The  - 128 -  - 129  mean f o r c e near c o n t a c t . the  As we  would expect f o r the d i p o l a r  f o r c e changes s i g n at ( r - d-Q)/d The  Figures  Li /F +  -  16,  17,  and  the  and  Cs /I~ +  i s the first  U /F~ +  18.  The  total  always being  i n the w a t e r - l i k e f i n d only  The  again  ( c f . Figures  i n width and  Cs /I  i n the w a t e r - l i k e  (r - dj_j)/d  g  and  Figures  17 and  = 1.  the w a t e r - l i k e  solvent  However, the r e l a t i v e  s i m i l a r w i t h both s o l v e n t s g  r  17,  structure.  Not  Hence, we  but  with ion importance  18 compare 3F^j  dipolar solvents, respectively.  l e s s a t t r a c t i v e i n the w a t e r - l i k e  ( r - d£j)/d  f°  = 1 i s much more  observe much more s t r u c t u r e f o r the d i p o l a r s o l v e n t .  is  the  16 and  be a t t r i b u t e d to the decrease i n r e l a t i v e  quadrupolar i n t e r a c t i o n .  "better" solvent.  large  r e l a t i v e l y more  These s t r u c t u r a l changes i n  o f the  is  of two  compare &F±j  some s i m i l a r i t y i n r e l a t i v e  d i p o l a r s t r u c t u r e at +  diameter can  sum  a t t r a c t i v e and  I f we  solvent  the  fo r c e p r o p o r t i o n a l l y more a t t r a c t i v e at c o n t a c t ,  apparent f o r C s / I ~ .  -  1.  f o r c e i s again  r e p u l s i v e peak i s s m a l l e r  repulsive.  +  =  dependent term always r e p u l s i v e .  r e s p e c t i v e l y ) , we only  s  solvent,  mean f o r c e s a c t i n g between p a i r s of u n l i k e i o n s are presented i n  terms w i t h the coulombic p a r t now solvent  -  The  for  We  contact  again value  i n d i c a t i n g that i t i s a  s t r u c t u r e i n the mean f o r c e s  showing a change of s i g n at view the  s o l v a t i o n of u n l i k e ions i n  model as becoming more d i p o l a r i n n a t u r e as i o n s i z e i s  increased.  4.  The D i e l e c t r i c C o n s t a n t o f S o l u t i o n  In Chapter IV we  have shown that the d i e l e c t r i c  c o n s i s t s of both dynamical and  e q u i l i b r i u m terms.  constant The  of s o l u t i o n  dielectric  -  130 -  Figure  16  The f o r c e a c t i n g between a L i / F p a i r i n the w a t e r - l i k e s o l v e n t . The dash-dot curve r e p r e s e n t s the mean f o r c e between the o p p o s i t e l y charged i o n p a i r . The dashed curve r e p r e s e n t s the a t t r a c t i v e coulombic component and the s o l i d curve r e p r e s e n t s the n e g a t i v e of the s o l v e n t dependent r e p u l s i v e term. +  -  - 131 -  -  132  Figure  -  17  The f o r c e a c t i n g between a C s / IL pp aa ii rr i n the w a t e r - l i k e curves are i d e n t i f i e d as i n F i g u r e 16. +  solvent.  The  - 133 -  -  134  Figure  -  18  The f o r c e a c t i n g between a C s / I ~ p a i r i n the d i p o l a r s o l v e n t . curves are i d e n t i f i e d as i n F i g u r e 16. +  The  - 135 -  - 136  c o n s t a n t of an e l e c t r o l y t e the pure s o l v e n t . calculating the l i m i t i n g study we  -  s o l u t i o n at i n f i n i t e d i l u t i o n i s j u s t  However, e x p r e s s i o n s have been developed  the c o n t r i b u t i o n s of the e q u i l i b r i u m and s l o p e of the d i e l e c t r i c decrement.  calculate  the l i m i t i n g  s l o p e s and  t h a t of  for  dynamical  Hence, i n the  terms to present  then compare these w i t h  e x p e r i m e n t a l data at low c o n c e n t r a t i o n . In order to o b t a i n the e q u i l i b r i u m c o n t r i b u t i o n , A e , p  d i e l e c t r i c decrement, we  A  e P  write equation  = % - %  = i  e  ( D i  to the  (4.19) i n the reduced  p  *  form  • • •  +  total  (  5  -  3  a  )  where  (Di  (  (Di  * - „ ^ A TO0 110* p. = p. d , and Inr. l i s OU N  like  D'  o9y  £  (I  00110* 00  1_ 3  11000*. 00 '  _ 00110,,6 , . ., , , ,11000*. = I-,. /d (similarly for I _ ). 00 s 00 T  n  to be a b l e to estimate decrements f o r r e a l e l e c t r o l y t e  which can be assumed to be c o m p l e t e l y d i s s o c i a t e d . s a l t of the g e n e r a l form A B^ C ... we a  N 10  d  )  c  have  3  i  ' ^  s  (5.3b)  ;  „ We  would  solutions  For a s o l u t i o n of a  -  where c is  s a  i^  i s the  t  concentration  Avogadro's number, d  angstroms and IQQ^^ (4.30).  -  of the  salt  solvent  i n moles per  liter,  diameter expressed i n  = a, b, c  and The  i s the  s  137  IQQ**^  are c a l c u l a t e d u s i n g  the  dependence of these q u a n t i t i e s on  expressions  (4.28)  and  i o n diameter i s 00110*  illustrated h^^  and  h?^,  SS  AI  Also, g  T  A  since  We  note that s i n c e IQQ  n o o o * ,depends .  T  (squared because h ^ ^ g  11000*, 2 . , , 00 i i " P 1  S  n  e  e  "  n A  e  n  + t  *  and  the present  model.  From F i g u r e  19 i t can  , o o o and. ,011 h,  and ss is would expect IQQ^^  °f -£* z  W  e  „  a l s o be  i  again not  In F i g u r e u  +  T  upon the  s i g n of the  p o s i t i v e , a l t h o u g h i t does become n e g a t i v e  we  scale  t o  see 1  1  0  0  emphasize that IQQ  seen that I Q Q ^ ^  * becomes l a r g e r q u i c k l y f o r d^ > 1.3.  present  19 we +u  *  on  , because ,  on h  i s squared).  depend only upon the magnitude but for  depends  be independent of i o n charge i n the  00 s c a l e s w i t h i o n charge, we  know h ^ ^ 2  19.  i t will  IS  +u theory.  with z  i n Figure  that 0  *  -11  will  ionic  charge  is relatively  f o r very  small  small ions  11000* IQQ i s r e l a t i v e l y large  and and  00110* negative * d^  K  0.85  diameter.  and  i s more s t r u c t u r e d  than IQQ  from which i t drops very  .  I t has  a l o c a l maximum at  r a p i d l y with decreasing  *  A f t e r a s h a l l o w l o c a l minimum at d^  » 1.1  ion  i t continues  to 11000*  become more p o s i t i v e w i t h i n c r e a s i n g i o n s i z e . may  appear s u r p r i s i n g at f i r s t ,  two  c o n t r i b u t i n g f a c t o r s which vary w i t h i o n s i z e .  ion  i s increased  we  know that  but  the  i t can be  The  i o n w i l l be  structure i n  explained As  by the  surrounded by  IQQ  considering s i z e of an  an  -  138  Figure The  terms c o n t r i b u t i n g to  of i o n diameter. i o n s and  the  represents theory.  The  e  solid  (  1  )  i  f°  r  19 water-like  curve r e p r e s e n t s  s o l i d dots are r e s u l t s 00110*  IQQ  t n e  -  s o l v e n t as a  ~IQQ  /Z^  for divalent ions.  for The  univalent dashed  which i s independent of i o n i c charge i n the  Note that two  s c a l e s are used i n the  plot.  function  line  present  -  i n c r e a s i n g number of n e i g h b o r i n g the  charge-dipole  -  s o l v e n t m o l e c u l e s , but  i n t e r a c t i o n s between the  m o l e c u l e s must d e c r e a s e . offset  140  each other  These two  to some e x t e n t .  these  tested  of I Q Q ^ ^  strength  of  solvent  competing f a c t o r s w i l l We  a t t e m p t i n g to r e c a l c u l a t e v a l u e s  i o n and  the  tend  this explanation  by  for several ions,  a.  to  11000* IQQ  i o n , 3,  f o r the  C.N. /C.N.g and  by  a  nearest  1/s  w i t h d^ = d  s  was  , where s i s an  s o l v e n t molecules around the  m u l t i p l i e d by  the  "average" s e p a r a t i o n  ion.  We  were able  to  ratio  of  for  the  roughly  reproduce the b e h a v i o r e x h i b i t e d by * Q Q ^ ^ * f o r 0.7 < d < 1.3. * * In F i g u r e 20 we have p l o t t e d Q\.|_ f u n c t i o n of d^ f o r e  a s  a  * univalent  ions  i s negative  i n the w a t e r - l i k e  ( i n d i c a t i n g that  concentration)  and  e  solvent. will  p  I t can  I t :  of  e  * (j^>  a s  i n t e r e s t i n g to note that  i s  ion  decreases i n magnitude w i t h i n c r e a s i n g i o n diameter,  11000* IQQ dominates the v a l u e e  seen t h a t  decrease w i t h i n c r e a s i n g  a c t u a l l y becoming p o s i t i v e f o r d^ > 1.9.  * of (i)i«  be  For I e  s  a l l but  large  obvious from the  * (^)^ I  s  nearly  ions, behavior  constant  in  the reduced ion diameter range 0.8 to 1.2 which encompasses many of the * * a l k a l i metal and h a l i d e i o n s . For l a r g e r Ions such as I , e,.. ' ( D i begins to show a marked i n c r e a s e  and  because I Q Q ^ ^  i s decreasing  00110* IQQ i s b e g i n n i n g to have a s i g n i f i c a n t It  i s found  [23]  t h a t , f o r ions  i n magnitude  affect.  s i m i l a r i n s i z e to the  solvent,  * £  (l)i  *  S  a  -*-  most  o  n  e  o r <  ler  of magnitude more n e g a t i v e  f o r the  dipolar  -  141  Figure The  dependence of  the w a t e r - l i k e  e  Q)^  solvent.  o n  -  20  i o n diameter f o r u n i v a l e n t  ions  - 142 -  -  solvent.  143 -  00110* IQQ is still  In the d i p o l a r s o l v e n t  relatively  unimportant  11000* f o r i o n s of t h i s s i z e , however, IQQ 3).  The other  i s much l a r g e r (by a f a c t o r of  l a r g e c o n t r i b u t i n g f a c t o r i s y ( p r o p o r t i o n a l to u ) 2  which appears i n the denominator o f e q u a t i o n times s m a l l e r with e  Q  due to the s m a l l e r  I t i s about 3  d i p o l e moment of the d i p o l a r  solvent  1  = 78.5.  In order we r e w r i t e  to c a l c u l a t e the k i n e t i c d i e l e c t r i c decrement, A e ^ ^ ,  equation  (4.42) i n the form  Ae  Ae  where A e ^  D D  '/  T  a D  i  KDD  i-  s  t  n  •  e  A  (  E  (  KDD  „ C. i ^ 1T L  i  )  V  In o b t a i n i n g  C  D  salt  ( 5  '  5 a )  d i m e n s i o n l e s s r a t i o given by  . KDD  can  (5.3b).  ST __  (5.5a) we have used the f a c t  that at low c o n c e n t r a t i o n s  we  write  °i  = salt c  V  i X°i  ( 5  '  5 c )  I t has been shown [35,38,41] that f o r equal v a l u e s of u and p, e i s much l a r g e r f o r a p u r e l y d i p o l a r system than i f a quadrupole i s i n c l u d e d i n the model.  Q  - 144  -  where X^ i s the molar i o n i c conductance at i n f i n i t e express  i n ST  -1  m  dilution.  If  we  then  -1  T  A =  = 0.7415  P 4 i r  g  ton  (5.5d)  o  where £  i s p e r m i t t i v i t y of f r e e space and  0  =  8.25  [99]  have used  12  —  T  we  x 10  s.  We  again  have been assumed i n o b t a i n i n g  note that  s l i p boundary  equations  conditions  (5.5b). KDD  In F i g u r e 21 we depends o n l y on  see  t h a t the r a t i o  ratio  D  from the p r e v i o u s  i n F i g u r e 21. dependent on present  T  r  r  ' hi h w  c  c o r r e c t l a r g e d^ l i m i t i n g  We the  Values of A e ^  observe that  the v a l u e s  D  / f o r  the  behavior.  have used the  of the  since  present  a r e  v e r v  Stokes r a d i i .  Stokes r a d i i g i v e n  i n Ref.  f o r s l i p boundary c o n d i t i o n s by m u l t i p l y i n g by 3/2. and  previously c a l c u l a t e d values  f a i r agreement, p a r t i c u l a r l y f o r l a r g e r i o n s . occurs f o r the  smallest  i o n compared, L i .  encouraging s i n c e the v a l u e s  +  were obtained  quite d i f f e r e n t i n o r i g i n .  The  study  are a l s o shown  ^^^1°^^  of  somewhat a r b i t a r y c h o i c e  comparison between the present  which are  a  r e s u l t s of Hubbard e t . a l . [82]  c a l c u l a t i o n we  corrected  ^ D i^ i^ i  i s independent of i o n charge as would be expected  U^ /N^ s c a l e s w i t h charge. and  (Ae^  the i o n i c c r y s t a l r a d i u s , r ^ , i s a smooth continuous  f u n c t i o n of i o n diameter w i t h the This  ST  largest  In  the  100 The produces  discrepancy  T h i s agreement i s q u i t e using  approximate methods  - 145 -  F i g u r e 21 The dependence on i o n diameter of the k i n e t i c d i e l e c t r i c the w a t e r - l i k e s o l v e n t . for  The s o l i d  line  represents  u n i v a l e n t i o n s and s o l i d dots are r e s u l t s  Results f o r  ^E^VO^T^  i n the present  (Ae^  decrement f o r ^ ± Y)^ ±^ ± a  T  T  T  for divalent ions.  theory f o r s e v e r a l p o s i t i v e and  n e g a t i v e i o n s are g i v e n by open c i r c l e s and s o l i d squares, r e s p e c t i v e l y , w h i l e the s t a r s r e p r e s e n t (from r i g h t to l e f t ) p r e v i o u s r e s u l t s of Hubbard e t . a l . [82] f o r L i , N a , C l ~ , C s . +  +  +  - 146 -  24.0  20.0  16.0 o  12.0 -(AeT/oiT )^ D  '(STR  8.0  4.0  0  /  0  Li NaVRl> CsT F" CI" Br +  J  I  .0  I  di  L  2.0  _l  3.0  L  - 147  In the p r e s e n t study we Ae  p  + Ae^jj.  d e f i n e the t o t a l d i e l e c t r i c decrement Ae =  In F i g u r e 22 we  v a l u e s of Ae, A e  p  -  compare f o r L i C l , KC1, CsCl  as g i v e n by e q u a t i o n ( 5 . 4 ) , and  using l i t e r a t u r e values  a l k a l i halides.  obtained  [4] of X° as g i v e n i n Table IV.  t h r e e s a l t s g i v e q u i t e s i m i l a r r e s u l t s as i s found all  Ae^Q  solutions  These  to be the case f o r  I t i s c l e a r from F i g u r e 22 t h a t the v a l u e of Ae f o r  any aqueous a l k a l i h a l i d e s o l u t i o n i s dominated by A e . p  This  t h e o r e t i c a l p r e d i c t i o n seems c o n s i s t e n t w i t h p r e v i o u s e x p e r i m e n t a l [80,83] a l t h o u g h i t c o n t r a d i c t s the t h e o r i e s of these  T a b l e IV.  Molar  i o n conductances  Cation Li  +  Na  +  authors.  [4] a t i n f i n i t e d i l u t i o n at 25°C.  X°_  Anion 38.68  F~  55.4  50.10  CI"  76.35  73.50  Br"  78.14  Rb+  77.81  I"  76.84  Cs  77.26  K  +  +  In F i g u r e s 23 and 24 we f o r d i l u t e s o l u t i o n s of 1:1  compare t h e o r e t i c a l and  decrement f o r CuCl2 i s observed  2:1  and e x p e r i m e n t a l  electrolytes, respectively.  to be about f o u r times l a r g e r  results The  than  those o b t a i n e d f o r a l k a l i h a l i d e s a t the same s a l t c o n c e n t r a t i o n . the t h e o r e t i c a l c a l c u l a t i o n s we 0.72A.  The  experimental  data  have taken the r a d i u s of Cu"*" to be  p o i n t s from Ref. 80 r e p r e s e n t  2  In  -  148  Figure  -  22  A comparison of the e q u i l i b r i u m and dynamical c o n t r i b u t i o n s to the d i e l e c t r i c decrement. T h e o r e t i c a l r e s u l t s f o r Ae , A s ^ p , <l Ae f o r s o l u t i o n s of L i C I , KC1, and C s C l i n the w a t e r - l i k e s o l v e n t are p r e s e n t e d . The i o n diameters are those g i v e n i n T a b l e I . ari  p  total  - 149 -  I  0  I  I  0.02  I  I  0.04  I  CsALT (moles/litre)  I  0.06  - 150  Figure  -  23  Comparing e x p e r i m e n t a l and t h e o r e t i c a l d i e l e c t r i c decrements f o r aqueous 1:1 e l e c t r o l y t e s o l u t i o n s . The open squares, open c i r c l e s , and s o l i d dots r e p r e s e n t e x p e r i m e n t a l r e s u l t s , A e p , as o b t a i n e d from Ref. 80 u s i n g e q u a t i o n (5.6a), f o r KC1, L i C l , and NaCl s o l u t i o n s , r e s p e c t i v e l y . The s t a r s are experimental r e s u l t s o b t a i n e d from Ref. 81 u s i n g e q u a t i o n (5.6b) f o r a KC1 s o l u t i o n . The t h e o r e t i c a l r e s u l t s , Ae^nj), A e , and Ae, are f o r the w a t e r - l i k e s o l v e n t model. The dashed l i n e r e p r e s e n t s the e q u i l i b r i u m c o n t r i b u t i o n to the d i e l e c t r i c decrement f o r a KC1 s o l u t i o n u s i n g the present d i p o l a r s o l v e n t model. e X  t  p  -  0.0  151  0.05 CsALT  -  0.1  (moles/litre)  - 152 -  Figure  24  Comparing e x p e r i m e n t a l and t h e o r e t i c a l d i e l e c t r i c decrements f o r aqueous CuCl2 s o l u t i o n s . The s o l i d dots r e p r e s e n t e x p e r i m e n t a l r e s u l t s , A e p , as o b t a i n e d from Ref. 80 u s i n g e q u a t i o n (5.6a). The t h e o r e t i c a l r e s u l t s , Ae^nn* A e , and Ae, are f o r the w a t e r - l i k e s o l v e n t model. e X  t  p  0  0.02  0.04  0.06  CsALT( °'es/Htre) m  0.08  - 154 -  A £  where A e  n p  expt  =  e  SOL ~ o ~ £  A £  DF  i s g i v e n by e q u a t i o n (4.37).  ( 5  '  6 a  >  '  6 b )  The e x p e r i m e n t a l p o i n t s  from Ref. 81 r e p r e s e n t  A £  expt  =  £  SOL - o £  ( 5  where we assume that these data p o i n t s do not c o n t a i n a DF c o n t r i b u t i o n s i n c e they were o b t a i n e d u s i n g h i g h frequency As noted  i n Chapter  techniques.  IV, we see t h a t i n both F i g u r e s 23 and 24 the  e x p e r i m e n t a l p o i n t s of van Beek [80] are a l l s h i f t e d  upward.  I f we  assume t h a t h i s e x p e r i m e n t a l p o i n t s obey a l i n e a r law ( i g n o r i n g the two p o i n t s a t lowest c o n c e n t r a t i o n f o r L i C l ) , we f i n d and t h e o r e t i c a l s l o p e s agree q u i t e w e l l . would not e x t r a p o l a t e to z e r o . Ae  observed.  However, the e x p e r i m e n t a l  I t i s obvious  i s too s m a l l to alone account  that the e x p e r i m e n t a l  from both p l o t s  f o r the d i e l e c t r i c  line  that  decrement  The e x p e r i m e n t a l r e s u l t s o f Weiss e t . a l . [81] i n d i c a t e a  s m a l l e r slope f o r the d i e l e c t r i c decrement.  Although these r e s u l t s ( o f  which o n l y the two lowest c o n c e n t r a t i o n p o i n t s a r e shown) form a n i c e smooth curve which e x t r a p o l a t e s to z e r o , we must be c o n s c i o u s o f the fact two  t h a t they were o b t a i n e d u s i n g h i g h frequency techniques w i t h o n l y frequency measurements made f o r each p o i n t .  The n e a r l y i d e a l  b e h a v i o r o f these r e s u l t s c o u l d e a s i l y be o n l y an a r t i f a c t techniques.  of f i t t i n g  We would expect r e l a t i v e l y l a r g e e r r o r s i n these  e s p e c i a l l y f o r the p o i n t s a t lowest c o n c e n t r a t i o n .  results  A l t h o u g h van Beek  -  [80]  predicted  evident  i n the  d i e l e c t r i c c o n s t a n t s f o r most of the  t r y i n g to account f o r the  h i s data f o r both 1:1 i s no  and  t h e o r e t i c a l basis  w e l l be due  errors.  studied  24 we  current  exist.  solutions.  there  to support t h i s a d d i t i o n a l increment and  l a c k of "good" e x p e r i m e n t a l data f o r  This  have i n c l u d e d  d i p o l a r model f o r a KC1 obviously  problem  At present  it  may  concentrations,  dielectric  i t would appear  present s o l u t i o n model i s i n r e a s o n a b l e agreement w i t h  r e s u l t s which do Figure  s a l t s he  increment which appears i n  electrolyte solutions.  c o n s t a n t s of aqueous s o l u t i o n s a t low the  scatter  There a l s o remains the  added d i e l e c t r i c  2:1  the  to e x p e r i m e n t a l problems.  D e s p i t e the  that  -  r e l a t i v e l y modest e r r o r s on h i s p o i n t s ,  appears to exceed h i s p r e d i c t e d of  155  the  i s c l e a r from both F i g u r e s  23 and  d i e l e c t r i c decrement p r e d i c t e d  solution.  g r e a t l y o v e r e s t i m a t e s the  A purely  d i p o l a r solvent  the 24.  by  the  model  d i e l e c t r i c decrement f o r aqueous  In  -  156 -  CHAPTER VT  CONCLUSIONS  In t h i s t h e s i s we have examined model aqueous e l e c t r o l y t e s o l u t i o n s at i n f i n i t e d i l u t i o n using  the LHNC theory.  A p o l a r i z a b l e hard-sphere  f l u i d w i t h embedded p o i n t d i p o l e s and t e t r a h e d r a l quadrupoles w i t h water-like  parameters was used to model the s o l v e n t  s p h e r i c a l ions were immersed.  Results  obtained  s o l v e n t model were compared with those g i v e n as w e l l as w i t h the p r i m i t i v e model. ion-solvent  c o r r e l a t i o n functions  infinite dilution limit.  by a p u r e l y  The LHNC theory  i n t e r a c t i o n energies,  dipolar  d i e l e c t r i c constant An e x p r e s s i o n KDD' defined  solvent  was used to o b t a i n  ion-solvent  numbers, and i o n - i o n p o t e n t i a l s of mean f o r c e .  of s o l u t i o n has been  The mean  Furthermore, the theory  f o r both e q u i l i b r i u m and dynamical c o n t r i b u t i o n s  theory  water-like  f o r e l e c t r o l y t e s o l u t i o n s i n the  f o r c e a c t i n g between i o n s was a l s o examined.  Ae  with t h i s  These c o r r e l a t i o n f u n c t i o n s were then used t o  determine average i o n - s o l v e n t coordination  i n t o which hard  to the apparent  discussed.  f o r the s o l u t e dependent dynamical c o n t r i b u t i o n ,  by Hubbard e t . a l . [82], was e v a l u a t e d  i n the i n f i n i t e d i l u t i o n l i m i t .  were c a l c u l a t e d u s i n g  the i o n - s o l v e n t  i n the present  V a l u e s f o r A e ^ ^ and A e  p  c o r r e l a t i o n f u n c t i o n s and both of  these terms, as w e l l as the t o t a l d i e l e c t r i c decrement, were compared w i t h e x p e r i m e n t a l r e s u l t s f o r 1:1 and 2:1 aqueous e l e c t r o l y t e s . The dependence on i o n s i z e and charge was of primary concern i n our i n v e s t i g a t i o n o f the s t r u c t u r a l and d i e l e c t r i c p r o p e r t i e s  of model  -  157 -  aqueous e l e c t r o l y t e s o l u t i o n s . The  ion-solvent  c o r r e l a t i o n functions  s e n s i t i v e to i o n s i z e and charge. correlation  functions  univalent  i o n s , the i o n - s o l v e n t  saturated.  s c a l e w i t h the i o n charge,  T h i s b e h a v i o r i n d i c a t e s that even f o r orientational structure i s e f f e c t i v e l y  model i n the LHNC theory  and i t i s a l s o c l e a r that  " s c a l i n g " could not occur a t f i n i t e  i o n concentrations Oil  "higher  order"  theories.  s c a l i n g i n other The  The s c a l i n g i n h ^  g  this  or i n  022 and h ^  g  l e a d s to  f u n c t i o n s and p r o p e r t i e s of the system.  i o n - i o n p o t e n t i a l s of mean f o r c e demonstrate a s t r o n g  upon diameter and t h e i r charge dependence d i f f e r s size.  these  We have noted that t h i s charge dependence i s not unexpected  f o r the present exact  However, i n the LHNC theory  are found to simply  i r r e s p e c t i v e of i o n diameter.  are found to be q u i t e  dependence  f o r ions of v a r y i n g  F o r s m a l l i o n s the p o t e n t i a l s of mean f o r c e s c a l e to a f a i r  a p p r o x i m a t i o n w i t h the product of the i o n charges whereas f o r l a r g e r i o n s t h i s behaviour i s not observed. solvent,  F o r i o n s i n the w a t e r - l i k e  8w^j e x h i b i t s h i g h l y dampened o s c i l l a t o r y  structure,  a p p r o a c h i n g the continuum l i m i t much more r a p i d l y than f o r i o n s i n a purely  dipolar solvent.  solvent, reducing ions at small very  important  The w a t e r - l i k e  to a g r e a t e r  separation.  extent  fluid  i s found to be a " b e t t e r "  the a t t r a c t i o n between two u n l i k e  Quadrupolar i n t e r a c t i o n s are found to p l a y a  role i n reducing  the s t r u c t u r e i n 8w^j and the  p o t e n t i a l s of mean f o r c e f o r r e l a t i v e l y s m a l l  ions  quadrupolar f e a t u r e s .  the average charge-  As i o n s i z e i n c r e a s e s ,  show d i s t i n c t  quadrupole energy decreases much more r a p i d l y than the average  -  c h a r g e - d i p o l e energy and d i p o l a r i n nature. dampen p u r e l y  -  the p o t e n t i a l s of mean f o r c e become more  However, the quadrupolar i n t e r a c t i o n s continue  dipolar features.  s i m i l a r conclusions. has  158  We  Comparisons of the mean f o r c e s  emphasize that  lead  i n the p r i m i t i v e model,  none of these s t r u c t u r a l f e a t u r e s which depend upon  to to  3w^j  ion-solvent  correlations. For small  the present model and  i o n s , but  larger ions.  increases  theory, Ae  i s found to be n e g a t i v e f o r  p  with i o n s i z e to a c t u a l l y become p o s i t i v e f o r  I n t e r e s t i n g l y , the e q u i l i b r i u m c o n t r i b u t i o n to  the  d i e l e c t r i c decrement i s r e l a t i v e l y c o n s t a n t f o r a l k a l i metal and ions. Ae  The  KDD'  a  r  e  v a l u e s c a l c u l a t e d f o r the k i n e t i c d i e l e c t r i c ^  o u n  ^  t o  b e  ^  n  r e a s o n a b l e agreement w i t h the  c a l c u l a t i o n s of Hubbard e t . a l .  This  only  about o n e - t h i r d  the  decrement, previous  dynamical c o n t r i b u t i o n to  t o t a l d i e l e c t r i c decrement i s a l s o n e g a t i v e ,  a l t h o u g h i t was  s i z e of the e q u i l i b r i u m  halide  the  found to  term f o r aqueous  be  alkali  halides. I t was  p o s s i b l e i n t h i s study to make comparisons between  d i e l e c t r i c decrements p r e d i c t e d experimentally. calculated  I t i s found that  electrolytes  the  The  limiting  from e x p e r i m e n t a l data  those measured  theoretical dielectric  slopes  f o r 1:1  to make the  i n the  decrements  relatively and  2:1  concentrations  i n t h i s study.  however, the u n c e r t a i n t i e s  s u f f i c i e n t l y l a r g e so as  alkali  present theory and  [80,81] at low  i n c o n s i s t a n t w i t h those p r e d i c t e d  concentrations, are  the  f o r aqueous a l k a l i h a l i d e s o l u t i o n s are  i n s e n s i t i v e to i o n s i z e .  not  by  the  At  are  low  experimental  points  trends observed f o r i n d i v i d u a l  halide solutions e s s e n t i a l l y indistinguishable.  Also, a l i n e a r  -  extrapolation  of the  concentration  does not  though we the  DF  159  -  e x p e r i m e n t a l data of van give  have attempted  Beek [80]  to  zero  the d i e l e c t r i c c o n s t a n t of pure water, even  to c o r r e c t  e f f e c t i s o n l y known i n the  f o r the  DF  effect.  The  magnitude of  infinite dilution limit.  Its  b e h a v i o r at f i n i t e c o n c e n t r a t i o n s i s unknown, a l t h o u g h i t has past been assumed that  the  i n f i n i t e d i l u t i o n r e s u l t could  c o n c e n t r a t i o n s of a t l e a s t 10 the  data of van  e x p e r i m e n t a l and of the  remains a p u z z l e and  theoretical investigation.  be  The  deserves  in  are needed not  Beek [80], but  also  only  to  b e h a v i o r of  further  More a c c u r a t e measurements  to c o n f i r m or r e f u t e  to f u r t h e r  the  used up  d i e l e c t r i c c o n s t a n t s of aqueous e l e c t r o l y t e s o l u t i o n s at  concentration of van  Beek [80]  e q u i v a l e n c e per mole.  true  t e s t the  low  the measurements  theoretical results  presented i n t h i s t h e s i s . The areas.  r e s u l t s of the The  present study may  average i o n - s o l v e n t  be a p p l i e d  i n t e r a c t i o n s can  c a l c u l a t i o n of s o l v a t i o n e n e r g i e s f o r ions solvent. solvent  The has  r e s u l t s and currently  be used i n present  already the  been c a r r i e d out  [72]  the  water-like  c a l c u l a t i o n of s o l v a t i o n e n e r g i e s i n a p u r e l y  taking  d i l u t i o n are  i n the  to many o t h e r  dipolar  with somewhat s u r p r i s i n g  a p p l i c a t i o n of t h i s method to the w a t e r - l i k e place.  The  also presently  average i o n - s o l v e n t  e n e r g i e s at  have been a p p l i e d  been used i n dynamical c a l c u l a t i o n s  The  potentials  to dynamical t h e o r y .  times can  [101,102] which p r e d i c t NMR  be measured e x p e r i m e n t a l l y and  for of  They have spin  r e l a x a t i o n times f o r diamagnetic paramagnetic i o n p a i r s i n s o l u t i o n . These r e l a x a t i o n  is  infinite  being used to c a l c u l a t e heat c a p a c i t i e s  aqueous e l e c t r o l y t e s o l u t i o n s at h i g h temperatures. mean f o r c e o b t a i n e d  solvent  are  very  -  160 -  s e n s i t i v e to the s t r u c t u r e i n the p o t e n t i a l s o f mean f o r c e ( i . e . t o the ion-ion correlations). dynamical  These comparisons p r o v i d e a good t e s t of both  and e q u i l i b r i u m t h e o r i e s .  Recent r e s u l t s  good agreement between theory and experiment.  [102] have shown  I t has a l s o been  found  t h a t i f the p r i m i t i v e model p o t e n t i a l o f mean f o r c e i s used, particularly  f o r u n l i k e i o n s , v e r y poor agreement i s o b t a i n e d .  A f u r t h e r study of e l e c t r o l y t e s o l u t i o n s c o u l d be attempted " b e t t e r " theory f o r the present s o l u t i o n model. t h e o r y we would not expect s c a l e w i t h i o n charge  the i o n - s o l v e n t c o r r e l a t i o n f u n c t i o n s t o  A finite  u s i n g a McMillan-Mayer  i n v o l v e s u s i n g an e f f e c t i v e would attempt  I n the QHNC or HNC  s i n c e the s o l v e n t packing w i l l now depend on the  ion-solvent interactions. performed  using a  c o n c e n t r a t i o n c a l c u l a t i o n c o u l d be  [103] l e v e l t h e o r y .  T h i s approach  ef f i o n - i o n i n t e r a c t i o n p o t e n t i a l , u ^ , which  to i n c l u d e a l l s o l v e n t e f f e c t s .  F o r such a c a l c u l a t i o n ,  ef f t h i s u^j  c o u l d be taken as a p o t e n t i a l o f mean f o r c e f o r the p r e s e n t  model a t i n f i n i t e  dilution.  studied at f i n i t e  i o n c o n c e n t r a t i o n by extending  this thesis.  The present  s o l u t i o n model c o u l d a l s o be the theory presented i n  T h i s would r e p r e s e n t a " t r u e " f i n i t e c o n c e n t r a t i o n  c a l c u l a t i o n , u n l i k e the McMillan-Mayer approach, t e s t of the McMillan-Mayer t h e o r y .  Although  and would serve as a  the theory and c a l c u l a t i o n  would be much more c o m p l i c a t e d , the r e s u l t s of such a study c o u l d be r e a d i l y compared w i t h e x p e r i m e n t a l d a t a .  At f i n i t e  concentration there  i s a wealth of a v a i l a b l e e x p e r i m e n t a l data f o r thermodynamic and d i e l e c t r i c p r o p e r t i e s w i t h which t o compare. improved by t a k i n g i n t o account ion  i n solution.  The p r e s e n t model c o u l d be  the p o l a r i z a t i o n o f the s o l v e n t by an  We would expect  t h i s t o i n c r e a s e the degree o f  -  161 -  o r i e n t a t i o n a l s a t u r a t i o n of the s o l v e n t around an i o n . A l s o , the s o l u t i o n model c o u l d be a l t e r e d to a l l o w  i o n s to have d i p o l e and/or  quadrupole moments (as i n the case of CN~). be made w i t h computer s i m u l a t i o n s present  t o ensure that the theory  model has not broken down.  c u r r e n t l y being In t h i s  F i n a l l y , comparisons  examined.  study i t has been shown that f o r aqueous e l e c t r o l y t e to d e s c r i b e the  q u a l i t a t i v e b e h a v i o r of the apparent d i e l e c t r i c constant  s o l u t i o n a t low c o n c e n t r a t i o n s . molecular species the  f o r the  Many of the above e x t e n s i o n s a r e  s o l u t i o n s a simple e l e c t r o s t a t i c model i s s u f f i c i e n t general  should  Including  of  the s o l v e n t as a t r u e  i n the model s o l u t i o n s i s found to g r e a t l y  short-range i o n - i o n c o r r e l a t i o n s p r e d i c t e d .  influence  T h i s t h e s i s has c l e a r l y  demonstrated that one can and should  go beyond the p r i m i t i v e model i n  the study o f e l e c t r o l y t e s o l u t i o n s .  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Brigham and R.E. Morrow, The F a s t F o u r i e r Transform, Spectrum, New York, 1967.  In p r e s s .  F.G. H e r r i n g and G.N. Patey, J .  L3, 276 (1945).  36_, 241 (1978).  IEEE  - 167 -  108.  F. Lado, J . Comp, Phys., 8, 417 (1971).  109.  A.A. B r o y l e s , J . Chem. Phys., 33, 456 (1960).  - 168 -  APPENDIX A  ROTATIONAL INVARIANTS  The  rotational invariants,  ^QO^'  a s  defined  be expressed i n terms of s p h e r i c a l harmonics,  by  e q u a t i o n (2.11),  Y^(9,<|)),  by  defining  Wigner m a t r i x elements, Dg^(fi), i n terms of s p h e r i c a l harmonics. forms of Wigner m a t r i x elements can explicit study.  e x p r e s s i o n s f o r the We  find  [104,105] of $ ^ ' uv  where x,, i  m n  A  is  1 and one  *  0 0 0  u n i t v e c t o r i n the  t n e  2.  Thus u s i n g the  cartesian  (12) =  f o r the  d i r e c t i o n of the  shift-up  = z  $  (12)  = - z  (12)  = 3(zj^ • r  112  2  • r  x  •  1  2  z , 2  )(z  2  • r  1 2  )  The  used i n t h i s representation unit and  segment j o i n i n g m o l e c u l e s  (down) o p e r a t o r s g i v e n i n [104,105]  ,  1 2  the  obtain  present model i n f i g u r e 2.2)  1,  (12)  to  z, denote the m o l e c u l e - f i x e d i  obtains  $  $  the  A  y., i  v e c t o r s of molecule i ( d e f i n e d r^2  generalized  rotational invariants,  i t convenient to use  A  then be  can  - (z  x  • z ), 2  - 169 *022  (12) =  /6  . ^^2  [ ( x 2  - (y •r 2  A  A  =4 l [x  *  211  (12)  *  213  ( 1 2 ) = /6{-5[  A  *2 z  A  ) ( x  +2[x  *  220  $  2 2 2  A  (12) = - 2 [  '12 " r  A  A  A  2  •r  A  2  A A 1  1  2  -(x «y )(x T 1  l ' 12 '  y  r  A  1 2  A  A  1 2  1 2  1 2  )  A A  A  •z ) (y • r ) ] } ,  l  1 2  ) ]  • r  2  A  ) - (y  2  A  x  A  1 2  A  A  •y ) - (x •y ) - ^  7 l  2  2  )(x T  2  y  2  2  AA  2 ^ ~ (yr* ) ] z  2  ) + ( y *y )(y * l 2 ^ 2 * 1 2 )  1 2  1  2  1  y  r  1  ) - ( y •x )(y T  1 2  • x ) ],  2  r  2  )(y T  2  L  AA  ^[(xj^ • x ) ( x T  2  } (  + (y^)* - (v  /  2  1  Z  A  1  2  (i2) = -2{[(x -x )  l '2  y  x  •x ) + (  X l  (  ) * - (y •r ) * ] ( z  •z )(x  A  }  A  1 2  A  x  A  A  l  •r  Xl  ) ],  1 2  2  1  1 2  )(x »r 2  1 2  ) ] },  and  A A *  224  (12) = | { 3 5 [ ( - r X l  AA 1 2  )  2  -  -20[(xj * x ) ( x • r 2  1  -(x «y )(x «r 1  1  2  AA  T  i  1  2  2  )  2  )(x «r 2  1 2  1 2  y  )(y T 2  1 2  AA  +2[( -x ) X l  (  AA ]  [(x -r 2  1 2  2  2  )  - (y -r  2  2  ) + (y *y ) ( y T  ) -  1  2  x  (y »x )(y « r 1  1  2  AA  + (y -y ) 1  1 2  2  AA  - (V 2 y  1 2  1 2  ) ( y *r 2  )(x *r 2  AA  ) 2  "  ) ] 2  1 2  (y!* ) n. x  2  2  1 2  1 2  ) ]  )  - 170  -  APPENDIX B  ITERATIVE PROCEDURES  Two  iterative  the p r e s e n t below, was how  procedures  study.  f o r o b t a i n i n g s o l u t i o n s were attempted i n  A modified  i t e r a t i v e method, which i s d i s c u s s e d  used as the a c t u a l l y working method.  one might a p p l y a r e c e n t l y developed  Newton-Raphson (NR)  method  We  have a l s o examined  [68] of the  t y p e , which had appeared to have  promising  characteristics. The  Newton-Raphson method has been s u c c e s s f u l l y used by G i l l a n  to o b t a i n q u i c k l y converging  s o l u t i o n s i n the study of i o n i c  fluids.  The method i s a c t u a l l y a h y b r i d of the t r a d i t i o n a l i t e r a t i v e o f the u s u a l NR  technique  procedure r e q u i r e s t h a t we thus  f o r f i n d i n g zeros of e q u a t i o n s . c a l c u l a t e and  r e p r e s e n t s an e x t e n s i v e c a l c u l a t i o n .  advantages to u s i n g t h i s method.  I t was  found  and  claimed  to g r e a t l y reduce  two  the  the method would  always converge i r r e s p e c t i v e of i t s s t a r t i n g p o i n t .  U n f o r t u n a t e l y , the NR method was system.  This  [68] had  number of i t e r a t i o n s r e q u i r e d to reach convergence and virtually  scheme and  i n v e r t a jacobian matrix Gillan  [68]  The  not e a s i l y a p p l i e d to the  systems G i l l a n s t u d i e d were d e f i n e d u s i n g o n l y  p o t e n t i a l s and  so the g e n e r a l i z e d c o n v o l u t i o n of the OZ  i n v o l v e s o n l y F o u r i e r transforms multicomponent  ( c f . equation  system, i n which "(12)  r o t a t i o n a l i n v a r i a n t s , the e x p r e s s i o n s i n v o l v i n g Hankel t r a n s f o r m s .  (3.15)).  present isotropic  equation For the  present  has been expanded i n terms of become much more c o m p l i c a t e d ,  As a r e s u l t each NR  cycle represents  a  now  - 171 -  very laborious numerical computational  c a l c u l a t i o n , r e q u i r i n g about 5 times as many  s t e p s as i n a normal i t e r a t i v e c y c l e .  For the present  system the NR method probably r e p r e s e n t e d a longer computation i n o b t a i n i n g convergence. technical d i f f i c u l t i e s present system. present  There a l s o appeared to p o s s i b l y be some i n the a p p l i c a t i o n o f the NR method to the  Thus, we d i d not a c t u a l l y use the NR method i n the  study.  A m o d i f i e d i t e r a t i v e method, d i a g r a m m a t i c a l l y  represented  i n Figure  25, was s u c c e s s f u l l y used to o b t a i n a l l s o l u t i o n s i n t h i s study. f o l l o w i n g i s a step-by-step Step  (1):  Input.  explanation of t h i s  The  procedure.  The input step i n v o l v e s the r e t r i e v i n g o f a  p r e v i o u s s o l u t i o n c o n s i s t i n g of a s e t o f n ^ ^ C r ) which has been s t o r e d is i n a computer f i l e from a p r e v i o u s c a l c u l a t i o n . In the case where no p r e v i o u s s o l u t i o n s e t e x i s t s , an i n i t i a l  "guess" i s used as a s t a r t i n g  HS p o i n t a l t h o u g h care i s r e q u i r e d . retrieved  The g ^ ( ) must be c a l c u l a t e d o r r  s  from a f i l e i f p r e v i o u s l y c a l c u l a t e d .  At t h i s time we a l s o  c a l c u l a t e any c o e f f i c i e n t s o r c o n s t a n t s , such as the a n a l y t i c a l p i e c e o f c ? ^ , which w i l l be needed, is Step  (2):  equations h  j, old m n  at  A p p l y i n g the LHNC C l o s u r e .  The LHNC c l o s u r e , as g i v e n by  (3.45), i s a p p l i e d to the c u r r e n t s o l u t i o n , r e p r e s e n t e d by  to c a l c u l a t e c ? . is n A  The long-range  term o f c ? ^ i s not i n c l u d e d is  this point. Step  (3):  Hankel Transforms.  performed by f i r s t functions.  The Hankel transforms  of c  m g  ^ are  i n t e g r a l t r a n s f o r m i n g then F o u r i e r t r a n s f o r m i n g the  The i n t e g r a l transforms a r e d e f i n e d by e q u a t i o n s  (3.46).  - 172  F i g u r e 25.  -  A diagrammatic r e p r e s e n t a t i o n of the i t e r a t i v e method used to s o l v e the i n t e g r a l equations.  -  The  i n t e g r a t i o n s a r e performed  173 -  u s i n g the t r a p e z o i d a l r u l e .  A Fast  F o u r i e r t r a n s f o r m (FFT) s u b r o u t i n e package [106] which can t r a n s f o r m two real  functions simultaneously  [107,108] was used  transforms d e f i n e d by e q u a t i o n s Step n  s  reduced  (3.47).  A p p l y i n g t h e Reduced OZ E q u a t i o n .  i s calculated  m n A  1  (4):  to e v a l u a t e the  u s i n g (3.42).  We note  A new s o l u t i o n s e t o f  t h a t b e f o r e a p p l y i n g the  OZ e q u a t i o n we must i n c l u d e the a n a l y t i c a l p i e c e o f c ™ ^ .  Step  (5):  performed  Back Hankel Transforms.  by f i r s t  back F o u r i e r t r a n s f o r m i n g and then r e v e r s i n g the  i n t e g r a l transforms. as i n Step  The back Hankel transforms are  The back F o u r i e r t r a n s f o r m uses the same  (3) where the i n t e g r a t i o n i s now over k.  procedure  The back i n t e g r a l 022  t r a n s f o r m s a r e d e f i n e d i n (3.46). the f i r s t  When back i n t e g r a l t r a n s f o r m i n g n ^  t e n p o i n t s o f the f u n c t i o n a r e f i t t e d  g  ,  to a p o l y n o m i a l of  degree <^ 5 and the r e q u i r e d i n t e g r a t i o n performed  analytically.  This  improves the n u m e r i c a l accuracy near r = 0, Step ( 6 ) : T e s t Convergence. The new s o l u t i o n s e t , n , is new m n A  f  w  compared w i t h the c u r r e n t s e t , used The  h  m  n  j .  old  The c o n t a c t v a l u e s ( r . = d is  f o r comparison s i n c e we would expect  ) are  them to converge most s l o w l y .  convergence c o n d i t i o n i s s a t i s f i e d when the o l d and new v a l u e s  d i f f e r by l e s s than 0.01% i n a l l c a s e s . Step  (7):  Mixing.  Direct iteration  ( r e p l a c i n g the c u r r e n t s o l u t i o n  by t h e new s o l u t i o n ) i s an u n s t a b l e method and w i l l d i v e r g e u n l e s s the initial [109]  trial  s o l u t i o n i s " c l o s e " to the " c o r r e c t " r e s u l t .  I t was found  t h a t the s o l u t i o n can be f o r c e d t o converge by mixing s u c c e s s i v e  iterates.  In order to determine  the new v a l u e of the c u r r e n t s o l u t i o n  - 174 -  we have used  n * - ( l - a) n m n  + a n *  m n £  (B.l)  m n  new  old  where  a = j  expH^n^  - n  o l d  )/n  o l d  |}.  (B.2)  The e x p o n e n t i a l f u n c t i o n was used i n (B.2) because i t r e p r e s e n t e d a convenient c h o i c e .  A l s o , the c o n s t a n t f a c t o r of 4 i n e q u a t i o n (B.2)  p r o v i d e s an "optimum" mixing parameter, a, f o r the p r e s e n t system. upper l i m i t of 0.975 was s e t on the p o s s i b l e v a l u e s of oc. v a l u e s of ™ n  £ g  were used to c a l c u l a t e  parameter was c a l c u l a t e d g r e a t l y improved allowed  f o r each ™ n  The c o n t a c t  ot. We note t h a t a separate m i x i n g £  g  a t each i t e r a t i o n .  This  the speed of convergence because each p r o j e c t i o n was  to converge i n d e p e n d e n t l y , some p r o j e c t i o n s c o n v e r g i n g more  r a p i d l y than o t h e r s .  T h i s method of mixing a l s o proved v e r y s t a b l e .  were always a b l e to converge a s o l u t i o n even a f t e r r e l a t i v e l y s t e p s , up o r down, i n diameter or charge. i t e r a t i o n s were r e q u i r e d represented  large  In a l l cases l e s s than 150  to o b t a i n a convergence of 99.99%.  This  l e s s than 60 seconds of CPU time on an Amdahl 470 V/8  running under an MTS o p e r a t i n g Step ( 8 ) : Output. current  An  system.  The output step i n v o l v e s the s a v i n g of the  s o l u t i o n s e t i n a computer f i l e  and the c a l c u l a t i o n of a l l  We  -  e q u i l i b r i u m system p r o p e r t i e s .  175 -  These v a l u e s  i n c l u d e e n e r g i e s of  • j . . . 00110* 11000* i n t e r a c t i o n , c o o r d i n a t i o n numbers, IQQ , IQQ , as w e l l as T  functions  T  such as the p o t e n t i a l s o f mean f o r c e .  - 176 -  APPENDIX C  NUMERICAL RESULTS  T a b l e s V and VI summarize some of the n u m e r i c a l d a t a o b t a i n e d i n the p r e s e n t study.  T a b l e V c o n t a i n s the data f o r u n i v a l e n t i o n s and  VI c o n t a i n s the data f o r d i v a l e n t  c h a r g e - d i p o l e , 3U"cn/Ni»  a n d  ions.  The  The  reduced  energies f o r a single ion, 3Uis/ i> N  (3.38a). was  reduced  calculated  IQQ^^ The  The  to  c o o r d i n a t i o n number, C.N.,  equation  (5.1).  r  e  u s i n g equations  were determined  (5.5b). a  3^0/%, (3.38b) and  average i o n - s o l v e n t i n t e r a c t i o n  kinetic dielectric  using equation  , contributing  average  average charge-quadrupole,  e n e r g i e s f o r a s i n g l e i o n were c a l c u l a t e d 3.38c), r e s p e c t i v e l y .  reduced  Table  using  decrement, (Ae  The  reduced  ST /a.T ) r . / r . , 1 JJ  terms, IQQ^*^  g i v e n by e q u a t i o n s  f o r a g i v e n i o n was  KJJD  equation  (4.28).  obtained  using  1  1  and  T a b l e V.  *  d. 1  0.44 0.68 0.84 0.88 0.96 1.00 1.08 1.12 1.16 1.20 1.28 1.40 1.44 1.56 1.80 2.00 2.52  PU  i  449.4 358.7 316.2 307.0 290.0 282.2 267.9 261.2 254.9 248.9 237.7 222.6 218.0 205.2 183.5 168.6 139.0  - is i 3U  - CD N  Numerical R e s u l t s f o r U n i v a l e n t  N  i  122.6 73.24 54.09 50.32 43.76 40.91 35.92 33.73 31.71 29.85 26.52 22.38 21.19 18.09 13.47 10.75 6.42  N  572.0 431.9 370.3 357.3 333.8 323.1 303.8 294.9 286.6 278.8 264.2 245.0 239.2 223.3 197.0 179.3 145.4  " W Vi TD A  n  r  '  4.14 5.10 5.55 5.65 5.82 5.90 6.05 6.12 6.18 6.25 6.36 6.52 6.56 6.69 6.91 7.05 7.33  i  ST r^  Ions.  11000* 00  -100.32 -61.50 -53.42 -53.32 -54.36 -55.06 -55.90 -55.87 -55.52 -54.86 -52.77 -48.51 -46.99 -42.76 -37.41 -35.55 -29.6  00110* 00  -0.54 -0.10 0.43 0.55 0.76 0.86 1.06 1.21 1.39 1.61 2.23 3.54 4.07 5.88 10.18 14.57 —  C.N.  7.32 8.97 10.38  -  -  11.55 12.33  14.4 16.25  13.15  17.6  -  -  Table VI.  * d.  1  -* CD N. l U  " CQ 3 U  N  i  Numerical R e s u l t s f o r D i v a l e n t  "  IS N. l  3 U  " V  A  i  W T_  D  r  '  i ST r  Ions.  11000* 00  00110* 00  C.N.  i  0.24  2303.6  808.5  3112.1  2.89  -568.0  -  -  0.48  1723.8  447.4  2271.2  4.33  -367.8  -  -  0.68  1434.7  293.0  1727.7  5.10  -246.0  -0.10  8.97  0.72  1388.3  270.9  1669.2  5.23  -232.6  -  -  0.96  1160.1  175.1  1335.2  5.82  -217.4  0.76  -  1.00  1128.9  163.6  1292.5  5.90  -220.2  0.86  11.55  1.28  950.7  106.1  1056.8  6.36  -211.1  2.23  14.4  

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