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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.Sc, The University of Lethbridge, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of (Chemistry) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1984 ® Peter Gerard Kusalik, 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f CAemiS"^ y The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date >E-6 (3/81) - i i -ABSTRACT 4 In the past, studies of e l e c t r o l y t e solutions have generally treated the solvent only as a d i e l e c t r i c continuum. This was the approach taken i n the theory of Debye and Huckel and i s s t i l l widely used today. A s i g n i f i c a n t improvement to t h i s approach would be to include the solvent as a true molecular species. This 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 e l e c t r o l y t e solutions considers a polarizable hard-sphere f l u i d with embedded point dipoles and tetrahedral quadrupoles with water-like parameters as i t s solvent. The model solutions are systems of hard sp h e r i c a l ions immersed i n t h i s water-like solvent at 25°C. The l i n e a r i z e d hypernetted-chain theory i s applied to these solutions i n the i n f i n i t e d i l u t i o n l i m i t . The properties of solution are studied as functions of ion size and charge. Both dynamical and equilibrium contributions to the apparent d i e l e c t r i c constant of solution are examined and compared with experimental measurements at low concentrations. In the present theory, the ion-solvent c o r r e l a t i o n functions for t h i s model e l e c t r o l y t e solution are found to scale exactly with charge. The ion-ion potentials of mean force demonstrate strong dependence on ion size and for small ions scale to a f a i r approximation with ion charge. For ions in the water-like solvent the potentials of mean force are observed to be less structured and approach the continuum l i m i t more r a p i d l y than for ions i n a simple dipolar solvent. The equilibrium contribution to the d i e l e c t r i c decrement for a l k a l i metal - i i i -and halide ions i s found to be negative but not strongly dependent upon ion s i z e . The values for the k i n e t i c d i e l e c t r i c decrement are also negative and are i n f a i r agreement with previous t h e o r e t i c a l r e s u l t s . The t o t a l d i e l e c t r i c decrement i s dominated by the equilibrium term and i s r e l a t i v e l y i n s e n s i t i v e to ion size for aqueous a l k a l i halides. The l i m i t i n g slopes for 1:1 and 2:1 e l e c t r o l y t e s at 25°C obtained from experimental data at low concentrations are found to be i n f a i r agreement with those predicted by the present theory. - i v -TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS i x CHAPTER I - INTRODUCTION 1 CHAPTER II - MODEL POTENTIALS . 6 1. Introduction 6 2. Water-like Solvent Model 9 3. E l e c t r o l y t e Solution Model 23 CHAPTER III - STATISTICAL MECHANICAL THEORY 28 1. Introduction 28 2. D i s t r i b u t i o n Functions 29 3. Integral Equation Methods 33 (a) The Ornstein-Zernike Equation 33 (b) Closure Approximations 35 4. Application of Integral Equation Methods to E l e c t r o l y t e Solutions 38 (a) A p p l i c a t i o n of the Ornstein-Zernike Equation.. 39 (b) The LHNC Closure Approximation 46 (c) Average Interaction Energy and Po t e n t i a l of Mean Force 51 (d) Method of Solution 54 CHAPTER IV - DIELECTRIC THEORY OF ELECTROLYTE SOLUTIONS 61 1. Introduction 61 2. Equilibrium Theory of the D i e l e c t r i c Constant 67 (a) The D i e l e c t r i c Constant of a Solution 69 (b) The Solute Dependent D i e l e c t r i c Decrement 75 - v -Page 3. Dynamical Theory of the D i e l e c t r i c Constant 80 (a) The Debye-Falkenhagen E f f e c t 85 (b) The K i n e t i c D i e l e c t r i c Decrement 88 CHAPTER V - RESULTS AND DISCUSSION 93 1. Introduction 93 2. Energies and Coordination Numbers 95 3. Co r r e l a t i o n Functions and Potentials of Mean Force 97 4. The D i e l e c t r i c Constant of Solution 129 CHAPTER VI - CONCLUSIONS 156 LIST OF REFERENCES 162 APPENDIX A - Rotational Invariants 168 APPENDIX B - I t e r a t i v e Procedures 170 APPENDIX C - Numerical Results 176 - v i -LIST OF TABLES Tables Page I Diameters, d^, for a l k a l i metal and halide ions.... 25 II Reduced parameters and d i e l e c t r i c constants for the water-like and dipolar solvents at 25°C with d = 2.8A 94 s II I Energies of i n t e r a c t i o n and coordination numbers for a l k a l i metal and halide ions 95 IV Molar ion conductances [4] at i n f i n i t e d i l u t i o n at 25°C 147 V Numerical r e s u l t s for univalent ions 177 VI Numerical results for divalent ions 178 - v i i -LIST OF FIGURES Figure Page 1 The o r i e n t a t i o n a l coordinate system where 6 , < j i , \ | ; are the Euler angles 10 2 Molecular axis system for the water-like solvent model 13 3 A charge d i s t r i b u t i o n possessing (a) a l i n e a r quadrupole and (b) a tetrahedral quadrupole.. 15 4 The d i e l e c t r i c constant of the water-like solvent as a function of temperature 22 5 Comparing n ^ ^ ( r ) for L i + and I - ions i n the water-like solvent model 98 f\0 0 6 Comparing h i g (r) for L i + and I - ions i n the water-like solvent model 100 7 A comparison of Sw^Cr) for ions of d i f f e r e n t s i z e 104 8 A comparison of 3w^j(r) for d i f f e r e n t pairs of oppositely charged ions 106 9 Comparing 3wij(r) for pairs of oppositely charged ions for which the d_^ _. values are equal.... 110 10 Comparing gw-^ j (r) / J z ^ z 2 j f o r pairs of oppositely charged univalent and divalent ions 112 11 A comparison of 3w^j(r) for pairs of oppositely charged ions i n d i f f e r e n t solvents 115 12 A comparison of Sw^Cr) for F"/F~ and I ~ / I ~ i n d i f f e r e n t solvents 117 13 The force acting between two I - ions i n the water-like solvent 123 14 The force acting between two L i + ions i n the water-like solvent 125 - v i i i -Figure Page 15 The force acting between two L i + ions i n the dipolar solvent 127 16 The force acting between a L i + / F ~ pair i n the water-like solvent 130 17 The force acting between a C s + / I ~ pair in the water-like solvent 132 18 The force acting between a C s + / I ~ pair i n the dipolar solvent 134 * 19 The terms contributing to e Q ) i f ° r t n e water-like solvent as a function of ion diameter 138 70 ^ The dependence of e ( ^ ) ^ o n i ° n diameter for univalent ions i n the water-like solvent 141 21 The dependence on ion diameter of the k i n e t i c d i e l e c t r i c decrement for the water-like solvent.... 145 22 A comparison of the equilibrium and dynamical contributions to the t o t a l d i e l e c t r i c decrement.... 148 23 Comparing experimental and t h e o r e t i c a l d i e l e c t r i c decrements for aqueous 1:1 e l e c t r o l y t e solutions... 150 24 Comparing experimental and t h e o r e t i c a l d i e l e c t r i c decrements for aqueous C u C l 2 solutions 152 25 A diagrammatic representation of the i t e r a t i v e method used to solve the i n t e g r a l equations 172 ACKNOWLEDGEMENTS I would l i k e to thank my academic advisor, Dr. G. Patey, for h i s guidance and support, and the Chemistry department of the University of B r i t i s h Columbia and the Natural Sciences and Engineering Research Council of Canada (NSERC) for t h e i r f i n a n c i a l assistance. I would also l i k e to express my sincere gratitude to my family and e s p e c i a l l y to my fiancee, Sheila Davis, for 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 solutions has been one of the most active areas of physical chemistry, p a r t i c u l a r l y i n the f i r s t few decades of t h i s century. This i n t e r e s t has been due, i n part, to the c e n t r a l r o l e charged species play i n many chemical reactions and processes, e s p e c i a l l y i n aqueous solutions. In r e a l l i f e we are confronted by e l e c t r o l y t e solutions many times each day. What do we mean by an e l e c t r o l y t e solution? For the purposes of t h i s study the term " e l e c t r o l y t e s o l u t i o n " i s synonymous with " i o n i c s o l u t i o n . " It was Faraday [1] who coined the term e l e c t r o l y t e . However, he was seeking to describe substances, such as water, which y i e l d t h e i r constituent elements upon e l e c t r o l y s i s . Today the term has two d i f f e r e n t i n t e r p r e t a t i o n s . In physical chemistry an e l e c t r o l y t e i s a substance which produces an i o n i c a l l y conducting medium when dissolved i n s p e c i f i c solvents. To the electrochemist or p h y s i c i s t the term usually refers to the conducting so l u t i o n i t s e l f . We use the former d e f i n i t i o n and consider only those e l e c t r o l y t e s which, for a l l p r a c t i c a l purposes, can be viewed as being completely ionized i n s o l u t i o n . P h y s i c a l l y an e l e c t r o l y t e s o l u t i o n i s a homogeneous l i q u i d consisting of an i o n i c s o l i d , a s a l t , dissolved i n a polar solvent, a l i q u i d characterized by a "large" d i e l e c t r i c constant such as water. Macroscopically the r e s u l t i n g l i q u i d i s well understood [2], An e l e c t r o l y t e s o l u t i o n i s often defined as a s o l u t i o n having a "high" - 2 -conductance [3], Many other macroscopic properties [2,4,5] such as density, vapour pressure, and apparent d i e l e c t r i c constant can be r e a d i l y measured and t h e i r dependence on s a l t concentration determined. M i c r o s c o p i c a l l y , e l e c t r o l y t e solutions are s t i l l poorly understood with many fundamental questions and puzzles remaining unanswered. We know that the solute exists as free i o n i c species in solution but the solvent-solvent and ion-solvent structure present i n solution i s s t i l l a matter of great controversy [4,6-9], A more complete microscopic understanding i s needed i n order to explain what can be observed and measured macroscopically. I t i s to t h i s point that t h i s thesis i s addressed. The study of e l e c t r o l y t e solutions began i n , and in fact dominated, the early days of what i s known as electrochemistry. Investigators were intrigued by the fact that matter was transported ln e l e c t r o l y t e conductors whereas t h i s was not the case with e l e c t r o n i c conductors. Clausius [10] noted that i o n i c solutions obeyed Ohm's law. He concluded that there must be e l e c t r i c a l l y charged p a r t i c l e s present to carry the current, possibly a small portion of the e l e c t r o l y t e was in such a form. In 1887 van't Hoff [11] published experimental r e s u l t s which showed that conducting solutions possess c o l l i g a t i v e properties d i s t i n c t from those of non-conducting solutions. Planck [12] interpreted these r e s u l t s as a possible i n d i c a t i o n of i o n i z a t i o n of the solute. However, Arrhenlus [13], who f i r s t published his theory i n 1887, i s usually credited for 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 s o l u t i o n . Although the idea of free ions i n so l u t i o n seemed r a d i c a l at that time, acceptance of the idea did come slowly with the support of a - 3 -wealth of data. By the turn of this century the basic concept was more or less u n i v e r s a l l y accepted and the f i e l d was entering a rapid growth period. In 1907 Lewis [14] introduced the concept of a c t i v i t y and a c t i v i t y c o e f f i c i e n t s as a to o l to handle the deviation i n behavior of r e a l solutions at higher concentrations from that of i d e a l solutions. Many attempts were made during the f i r s t decades of this century to develop equilibrium and l a t e r dynamical theories to describe e l e c t r o l y t e solutions and t h e i r behavior. The long-range nature of the ion-ion i n t e r a c t i o n s made i t possible to derive exact l i m i t i n g laws for d i l u t e i o n i c solutions [15]. 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 very good agreement with experiment. The theories of Debye and Huckel [16] and of Onsager [17] stand out today as landmarks. The work of Onsager led to the development of a dynamical theory for e l e c t r o l y t e solutions [8]. The theory of Debye and Huckel (DH) for the equilibrium structure of e l e c t r o l y t e solutions s t i l l provides a common approach used i n the desc r i p t i o n and discussion of i o n i c solutions [2,4,18], Debye and Huckel only considered i n t e r i o n i c i n t e r a c t i o n s as influenced by the solvent d i e l e c t r i c constant and examined t h e i r e f f e c t upon i o n i c behavior. Many theories have b u i l t on t h e i r general approach. E l e c t r o l y t e solutions have been i n the past t r a d i t i o n a l l y studied using t h i s "primitive model" (PM) approach which treats the solvent only as a d i e l e c t r i c continuum. The ion-ion i n t e r a c t i o n i n the PM i s ju s t the coulombic r e l a t i o n , q ^ q j / e Q r . The a b i l i t y of the continuum solvent to screen i o n i c charges [2,15] i s e n t i r e l y represented by i t s d i e l e c t r i c constant, e Q. The PM and var i a t i o n s of i t are s t i l l commonly used today to study [19,20] i o n i c solutions because of the - 4 -a t t r a c t i v e s i m p l i c i t y of these models. To improve the r e s u l t s obtained using these models, improvement of the theories applied has been continually sought [19,20]. Recent attempts [7,9,21-24] to introduce the molecular nature of the solvent i n the treatment of e l e c t r o l y t e solutions have served to point out the complexities within i o n i c solutions and the inadequacies of the PM. The presence of enormous l o c a l e l e c t r i c f i e l d s generated by the free ions i n s o l u t i o n implies that the solvent i n the neighborhood of the ions might be i n a quite d i f f e r e n t state from that of the bulk solvent. The solvent d i e l e c t r i c constant i s an i n d i r e c t probe of such solva t i o n e f f e c t s . Instead of t r y i n g to work within the PM framework, we have chosen to study a "better" model, a model which includes the solvent as a true molecular species. In modelling aqueous e l e c t r o l y t e solutions, i t i s important that the model solvent have the d i e l e c t r i c constant of water. The search for better models has been the aim of a number of recent t h e o r e t i c a l studies [21-23,25-30]. The i n t e g r a l equation approach, commonly used i n l i q u i d state theory [31-33], has been shown to be very useful [21-30, 35-41] i n such studies. Calculations of t h i s type should i n p r i n c i p l e y i e l d a l l equilibrium properties of the system. From several studies [21-23,35] i t has become clear that a simple dipolar hard-shere f l u i d i s a very poor model for water. A f l u i d of p o l a r i z a b l e hard spheres with embedded point dipoles and tetrahedral quadrupoles has been found [41] to be a f a r superior model for water. This "water-like" 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 study of aqueous e l e c t r o l y t e s i n that the d i e l e c t r i c constants calculated for such a f l u i d system show - 5 -good agreement with experimental r e s u l t s for water over a large temperature and pressure range [34,41], A recent study [23] has considered a so l u t i o n of hard spherical ions immersed i n t h i s water-like solvent i n the i n f i n i t e d i l u t i o n l i m i t . This study examined only univalent ions with the same diameters as the solvent. It focussed upon ion-ion potentials of mean force and the equilibrium d i e l e c t r i c properties of s o l u t i o n . In the present study, part of which has already been published [42], we extend the work i n i t i a t e d i n Ref. 23. We investigate the influence of ion size and charge upon the s t r u c t u r a l and equilibrium d i e l e c t r i c properties of s o l u t i o n . An examination of dynamical d i e l e c t r i c properties i s c a r r i e d out; d i f f e r e n t solvent models and previous t h e o r e t i c a l r e s u l t s are compared and contrasted. Also, the t h e o r e t i c a l r e s u l t s are compared with experimental data wherever possible. Predictions and comparisons w i l l tend to be rather q u a l i t a t i v e i n nature, with the basic purpose being to demonstrate the usefulness of the present so l u t i o n model i n examining microscopic properties and t h e i r macroscopic implications. In Chapter II we present the current solution model i n d e t a i l . The c l a s s i c a l s t a t i s t i c a l mechanical theory applied to our so l u t i o n model i s discussed i n Chapter I I I , while i n Chapter IV we examine the d i e l e c t r i c theory of solutions. The r e s u l t s obtained are discussed i n Chapter V and our conclusions are given i n Chapter VI. - 6 -CHAPTER II MODEL POTENTIALS 1. Introduction In the study of r e a l systems and the physical properties that characterize them, the development of "useful" models i s e s s e n t i a l . We could choose to model the systems of i n t e r e s t i n t h i s study i n a great many ways; we could describe our water-like solvent, in the simplest case, by a d i e l e c t r i c continuum ( i . e . PM) or in the most extreme case we could attempt to solve the f u l l quantum mechanical problem. For a model to be u s e f u l , i t must be simple enough to enable us to produce meaningful r e s u l t s , but i t must also have a s u f f i c i e n t degree of s o p h i s t i c a t i o n so as to represent adequately our system (or at least the properties of primary i n t e r e s t ) . There i s , however, a basic tradeoff between how r e a l i s t i c a model might be and how e a s i l y i t lends i t s e l f to s o l u t i o n . The desire to work with the simplest model possible that y i e l d s r e a l i s t i c r e s u l t s has made the development of such models an important step i n s c i e n t i f i c advancement. In any s t a t i s t i c a l mechanical study, the i n t e r a c t i o n s within the system are described by a model p o t e n t i a l . Hence, the p o t e n t i a l mathematically defines (or represents) the model. We can then define our s t a t i s t i c a l mechanical system in terms of some energy, Ufj, the sum of a l l i n t e r a c t i o n s of the system. In developing a useful model, we need to determine and r e t a i n those terms of the f u l l i n t e r a c t i o n p o t e n t i a l that make important contributions to the properties of Interest. It i s sometimes found that - 7 -we can ignore otherwise large factors because they have l i t t l e influence on properties we wish to i n v e s t i g a t e . 1 As a f i r s t approximation to our p o t e n t i a l , we w i l l l i m i t our treatment of 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 , etc., which are assumed to be small, and a l l intramolecular in t e r a c t i o n s w i l l also be ignored. The molecules w i l l be treated as r i g i d p a r t i c l e s (Born-Oppenheimer approximation) and average values for such quantities as the dipole moment and p o l a r i z a b i l i t y w i l l be used. Such approximations are convenient for our l e v e l of study. The t o t a l i n t e r a c t i o n p o t e n t i a l w i l l thus depend only upon the positions and orientations of the molecules of the system. For the t o t a l energy of i n t e r a c t i o n of the system we write UJJ = U(X^ , 5^ 2» 2^ 3 »• • • >X^) (2.1) where represents the coordinates of molecule i and N i s the number of molecules i n the system. In general, t h i s N-body i n t e r a c t i o n , p a r t i c u l a r l y f o r l i q u i d systems, i s very complicated and the N-body problem would be very d i f f i c u l t to solve. We are able to make a second major s i m p l i f i c a t i o n by assuming that we can express the t o t a l i n t e r a c t i o n energy of equation (2.1) as the sum of terms 1 Sometimes p r i o r knowledge from previous studies can be used to support such approximations or we may be able to show that they are rigorously true. - 8 -U„ = I u(X., X.) + Z u'(X., X., X.) + ... (2.2) i<j _ 1 ~ 2 i<j<k _ 1 ~ 2 ~* where the f i r s t term represents the sum of a l l unique pair i n t e r a c t i o n s , the second term represents the sum of a l l unique 3-body i n t e r a c t i o n s , and so on. It i s generally agreed that for most l i q u i d s the pair i n t e r a c t i o n p o t e n t i a l i s a dominate term. Also, to include anything more than only the f i r s t term of the expansion in equation (2.2) would complicate the problem, i n most cases, to the degree of rendering i t unsolvable. As a r e s u l t , in almost a l l cases higher order terms are neglected (they are assumed to be small) or the pair p o t e n t i a l i s modified in an attempt to take into account higher order terms r e s u l t i n g In what i s usually c a l l e d an " e f f e c t i v e " pair p o t e n t i a l . Therefore, we write U = I u(X X ) (2.3) i<j 2 where the pair i n t e r a c t i o n p o t e n t i a l between molecules 1 and 2 can be 2 expressed as u ( X l f X 2) = u(12) = u ( r 1 2 , ftj, fi2) (2.4) 2 Here we choose the centre of mass of p a r t i c l e 1 as the o r i g i n of the coordinate system. - 9 -where r_i2 i s the vector from the centre of p a r t i c l e 1 to the centre of p a r t i c l e 2 and fij, ft2 represent the molecular o r i e n t a t i o n s . In our coordinate system, the ori e n t a t i o n fi^ i s described, as i l l u s t r a t e d i n Figure 1, by the Euler angles, 9, (j), ty. The use of only a pair p o t e n t i a l [31,43], or an e f f e c t i v e pair p o t e n t i a l , predominates within current s t a t i s t i c a l mechanical studies, although i t s v a l i d i t y , p a r t i c u l a r l y for systems of l i q u i d density, has been questioned. The pair i n t e r a c t i o n p o t e n t i a l i s generally much longer ranged than other higher order many-body p o t e n t i a l s . Hence, i t would be expected that errors r e s u l t i n g from the use of only a pair p o t e n t i a l should be most evident i n short-range structure i n most cases. 2. Water-Like Solvent Model There have been many d i f f e r e n t pair potentials used to model water-like f l u i d s [6,35,41,44,45] or water-like solvents [21-23]. For many model p o t e n t i a l s , i t i s sometimes hard to find j u s t i f i c a t i o n for terms used and t h e i r respective values. Often parameters used i n model pot e n t i a l s are determined through f i t t i n g techniques; the pot e n t i a l parameters are varied so as to reproduce some r e a l physical properties of the system. However, i n trying to model an e l e c t r o l y t e s o l u t i o n , we find that some basic molecular properties of i t s components are known ( i . e . they have been measured). The incorporation of some of these "measured" values into our model should not only give us a more r e a l i s t i c picture but also should demonstrate to some degree t h e i r r e l a t i v e importance. - 10 -z y Figure 1 - The o r i e n t a t i o n a l coordinate system where 9 , <j>, are the Euler angles. - 11 -The pair d i s t r i b u t i o n functions for water has been measured both by X-ray [46] and neutron [47] d i f f r a c t i o n , though the l a t t e r i s considered to be a more accurate method. Most r e s u l t s are i n general o v e r a l l agreement. The f i r s t peak of the 0-0 r a d i a l d i s t r i b u t i o n function i s found to occur near 2.8A. The dipole moment, the quadrupole tensor [48], and the p o l a r i z a b i l i t y tensor [49] for the water molecule are known to reasonable accuracy. In developing a model for the present study we follow previous work [23,41], We 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 i n t e r a c t i o n s . We would expect properties such as the d i e l e c t r i c constant and p o t e n t i a l of mean force 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 properties should not depend strongly on the short-range part of the model p o t e n t i a l . The oxygen atom of the water molecule i s large compared to the protons, hence the molecule i s roughly s p h e r i c a l . Therefore, we choose to treat the molecule as a sphere since a non-spherical model would involve rather complicated short-range i n t e r a c t i o n s . There are many s p h e r i c a l l y symmetric short-range potentials [31,43] that could be used, but since the choice i s somewhat a r b i t r a r y , the molecules w i l l be 4 HS treated simply as hard spheres. The hard sphere i n t e r a c t i o n , u a g ( r ) , i s given by We note that e l e c t r o s t a t i c interactions tend to be considered long-ranged, p a r t i c u l a r l y charge-charge, charge-dipole, and dipole-dipole i n t e r a c t i o n s . 4 Unlike s o f t - p o t e n t i a l s which usually involve 2 or more parameters, the hard sphere p o t e n t i a l has only one parameter, the hard sphere diameter. - 12 -u„J(r) = -< , (2.5a) P 7 0 , r > d a d a + d 3 where d ^ = ^ » (2.5b) with r being the separation between the centres of the molecules a and 3, and d a , dg being t h e i r respective diameters. The exact choice of a hard sphere diameter for a molecule i s again somewhat a r b i t r a r y , but f o r our water-like model a value of d s " 2.8A i s an obvious choice consistant with the d i f f r a c t i o n data [47] mentioned above. The permanent (gas phase) dipole moment, Pp = 1.855D (1 Debye = 10 esu cm) where we define the axis system as shown i n Figure 2 and jip l i e s along the z-axis. The quadrupole tensor, Q, has the general form Q = yy zz (2.6a) where = J I e i ( 3 r i a r i 3 " r l 6at? (2.6b) - 13 -2 -> X Figure 2 - Molecular axis system for the water-like solvent model. - 14 -, and r ^ a denotes the a-component of the vector r± which i t s e l f gives the p o s i t i o n of the charge e^ with respect to the molecular centre of mass. We know that Q must be traceless ( i . e . Cr +Q +Q = 0) i f -"• TCX yy z z our molecule i s e l e c t r i c a l l y n e u t r a l . For molecules that have a x i a l l y symmetric or l i n e a r quadrupoles, Q = Q = -QT/2 and Q = Q where xx yy L. zz Lt i s known as a l i n e a r quadrupole moment. Water i s not a x i a l l y symmetric and one finds [48] Q = 2.63B, Q = -2.50B, and Q = -0.13B xx yy zz (IB = 10" esu cm ). We note, however, that Q z z " 0 , so we approximate [48] Q, the quadruple tensor, (to within 5%) by Q T 0 0 Q = 0 -QT 0 • (2.7) 0 u . 0 0 We s h a l l r efer to Qj, as a tetrahedral quadrupole moment [50]. We can more f u l l y understand the physical d i f f e r e n c e between a l i n e a r and a tetrahedral quadrupole by examining the implications of t h e i r geometries more c l o s e l y (see Figure 3). A tetrahedral quadrupole w i l l i n t e r a c t with p o s i t i v e and negative charges i n an equivalent fashion, whereas a l i n e a r quadrupole w i l l i n t e r a c t with opposite charges in a much d i f f e r e n t manner. Ions w i l l be solvated i n a symmetric fashion by a solvent with a tetrahedral quadrupole. A solvent with a l i n e a r quadrupole w i l l give r i s e to asymmetric solvation with ions of one charge being solvated d i f f e r e n t l y than ions of the opposite sign. If we now look at a tetrahedral charge d i s t r i b u t i o n , i . e . equivalent charges located at the v e r t i c e s of a tetrahedron (see Figure 3), we find 15 -as-a ) Linear Quadrupole b ) lehrxxhedral quadrupole Figure 3 - A charge d i s t r i b u t i o n possessing (a) a l i n e a r quadrupole and (b) a tetrahedral quadrupole. - 16 -that i t possesses a dipole and tetrahedral quadrupole as Its two lowest order multipolar moments. We note that the bond angle of a water molecule i n the l i q u i d state, 1 0 4 . 5 ° , i s r e l a t i v e l y close to the tetrahedral angle. Hence, we can see a correspondence between the two p o s i t i v e charges and the two protons of a water molecule, and another between the two negative charges and the lone pairs of the water molecule. These multipole moments are added to our water-like model as point moments embedded at the centre of hard spheres. Qp i s taken as 2.5B. However, the value for the permanent dipole moment of 1.855D i s not used. Instead, an " e f f e c t i v e " dipole moment, which i s larger than Pp due to the p o l a r i z a t i o n of the model, i s determined and used in our system. The p o l a r i z a b i l i t y of any molecule i s defined by i t s p o l a r i z a b l i t y tensor, a, but since the water molecule i s nearly 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 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 3 . (2.8) The " e f f e c t i v e " dipole moment, nig, i s determined using the s e l f - c o n s i s t e n t mean f i e l d (SCMF) theory of Carnie and Patey [41]. The many-body e f f e c t of p o l a r i z a t i o n i s reduced i n t h i s theory 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 attempting to incorporate the t o t a l e f f e c t due to p o l a r i z a t i o n into an e f f e c t i v e dipole moment. In the SCMF theory for a homogenous f l u i d the average t o t a l dipole moment, m', i s given by 5Note that were we to take the centre of the oxygen atom as our o r i g i n instead of taking the centre of mass as our o r i g i n , Qj would be changed by le s s than 2%. (2.9a) where the induced dipole moment p = a • < E L > (2.9b) and < ET, > i s the average l o c a l e l e c t r i c f i e l d which w i l l have both dipolar and quadrupolar components. 6 In p r i n c i p l e , determination of the average l o c a l f i e l d w i l l be a many-body problem, depending on a l l molecules i n the system. < Ej, > for an i s o t r o p i c f l u i d of a x i a l or C 2 v symmetry w i l l be non-zero only i n the d i r e c t i o n of jjp. Hence 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 the vector notation, understanding that a l l vectors w i l l l i e along the z-axis. In determining mg, the e f f e c t i v e dipole moment, flu c t u a t i o n s i n the e l e c t r i c f i e l d and in m^ , the instantaneous dipole moment of p a r t i c l e i , are ignored. For d s = 2.8A at 25°C, a value of m' = 2.56D i s obtained which agrees well with values that have been calculated for ice [51] and estimated for water [6], although the value of m' does vary somewhat with choice of diameter. The determined value of the e f f e c t i v e permanent dipole moment, 6 I t has been found [41] that the quadrupolar f i e l d constitutes approximately 25% of the t o t a l f i e l d . 7 2 2 I I By s e t t i n g m^nu = <m > = , = pH^I = m ' • - 18 -me * m' , w i l l be used i n a l l following c a l c u l a t i o n s as the permanent dipole moment for our water-like solvent. The pair i n t e r a c t i o n , u s s , between two solvent molecules i s given by U s s ( 1 2 ) = U s s ( r ) + U D D ( 1 2 > + U D Q ( 1 2 ) + U Q Q ( 1 2 ) < 2 ' 1 0 > HS where u (r) i s the previously defined solvent-solvent hard sphere s s p o t e n t i a l , 11^(12) i s the dipole-dipole i n t e r a c t i o n , 11^(12) i s the dipole-quadrupole i n t e r a c t i o n , and U Q Q ( 1 2 ) i s the quadrupole-quadrupole i n t e r a c t i o n . We now employ a method f i r s t exploited by Blum [52-54] for expanding multipolar potentials i n terms of r o t a t i o n a l i n v a r i a n t s , $ ™ ^ ( 1 2 ) , defined by Amn&/-,0\ r j-mnJl/m n SL x m^ , „ N _ n , „ x „ Jl , _ N , „ ... •yv ( 1 2 ) " , E „ i t f (y'VA»> D y p ' ( V Dvv'<V D 0 X ' ( a i 2 ) ( 2 ' U ) y v A where fi^ represents the or i e n t a t i o n of the vector r ^ = £2 ~ —1 a n c * ^1 , ^2 describe the orientations of p a r t i c l e s 1 and 2, re s p e c t i v e l y , with respect to a laboratory fixed frame of reference. The Wigner matrix elements, D * v ( f i ) , and the 3-j symbols, (™ " are defined by me i s approximately 4% larger than m' for the current model, m2 = m'2 + 3a'kT e where a' i s a renormalized p o l a r i z a b i l i t y [41]. - 19 -Rotenberg et a l . [55] and following the usual convention [38], we have the c o e f f i c i e n t s f m n ^ given by fmnZ = A | / (m n A } ( 2 > 1 2 ) 220 2 2 A with the exception of f = -2/5~ and f = 8/35/2. In Appendix A, these r o t a t i o n a l invariants are expressed in a more e x p l i c i t manner. In general, the i n f i n i t e set of r o t a t i o n a l invariants form a complete set which w i l l span the space of Eulerian angles denoting the orientations of p a r t i c l e s 1 and 2. Hence, we see how a unique set of r o t a t i o n a l i n v a r i a n t s , $ m ^ ( 1 2 ) , can be used to describe the r o t a t i o n a l symmetry of multipolar i n t e r a c t i o n s . We f i n d that we can write the components of the pair p o t e n t i a l i n terms of some set of $™"^(12). Symmetry imposes c e r t a i n r e s t r i c t i o n s upon u (12) and hence only c e r t a i n c o e f f i c i e n t s , u m n ^ ( r ) are allowed. ss J uv For a x i a l symmetry, only c o e f f i c i e n t s where u=v=0 are non-zero. Molecules of C 2 v symmetry, such as water, require that p and v must be even and the condition mn£/ x mn£v \ mnZ, » mn.lt * \ /<« io\ u u v ( r ) = u - u v ( r ) = u u - v ( r ) = u - u - v ( r ) ( 2 * 1 3 ) must be s a t i s f i e d . For the water-like solvent model we are considering, these conditions must also be obeyed. In addition, the s i m p l i f i c a t i o n - 20 -to a tetrahedral quadrupole requires a further r e s t r i c t i o n i n that p + v + 2% = 0 (mod 4) (2.14) which does not hold for general C2 V symmetry. In view of these r e s t r i c t i o n s , i t i s convenient to define the functions * 1 2 £ ( 1 2 ) = * J 2*(12) + $ J - 2 ( 1 2 ) ' (2.15a) * 2 U ( 1 2 ) = * 2 0 * ( 1 2 ) + * - 2 0 ( 1 2 ) ' (2.15b) and * 2 2 £ ( 1 2 ) = * 2 2 £ ( 1 2 ) + $ ? 2 2 ( 1 2 ) + * 2 - 2 ( 1 2 ) + $ - 2 - 2 ( 1 2 ) * (2.15c) We can now write [41] the multipolar interactions of equation (2.10) as and u D D ( 1 2 ) = u 1 1 2 ( r ) $ 1 1 2 ( 1 2 ) , (2.16a) u D Q ( 1 2 ) = u 1 2 3 ( r ) * 1 2 3 ( 1 2 ) + u 2 1 3 ( r ) $ 2 1 3 ( 1 2 ) , (2.16b) u Q Q ( 1 2 ) = u 2 2 4 ( r ) * 2 2 4 ( 1 2 ) , (2.16c) where u 1 1 2 ( r ) = - p 2 / r 3 , (2.17a) u 1 2 3 ( r ) = - u 2 1 3 ( r ) = "~™— r* , (2.17b) /6 r 4 - 21 -and 224. v VT u (r) = — j 2r _ (2.17c) have the required r dependence. It should be noted that we have dropped the subscripts 00 from $QQ^ assuming these to be understood when not otherwise indicated. The r o t a t i o n a l invariant $^^(12) = 1 for i s o t r o p i c potentials such as the hard sphere i n t e r a c t i o n of (2.10) can also be omitted. Moreover, the pair p o t e n t i a l expressed by equation (2.10) w i l l hold for molecules possessing l i n e a r quadrupoles. One has only to replace 0^ with ST/l Q L i n (2.17) and * m n £ ( 1 2 ) with $ m n*(12) in (2.6). This water-like solvent model has been found to be very successful g i n describing the d i e l e c t r i c properties of water [41]. The tetrahedral quadrupole was shown to be very important i n determining the general c h a r a c t e r i s t i c s of the system and p o l a r i z a b i l i t y was found to be e s s e n t i a l . The exact choice of hard sphere diameter proved not to be c r u c i a l ; the values of 2.7, 2.8 and 2.9A were examined and a l l gave very s i m i l a r r e s u l t s [41], Using a diameter of 2.8A, the model was found to give d i e l e c t r i c constants i n good agreement with experimental points on the vapour pressure curve over the temperature range 25 - 300°C (see Figure 4) as well as with points above the c r i t i c a l temperature [34]. The structure of the system, as expressed by the r a d i a l d i s t r i b u t i o n function, g^OO(r) , gave only general q u a l i t a t i v e agreement with Physical q u a n t i t i e s , such as the density and d i e l e c t r i c constant, of water are well tabulated as functions of temperature [18,56]. 100 T(°C) 200 300 Figure 4 - The d i e l e c t r i c constant of the water-like solvent as a function of temperature [41], The re s u l t s for the water-like model using the SCMF theory are indicated by the s o l i d l i n e ; the results from experiment are indicated by the dots. - 23 -experiment, but t h i s i s probably greatly influenced by the u n r e a l i s t i c , repulsive hard core p o t e n t i a l . However, the a b i l i t y of t h i s model to duplicate the d i e l e c t r i c properties 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. Electrolyte Solution Model Solutions containing symmetric ions, p r i m a r i l y a l k a l i halides, w i l l be the focus of t h i s study. Charged hard spheres have been used extensively to model e l e c t r o l y t e s [6,7] both i n solution and as molten s a l t s . They are an obvious choice for modelling ions i n solution given our choice of solvent model. The charge on an ion w i l l be represented by a point charge at i t s centre. The ions w i l l not be considered to be p o l a r i z a b l e . Their p o l a r i z a b i l i t y i s s p h e r i c a l l y symmetric and we assume that a solvated ion experiences no net average p o l a r i z a t i o n since the e l e c t r i c f i e l d components due to the surrounding solvent molecules w i l l cancel. Again we are faced with choices for hard sphere diameters. Though i o n i c r a d i i can be defined i n both the s o l i d and gas phases, there i s at present no d i r e c t method of estimating the r a d i i of ions i n s o l u t i o n . The r a d i i of gaseous ions are found to be much larger than t h e i r c y r s t a l r a d i i . This can be explained simply by noting that i n an i o n i c c r y s t a l , - the l a t t i c e i s highly compressed by coulombic a t t r a c t i o n and hence ions possess much smaller apparent r a d i i . An ion i n s o l u t i o n w i l l have a radius of intermediate value, somewhere between i t s c r y s t a l and gas values. The so l u t i o n value i s generally viewed as being very close to - 24 -the c r y s t a l radius and thus c r y s t a l r a d i i are often used for aqueous e l e c t r o l y t e s . We have several choices for c r y s t a l r a d i i , those of Pauling [57] being most widely used. Recently, X-ray electron density measurements of i o n i c c r y s t a l s have been used to determine i o n i c r a d i i . Such measurements seem to give a more p h y s i c a l l y r e a l i s t i c method of d e f i n i n g i o n i c r a d i i . We have chosen to use these r a d i i , i n p a r t i c u l a r those described by Morris [58], Table I summarizes the values of the ion Table I Diameters, d^, for a l k a l i metal and halide ions. Both those of Morris [58] and Pauling [59] are included. Also given are the reduced diameters, d. = d./d (with d = 2.8A; rounded to the nearest 0.04 to 1 i s s accommodate a g r i d width of 0.02d s), used in a l l numerical c a l c u l a t i o n s . T± ( i n A) Ion Pauling Morris d. L i + 0.60 0.93 0.68 Na + 0.95 1.17 0.84 K+ 1.33 1.49 1.08 Rb + 1.48 1.64 1.16 C s + 1.69 1.83 1.28 F~ 1.36 1.16 0.84 C l ~ 1.81 1.64 1.16 Br" 1.95 1.80 1.28 I " 2.16 2.04 1.44 - 25 -diameters for the a l k a l i halides and Pauling r a d i i are also included f o r comparison. We fi n d that for cations the values used are somewhat la r g e r , and for anions they are somewhat smaller than the Pauling r a d i i . We define the pair i n t e r a c t i o n p o t e n t i a l , UJJ(12), between two xons as , 1 0 . 000. <. HS, v , q i q j , 0 1 Q . u ±j(12) = u (r) = u ± j ( r ) + — ^ (2.18) where and q^ are the i o n i c charges and u ^ ( r ) i s just the ion-ion hard sphere i n t e r a c t i o n . The ion-ion pair p o t e n t i a l i s s p h e r i c a l l y symmetric with $000(12) = 1 again being understood. In defining the ion-solvent p o t e n t i a l , we are able to make some s i m p l i f i c a t i o n s that r e s u l t from the i n f i n i t e d i l u t i o n assumption. In the i n f i n i t e d i l u t i o n l i m i t , the bulk solvent remains unchanged. Hence the e f f e c t i v e dipole moment of the water-like solvent i s used as the permanent dipole moment, as opposed to using the gas phase value i n determining the charge-dipole i n t e r a c t i o n . The ion-solvent i n t e r a c t i o n i s thus also an e f f e c t i v e pair p o t e n t i a l i n which we are attempting to account for the p o l a r i z a t i o n of a solvent molecule by a l l other solvent molecules. The p o l a r i z a t i o n of the solvent molecules by an ion i s not taken into account. However, we need only worry about one such ion because we are considering the i n f i n i t e d i l u t i o n l i m i t . A single ion w i l l only generate a s p h e r i c a l l y symmetric e l e c t r i c f i e l d i n the solvent. For the model we consider here, t h i s can only contribute - 26 -s p h e r i c a l l y symmetric terms to the ion-solvent i n t e r a c t i o n and we choose to ignore a l l such terms except those that might be included i n the hard sphere p o t e n t i a l . Thus, for the present model the ion-solvent pair i n t e r a c t i o n , u i s ( 1 2 ) , i s given [23] by , 1 0v 000, . . O i l , v A011 / 1 ov , 022 -022, 1 0. , 0 1 Q . u i g ( 1 2 ) = u i g (r) + u i g (r) $ (12) + u ± g $ (12) (2.19a) where 000. . HS, . ,„ .... u i s = U i s ^ » (2.19b) U i s 1 ( r ) = 'V 7^' (2.19c) 022, . ^ ( 2 ' 1 9 d > and we define * 0 2 2 ( 1 2 ) = $° 2 2(12) + * [ J " ( 1 2 ) . ( 2 ' 2 0 ) The second term of (2.19a) gives the charge-dipole i n t e r a c t i o n and the t h i r d term gives the charge-quadrupole i n t e r a c t i o n . Also referred to i n t h i s study w i l l be a dipolar hard sphere solvent. A l l pair potentials needed for t h i s solvent can be obtained from those of the water-like solvent by s e t t i n g Qp = 0. - 27 -Having now defined our model i n terms of pair i n t e r a c t i o n p o t e n t i a l s , we w i l l i n Chapter V study t h i s system using the s t a t i s t i c a l mechanical theory developed i n Chapter III and the d i e l e c t r i c theory 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 provides several d i f f e r e n t approaches by which to study f l u i d s [31,59,60]. D i s t r i b u t i o n function language i s frequently used and i s p a r t i c u l a r l y useful i n studies of f l u i d s at the molecular l e v e l . D i s t r i b u t i o n functions allow a complete but compact de s c r i p t i o n of the microscopic structure of f l u i d s [32,61]. Knowledge of even the lower order d i s t r i b u t i o n functions i s s u f f i c i e n t , i n general, to determine most equilibrium (macroscopic or thermodynamic) properties of a system defined using the pair p o t e n t i a l assumption. A model p o t e n t i a l uniquely determines some d i s t r i b u t i o n function. We can depict t h i s schematically a f t e r Rasaiah [24]: u(12) mechanical^ ^ S(12) ) thermodynamic properties model ^, theory where g(12) i s the pair d i s t r i b u t i o n function as defined below. The required sum over configurations ( 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) to g(12) i s usually performed using c l a s s i c a l s t a t i s t i c a l mechanics, since at ordinary temperatures most molecular f l u i d s can be treated i n a c l a s s i c a l manner. This approximation i s of course not v a l i d i n cases such as l i q u i d helium or hydrogen at low temperature. The sampling of phase space can sometimes be performed d i r e c t l y using Monte Carlo or Molecular Dynamics techniques which give - 29 -e s s e n t i a l l y "exact" r e s u l t s . Approximate methods, i n t e g r a l equation methods being one such general approach, can be used for some simple models. The Percus-Yevick (PY), Mean Spherical Approximation (MSA), and Hypernetted-Chain (HNC) methods are among the most frequently used i n t e g r a l equation theories. D i s t r i b u t i o n functions play primary roles i n the formulation of these theories. 2, D i s t r i b u t i o n Functions The pair d i s t r i b u t i o n function, g(12), can be defined [32] by f i r s t (N) constructing a normalized canonical p r o b a b i l i t y , P^ , for a homogenous system of N non-spherical molecules of a volume V and at a temperature T. The p r o b a b i l i t y of simultaneously f i n d i n g molecule 1 i n dXi at X_i, molecule 2 i n dXj> at 2^2 > e t c » I s given by pnN ) ( - 1 » - 2 , * , * » 2 N ) = \ exp[-$U N(X 1,X 2,...,X N)]dX 1 d ^ . - . d ) ^ (3.1) N where 3 = 1/kT, k i s the Boltzmann constant, and i s the co n f i g u r a t i o n a l i n t e g r a l (the normalization factor) defined by ZJJ = / . . . / exp [ - 3U N (X^ ,5^ 2 , • • •, XJJ ) ] dX_i dX^.t-dX^. (3.2) (n) The n-body p r o b a b i l i t y density, P^ , the p r o b a b i l i t y that any molecule i s i n dX] at X j , • • • » and any molecule i s i n dX^ at XQJ i s obtained by i n t e g r a t i n g over the coordinates of the remaining N-n - 30 -molecules: I t can be shown [32] that f o r a single molecule P^V^ - N/V = p (3.4) where p i s the number density of the system. We now examine the l i m i t of pf.11^ as the mutual distances between N the n p a r t i c l e s become large. As these separations increase, the c o r r e l a t i o n between the p a r t i c l e positions can be expected to decrease. Hence, i n the l i m i t the n-body p r o b a b i l i t y density can be fac t o r i z e d into the product of n single-body p r o b a b i l i t y d e n s i t i e s : 4 »V " P N 1 ) ( V PN 1 ) ( V"- PN 1 ) ( V = ( 3' 5 ) The n - p a r t i c l e d i s t r i b u t i o n function, g^1^, i s then defined by t \ (^1 > • • • »X ) 4n)(Xl,...,X ) = =2- (3.6) N — i —n n , . x n p ; i ; ( X i ) i = l which expresses the p r o b a b i l i t y density of observing d i f f e r e n t - 31 -configurations for a set of n molecules i n a system containing N molecules i n t o t a l . If a model for a system i s defined using only pair p o t e n t i a l s , i t (2) can be shown that a pair d i s t r i b u t i o n function, g^ ( X i , X2^» w i l l completely describe the equilibrium thermodynamics of that system. We write g(12) = iWtll /.../ e Xp[-3U N]dX 3...dX N (3.7) P ZN where g(12) i s a s i m p l i f i e d notation for the pair d i s t r i b u t i o n function. The r a d i a l d i s t r i b u t i o n function, g ( r ) , the p r o b a b i l i t y density of fi n d i n g a pair of molecules a distance r apart, i s obtained by i n t e g r a t i n g the pair d i s t r i b u t i o n function over a l l orientations of molecules 1 and 2: g(r) = // g(12) dfi.dH, ( 3- 8) (8TT r 1 1 where d^ = s i n 9d9d<j)d^.1 This angle-averaged pair d i s t r i b u t i o n function can be obtained from X-ray and neutron scattering experiments. For g ( r ) , and likewise for a l l n-body d i s t r i b u t i o n functions, the ^ h e p r o b a b i l i t y , p ( r ) , of finding a molecule a distance r from any other molecule i s given by p ( r ) = ^ g(D dr . - 32 -following must hold: g(r) •»• 1 as r -»• 0 0 . (3.9) E x p l i c i t formulae f o r the thermodynamic properties of f l u i d s can always be written as functions of u(12) and g(12). 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 studies, i s defined as 1 1 N U T 0 T = Y~ /.../ exp[-8U N][|. ^  2 u ( i j ) ] d X 1 . . . d X N (3.10a) N i * j which can be s i m p l i f i e d [32] to give 2 U, P T 0 T ( S * 2 ) 2 / u(12) g(12) dXj dX 2. (3.10b) For a multicomponent system, such as e l e c t r o l y t e s o l u t i o n , we write the t o t a l i n t e r n a l energy per molecule as U/N P . I X„X f t / g„ R(12) u (12) dr dfi.dfi, (3.11a) 2 ( 8 T T 2 ) 2 a3 a 3 a e a B - 1 2 where P = S P a (3.11b) a and X a = Pa/P> (3.11c) - 33 -p a being the number of density of component a. A discussion of the d i e l e c t r i c properties of e l e c t r o l y t e solutions and how they may be obtained from pair d i s t r i b u t i o n functions i s given i n Chapter IV. 3. Integral Equation Methods (a) The Ornstein—Zernike Equation In the above discussion, the pair d i s t r i b u t i o n function, g(12), has been defined. In the development of i n t e g r a l equation methods, i t becomes e s s e n t i a l to introduce the pair c o r r e l a t i o n function, h(12), defined by h(12) = g(12)-l. (3.12) It i s c l e a r that the function h(12) describes the departure of the d i s t r i b u t i o n function from i t s l i m i t i n g value of 1. Ornstein and Zernike [62] developed a r e l a t i o n s h i p i n which h(12), the t o t a l pair c o r r e l a t i o n , i s expressed as a sum of a d i r e c t part i n v o l v i n g p a r t i c l e s 1 and 2 only, and an i n d i r e c t part which takes into account a l l c o r r e l a t i o n s involving other p a r t i c l e s . This r e l a t i o n s h i p i s known as the Ornstein-Zernike (OZ) equation and can be written i n the form h(12) = c(12) + - P y / c(13) h(32) dX 3 (3.13) 8TT - 34 -where the i n t e g r a t i o n i s over a l l coordinates of p a r t i c l e 3. c(12) was c a l l e d the d i r e c t c o r r e l a t i o n function by the o r i g i n a l authors who had assumed ( i n c o r r e c t l y as i t turns out) that c(12) would depend only upon the p a i r i n t e r a c t i o n , u(12). The second term of equation (3.13) can be more f u l l y understood i f we i t e r a t e (3.13) to obtain the expansion h(12) = c(12) + - H y / c(13) c(32) dX 3 + C - ^ ) 2 / / c(13) c(34) c(42) 8ir 8ir dX 3 dX^ + ... (3.14) Thus, we see that the i n d i r e c t part of the t o t a l p a i r c o r r e l a t i o n involves c o r r e l a t i o n s between p a r t i c l e s 1 and 2 but only those through an increasing number of intermediate p a r t i c l e s 3,4, etc. The OZ equation (3.13) i s a basic r e l a t i o n s h i p i n the equilibrium theory of f l u i d s . It i s an exact r e l a t i o n s h i p which can be derived [32] through c l u s t e r diagram expansion methods. The d i r e c t c o r r e l a t i o n , c(12), can be viewed as being defined by the OZ equation and has no simple physical i n t e r p r e t a t i o n . In general, for large separations r , c(12) -»• -3u(12). In most cases, c(12) approaches i t s asymptote more quickly than does h(12) which has s i m i l a r o s c i l l a t o r y behavior to that of 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 function, g ( r ) . The convolution of the second term of the OZ equation (3.13) can pose many problems i n t r y i n g to solve for h(12). Fourier transform proves to be very useful . As the simplest case, i f one i s dealing only with a s p h e r i c a l l y symmetric p o t e n t i a l , the OZ equation can be Fourier - 35 -transformed immediately to give the k-space equation h(k) = c(k) + ph(k) c ( k ) . (3.15) The algebraic expression that r e s u l t s upon rearrangement of equation (3.15) expresses h(k) as a simple function of c(k) and p. (b) Closure Approximations The OZ equation (3.13) serves as the f i r s t equation i n any i n t e g r a l equation theory for f l u i d s . A second equation or r e l a t i o n s h i p i s necessary i n order to solve for the two unknowns, h(12) and c(12). Such a r e l a t i o n s h i p i s known as a closure for the OZ equation. However, th i s second r e l a t i o n s h i p which attempts to r e l a t e the pair c o r r e l a t i o n function, h(12), and the d i r e c t c o r r e l a t i o n function, c(12), by including the pair p o t e n t i a l , u(12), i s only (at present) an approximate r e l a t i o n s h i p . It i s the accuracy of the closure approximation used which l i m i t s the accuracy of an i n t e g r a l equation theory. Hence an i n t e g r a l equation theory i s i n v a r i a b l y known by the closure r e l a t i o n s h i p i t employs. The most commonly used closures, as mentioned above, are the Mean Spherical, Percus-Yevick, and Hypernetted-Chain approximations. The MSA works from the premise ( o r i g i n a l l y that of Ornstein and Zernike) that the d i r e c t c o r r e l a t i o n function, c(12), w i l l depend only on the pair p o t e n t i a l , u(12). The closure i s defined by two equations: - 36 -c(12) = -3u(12) for r > d (3.16a) and g(12) = 0 for r < d (3.16b) where d i s a hard-sphere diameter. The MSA closure always in s e r t s a hard core into a molecule (equation (3.16b) states that molecules never interpenetrate). An i m p l i c i t hard sphere p o t e n t i a l i s always implied even though u(12) may contain other s p h e r i c a l l y symmetric terms. The MSA does have the correct asymptotic form i n that equation (3.16a) i s exact i n the l i m i t r •*• °°. It i s not s u r p r i s i n g then that the MSA gives good r e s u l t s for monatomic f l u i d s at very low d e n s i t i e s . At higher d e n s i t i e s where short-range c o r r e l a t i o n s become much more important, the MSA generally gives poor r e s u l t s . This i s not the case for hard sphere systems where the MSA i s equivalent to the PY closure. The PY and HNC closures can be derived from functional Taylor series or c l u s t e r series expansions for c(12) by neglecting c e r t a i n terms. The PY closure i s given by: which also has the correct large r behavior. The PY theory gives good agreement with computer simulation r e s u l t s for some monatomic f l u i d s . I t gives very good r e s u l t s , p a r t i c u l a r l y at high d e n s i t i e s , for hard sphere, Lennard-Jones, and square well potentials [32]. The HNC closure, given by c(12) = g(12)[l - exp(Bu(12))] (3.17) c(12) = h(12) - Jin g(12) - 3u(12), (3.18) - 37 -i s superior to the PY closure for most other systems [32], I t s s u p e r i o r i t y for models possessing coulombic i n t e r a c t i o n s i s well known [32,35], For 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 been found [63] to be superior to both the PY and MSA theories through comparison with Monte Carlo r e s u l t s . The HNC closure also has the correct large r behavior; as r -*• 0 0, c(12) -> -3u(12). Integral equation theories can be solved a n a l y t i c a l l y i n some cases. The MSA, or equivalently the PY approximation, has been solved a n a l y t i c a l l y for hard sphere potentials [64-66], Wertheim [67] has solved the MSA theory exactly for dipolar hard spheres. Blum [54], using r o t a t i o n a l invariant expansions, has t r i e d to generalize the mean spheri c a l model for any hard sphere 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 obtained formal solutions which must be evaluated numerically i n a l l but the simplest cases. However, solutions to i n t e g r a l equation theories are generally obtained numerically. The numerical techniques used involve i t e r a t i v e procedures. The t r a d i t i o n a l method of determining h(12) and c(12) t y p i c a l l y proceeds as follows: an estimate for one of the functions, say h(12), i s used i n the closure r e l a t i o n s h i p to obtain c(12); the OZ equation i s then used to obtain a new estimate for h(12). This cycle can be i t e r a t e d u n t i l a desired degree of convergence i s obtained. A method known as the Newton-Raphson technique, recently developed by G i l l a n [68], has also proved very useful for some simple models. A further discussion of 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 of Int e g r a l Equation Methods to E l e c t r o l y t e Solutions Integral equation methods have been used extensively to study simple solvent models [35], E l e c t r o l y t e solutions, both at i n f i n i t e d i l u t i o n [22] and at f i n i t e concentrations [21], using only a dipolar solvent have also been studied. For these systems, the f u l l HNC theory can not 2 be solved at present and so further approximation to the HNC closure i s required i n order to obtain a tractable theory. One such closure, the l i n e a r i z e d hypernetted chain (LHNC) [36] approximation, has been shown to given good r e s u l t s f o r multipolar f l u i d s [35]. The LHNC closure i s obtained from the HNC closure by ret a i n i n g only the l i n e a r term of i t s expanded logarithm. The LHNC closure w i l l be e x p l i c i t e l y derived below. A s i m i l a r theory, the quadratic hypernetted chain (QHNC) [37], r e s u l t s from r e t a i n i n g terms of up to second order. For the present solvent model, the LHNC theory gives r e s u l t s for the d i e l e c t r i c properties that agree very well with r e s u l t s from the QHNC theory, and both agree very well with experiment [41], as was noted e a r l i e r . The LHNC and QHNC theories are also i n q u a l i t a t i v e agreement on s t r u c t u r a l properties. Recent work [69] has shown that r e s u l t s from both theories (although the QHNC i s somewhat superior) agree well with computer simulation (MD) re s u l t s for s t r u c t u r a l properties f o r a si m i l a r solvent model, a generalized Stockmayer f l u i d c o n s i s t i n g of Very recent work (Fries and Patey, to be published) has shown that the f u l l HNC theory can be solved for simple single component systems and i t s a p p l i c a t i o n to the present system looks promising. - 39 -Lennard-Jones spheres with dipole and quadrupole moments s i m i l a r to those of our water-like solvent. These r e s u l t s are even more emphatic when one considers that both theories are known [35] to give better r e s u l t s for systems with hard, as opposed to s o f t , s p h e r i c a l p o t e n t i a l s . The QHNC theory i s a more d i f f i c u l t theory to solve, though i t has been repeatedly shown to give superior r e s u l t s p a r t i c u l a r l y for multipolar systems at low density [35,37], However, unlike the LHNC closure, the QHNC closure i s not a se l f - c o n s i s t a n t approximation [37], The LHNC theory w i l l thus be used i n the present study because of i t s s i m p l i c i t y , self-consistancy, and i t s a b i l i t y to give good r e s u l t s f o r the d i e l e c t r i c properties for the solvent model. For mixtures, such as e l e c t r o l y t e solutions, the general solution to the OZ equation becomes much more complicated, however we fi n d that t h i s task i s much simpler at i n f i n i t e d i l u t i o n . (a) A p p l i c a t i o n of the Ornstein-Zernike Equation The following development c l o s e l y follows that of previous work [21,23,41], We begin with the OZ equation for a mixture, a generali z a t i o n of equation (3.13), which can be written as V 1 2 ) - C a 3 ( 1 2 ) = 72 1 P Y J V 1 3 ) C Y 3 ( 3 2 ) ^ 3 ( 3 ' 1 9 ) 8n y where py i s the number density of species Y, the int e g r a t i o n i s over a l l coordinates of p a r t i c l e 3 of species y, and hag(12) i s the pair c o r r e l a t i o n function between molecule 1 of species a and molecule 2 of - 40 -species 3. The sum i n equation (3.19) i s over a l l species y of the mixture. In the present study we w i l l consider only single component e l e c t r o l y t e s o lutions. Hence we can always consider a solution to be a three component mixture where + and - w i l l designate the i o n i c species and s the solvent. In the most general case for a three component system, equation (3.19) can be expressed as a system of 27 coupled equations. We have, however, chosen to make the assumption that i n a l l cases our s o l u t i o n i s i n f i n i t e l y d i l u t e , that i s p+ = p_ = 0. As a r e s u l t , f or the present study we have only to consider 9 equations. Furthermore, the solvent-solvent equation, given by completely decouples; that i s to say the solvent remains completely unchanged from i t s pure (or bulk) solvent state. The pure solvent system has been solved [41] and the solutions for h s s(12) and h (12) - c (12) = -Ar / h (13) c (32) dX_ ss ss 0 2 J ss ss —3 (3.20a) c s g ( 1 2 ) are r e a d i l y a v a i l a b l e . There are only 5 unique pairs of h ag(12) and c ag(12) for which to solve; we choose to solve for those given by h J,(12) - c._.(12) (3.20b) h (12) - c (12) - ^ j / h_ (13) c _(32) dX, 8ir S S (3.20c) - 41 -P S r V ( 1 2 ) " c + - ( 1 2 ) = ~ T $ h + s ( 1 3 ) c s - ( 3 2 ) d-3' (3.20d) 8TT P S r h + (12) - c (12) - -=2 / h + s ( 1 3 ) c S s ( 3 2 ) d* 3> (3.20e) 8TT and P S r h (12) - c (12) =-5- / h (13) c (32) dX„. (3.20f) - S — S o £• ~ S S S — J 8ir The sphe r i c a l symmetry of the ion-ion p o t e n t i a l requires that h+_(12) = h_+(12) = h ^ j ( r ) , and s i m i l a r l y for c ^ j ( r ) . An OZ expression f o r a$ = - + would thus be redundant since equation (3.20d) completely describes the unlike ion-ion c o r r e l a t i o n s . Similar symmetry arguments (as presented below) can be used to show that the OZ expressions for ag = s+, s- are likewise redundant. They would give no further information about the microscopic structure of the system. Following Blum [52], we expand h ag(12) and c ag(12) i n the same set of r o t a t i o n a l invariants previously defined to obtain h a g ( 1 2 ) = I h ^ y v ( r ) ^ ( 1 2 ) (3.21a) mn£ where - 42 -/h .(12) $ m n £ ( 1 2 ) d O - d O , n„ f t ( r ; — 5 ~ . (3.21b) J[*™*(12 ) r d O j d n 2 The denominator of equation (3.21b) serves to normalize the c o e f f i c i e n t s h ™ ^ ^ v ( r ) . The integrations i n equation (3.21b) are over a l l angular o r i e n t a t i o n and we note that J d B - ljl0 / J = 0 s i n e d B d W = STT2. (3.22) The function c ag(12) i s also expanded i n th i s manner with s i m i l a r expressions for the c o e f f i c i e n t s c m ^ ( r ) . v ctg,uv In general, these expansions w i l l have an i n f i n i t e number of terms, but i n the MSA and the LHNC theories only a f i n i t e number of r o t a t i o n a l invariants are required to form a closed set under the generalized convolution of the OZ equation (as described below). It can be equivalently stated that t h i s p a r t i c u l a r set o f *™*(12) w i l l generate only themselves and one another when the angular integrations of equation (3.19) are performed. Under the LHNC (or the MSA) closure, we fin d that only r o t a t i o n a l invariants belonging to th i s set can appear i n the closure expressions. In the LHNC (or MSA) theory this f i n i t e set constitutes an exact s o l u t i o n . This i s not the case i n the QHNC closure from which, i n p r i n c i p l e , an i n f i n i t e set of r o t a t i o n a l invariants i s generated. The functions h ag(12) and c ag(12) w i l l conform to the - 43 -same symmetry as that contained i n the terms of the expansion of u ag(12). We know then that at le a s t those r o t a t i o n a l invariants that appear In the expansion of u ag(12) w i l l appear i n the expansions of h ag(12) and c ag(12). Additional terms may be required so as to form a closed set, although t h i s i s not the case for the ion-solvent and ion-ion c o r r e l a t i o n functions. We can then write the ion-solvent pair c o r r e l a t i o n function as u / - I ' M - u 0 0 0 f \ _i_ 1.011/ . *011, l o. , ,022. . .022.... 0 , N h l g(12) = h ± g (r) + h i g (r) * (12) + h i g (r) * (12) (3.23) 022 with i = + or - and * (12) as previously defined ( c f . Chapter II equation (2.20)), and where h ? i ^ r > = 3 o 9 / \ d 2 ) * 0 U ( 1 2 ) dfi.dfi, (3.24a) i s ( 8 7 r2 }2 i s 1 2 , u 0 2 2 . . _ l r , 0 2 2 , . . , 0 2 2 , N 1 and h l g (r) - 2 - l h l s , 0 2 ( r ) + h i s , 0 - 2 ( r ) ] = 5/8 1 - / h. (12) $ 0 2 2 ( 1 2 ) d f l d n 9 (3.24b) (8rr ) Z 1 S 1 Z follow from equation (3.21b). The ion-ion pair c o r r e l a t i o n function, h^j(12), i s s p h e r i c a l l y symmetric as required by the s p h e r i c a l l y symmetric ion-ion pair p o t e n t i a l and hence, (3.25) The solvent-solvent p a i r c o r r e l a t i o n function w i l l be the same as that of the pure solvent and i t s expansion i s given by Carnie and Patey [41] (c f . equation (2.11) of Ref. 41). This expansion does contain terms that do not appear i n the solvent-solvent pair p o t e n t i a l . Expansions of the ion-solvent and ion-ion d i r e c t c o r r e l a t i o n functions are equivalent i n form to t h e i r respective p a i r c o r r e l a t i o n functions. The expansions f o r c a^(12) and h ag(12) can then be substituted i n t o equations (3.20) and the r e s u l t i n g expressions Fourier transformed [52], 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 I Z 2 1 h l 1(10 c 1 2 ( k ) (3.26a) cip ap s „ „ mnn, as s (J n l V 2 1 with t , M ™Vl n l n * 2 . m+n+n. * I % I I SL Z 2 1 = * f <?* + 1 > ("I) 1 { } ( * 2 n ) (3.26b) mnn^ m^nx. (2n^ + 1) Lm n n^ J 0 0 0 where g fl i s a 6-j symbol [70] and f m n ^ has been previously defined ( c f . Chapter II equation (2.12)). The sum over n^, %^ i s always f i n i t e , l i m i t e d by the non-zero values of Z . This generalized ' m n n^ b convolution of the OZ equation i s e s s e n t i a l l y that of Blum [54]. The - 45 -c o e f f i c i e n t s ( i n k space) of the transformed c o r r e l a t i o n functions are given by the Hankel transforms where the t i l d e denotes a Hankel transform, i = / - l , and j ^ ( k r ) i s the spher i c a l Bessel function of order I. I t w i l l now be convenient to introduce the function n(12) defined by which w i l l also be expanded i n the same manner as h(12) and c(12). The function h(12) i s p r e f e r e n t i a l l y used since i t i s a smooth, continuous, and well behaved function of r, unlike the function h(12) which i s discontinuous at r = d. The coe f f i c i e n t s n " (r) are thus easier to treat i n numerical transformations. h (k) = 4TT1 J r j ^ (kr) h (r) dr (3.27) TI(12) = h(12) - c(12) (3.28) Then evaluating the Z mnn i n equation (3.26a) we are l e f t with the 1 expressions ( i n k-space) 1_ P s ~000 "000 1 roll "Oil , 8 "022 ~022 ti. c . — - s - n. c. + - B - n . c. , i s sj 3 i s j s 5 i s j s (3.29a) 1_ P 000 ~000 (3.29b) • • c , i s ss s - 46 -P and 1 ~011 1 ~011 ."110 . 0 ~112. , 4 ~022 .~211 , , ~213N „ Q . — n. =-s-h. (c + 2 c )+--=- h. (c + 6 c ), (3.29c) p i s 3 i s ss ss 5 i s ss ss 1 "022 rOH A "121 . ~123. . 4 -022 .-222 ~220 . , ~224. , 0 O Q J . — r\ = h. (^ c + c ) + -g- h. (c - c + 4c ) (3.29d) p i s i s 6 ss ss 5 i s ss ss ss s , . , "101 "Oil . "202 where we have applied the symmetry requirements c g ^ = - c ^ s and c g ^ = ~Q22 c , and s i m i l a r l y for h(12). Certain projections have been set to zero 1 s (and hence do not appear) i f that projection does not occur i n a p a r t i c u l a r c o r r e l a t i o n function. For example, c ^ ^ ( k ) = 0 and s s n^ g*(k) = 0 since the solvent has no charge and the ions have no dipo l e . The ion-solvent c o r r e l a t i o n s d i r e c t l y , and the ion-ion c o r r e l a t i o n s i n d i r e c t l y , depend on the solvent-solvent c o r r e l a t i o n s as indicated by (3.29). The solvent-solvent c o r r e l a t i o n functions, as shown by the fact that equation (3.20a) completely decouples, can be thought of as constants depending only upon the pure solvent system [41]. (b) The LHNC Closure Approximation The LHNC closure was chosen i n the present study. There has been no evidence that the QHNC theory gives better r e s u l t s f o r the ion-ion c o r r e l a t i o n functions [21,22], or for d i e l e c t r i c properties [41,69] and i t i s these properties i n which we are most inte r e s t e d . The LHNC closure may be derived i n several ways and may appear i n d i f f e r e n t forms. We - 47 -note that the LHNC approximation i s e s s e n t i a l l y equivalent to the single superchain (SSC) approximation of Wertheim [71]. We st a r t with the HNC equation (3.18). Using the d e f i n i t i o n of h(12) given by equation (3.12) and the s i m i l a r r e l a t i o n s h i p for the s p h e r i c a l l y symmetric projection ,000, . 000, N , hctg ( r ) = gcxg ( r ) " 1» (3.30) we rearrange (3.18) to obtain c a e ( 1 2 ) = h a p ( 1 2 ) - In g°°°(r) - i n [ l + X a p ( 1 2 ) ] - 3u a g(12) (3.31a) where h r a R(12) - h°°R°(r) Xag< 1 2> - B 000, " * ( 3 ' 3 1 b ) 3For rani * 000, h ™ * ( r ) - g ^ * ( r ) by d e f i n i t i o n . - 48 -We now expand £n[l + X a R ( 1 2 ) ] i n (3.31a) as a Taylor s e r i e s . In the LHNC theory only the l i n e a r term i n X a3(12) i s retained ( f o r the QHNC closure, terms to the order of X 2 are retained) to give C<XB(12) = V 1 2 ) - *N CB ( R ) - W 1 2 ) - 0 u a B < 1 2 ) ( 3 - 3 2 ) We now r e c a l l the d e f i n i t i o n ( c f . equation (3.21b)) for the c o e f f i c i e n t s . J c Q(12) $ m n*(12) d n . d f l 0 - ~a , , , ,\1-/ n o • 'aB,uv / [ $ ^ ( 1 2 ) ] 2 d ^ d f i 2 (3.33) The LHNC closure expressions for c . (r) are obtained by s u b s t i t u t i n g Olp the l i n e a r i z e d HNC equation (3.32) into (3.33), expanding both h ag(12) and u a3(12), and performing the required i n t e g r a t i o n s . The integrations are greatly s i m p l i f i e d by using the orthogonality of the It should be noted that such an expansion w i l l rigorously hold only f o r -1 < X ag(12) j< 1. It i s not clear then why the expansion should work. For any systems possessing 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 anisotropic (e.g. multipolar) terms of the pote n t i a l become small. The expansion w i l l also, i n most systems, be v a l i d at large separations, r, since i n general g^g^C 1) * 1 as r ->- 0 0 and the dif f e r e n c e , h a f t(12) - h ^ ^ ( r ) , w i l l very quickly go to zero. Near contact ( i . e . r " d) however, we can i n most cases expect |X > 1. This fact can be tolerated simply because even f o r large anisotropic p o t e n t i a l s , the theories that r e s u l t from truncation of the expansion s t i l l give r e s u l t s that agree well [35,69] with exact solutions (computer simulations). Also as mentioned e a r l i e r , the LHNC approximation can be obtained by other methods which do not involve truncation of the logarithmic expansion. - 49 -r o t a t i o n a l i n v a r i a n t s . We then have 000/ >. ,000, x „ 000, \ a 000, s **\/ \ c a 3 ( r ) = h a 3 ( r ) " Z n g a 3 ( r ) " 3 u a 3 ( r ) (3.34a) ,mn£, . , maZ, . ,mn^ / N q mn£, N a3 K ' (3.34b) and c a 3 (r) = h a g (r) - 3 u a j 3 (r) - Q Q 0 g a 3 ( r ) for a l l other projections. It i s i n t e r e s t i n g to note that equation (3.34a), the closure r e l a t i o n s h i p for the s p h e r i c a l l y symmetric term, i s exactly the same as that given by the f u l l HNC equation. We rearrange these further to obtain 000/ -, rr,0()0, . 000, *, 000, . , 0 o c . ccx3 ( r ) = e x p t n a 3 ( r ) " 3 u a 3 ( r ) ] " nc * 3 ( r ) " 1 ( 3 * 3 5 a ) , mn£, \ 000, x. r„mn£, N „ mn&, », „mnJl, N , 0 ocl_v and c a g (r) = g ^ ( r ) [ n a 0 (r) - 3 u a e (r)] - ( r) , (3.35b) where these closures are expressed as functions of n(12) instead h(12). The closure r e l a t i o n s h i p s given by (3.35) are, i n general, only approximate. However, i n the present model for r < da$ we f i n d we have a s p e c i a l case because the hard sphere p o t e n t i a l i s i n f i n i t e . Rearranging (3.35) we can show that for hard sphere models c°a03°(r) + T r ™ ( r ) 4- 1 = gJ00( r) = exp[ (r) - 0u°a°O(r)] - 0 (3.36a) for r < d a e - 50 -and mnJi/_\ , ^ mn-fc/x _ _mn£/_\ rjamHf •> 0 mn£/ \i 000/ N n ,~ ^r,\ :a3 ( r ) + na3 ( r ) ga3 ( r ) = [ na3 ( r ) " 0 ua3 ( r ) ] ga3 ( r ) = 0 <3-36b> for r < d „. a3 We have from (3.36) that g(12) = 0 for r < d a3» 5 which i s an exact r e s u l t for hard sphere systems. We then can write the closures 000/ , /, , 000/ / 0 „, . ca3 ( r ) = " ( 1 na3 ( r ) ) (3.37a) and c ^ £ ( r ) = " ^ ( r ) (3.37b) fo r r < d ag. The expressions (3.37) can also be obtained by applying the condition g(12) = 0 for r < d ag d i r e c t l y to the d e f i n i t i o n (3.28) of n(12) and rearranging. The closures given by (3.37) are thus exact. Together, the equations of (3.35) and (3.37) constitute the f u l l LHNC closure for the present model. We fi n d that c(12), l i k e h(12), w i l l be a discontinuous function. However, unlike h(12) which equals zero for r < d a B , c(12) (and hence n(12)) w i l l have non-zero values 5 Note that t h i s i s the same expression as equation (3.16b) of the MSA closure expressions. The MSA closure (3.16) can thus be obtained from the LHNC closure by set t i n g = 1 f ° r r > d (which i s exact fo r large r ) . - 51 -fo r r < d a 3 . (c) Average Interaction Energy and P o t e n t i a l of Mean Force We are very r e s t r i c t e d i n the number of thermodynamic properties of an e l e c t r o l y t e s o l u t i o n we are able to investigate because we have chosen to examine only the i n f i n i t e d i l u t i o n l i m i t . Any values we are able to ca l c u l a t e are always the l i m i t i n g values at 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 unity. The osmotic pressure w i l l be zero. The solvent w i l l r e t a i n a l l the properties of the pure solvent. We can, however, calculate the average ion-solvent i n t e r a c t i o n energy for a single ion U I S / N i = f uCD + U C Q ^ / N i (3.38a) where Urjjj, UQQ are the average charge-dipole and average charge-quadrupole energies, r e s p e c t i v e l y , and i s the number of ions of species i . Ufjp and UQQ are determined by su b s t i t u t i n g the expansions f o r u i s(12) and g i g(12) into equation (3.11a) and taking the l i m i t +.O. Simplifying, again using the orthogonality of the r o t a t i o n a l i n v a r i a n t s , we have ir r p s q i y V h i s ( r ) d r ( 3 - 3 8 b ) i i s - 52 -and UCQ 16TT _ f» ,022, v j z- _ Q N T = — p s q i °T V h i s ( r ) d r ' ( 3 ' 3 8 C i i s The terms of the average ion-solvent i n t e r a c t i o n are not only important in themselves and i n giving a r e l a t i v e importance of the multipole moments, but we f i n d that the expression for w i l l also be needed i n the d i e l e c t r i c theory to be described in Chapter IV. The average ion-solvent i n t e r a c t i o n energies for a single ion can also be used i n c a l c u l a t i n g solvation energies for ions. In the commonly used Born theory, the solvation energy of a single ion at i n f i n i t e d i l u t i o n i s given by the energy gained by charging an ion immersed i n a continuum s o l v e n t 6 (completely described by s p e c i f y i n g a d i e l e c t r i c constant). The Born theory, however, makes no attempt to take into account changes in the solvent structure due to the presence of the ion. We have recently developed a theory [72] for determining the change in energy of a dipolar hard sphere solvent due to the presence of an ion at i n f i n i t e d i l u t i o n by c a l c u l a t i n g the changes i n solvent-solvent c o r r e l a t i o n functions. It was found that the presence of an ion has a very long-ranged ( i . e . a 1/r dependence) e f f e c t on solvent-solvent c o r r e l a t i o n s . For the dipolar hard sphere system, the solvent-solvent term was found to be opposite i n sign and roughly h a l f Equivalently, the solvation energy i s given by the energy released i n t r a n s f e r r i n g an ion from a near vacuum (the i d e a l gas phase) to being completely immersed in the solvent. - 53 -the magnitude of the ion-solvent i n t e r a c t i o n . The r e s u l t i n g s o l v a t i o n energies for ions i n a dipolar hard sphere solvent having a d i e l e c t r i c constant equal to that of water (using c r y s t a l r a d i i to determine ion diameters) are well within the correct order of magnitude of experimental values. This theory i s currently being applied to the water-like solvent. I n i t i a l r e s u l t s look promising, and are s u r p r i s i n g l y s i m i l a r to those for the simple dipolar solvent. It i s also convenient for us to introduce an ion-ion p o t e n t i a l of mean force at i n f i n i t e d i l u t i o n , w ij(12). w ij(12) i s defined by g i j ( 1 2 ) = exp [ - e W i j(12)l (3.39) where g^j(12) i s the ion-ion pair d i s t r i b u t i o n function. w^j(12) i s the p o t e n t i a l associated with the average force between two ions i and j at i n f i n i t e d i l u t i o n . (It i s the po t e n t i a l of the force acting between i and j i n the pure solvent.) It includes a l l solvent e f f e c t s that influence the ion-ion c o r r e l a t i o n s . Rewriting equation (3.39) and using the f u l l HNC approximation (3.18), we have 3w ± j(12) = -in g i ; j(12) - 011^(12) - 11^(12) . (3.40a) Noting that any ion-ion p o t e n t i a l must be s p h e r i c a l l y symmetric (for the present model), we obtain for r > d^j 8w (r) = ^ 1 - nj°°(r) (3.40b) - 54 -where - T l ^ j ^ ( r ) ^ s t h e contribution to the ion-ion p o t e n t i a l of mean force which depends upon the solvent. The p o t e n t i a l of mean force i s e a s i l y obtained once r|000(r) ^ as D e e n determined ( c f . equation (3.29)). It can be shown [21,25,25] that as r •* 0 0 q i q i where e Q i s the pure solvent d i e l e c t r i c constant. Expression (3.41) i s the correct continuum l i m i t . (d) Method of Solution Solutions for the present system are obtained by solving the equations given by (3.29) subject to the closures given by (3.35) and (3.37). Upon closer inspection of equation (3.29), we fi n d that n^_.(12) = n^ j^(r) depends d i r e c t l y only upon ion-solvent c o r r e l a t i o n functions. Hence, once solutions have been obtained for the ion-solvent c o r r e l a t i o n s , the value of the ion-ion c o r r e l a t i o n functions are fi x e d by (3.29a). We f i n d also that i n the LHNC theory n?°°(r) depends only i s on other s p h e r i c a l l y symmetric c o r r e l a t i o n s ( c f . equation (3.29b)). It decouples from the expressions for the other projections of r i i s ( 1 2 ) , O i l 022 namely n. (r) and n. ( r ) , given by the two remaining equations of IS X s (3.29). The major task thus becomes to obtain consistant solutions to - 55 -the coupled equations (3.29c) and (3.29d). For computational purposes, i t i s convenient to express a l l equations as functions of two unknowns; we have chosen h(12) and c(12). The closure r e l a t i o n s h i p s (3.35) and (3.37) have already been expressed i n t h i s form. We have, however, s t i l l to remove the dependence on h(12) from the reduced OZ equations (3.29). Replacing h ™ ^ by C1"!™^ + c^ g^ ) and rearranging, equations (3.29c) and (3.29d) become .- ~011 . - ~ 0 2 2 W l / - . . - ,~ -022 . ~ - O i l . - O i l ( c l c i s + c 2 c i s ) ( 1 / P s " c4> + c 2 ( c 4 c i s + C 3 c i s ) h, (k) = ; = = — (3.42a) ^ ( l / p a - c 4 ) ( l / P s - c x ) - c 2 c 3 and f - ~022 . ~ -OIK,., - . . ~ ,~ ~011 . " -022. ~022 ( c 4 c i s + c 3 C i s ) ( 1 / P s " C l } + e 3 ( c l c i s + C 2 c i s } TI. (k) = = = — (3.42b) i s ( 1 / P S - c 4 ) ( l / p 8 - C l ) - c 2 c 3 1 -110 "112 where c l = 3" < c s s + 2 c s s } (3.42c) 4 ,~211 , , ~213. ,„ / O J . c2 = J ( c s s + 6 C s s } ( 3 * 4 2 d ) When evaluating these expressions numerically, we must take into account the i 2 = ~1 that r e s u l t s from and product of any two projections n I B a A , ( k ) and (r) where both V and £" are odd. For odd V and i" these k-space functions are purely imaginary. - 56 -~ _ ,1 "121 , "123, c„ = (T- c + c ) 3 6 ss ss (3.42e) and 4 ,~222 ~220 . , ~224N c, = - F - (c - c + 4 c ), 4 5 ss ss ss (3.42f) We also rewrite equation (3.29b) as -000 " i s (10 -„ rooo rooo P h . h s i s ss 1 + p h s ss (3.43a) 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 (3.43b) which comes from solving (3.20a), the pure solvent OZ equation. Using (3.43a), we can show that , 000 "000 h i s c j s ,000 000 n. h. i s j s 1 + p h 0 0 0 s ss (3.44a) We then i n s e r t t h i s r e s u l t into (3.29a) which becomes 000 n.. "000 "000 i s j s , 1,~011 , "011 "011 . 8 "022 . "022 "022 000 3 i s i s j s 5 i s i s 1 + P h s ss (3.44b) - 57 -The r e l a t i o n s h i p s given by (3.35) and (3.37) constitute complete closures and could be solved as they are. However, at high d e n s i t i e s some inaccuracy would be introduced from the HNC treatment of the hard sphere p o t e n t i a l . To reduce t h i s e r r or, we apply a perturbation technique f i r s t suggested by Lado [73]. This technique ensures that f o r 000 HS charge-multipolar hard sphere systems g a R (r) •*• g a g ( r ) * n the l i m i t HS of vanishing charge and multipoles, where g 0 ( r ) i s the "exact" hard ctp sphere r e s u l t . The a p p l i c a t i o n of t h i s method i s very straightforward i n the LHNC theory. In the LHNC closure expression for c^g^( r)> only the s p h e r i c a l l y symmetric projections of n Q(12) and u Q(12) appear. Hence c^^P ( r ) Otp Ctp dp can only depend upon the i s o t r o p i c part of the pair p o t e n t i a l ( c f . equation (3.34a)). If u ^ g ^ ( r ) contains only the hard sphere 000 HS i n t e r a c t i o n , we have only to replace g a f t (r) by (r) i n a l l closure r e l a t i o n s h i p s for c™g^ ( r ) . For the present study, g^g ( r ) , the "exact" hard sphere r a d i a l d i s t r i b u t i o n function, i s taken to be the Verlet-Weis [74] f i t to Monte Carlo data. We can then solve equations 011 022 (3.42) f o r n (r) and n ( r ) , subject to the closure r e l a t i o n s i s Is c i s ( r ) g i s ( r ) [ n i s ( r ) " 3 u i s ( r ) 1 " n i s ( r )> r > d i s ( 3 ' 4 5 a ) and c. (r) = -n. ( r ) , r < d. (3.45b) Is i s i s - 58 -f o r (mnJO = ( O i l ) and (022). In obtaining solutions for n a g ( 1 2 ) , we must be prepared to perform Hankel transforms (as given by (3.27)) since the reduced OZ equation (3.29) has been solved i n k-space and the closure expression (3.35) and (3.37) are i n r e a l space. Hankel transforms using spherical Bessel functions of second-order or higher ( i . e . I 2) are very d i f f i c u l t to d i r e c t l y evaluate numerically. For t h i s purpose i t i s useful to introduce i n t e g r a l transforms as discussed by Blum [54], The i n t e g r a l transforms, and t h e i r inverses, needed i n the present study 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 m n 2 ( s ) , / o / ^ \ c (r) = c (r) - 3 J i — ^ ds, (3.46c) mn2, v ~mn2, s 3 rr 2 ~mn2 • % , t t i r * \ c (r) = c (r) J s c (s) ds (3.46d) J o r where the hat (~) indicates the i n t e g r a l transformed function. We note that the i n t e g r a l transform used i n obtaining c m n ^ ( r ) depends only on I. The c a l c u l a t i o n of c m n-^(k) i s then reduced to taking the zeroth-order Hankel (Fourier) transform of c m n ^ i f % i s even, or the f i r s t - o r d e r Hankel transform i f I i s odd. We write - 59 -c ^ O c ) = 4IT J " r 2 j Q ( k r ) cmn\r) dr, (3.47a) c m n £ ( r ) = _1_ j» k2 ~mn*(k) d k ( 3 > 4 ? b ) 2ir when % i s even and c m n J l ( k ) = 4rri r 2 j x ( k r ) c m n £ ( r ) dr, (3.48a) cmn* ( r ) = j» k Z r ) r m n £ ( k ) d f c ( 3 > 4 g b ) 2ir when % i s odd. Similar expressions can also be written for n m n ^ and h™1-^. The i n t e g r a l transforms are e a s i l y performed numerically, as are the zeroth (Fourier) and f i r s t - o r d e r Hankel transforms, which can be done by Fast Fourier transform techniques (see Appendix B). Cal c u l a t i n g the Hankel transforms i n t h i s manner has a second advantage i n that i t allows long-range parts of ce r t a i n projections to be treated exactly. c £ s ^ r ) w i l l have a long-range term (due to u ^ ^ ( r ) ) with a 1/r 2 dependence. It must be treated with great care X s since i t becomes a 1/r term i n the Hankel transform i n t e g r a l . As a r e s u l t , t h i s term i s always subtracted from c ^ g ^ ( r ) and Fourier transformed a n a l y t i c a l l y . What remains of c ^ ^ ( r ) can then be 1 s - 60 -numerically transformed. Special attention must be paid i n i n v e r t i n g n ^ j ^ ( k ) , as given by equation (3.44b). We can show that i f 3w^(r) i s to have the correct 1/r behavior for larger r, then as k ->• 0 ~ 0 0 0 / , >. / £o ~ 1 1 /, / f t x n (k) 4TT q ^ — 2 . (3.49) J o k Thus upon numerical transformation of T\j*?^(k), the k = 0 co n t r i b u t i o n to the i n t e g r a l must be taken into account a n a l y t i c a l l y . O i l 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 also be handled with i s i s 022 care. c. (r) must be treated as being dual valued at r = d. when i s i s performing i t s i n t e g r a l transform. In using Fast Fourier transform techniques, we find i t convenient to perform the integrations i n (3.47) Q and (3.48) using the trapezoidal r u l e , and hence we have only to average the dual valued points at the d i s c o n t i n u i t i e s . Solutions for the present model are obtained by the i t e r a t i v e procedure described i n Appendix B. In a l l c a l c u l a t i o n s functions were represented numerically using 512 points with a grid width Ar = 0.02d s The r e s u l t s obtained w i l l be presented and discussed in Chapter V. A l l integrations are performed using the trapezoidal r u l e . As a check, Simpson's rule i s used to repeat the integrations i n the determination of i n t e r a c t i o n energies of the system. Good agreement was always obtained. - 61 -CHAPTER IV DIELECTRIC THEORY OF ELECTROLYTE SOLUTIONS 1. Introduction Electrodynaniic and e l e c t r o s t a t i c properties are among the most d i s t i n c t i v e used i n recognizing an e l e c t r o l y t e solution [4-6,8,75], An e l e c t r o l y t e solution i s characterized by a large e l e c t r i c conductivity. The solvent possesses a large d i e l e c t r i c constant which decreases with i o n i c concentration. The d i e l e c t r i c properties of any solution (any substance) depend on the e l e c t r i c moments, both permanent and induced, of the molecules which compose i t . The s o l v a t i o n of an ion by a solvent occurs because of large 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 between the ion and the solvent molecules. These int e r a c t i o n s w i l l generally be dominated by the charge-dipole p o t e n t i a l . Hence in e l e c t r o l y t e solutions, ion solvation and the d i e l e c t r i c properties of solution have t h e i r basis i n r e l a t e d processes at the microscopic l e v e l . A more complete understanding of t h e i r d i e l e c t r i c properties would improve the general understanding of e l e c t r o l y t e solutions. The d i e l e c t r i c response of an e l e c t r o l y t e solution i s represented by i t s "apparent" d i e l e c t r i c constant. To understand what i s meant by the apparent d i e l e c t r i c constant of a s o l u t i o n , we must f i r s t understand the notion of a s t a t i c d i e l e c t r i c constant. For a non-conducting material the s t a t i c d i e l e c t r i c constant i s well defined. It i s related to the p o l a r i z a t i o n , P (the mean dipole moment per unit volume), of an - 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 f i e l d , E_, by the equation ( e - l ) E = 4TTP (4.1) where e i s the s t a t i c d i e l e c t r i c constant. The net e l e c t r i c f i e l d that e x i s t s within the material, which takes into 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 induced d i p o l e s ) , can be r e l a t e d [76] to i t s capacitance. The capacitance, C, for a p a r a l l e l plate capacitor can be expressed as where A i s the area of the plates of the capacitor, d i s the distance between the plates, and £ i s the e l e c t r i c p e r m i t t i v i t y of the material. The d i e l e c t r i c constant, e, i s defined as the r e l a t i v e p e r m i t t i v i t y , £/5o» where 5o i s the e l e c t r i c p e r m i t t i v i t y of free space. In p r i n c i p l e , e w i l l depend on f i e l d strength, but we consider only the low f i e l d l i m i t . Hence, the d i e l e c t r i c constant of a material can be 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 c i r c u i t . It i s useful [78] to now introduce a frequency dependent complex d i e l e c t r i c constant, e*(w), which for both conducting and non-conducting i . e . between two oppositely charged i n f i n i t e p a r a l l e l plates as i n a capacitor. - 63 -systems i s given by e*(a)) = e'(u)) - ie"(uj). (4.3) The r e a l part, e'(u), represents the d i e l e c t r i c constant discussed above. It approaches the s t a t i c value as w ->• 0. e' depends on frequency since the p o l a r i z a t i o n process requires a f i n i t e time period ( i . e . for r e o r i e n t a t i o n of molecular d i p o l e s ) . e"(to), the imaginary part, i s known as the d i e l e c t r i c l o s s . This loss term r e s u l t s from the motion of e l e c t r i c a l charge; either from the impedence of dipole o r i e n t a t i o n by intermolecular forces or from actual charge transport (conduction). For a conducting s o l u t i o n , E*(UJ) w i l l diverge at low frequency according to the r e l a t i o n s h i p [8] where a i s the conductance of the s o l u t i o n . Hence, one defines the apparent d i e l e c t r i c constant, £ c m , of a conducting solution by E*(U)) •*• 4Tro/io) as OJ ->- 0, (4.4a) e SOL l i m i t io 0 [e*(u) -4TTQ-ioj ] (4.4b) where cr i s taken as the measured zero-frequency conductivity. egoL represents the measured d i e l e c t r i c constant of a s o l u t i o n which w i l l remain f i n i t e as a) •*• 0. - 64 -The conductivity of an e l e c t r o l y t e solution has made the determination of i t s d i e l e c t r i c constant experimentally d i f f i c u l t . The measurement of the pure solvent d i e l e c t r i c constant i s r e l a t i v e l y easy to perform (at low frequencies) 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 concentrations increase, higher frequencies are required to o f f s e t the influence of increased conductivity and other i o n i c e f f e c t s . Early measurements of the "apparent" d i e l e c t r i c constants of e l e c t r o l y t e solutions were a l l ca r r i e d out at low concentrations and low frequencies ( 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 early l i t e r a t u r e , some ca r e f u l measurements[15,79] gave strong evidence of the Debye-Falkenhagen (DF) e f f e c t (discussed below) at low concentrations. Later measurements have generally a l l been at higher concentrations and at higher frequencies ( i . e . > 100 MHz) [6,75]. Co-axial transmission l i n e s , waveguides, and microwave i n f e r o -meter techniques have been applied i n high frequency measurements. In these high frequency techniques, Cole-Cole [95] diagrams are usually constructed showing the frequency dependence of e1 and e", the r e a l and imaginary parts, respectively, of the complex d i e l e c t r i c constant. However, determination of the zero frequency value usually requires a long extrapolation from at best several points and at least one assumed value. Usually i t i s the value of the high frequency l i m i t , e<x» the d i e l e c t r i c constant that arises only from intramolecular p o l a r i z a t i o n e f f e c t s , that i s assumed. Hence even moderate ( i . e . 1-2%) accuracy has been hard to obtain. At moderate to high concentrations t h i s accuracy has proved to be more than adequate to follow the large trends that - 65 -e x i s t . High frequency methods have generally not been used at low i o n i c concentrations where trends w i l l be less pronounced. Only recently [80,81] has an attempt again been made at measuring d i e l e c t r i c constants of solutions at low i o n i c concentrations. These r e s u l t s w i l l be useful for comparison with the present study. We may also define [76] the d i e l e c t r i c constant of a system using Coulomb's law. The force, F ^ j ( r ) , between two charges i n a continuum medium with a d i e l e c t r i c constant £g (the subscript E i s used here 2 to denote the equilibrium contribution as discussed below) i s given by F ± , ( r ) = (4.5a) e E r and hence the p o t e n t i a l , u ^ j ( r ) , i s q i q i u..(r) = - i - i . (4.5b) As we w i l l describe below, t h i s expression (4.5b) also defines the large r asymptotic form of an e f f e c t i v e ion-ion p a i r p o t e n t i a l for a s o l u t i o n containing a f i n i t e concentration of i o n i c solute p a r t i c l e s . i s the d i e l e c t r i c constant for a s o l u t i o n which may not be equal to that of the pure solvent. Here the d i e l e c t r i c constant represents the long-range s h i e l d i n g of e l e c t r i c charges by the solvent. Therefore, £g One must always be mindful of which system of units 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 unrationalized [77]. We have chosen to use the unrationalized system where the factor 4TT w i l l appear i n Maxwell's equations and not i n Coulomb's law. - 66 -w i l l depend on the equilibrium structure i n the solu t i o n and t h i s structure can be described by molecular c o r r e l a t i o n functions. Thus for a s o l u t i o n , £„ i s an exact analogue (describing the same processes or responses) to the s t a t i c d i e l e c t r i c constant defined e a r l i e r of a pure solvent. The apparent d i e l e c t r i c constant of a conducting s o l u t i o n , egoL» i s not a true equilibrium quantity and w i l l not equal e^ ,. Several authors [8,15,82,83] have i d e n t i f i e d dynamical contributions, en, to eSOL a n d ^ n §eneral eS0L = + €~D' These dynamical contributions a r i s e because the conductivity, a( co), that appears i n the d e f i n i t i o n of EgQ^* equation (4.4), i s a complex function of frequency. The dynamical contributions to e g Q L a r i s e then r e a l l y as imaginary conductivity terms which have not been taken into acount. The f i r s t of these terms, which increases e g n T j » w a s i d e n t i f i e d and estimated many years ago i n the theory of Debye and Falkenhagen [15]. A second term, which decreases E g Q j / w a s recently i d e n t i f i e d by Hubbard and Onsager [84], This term, known as the k i n e t i c d i e l e c t r i c decrement (KDD), has been investigated further from a microscopic point of view by Hubbard, Colonomos, and Wolynes [82], In the present study we w i l l examine both the equilibrium and dynamic terms of e g Q T / Although the present study w i l l be ca r r i e d out at i n f i n i t e d i l u t i o n , we are able to determine l i m i t i n g laws for the d i f f e r e n t contributions to the apparent d i e l e c t r i c constant of a so l u t i o n at low concentration. - 67 -2. Equllbrium Theory of the D i e l e c t r i c Constant Empirical and semi-empirical equilibrium theories have proven successful i n accounting for much of the v a r i a t i o n observed [4-6,75] i n the d i e l e c t r i c constant of a solution as the i o n i c concentration i s increased. The d i e l e c t r i c constant decreases roughly l i n e a r l y with concentration at low to moderate concentrations ( i . e . < 1M for most strong e l e c t r o l y t e s ) . At very low concentrations the DF e f f e c t , which has a square root dependence upon concentration, can be observed although i t i s r e l a t i v e l y small. The rate of decrease of the d i e l e c t r i c constant becomes smaller as the concentration increases. Equilibrium theories have placed much emphasis on explaining the l i n e a r behavior of the d i e l e c t r i c decrement and arguments have been put forward [4,6,75] which supply at least q u a l i t a t i v e explanations. Many strong e l e c t r o l y t e s (eg. the a l k a l i halides) possess no permanent dipole moments. Hence, the " s t a t i c " d i e l e c t r i c constant of a solut i o n i s lowered by a simple volume e f f e c t due to the displacement of polar solvent molecules by non-polar solute p a r t i c l e s . A further decrement ( i n most cases the dominant term) results from what i s sometimes known as d i e l e c t r i c saturation. This l o c a l h i g h - f i e l d e f f e c t r e s u l t s from the intense e l e c t r i c f i e l d (=10 V/cm at the nearest neighbor separation) which surrounds an ion. This f i e l d w i l l greatly This assumes a small density change r e l a t i v e to the drop i n d i e l e c t r i c constant, which i s the case. - 68 -reduce the a b i l i t y of solvent molecules close to the ion to orient themselves with an applied f i e l d . For a small ion, the solvent molecules immediately surrounding i t (for a monovalent ion, i t s nearest neighbor s h e l l ) are said to be i r r o t a t i o n a l l y bound and, therefore, unable to orient with an applied f i e l d . C l e a r l y , the volume exclusion of the solvent w i l l r e s u l t i n a l i n e a r concentration 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 of the d i e l e c t r i c decrement due to d i e l e c t r i c saturation has only been recently proven [72,86] for non-polarizable systems using microscopic theory. One should be able to express both e f f e c t s i n terms of the molecular c o r r e l a t i o n functions of the s o l u t i o n . This implies that one know how these functions depend upon i o n i c concentration. Relationships for determining the d i e l e c t r i c constant of a pure solvent [35,75,85] or of a so l u t i o n of non-polarizable p a r t i c l e s [21], from solvent-solvent c o r r e l a t i o n functions have been obtained. The concentration dependence of the d i e l e c t r i c constant could be examined by repeating a f i n i t e These molecules should not be confused with those that make up what i s known as the hydration s h e l l , the proposed s h e l l of solvent molecules that surrounds and moves with an ion i n s o l u t i o n . The molecules of the hydration s h e l l , i f i t e x i s t s , must be a subset of these molecules. Although the notion of a hydration s h e l l seems to be supported by thermodynamic data, there has been no clear microscopic evidence (except for ions that covalently bond solvent molecules or possibly for very small ions, eg. L i + ) of i t s actual existence or the time scale on which i t may e x i s t . I t may be a "structure" which experiences very rapid exchange of solvent molecules. 5 P o l a r i z a b i l i t y may be included i f expressed as part of an e f f e c t i v e p o t e n t i a l . - 69 -concentration c a l c u l a t i o n at a number of d i f f e r e n t concentrations. However, at i n f i n i t e d i l u t i o n , as i n the present study, the d i e l e c t r i c constant of the s o l u t i o n w i l l always be that of the pure solvent. Friedman [86] has recently obtained an expression (as a function of the solvent-solvent d i r e c t c o r r e l a t i o n ) for the l i m i t i n g concentration dependence of the d i e l e c t r i c constant at i n f i n i t e d i l u t i o n . Using t h i s expression (some correction being necessary [23]), we are able to calculate the l i m i t i n g slope which determines the decrement i n e^ , at low concentration. (a) The D i e l e c t r i c Constant of a Solution The determination of the d i e l e c t r i c constant, e^, of a s o l u t i o n using equilibrium theory i s best understood by f i r s t examining the pure solvent case. We develop the expression for e 0, the d i e l e c t r i c constant of a pure solvent system, f i r s t proposed by Kirkwood [85], We consider an i n f i n i t e solvent composed of non-polarizable dipolar molecules and examine a spherical sample cavity i n our system. Using equation (4.1), i t i s e a s i l y shown [6,77] that the e l e c t r i c f i e l d i n s i d e the sample, _E, i s given by — = 7 T T ^o ( 4 ' 6 ) o where i s the homogeneous external applied f i e l d . We l e t M represent the average t o t a l dipole moment of the system. Using c l a s s i c a l s t a t i s t i c a l mechanics, we determine <Uj_*'e>, the average - 70 -moment of the i t n molecule i n the d i r e c t i o n , e^  (a unit vector), of the applied f i e l d . Assuming M«E_0/kT « 1, averaging over a l l d i r e c t i o n s and 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 of the system when only the i t Q molecule i s fixed and <y_j/M_l> i s the z e r o - f i e l d average. We define the dipole moment y_* to be that of the spherical sample when a cen t r a l ( i t n ) molecule i s held f i x e d . Our sphere must be s u f f i c i e n t l y large so that the system outside can be regarded as being e l e c t r i c a l l y homogeneous from the point of view of the i t n molecule, _y* w i l l depend on the alignments of a l l the molecules within our spherical sample. We can show [77] that 9e 2 = ( e + 2 ) ( 2 e + l ) ±*' ( 4 ' 7 b ) o o Thus i t follows that 9e E <V^ > = TTWuTT+TY Ji'Ji* ife ' ( 4 * 7 c ) o o and from equations (4.1) and (4.6) we have - 71 -4 ^ 4 * ( £ o + 2 ) M 4 * (e o+2)H <P±.e> Eo VE 3VE 3VE { * 8 ) o o where V Is the volume of the system and N i s the number of molecules. Substituting the value f o r <ui«e> from (4.7c) into (4.8) and rearranging we obtain (e - l ) ( 2 e +1) . v o o 4 up o , 2 where y = ^ L £ , (4.9b) p i s the molecular dipole moment and g i s known as the Kirkwood c o r r e l a t i o n parameter or g-factor. In order to determine a more e x p l i c i t expression for g, we follow a s i m i l a r derivation for the same system proposed by F r o h l i c h [87], It 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 given by <M'e> = <M2> (4.10) 2 where <M > i s the mean squared moment i n the absence of an applied f i e l d . It i s then easy to show that - 72 -g = <*!| = 1 + ^ ± < y 1 - M 2 > . (4.11a) Ny y Noting that we can integrate over the appropriate projection of the p a i r c o r r e l a t i o n function (see Appendix A) i n order to obtain the average c o r r e l a t i o n between dipole o r i e n t a t i o n s , we have [35,36] g = 1 + 4TT P /" h 1 1 0 ( r ) r 2 dr (4.11b) J d ss ss The Onsager expression [88] for the s t a t i c d i e l e c t r i c constant can be obtained from (4.9) by s e t t i n g g = 1. The Onsager formula w i l l , i n general, underestimate e Q since g > 1 for systems where dipolar c o r r e l a t i o n s strongly influence the o r i e n t a t i o n a l structure. It can be shown [97] that the d i e l e c t r i c constant can be equivalently obtained through the l i m i t 11? ( £ n _ 1 ) h (r) •»• - as r •> ». (4.12) ss , -5 4ire pyr o ~112 This expression i s derived by examining the k-K) l i m i t of h g g (k). We note that i n i n t e g r a l equation theories equations (4.9) and (4.12) must give the same r e s u l t . In the LHNC and QHNC theories, the above relationships for e Q have been shown [35] to give good agreement with the values f o r e Q determined by computer simulation for dipole-quadrupole systems. The theories become less accurate, overestimating e Q , as y becomes large r e l a t i v e to Q. We can use (4.9) to calculate the d i e l e c t r i c constant - 73 -fo r the current solvent model even though we have allowed the solvent molecules to be p o l a r i z a b l e . This 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 are a l l absorded into the e f f e c t i v e dipole moment, nig, and e Q i s then determined for the e f f e c t i v e system. We now consider a solvent system into which there has been added a f i n i t e concentration of solute p a r t i c l e s which are assumed to be non-polarizable. It i s possible [25,26] to write the solute-solute component of the general OZ equations describing solute-solute c o r r e l a t i o n functions as e f f e c t i v e OZ equations of the form, h i j ( k ) - c ^ f f ( k ) - E p a h l a ( k ) ~cf (k), (4.13) where the sum i s only over solute species. We can then determine [21,35] e E by considering the e f f e c t i v e solute-solute d i r e c t c o r r e l a t i o n ef f function, c ^ (r) (which includes a l l solvent e f f e c t s ) , as defined by ef f (4.13). c ? r(r) can be related [21,25] to the e f f e c t i v e solute-solute ef f p a i r p o t e n t i a l , u^. ( r ) , by c e . ( r ) + -Su.. (r) * — - i - l as r + ~. (4.14) As mentioned e a r l i e r ( c f . equation (4.5b)), (4.14) can serve as - 74 -~ef f d e f i n i t i o n of E g . Equation (4.13) can be solved for c ^ and the k 0 ~ef f l i m i t of c ^ ( k ) i s determined by examining the k 0 behavior of the c o r r e l a t i o n functions contributing to i t . From (4.14) we have that 'eff „ , ~ ^ ± ^ ( I F 1 -A- -* c ± , k) »  f  as k •»• 0, (4.15) E K which i s then used to obtain an expression r e l a t i n g e^ , and g. It i s found [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 (p^ = p.. =0) the Kirkwood expression, (4.9) and (4.11), i s again obtained with £g = e Q. This constitutes a microscopic derivation of the Kirkwood r e l a t i o n s h i p . At f i n i t e concentrations (p^ > 0), however, the Kirkwood r e l a t i o n s h i p does not hold because of the Debye screening of the long-range dipole-dipole c o r r e l a t i o n s . In t h i s case, one obtains [21] -3 yg = y (1 + j P s h g s ) (4.16) where h * 1 0 ^ i s the k •>• 0 l i m i t . Thus (4.12) and (4.9) w i l l only hold i n the i n f i n i t e d i l u t i o n l i m i t . £g i s also related [21] to the ion-ion pair c o r r e l a t i o n function by £ E = 4 7 r 3 ^ . P i p j q i q j h i j } ( 4 a 7 ) - 75 -where ~(2) 1 r°° 2 hij = V r ) r dr- (4-18) e„ should be given c o n s i s t e n t l y by both (4.16) and (4.17) i n the C i MSA, LHNC, and QHNC theories. (b) The Solute Dependent D i e l e c t r i c Decrement At low concentrations, e„ = e p can be expressed [64] by the expansion ep = eo + I E(l)i pi + ( 4* 1 9 ) where p - Ep^ and i s concentration of i o n i c species i . Friedman [86] has proven that the leading term of t h i s expansion Is l i n e a r i n p^. It i s t h i s l i n e a r term which w i l l dominate at low ef f concentrations. By examining and formulating a theory for c ^ ( s i m i l a r to that used i n determining equation (4.17) for e E; i . e . working with the OZ equation at the McMillan-Mayer l e v e l ) , Friedman [86] has formally shown that as p •> 0 ( V 1 ) 2 ~iio £ ( l ) i = - 9 y — P s 6 i c s s ( 4 ' 2 0 a ) where - 76 -- "110 r 3 "110,. _ n. . o.c = [-5— c (k)]. „ _ (4.20b) i ss 3p^ ss k=0, (^ ,=0 and p s i s the number density of the solvent. Applying c l u s t e r expansion methods, he then proved 6 that i n the HNC theory * 110 l i m i t -110,, x ,, 0 1 x Vss = k - 0 F ( k ) ' ( 4 * 2 1 a ) F ^ ^ ( r ) i s the (110) projection of the function F(12) = / h s s ( 1 2 ) h i s ( 3 2 ) h s i ( 1 3 ) d r 3 (4.21b) where p a r t i c l e s 1 and 2 are solvents and p a r t i c l e 3 i s an ion. We note that F(12), and hence £(i)i» depends on the solvent-solvent c o r r e l a t i o n convoluted with solvent-ion and ion-solvent c o r r e l a t i o n s over a l l positions of the ion. F(12) w i l l thus depend not only on the solvent-solvent pair c o r r e l a t i o n s , but w i l l also depend on the c o r r e l a t i o n between two solvent molecules that exists through ( i . e . depends on) the ion present i n s o l u t i o n . Since h s s , h^ s, and h s ^ are the appropriate c o r r e l a t i o n functions at i n f i n i t e d i l u t i o n , we should then be able to evaluate the c o e f f i c i e n t s , ^ ( i ) ± t and hence determine the l i m i t i n g behavior of e p i n the present study. The r e s u l t s of Friedman are correct only to this point [23] and subsequent re l a t i o n s h i p s are given i n Ref. 23. - 77 -Using the d e f i n i t i o n [23] ( s i m i l a r to (3.21b) and (3.33)) F U ° ( r ) = ^ / F(12) $ 1 1 0 ( 1 2 ) dn.dn9, (4.22) (8rr r we can rewrite (4.21a) as S. c 1 1 0= 3-^- T / h (12)h. (32)h ,(13) $ 1 1 0 ( 1 2 ) dX.dX0dX_ (4.23) l ss y^g^Z^J ss i s s i —1 —l —3 where V i s the volume of the system. Following Patey and Carnie [23], we can prove that V « = -i—•* / % o -x / dndaiQ $ 1 1 0 ( i 2 ) h c(n l fn k) ( 2 T T ) J ( 8 T I ) ss i z -x h±8(n2,k) hsl(fl1,k), (4.24) where we have taken the ion, which i s l a b e l l e d as p a r t i c l e 3, as our o r i g i n . In equation (4.24), h ag(&i,fl2,k) i s just the Fourier transform of h a g (fti,&2>r_) and we note that h^ g and h g ^ do not depend on the ori e n t a t i o n of the ion. As before, we now expand h , h. , and h , S S I S S X i n terms of r o t a t i o n a l i n v a r i a n t s . Then by sub s t i t u t i n g these expansions i n (4.24) and performing the angular integrations we obtain - 78 ,-, "110 . „ 110_Jc.0£.0&' £' MNL , 101.,0 V V w 1XM W 1£'N . Vss = MMT f f f f ( 0 0 0 ) ( 00 0 )(0m-m)(0m'-m') MNL U'mm' x [ ZX Ux0 ) (X-X0 ) (-X0X ) (X0-X ) ] Imm' ' ( 4 ' 2 5 ) where we have used standard properties [73] of the 3-j symbols and Wigner matrix elements and use the d e f i n i t i o n I , = — r r Jn k h . (k) h. n , (k) h . ~(k) dk. (4.26) mm' _ 2 J0 ss.-m-m' is.Om' si.mO The constant f a c t o r s , f™1"'-, of equation (4.25), which r e s u l t from our mnl d e f i n i t i o n of , are given by equation (2.12). In general, the c a l c u l a t i o n of 6. c ^ ^ would appear to be a very X s s d i f f i c u l t task. However, for the present model, many s i m p l i f i c a t i o n s are possible. The orthogonality of the r o t a t i o n a l invariants greatly l i m i t s the possible non-zero terms of (4.25). For the dipole-tetrahedral quadrupole solvent, there i s also some c a n c e l l a t i o n of terms. We obtain * 110 _ T00110 . 1 T11000 0 7 x Vss = ^ 0 + T I 0 0 ( 4 ' 2 7 ) which i s also the expression found [23] for a dipolar solvent. Using equation (4.26), we have 79 -T00110 1 r°° , 2 r, 000,, N 12 ,110,, >, ,, ,, O Q . x o o = 7 T 'o k [ h i s ( k ) 1 h s s ( k ) d k ( 4' 2 8 a ) 2TT and T11000 -1 rm , 2r,011,, N 12 ,000,, N O Q u N -Inn = — y Jo k l h 4 „ ^ k ) j h „ ( k) d k « (4.28b) UO „ I '0 i s ss 2TT IQQ^^ can be viewed as e s s e n t i a l l y an e l e c t r o s t a t i c term since i t depends on h ^ g l ( k ) , which i t s e l f w i l l depend on the charge-dipole i n t e r a c t i o n between the ion and the solvent. S i m i l a r l y , IQQ**^ can be viewed as being predominately a packing term since i t depends of the ion-solvent r a d i a l d i s t r i b u t i o n , which w i l l depend on the packing within the system. Some care must be used i n the determination of IQQ^^ since the integrand of (4.28a) w i l l remain f i n i t e as k -*• 0. In the present theory, i t can be shown [21] (by examining, i n part, the l i m i t i n g behavior of c^**) that e -1 h 0 1 1 ( k ) > 4Tr[3q.u ( °_) i as k * 0 (4.29) i s I 3y EQ k where i Hence, the small k behavior of the integrand i s given by - 80 -. 2 " O i l , , . . 2 "000,. . ,. . .2 , _fb~* 2 J000, n . , . o n . k [ h i s ( k ) ] h s s ( k ) * < 4 7 r e V } <3^> h s s ( 0 ) ' ( 4 ' 3 0 ) We note that h^^^(k) i s well behaved i n the l i m i t k-H). This a n a l y t i c ss term for the value of the integrand i n the l i m i t k-K) has been found [23] for the present model to be large and negative i n value. In fact the small k behavior tends to dominate the i n t e g r a l . The value of obtained, using (4.27) and (4.28), for a dipolar solvent system was found [23] to agree very well with the l i m i t i n g slope of £g from f i n i t e concentration r e s u l t s [21], 3. Dynamical Theory of the D i e l e c t r i c Constant The apparent d i e l e c t r i c constant of an e l e c t r o l y t e s o l u t i o n , e o r. T = € „ + £ , as measured by experiments, has been found to be the sum of both equilibrium and dynamical terms. The analogue of the s t a t i c d i e l e c t r i c constant, e E , defined by microscopic theories, can not at present be r e a d i l y measured experimently. The dynamical contributions to the apparent d i e l e c t r i c constant a r i s e because measurements must be made using o s c i l l a t i n g e l e c t r i c f i e l d s . In p r i n c i p l e one could determine d i r e c t l y i f one could measure the ion-ion structure factors using neutron or X-ray s c a t t e r i n g experiments (c f . equation (4.18)). Unfortunately these experimental probes remain [6] t e c h n i c a l l y d i f f i c u l t to carry out. A knowledge of e n would not - 81 -only f a c i l i t a t e the determination of from the measured quantity e , but would also improve the understanding of dynamical O U L J processes taking place In sol u t i o n . The dynamical contribution to the d i e l e c t r i c constant of a s o l u t i o n , £Q, arises from the d e f i n i t i o n of EgQL (4.4) i n which the imaginary term of the complex conductivity i s assumed to be zero i n the zero-frequency l i m i t . At any f i n i t e frequency, there w i l l be a phase lag between a (GO) and the applied e l e c t r i c f i e l d and t h i s lag remains f i n i t e even i n the zero-frequency l i m i t . It i s t h i s non-zero value of the imaginary term of o (u) as u + 0 which appears as a dynamical term o f eSOL' The Debye-Falkenhagen theory [15] was the f i r s t theory to deal with the frequency dependent properties of an e l e c t r o l y t e s o l u t i o n . The DF theory was o r i g i n a l l y developed to explain the frequency and concentration dependence of the molar conductivity, A, of e l e c t r o l y t e s o l u t i o n s . It proved very successful [15] i n explaining the frequency and concentration dependence of conductivity data at low concentrations. At low concentration, the zero-frequency l i m i t of the molar conductivity, A 0, was shown to obey A = A + A c 1 / 2 (4.31) O oo where Aoo i s the molar conductivity at i n f i n i t e d i l u t i o n , c i s the solute concentration, and A i s a constant dependent on the nature of the e l e c t r o l y t e . The DF theory also predicted a d i e l e c t r i c increment now - 82 -c a l l e d the DF e f f e c t . The increase i n the d i e l e c t r i c constant at low concentration i s predicted to have a /c dependence. Measurements at low concentration [15,79,80] have shown both the concentration and frequency dependence predicted by the DF theory. The DF theory was developed [3] by examining how the complex 7 conductivity i s affected by the relaxation of the i o n i c atmosphere surrounding a c e n t r a l ion. The DF e f f e c t w i l l thus depend, i n part, on ion-ion i n t e r a c t i o n s . In the formulation of the DF theory, the i n f i n i t e d i l u t i o n l i m i t was assumed i n defining the i n t e r a c t i o n between an ion and i t s i o n i c atmosphere. The o r i g i n a l authors had predicted that the DF theory would hold up to concentrations of about 10 equivalence per mole and conductivity measurements [15] have supported t h i s claim. The DF e f f e c t i s most evident at low concentrations. Its -/c 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 low concentration. At higher concentrations the DF e f f e c t i s usually Q ignored since the DF e f f e c t i s only understood i n the low concentration l i m i t , although i t i s assumed that i t s /c" behavior i s v a l i d at f i n i t e concentrations. Recently a second dynamical contribution to the d i e l e c t r i c constant has been predicted [82,84] I t i s known as the k i n e t i c d i e l e c t r i c decrement (KDD). The KDD i s i n e f f e c t an analogue to the DF e f f e c t that The d i s t r i b u t i o n of ions, equal and opposite i n charge to the c e n t r a l ion, that surrounds a central ion. Q Also, at the higher frequencies generally used [6,80] i n measuring eS0L a t h i S n e r concentrations, we assume the DF e f f e c t w i l l not be evident. The relaxation 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 occur. - 83 -ar i s e s from solvent-ion i n t e r a c t i o n s . Called 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 by the o r i g i n a l authors [84], i t was f i r s t examined within the p r i m i t i v e model framework. A microscopic theory f o r the KDD has also been developed and examined [82] for ions i n water. Theories f o r the KDD predict a l i n e a r decrease i n the d i e l e c t r i c constant as a function of the conductivity of the so l u t i o n . At low concentration, to a f i r s t approximation ( c f . 4.31), a = cAa>« Thus the KDD predicts a decrease i n the apparent d i e l e c t r i c constant of an e l e c t r o l y t e solution which, for d i l u t e solutions, 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 [80,83] to account for the observed behavior of the d i e l e c t r i c constants of aqueous a l k a l i halide solutions at low concentrations e n t i r e l y i n terms of the two dynamical e f f e c t s mentioned above. However, the re s u l t s are far from conclusive. Although the measured r e s u l t s appear to have been c a r e f u l l y obtained, there i s substantial scatter i n the data points, i n d i c a t i n g that the accuracy of the measurements was not s u f f i c i e n t to follow the r e l a t i v e l y small e f f e c t s at low concentration. Also, van Beek [80] seemed unable to accurately reproduce (when compared with the r e s u l t s of Wien [79]) the frequency dependence of the DF e f f e c t . The increases i n the measured d i e l e c t r i c constants at very low concentrations were found to be larger than that predicted by the DF theory (unless one ignored a l l other known e f f e c t s that should s t i l l be present at these concentrations). The observed decrease [80] i n the measured d i e l e c t r i c constant, a f t e r removal of the DF term, was expressed as a l i n e a r function of cr, although the scatter i n the data points leaves much - 84 -ambiguity i n t h i s operation. Moreover, the slope predicted by continuum theory [83] for the KDD i s only about one-third that of the o r i g i n a l slope obtained from the experimental data for aqueous so l u t i o n s . The r e s u l t s of van Beek [80] w i l l be discussed i n more d e t a i l i n Chapter V. The KDD i s also a l i n e a r function of T d the d i e l e c t r i c (Debye) rel a x a t i o n time [6,75], i n continuum and microscopic theories. One would expect then that the KDD would be more evident, or even a dominant Q f a c t o r , for polar solvents with long relaxation times. This conclusion seems to be consistent with recent results for solutions of ions i n methanol [89] and for s u l f u r i c acid [90]. The relaxation times for the KDD of d i l u t e aqueous and methanol solutions have been found [89] to be the same order of magnitude as (approximately equal to at low concentrations) the solvent d i e l e c t r i c r e l axation times. The proximity of the relaxation times would further complicate the behavior observed i n Cole-Cole diagrams 1 0 and would help to explain such experimentally observed behavior which remains poorly understood. There i s s t i l l much to be learned, both through theory and experiment, about the concentration and frequency dependence of the KDD. Solvents consisting of large polar molecules would have longer d i e l e c t r i c r e l axation times. Their size would r e s t r i c t t h e i r o r i e n t a t i o n freedom and hence they would take longer to orientate with an applied f i e l d ; e.g. Water - - 9ps, methanol - Tn ~ 48ps, H 2S0 l t - T d = 400ps (Ref. 89,90). 1 0 P l o t s [95] of e"(oj) against e'(co) are frequently used by experimentalists [6,75] to determine d i e l e c t r i c constants and d i e l e c t r i c r e l a x a t i o n times. - 85 -(a) The Debye-Falkenhagen E f f e c t The theory of Debye and Falkenhagen [91,92] b u i l t on the i n i t i a l work of Debye and Huckel [93] and of Onsager [94]. This early work attempted to provide 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 solution (using only a continuum model approach) subjected to a s t a t i c e l e c t r i c f i e l d . An ion i n s o l u t i o n w i l l be caused to move by such a s t a t i c f i e l d . I t s motion w i l l not only be r e s i s t e d by the f r i c t i o n a l force due to the solvent, but also by two factors due to the i o n i c atmosphere surrounding an ion and the long-range e l e c t r o s t a t i c forces that exist between ions. The f i r s t e f f e c t , known as electrophoresis, i s an a d d i t i o n a l f r i c t i o n a l force. It r e s u l t s from the motion of the oppositely charged i o n i c atmosphere i n a d i r e c t i o n opposite to that of the ion. This motion tends to drag the solvent along with i t , thus increasing ordinary resistance to motion. The second e f f e c t r e s u l t s from the motion of the ion. At equilibrium, the charge d i s t r i b u t i o n around a central ion i s i s o t r o p i c . However, an Ion moving with 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 front of i t , while behind i t , the charge d i s t r i b u t i o n must relax. The assymmetry of the i o n i c atmosphere w i l l increase with increasing ion v e l o c i t y . This assymmetry gives r i s e to a retarding force, sometimes c a l l e d the relaxation force. Both e f f e c t s are proportional to the square root of i o n i c concentration. Debye and Falkenhagen [91,92] extended t h i s approach to an e l e c t r o l y t e s o l u t i o n placed i n an a l t e r n a t i n g e l e c t r i c f i e d . At low frequencies the assymmetry of the i o n i c atmosphere w i l l depend on the instantaneous v e l o c i t y of the ion as i t moves to and fro i n response to the applied f i e l d . If however, the period of o s c i l l a t i o n i s comparable - 86 -to (or smaller than) the c h a r a c t e r i s t i c relaxation time for the i o n i c atmosphere 1 1 then p r a c t i c a l l y no assymmetry in the charge d i s t r i b u t i o n can be set up. Thus the relaxation force disappears almost completely. It i s th i s frequency dependent relaxation force which gives r i s e to a dispersion i n the conductivity and p e r m i t 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 , the Debye-Falkenhagen e f f e c t . An expression f o r the mean v e l o c i t y was found [91,92] by requiring that under stationary conditions, the four forces (those considered above) acting on an ion i n solu t i o n must sum to zero. In obtaining t h i s expression several approximations were necessary, many assuming conditions present only i n the i n f i n i t e d i l u t i o n l i m i t . Debye and Falkenhagen [91,92] showed that the t o t a l current density, J , must equal the sum of the conduction and the displacement current, T l ^ t , ico ,, s„ iwt 0 0 N J = aEe + e(io)Ee , (4.32a) where E i s the e l e c t r i c f i e l d strength. They [15] expand to obtain 2 o . „ itot . j _ 2 , q l q 2 _ 2 _ n j q j i _ icot N J = — icoEe + {En.q.co + ~—:-=• K En.q.tox - £ 7 J | } Ee (4.32b) 4TT . 1 1 3ekT . 1 1 6TTTI J J J J o J J o where n Q i s the solvent v i s c o s i t y , K i s the r e c i p r o c a l Debye radius of Although concentration d e p e n d e n t , the relaxation time for the i o n i c atmosphere (" 20 ns for a 10" M aqueous solution) i s several orders of magnitude longer than the solvent d i e l e c t r i c r e l axation time. the i o n i c atmosphere (proportional to / c ) , and x i s a function which re l a t e s ion v e l o c i t y to the f i e l d frequency, OJ. The function x i s given by X(b,oj0 ) /b{(l + i(Q0 )1 / 2 - l / / b | 1 + iO)0 - l//b (4.33) where 9 i s the relaxation time of the i o n i c atmosphere (given by equation (5.35) of Ref. 15) and b i s a dimensionless parameter ( c f . equation (5.36) of Ref. 15) depending on the charge and equivalent i o n i c 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 present. The r e a l part of (4.32b) can be rearranged to obtain an expression for the frequency dependent s p e c i f i c conductivity, CT( OJ) , while the imaginary part represents an extra contribution to the d i e l e c t r i c constant of the s o l u t i o n . General formulae for the conductivity and the apparent increase i n d i e l e c t r i c constant have been derived [15]. It i s convenient to define the dimensionless quantities R = _ 1 [(1 + a > 2 0 2 ) 1 / 2 + 1] 1/2 (4.34a) /2 and Q = — [(1 + c o 2 © 2 ) 1 / 2 - l ] 1 ' 2 . (4.34b) /2 The apparent increase i n the d i e l e c t r i c constant for a very d i l u t e s o l u t i o n , as described by Debye and Falkenhagen [15], i s given by - 88 -4 7 T q l q 2 e ( a , ) " £o = JTYT e /b{Q(l ~ 1/b) - io9 (R - l//b)} O)0{(1 1/b) 2 + co 2© 2} (4.35) where Oc i s the s p e c i f i c conductivity at i n f i n i t e d i l u t i o n . This increase w i l l be proportional to /c (concentration dependence coming through K) and i t s frequency dependence has been studied. The increase i n d i e l e c t r i c constant i s greatest f o r co = 0. It i s the increase i n the d i e l e c t r i c constant given by (4.35) i n the zero-frequency l i m i t that i s usually known as the DF e f f e c t . In the l i m i t co + 0 we can show that Q -> co0/2 and R ->- 1, and hence l i m i t ; , . i 4 i r q l q 2 0 /b • * * - . • o ^ - ' J - % - -^r- 2 ( 1 + u r b ) 2 H-36> Thus for a given e l e c t r o l y t e at constant temperature T, we can write f o r very d i l u t e solutions A e D p = A/c (4.37) where A i s a constant depending upon the temperature and the natures of the solvent and e l e c t r o l y t e . However, at ordinary temperatures values of A for d i f f e r e n t e l e c t r o l y t e s are available i n tables [2,15], (b) 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 developing theories to study the motions of ions i n a continuum solvent, i d e n t i f i e d s p e c i f i c dynamical - 89 -mechanisms which r e s u l t i n an apparent reduction i n the d i e l e c t r i c constant of a s o l u t i o n . 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 i t s basis i n how ion-solvent i n t e r a c t i o n s w i l l dynamically a f f e c t the p o l a r i z a t i o n of the solvent. Two apparently d i s t i n c t , but c l o s e l y r e l a t e d , e f f e c t s were found. As an ion migrates through the solution under the influence of an applied external f i e l d , i t tends to cause solvent molecules i n i t s v i c i n i t y to rotate i n the opposite d i r e c t i o n to that favoured by the applied f i e l d . Although d i e l e c t r i c relaxation tends to restore the solvent p o l a r i z a t i o n appropriate to the l o c a l f i e l d , t h i s relaxation process lags behind the f i e l d by the d i e l e c t r i c relaxation time. The second e f f e c t r e s u l t s from the retardation force experienced by an ion due to the p o l a r i z a t i o n of the solvent. As solvent molecules rotate i n response to the applied f i e l d over a period equal to the d i e l e c t r i c r e l axation time, an ion i n s o l u t i o n w i l l 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 solution to an applied a l t e r n a t i n g e l e c t r i c f i e l d . Approximations for the KDD have been obtained only for d i l u t e solutions. However, unlike the DF e f f e c t , we would expect the KDD to disappear only at very high frequencies ( i . e . as oo ->- 1 / T D ) . The KDD has been shown [82-84,96] to be proportional to the product of the d i e l e c t r i c relaxation time, x D, of the solvent and the low frequency conductivity, a, of the s o l u t i o n . For a continuum solvent, the KDD has been expressed [82] (assuming s l i p boundary conditions between the ion and solvent) by - 90 -Ae, KDD e — e -8ir , o °°N — ( — — } V (4.38) which d i f f e r s s l i g h t l y from the expression of the o r i g i n a l authors [84], Felderhof [96], also using the continuum model, has recently made a further refinement to this r e s u l t , arguing that not a l l the assumptions of the o r i g i n a l authors hold. His expression d i f f e r s only s l i g h t l y from (4.38) and must give smaller values for the decrement. An expression for Ae^-p ^ias also been derived using molecular theory (which requires the s o l u t i o n of two coupled Langevin equations). In t h e i r approximate theory, Hubbard et. a l . [82] obtained . v -8TT f a i T D i „ ST . — * — . KDD - J — h 7 ~ } s r i < 2~~ > ( 4 ' 3 9 ) i i r where the sum i s over a l l ion species, i , i s the conductivity due to species i , N g i s the number of solvent molecules, r S T represents Stokes r a d i i obtained assuming s l i p boundary conditions [82], and the angular brackets denote an equilibrium average. Within the equilibrium average, j£ represents a solvent dipole vector and _r the unit vector i n d i c a t i n g the d i r e c t i o n from an ion to the solvent molecule. We can then write = y cos6 = u$ 0 1 1(12). (4.40) and by d e f i n i t i o n - 91 -> = 2 2 /// g l s ( 1 2 ) dr d f l j d ^ (4.41a) where V i s the sample volume. Then expanding g£ S(12), using the r e l a t i o n s h i p s (3.23) and (4.40), and s i m p l i f y i n g (again u t i l i z i n g the orthogonality of r o t a t i o n a l invariants) we have y • r < ^—f1 > ^ - 2 — /// h j 8 1 ( r ) [ $ ° 1 1 ( 1 2 ) ] 2 dr d^dO, (4.41b) r (8TT ) V r and i t i s not d i f f i c u l t to show that t h i s further s i m p l i f i e s to ^ ' - N 4TT y f» O i l , . 2~~ ~3 V •'d i s ^ (4.41c) r i s We now r e c a l l that the value of the i n t e g r a l of (4.41c) i s proportional to the average charge-dipole energy ( c f . equation (3.38b)). It immediately follows that < — > = ITqTV T • ( 4' 4 1 d ) r s i I Inserting t h i s r e s u l t into (4.39), we obtain - 92 -KDD 8TT 3 (4.42) Values f o r the KDD are r e a d i l y obtained i n the present c a l c u l a t i o n since we can view equation (4.42) as depending only upon the average charge-dipole energy, and other values can be treated as system parameters. It i s important to note that two d i f f e r e n t i o n i c r a d i i are used i n the c a l c u l a t i o n of Ae^^. The Stokes radius enters the expression through the dynamical theory of Hubbard et. a l . [82] and the c r y s t a l radius enters i n the c a l c u l a t i o n of UQQ/N^. This greatly e f f e c t s the behavior of AG^QQ as a function of ion size for small ions, although the r a d i i are roughly equal for larger ions. In the l i m i t of i n f i n i t e l y large ions, we can use the long-range asymptotic form of h?** ( c f . Ref. 35) i n order to determine U.^ /N., i s CD i a n a l y t i c a l l y ( c f . equation (3.38b)). If we then take the high frequency d i e l e c t r i c constant to be one, we obtain the continuum re s u l t as given by equation (4.38). - 93 -CHAPTER V RESULTS AND DISCUSSION 1. Introduction In Chapter II we have defined a model p o t e n t i a l for an e l e c t r o l y t e s o l u t i o n . In Chapter III we have derived the LHNC approximation for the system characterized by t h i s p o t e n t i a l at i n f i n i t e ion d i l u t i o n . Then using the s t a t i s t i c a l mechanical theory introduced i n Chapter I I I , we have described, i n Chapter IV, dynamical and equilibrium contributions to £gOL f ° r a n i n f i n i t e l y d i l u t e e l e c t r o l y e s o l u t i o n . In the present chapter we w i l l discuss the r e s u l t s obtained from solving the LHNC theory for the present model so l u t i o n . Both s t r u c t u r a l and d i e l e c t r i c properties and t h e i r dependence on ion size and charge w i l l be examined. In the present study temperature dependence was not investigated. The temperature dependence of Ae p and $w^j(r) have been previously examined for equal diameter 1:1 e l e c t r o l y t e solutions by Patey and Carnie [23], We w i l l , however, make comparisons, as did Patey and Carnie, [23] between water-like and dipolar solvents. Our discussion w i l l focus mainly on a l k a l i halide s a l t solutions although r e s u l t s were obtained for a wide range of diameters for both univalent and divalent ions. In our c a l c u l a t i o n s and in our discussion i t i s convenient to express a l l parameters i n reduced u n i t s . The present so l u t i o n model at i n f i n i t e d i l u t i o n can be t o t a l l y characterized by the reduced parameters - 94 -P*s - P gd 3, d* = d l / d s , y* = (V/dJ)172, Q* = (3Q2/dJ)1/2 and q? = (3q 2/d ) 1 / 2 where 3 = 1/kT. A solvent diameter of d c = 2.8A and l i s => a temperature of 25°C are used throughout t h i s study. We use the values of Qr and y = me as described i n Chapter I I . For the water-like solvent these parameters give e 0 = 77.47 which i s close to the experimental value, e Q = 78.5, for water at 25°C. For the dipolar solvent examined, the dipole moment and density have been adjusted such that the t h e o r e t i c a l d i e l e c t r i c constant i s very close to that of water at 25°C. Table II contains the reduced parameters for the water-like and d i p o l a r solvents. The diameters assigned to the a l k a l i metal and halide ions were discussed i n Chapter II and are summarized i n Table I. * The reduced charge, = 14.1527(z^) where z^ i s the valency associated with the ion. Table I I . Reduced parameters and d i e l e c t r i c constants for the water-like and dipolar solvents at 25°C with d s = 2.8A. Solvent * * p s P * Q T e o Water-like 0.732 2.751 0.94 77.47 Dipolar 0.800 1.5 0 78.5 For the present model the res u l t s obtained for ions of the same size but opposite charge are i d e n t i c a l i n a l l cases except for the - 95 -ion-solvent c o r r e l a t i o n functions h ^ g and which change i n sign with ion charge. Detailed numerical r e s u l t s for a l l ions studied are presented i n Appendix C. 2. Energies and Coordination Numbers Table I I I summarizes the reduced energies of i n t e r a c t i o n experienced by an ion i n the water-like model. We note that since the sol u t i o n i s at i n f i n i t e d i l u t i o n there i s no ion-ion Interaction energy and the ion-solvent i n t e r a c t i o n energies are for a single ion. The average ion-solvent energy and i t s charge-dipole and charge-quadrupole Table I I I . Energies of fo r a l k a l i i n t e r a c t i o n and metal and halide coordination ions. numbers - 3 U C D / N i -BU C Q/ N i -3U I S/N. C.N. L i + 358.7 73.24 431.9 8.97 Na +, F - 316.2 54.09 370.3 10.38 K+ 267.9 35.92 303.8 12.33 Rb +, C l " 254.9 31.71 286.6 13.15 Cs +, Br~ 237.7 26.52 264.2 14.40 I - 218.0 21.19 239.2 16.25 components were calculated using equations (3.38). We f i n d that as the ion size increases the average charge-quadrupole i n t e r a c t i o n decreases much more r a p i d l y than the charge-dipole i n t e r a c t i o n . This behavior i s - 96 -due simply to the d i f f e r e n t r dependence of the charge-quadrupole and charge-dipole potentials (U CQ i s proportional to 1/r 3 whereas u C D i s proportional to 1/r ). These trends i n the energies of i n t e r a c t i o n w i l l prove useful i n discussing s t r u c t u r a l features. 2 It was also found that the average energies scale with z^ (to the accuracy we have ca r r i e d our c a l c u l a t i o n s ) regardless of ion s i z e . This i s a very i n t e r e s t i n g r e s u l t since through equations (3.38) i t indicates O i l 022 that the i n t e g r a l s over h. and h. must scale with the charge of the i s i s ion . A further discussion of t h i s s c a l i n g behavior i s given below. Also included i n Table III are the ion-solvent coordination numbers (C.N.) calculated using n XT / fR 2 000, v , /c i x C.N. = 4TTP J r g. (r) dr (5.1) d^ i s where R i s the f i r s t minimum i n the r a d i a l d i s t r i b u t i o n function. In 000 HS the LHNC theory g^T (r) = g?g(r). Thus the coordination number of an ion depends only on i t s size and not on i t s charge. Ion-solvent i n t e r a c t i o n s do not influence g ? ^ ( r ) i n the LHNC theory. However, t h i s 1 s i s not seen as a s i g n i f i c a n t drawback to the present study since our system i s at r e l a t i v e l y high density. For a high density hard-sphere 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 have a larger influence on the o v e r a l l packing structure. Inclusion of the ion-solvent i n t e r a c t i o n e f f e c t s would cause the f i r s t minimum of the r a d i a l d i s t r i b u t i o n function to s h i f t i n , and at the same time cause i t s contact value to increase. These e f f e c t s tend to cancel and so the value of the - 97 -coordination number remains r e l a t i v e l y unchanged. As we would expect, the C.N. were found to increase with ion s i z e . However, closer inspection of values reveals that the C.N. do not follow a smooth curve as a function of ion 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 of Mean Force C o r r e l a t i o n functions describe the structure of a system. In the present study the ion-ion c o r r e l a t i o n s are fixed by the ion-solverit and solvent-solvent c o r r e l a t i o n s . The solvent-solvent c o r r e l a t i o n s are those of the pure solvent [41] since the solution i s at i n f i n i t e d i l u t i o n . The Ion-solvent pair c o r r e l a t i o n function, h^ g(12), defined i n the present theory by equation (3.23), contains only three terms, u000 .011 . u022 A _ _ , J „ h ^ g , h ^ g , and h ^ g . As discussed e a r l i e r , the s p h e r i c a l l y symmetric part, h ^ ^ , i s , i n the LHNC theory, just the hard-sphere r e s u l t and so 1 s not of great i n t e r e s t . The p r o j e c t i o n h^ g^ has the same symmetry as the charge-dipole i n t e r a c t i o n and so e s s e n t i a l l y describes that part of the ion-solvent c o r r e l a t i o n . In Figure 5 we have plotted h^ g^ for L i + and ~h^ g* for - . . . , .011 ,011 - ,022 I . As mentioned above, for the present model h + g = "h_ g and h + g = 022 -h_ g . It i s clear from the figure that although the magnitude of the s t r u c t u r a l features i s greater for smaller i o n s , i n d i c a t i n g t h e i r greater a b i l i t y to order solvent p a r t i c l e s , the pattern of the o s c i l l a t i o n s i n h^ g^ i s r e l a t i v e l y independent of ion s i z e . Also, we see in Figure 5 that the very rapid r i s e of the functions near contact i s a prominent feature. - 98 -Figure 5 Comparing h ? g * ( r ) for L i + and I - ions i n the water-like solvent model. The s o l i d curve i s h 0 1 1 ( r ) for L i + and the dashed curve i s - h 0 1 1 ( r ) i s i s f o r I - . The ion diameters are those given i n Table I. Note that two scales are used i n the p l o t . - 100 -Figure 6 Comparing h ? 2 2 ( r ) for L i + and I - ions in the water-like solvent model. is The 022 022 s o l i d curve i s -h, (r) for Li"1" and the dashed curve i s h. (r) for i s i s I - . The ion diameters are those given i n Table I. Note that two scales are used i n the p l o t . - 102 -022 The projection h e s s e n t i a l l y describes the charge-quadrupole part 1 s 022 + of the ion-solvent c o r r e l a t i o n . Figure 6 compares - h + g (r) for L i 022 - 022 and h (r) for I . We fi n d that not only does h. (r) have the same -s i s general o s c i l l a t o r y patterns as h ^ ^ ( r ) , but i t s dependence upon ion si z e i s also very s i m i l a r to that of h ^ ^ ( r ) . The contact values of 022 h^ g are again observed to be the most prominent features. For divalent ions i t was found that for the diameter range investigated the anisotropic ion-solvent c o r r e l a t i o n functions are, to the numerical accuracy obtained, simply twice those of the univalent ion of the same diameter. This s c a l i n g with charge i s i n i t i a l l y somewhat su r p r i s i n g since p h y s i c a l l y i t means that for the present model and theory the o r i e n t a t i o n a l structure i s e f f e c t i v e l y saturated for univalent ions. As a d i r e c t r e s u l t we see that the i n t e g r a l s over h?** i s 022 and h must also scale with ion charge, and hence from equation (3.38) so w i l l the average ion-solvent energies of i n t e r a c t i o n . The s c a l i n g behavior of the ion-solvent c o r r e l a t i o n functions i s not unexpected f o r the present model and theory at i n f i n i t e d i l u t i o n . We 011 022 know that both u^ g (r) and u^ g (r) w i l l scale with charge and 000 HS g^ g = g^ g i s always independent of q. Then from equations (3.45), , r mn£ , , , , mn£ , , , mn£, N i f n scales with charge, c. must also scale. We note that c. (r) i s ° i s i s w i l l always scale for larger r. From equations (3.42) we see that Ti? g^(k) and n9g2(k) are simply l i n e a r combinations of cjjg^(k) and - 103 -022. ' O i l j 022 . . . . O i l c. (k). Thus i f both c, and c, scale with charge, then n. and xs xs xs xs 022 h^ likewise must scale. Therefore, for the present model and theory there should exist a solution which w i l l scale with the i o n i c charge. This cannot be the case at f i n i t e concentrations or i n higher order theories, but i s true only for the LHNC treatment of the present model at i n f i n i t e d i l u t i o n . However, although the h m n ^ ( r ) w i l l not scale for X S small r i n higher order theories, they must always scale with charge for large r. Therefore, we would expect the d i e l e c t r i c properties and ion-ion structure predicted by the LHNC theory not to d i f f e r g r e atly from that predicted by the QHNC or HNC theory. Furthermore, i t has been shown [22] that for an i n f i n i t e l y d i l u t e solution of hard spherical ions immersed i n a dipolar solvent the p o t e n t i a l of mean force calculated in the LHNC theory i s i n good agreement with Monte Carlo computations. Potentials of mean force as defined by (3.39) describe the ion-ion pair 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 for ions of the same s i z e , 3w++(r) = gw ( r ) . Therefore, i t i s only necessary to know whether the p o t e n t i a l of mean force i s between a l i k e or an unlike ion p a i r . Potentials of mean force spanning the a l k a l i metal and halide ion diameter range for l i k e and unlike ion pairs are shown i n Figures 7 and 8, r e s p e c t i v e l y . It i s c l e a r l y evident i n both figures that both gwn and gw^ exhibit the most d i s t i n c t i v e s t r u c t u r a l features for p a i r s of r e l a t i v e l y small ions. This perhaps i s not a s u r p r i s i n g r e s u l t since we have already seen that the smaller the ion, the more strongly i t w i l l i n t e r a c t with solvent p a r t i c l e s . Thus smaller ions w i l l have - 104 -Figure 7 A comparison of 3w£^(r) for ions 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 for L i + / L i + , K +/K +, and T~/I~ p a i r s , r e s p e c t i v e l y , i n the water-like solvent. The ion diameters are those given i n Table I. - 105 -- 106 -Figure 8 A comparison of 3w^j(r) for d i f f e r e n t pairs of oppositely charged ions. The s o l i d , dashed, and dash-dot curves represent 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 water-like solvent. The ion diameters are those given i n Table I. - 107 -- 108 -l a r g e r ordering e f f e c t s upon the solvent-ion structure. This, i n turn, must influence how an ion pair w i l l i n t e r a c t . In Figure 7 we observe that for l i k e pairs of r e l a t i v e l y small ions the potentials are strongly repulsive at contact. They then very quickly drop to a f i r s t minimum at (r - d^£)/d s « 0.25 which, for smaller 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 ( i . e . negative i n s i g n ) . This i s a s u r p r i s i n g r e s u l t i n that i t gives a r e l a t i v e l y high p r o b a b i l i t y of f i n d i n g pairs of small l i k e ions at quite small separations. We also see that as ion size i s increased the repulsion at contact very r a p i d l y decreases and the f i r s t minimum s h i f t s to smaller separations. If the potentials of mean force of s t i l l larger ions are examined, the f i r s t minimum i s a c t u a l l y found to move right up to contact and the contact value becomes a t t r a c t i v e at a 1,8. We have from equation (3.40a) that the p o t e n t i a l must be the sum of an HS e l e c t r o s t a t i c term and a hard sphere packing term, -Jin g ^ ( r ) . The e l e c t r o s t a t i c term w i l l decrease i n magnitude with increasing ion size and the hard sphere term w i l l increase i n magnitude. For l i k e ion p a i r s * with d^ > 1.8, the negative hard sphere term dominates the value of 3w^(r) at contact. As shown i n Figure 8, 3w i ;j(r) for ion pairs of opposite charge i s strongly a t t r a c t i v e at contact and then quickly r i s e s to a f i r s t maximum at (r - d ^ j ) / d g » 0.35. We observe that as ion size increases the magnitude of 3w^j(r) at contact decreases. Although the f i r s t maximum decreases i n value with increasing ion s i z e , i t s p o s i t i o n remains r e l a t i v e l y unchanged. It i s i n t e r e s t i n g to note that for a pair of very - 109 -small unlike ions such as L i + / F - , the f i r s t maximum of the po t e n t i a l of mean force a c t u a l l y becomes repulsive. The behavior exhibited by the second minimum of Bwj_j(r) w i l l be discussed below. The p o t e n t i a l s of mean force for L i + / I - and C s + / F - are compared i n Figure 9. These two pairs are such that the sum of the ion diameters i s equal i n each case. The potentials are found to be almost i d e n t i c a l . This indicates that at least for the a l k a l i metal and halide ions, Bw^j(r) depends strongly on d^j but i s r e l a t i v e l y independent of the i n d i v i d u a l ion diameters. We would expect t h i s to break down, however, for a pair c o n s i s t i n g of a very small and a very large ion. In Figure 10 we have plotted 3 W ^ J / | Z ^ Z J | for a pair of univalent and a pair of divalent ions where a l l the ions have the same diameter as the solvent. We f i n d that to a f a i r approximation the p o t e n t i a l of mean force scales with product of the charge r a t i o s of both ions. This approximate s c a l i n g has been found to hold for both l i k e and unlike ion pairs where d^ _< d g . For larger ions the potentials do not scale so w e l l , p a r t i c u l a r l y near contact. This behavior i s explained by r e c a l l i n g that we can express the po t e n t i a l of mean force 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. The hard sphere term w i l l be unaffected by i o n i c charge i n the LHNC theory. We have found f o r the present model and theory that even f o r univalent ions the ion-solvent structure i s o r i e n t a t i o n a l l y saturated and so we would expect the e l e c t r o s t a t i c term to scale with i o n i c charge. For small ions the hard sphere term i s small and hence the po t e n t i a l of mean force appears to approximately scale with charge. However, f or larger - 110 -Figure 9 Comparing gw^j(r) for pairs of oppositely charged ions for which the d^A values are equal. The s o l i d and dashed curves represent 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 water-like solvent. The ion diameters are those given i n Table I. - Ill -CO o - 112 -Figure 10 Comparing 3 w ^ ^ ( r ) / I f ° r pairs of oppositely charged univalent and divalent ions. The s o l i d and dashed curves represent univalent and divalent ion p a i r s , r e s p e c t i v e l y , i n our water-like solvent. The ion diameters are a l l equal to the solvent diameter. - 113 -- 114 -ions the hard sphere term becomes important near contact and so scaling cannot occur. The p o t e n t i a l of mean force of ions i n a pri m i t i v e model solvent, a dipolar solvent and the present water-like solvent are compared i n Figures 11 and 12. The potentials of mean force for ions i n a dipolar solvent with parameters as given i n Table II are determined i n the same manner as those f o r ions i n the water-like solvent. The po t e n t i a l of mean force between two ions, a and 3, i n a PM solvent i s just the PM continuum l i m i t , 3w Q ( r ) = 3q q J e r, where e i s the solvent d i e l e c t r i c a3 o r 3 o o constant. This expression i s exact for large r and therefore 3w ag(r) PM, for any other solvent must approach aQ\r~) a s r + °°* It becomes obvious from both figures that the water-like f l u i d i s a "better" solvent than the dipolar system. For a given ion p a i r , l i k e or unlike, the magnitude of the pot e n t i a l of mean force at contact for the water-like solvent i s always much less than for the dipolar solvent. Also 3wag for the present water-like solvent i s much less structured and approaches the continuum l i m i t much more r a p i d l y . The dipolar solvent would be expected to be highly structured since only a small range of orientations of the solvent with respect to an ion are highly favoured e n e r g e t i c a l l y . Addition of a tetrahedral quadrupole moment would allow a larger range of angles to be favourable. The quadrupole of the water-like solvent thus must act to "soften" much of ion-solvent structure despite i t s large dipole moment. In Figure 11 we have plotted 3w^j for L i + / F ~ and C s + / I " i n the water-like and the dipolar solvents. For the dipolar solvent the po t e n t i a l i s always characterized by a deep well at (r - d..)/d = 1 . - 115 -Figure 11 A comparison of 3w^j(r) for pairs of oppositely charged ions i n d i f f e r e n t solvents. The s o l i d and short-dashed curves represent the C s + / I ~ pair i n the water-like and dipolar solvents, r e s p e c t i v e l y . The dash-dot, long-dashed, and dotted curves represent the L i + / F - pair i n the water-like, d i p o l a r , and continuum solvents, r e s p e c t i v e l y . - 116 -- 117 -Figure 12 A comparison of pV-^Cr) for F /F~ and I ' l l " i n d i f f e r e n t solvents. The s o l i d and short-dashed curves are for the I ~ / I ~ pair i n the water-like and dipolar solvents, r e s p e c t i v e l y . The dash-dot, long-dashed, and dotted curves are for the F -/F~ pair i n the water-like d i p o l a r , and continuum solvents, r e s p e c t i v e l y . - 118 -- 119 -This must correspond to the e n e r g e t i c a l l y very favourable structure of two oppositely charged ions separated by a solvent p a r t i c l e . 3WJ[J f o r L i + / F ~ i n the water-like solvent has i t s second minimum at (r - dj:j)/d s = 0.75 although i t does have a s l i g h t dip at 1.0. This second minimum appears to be a purely quadrupolar feature since i t does not appear i n 3w^j for the dipolar solvent and the ion-ion separation roughly corresponds to the separation of ions place at tetrahedral "corners" of a solvent molecule. The tetrahedral corners would correspond to s i t e s on the surface of the solvent molecule ( i . e . 109° apart) of most favourable charge-quadrupole i n t e r a c t i o n . No d e f i n i t e solvent structure around two ions at such a separation can be r e a d i l y assigned. For C s + / I ~ i n the water-like solvent the second minimum i n 3w-£j i s again at (r - d ^ j ) / d s = 1.0 corresponding to a solvent separated ion pair 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 of the charge-quadrupole i n t e r a c t i o n as the ion size i s increased. Hence, the second minimum for C s + / I ~ i s i n the same p o s i t i o n f o r both the dipolar and water-like solvents although i n the l a t t e r the quadrupole i s s t i l l acting to greatly dampen the structure i n the p o t e n t i a l of mean force. In Figure 12 we have plotted pwi±(r) for F~/F~ and I ~ / I ~ i n the water-like and dip o l a r solvents. For the dipolar solvent the po t e n t i a l i s characterized by a peak at (r - d i : L ) / d s = 1, which corresponds to the e n e r g e t i c a l l y unfavourable structure of two l i k e ions separated by a solvent molecule. No peak appears at t h i s separation for the water-like solvent. We observe that for F~/F~ i n both solvents the - 120 -f i r s t minimum of &w±± i s so deep as to be a t t r a c t i v e , more so for the dip o l a r solvent. This could possibly be a dipolar feature due to some solvent structure which acts to counteract the coulombic repulsion between the ions. However, there i s no obvious form for such a hypothetical structure. This feature may simply arise from the ions having to go somewhere, so they choose the least unfavourable separation. It i s important to bear i n mind that these are pot e n t i a l s of mean force at i n f i n i t e d i l u t i o n where the average solvent-solvent structure remains unchanged. I t would be i n t e r e s t i n g to see i f some of the above structures p e r s i s t at f i n i t e ion concentrations. For both molecular solvents the potentials of mean force near contact have been found to be very structured. C l e a r l y , the PM i s unable to represent the ion-ion c o r r e l a t i o n s for small separations. The continuum l i m i t i s approached very rapidly by the potentials of mean force for ions i n the water-like solvent. The continuum l i m i t i s e s s e n t i a l l y reached within three solvent diameters for most ions except i n the case of small multivalent ions which w i l l more r e a d i l y structure the solvent. Such r e s u l t s for 3w a R can perhaps explain why the PM can give good agreement with some experimental data for aqueous solutions [4,98].particularly i f the e f f e c t i v e ion diameters are increased by using solvent cospheres. In the PM framework the use of solvent cospheres i s a very common approach. Many researchers [7,18,24,75] have t r i e d to improve t h i s approach by defining a separation dependent d i e l e c t r i c constant to represent the solvent within the immediate v i c i n i t y of an ion. Many approximations have been put forward, including the assumption that the d i e l e c t r i c constant at the nearest - 121 -neighbor separation w i l l approach e<», the high-frequency l i m i t . However, the d i e l e c t r i c constant i s a macroscopic property of a system and can be defined by the pot e n t i a l of mean force between two ions at large separations, as given by (4.5b). At small separations one can no longer e f f e c t i v e l y use a d i e l e c t r i c constant to describe the e f f e c t of the solvent on i o n i c i n t e r a c t i o n s . For example, i f we t r i e d to use an r dependent d i e l e c t r i c constant to describe 3w^j at small separations f o r L i + / F ~ ( see Figure 11), we would find that t h i s d i e l e c t r i c constant must go through i n f i n i t y and then becomes negative at the separations where the p o t e n t i a l i s repulsive. Also, the behavior of t h i s d i e l e c t r i c constant would change with each ion. We see then that the usefulness of t h i s type of cor r e c t i o n to the PM solvent i s indeed l i m i t e d . It i s clear that i n order to study the short-range solvent 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 force which takes ion-solvent and solvent-solvent structure into account must be employed. The ion-ion c o r r e l a t i o n s within solution may also be investigated by examining the mean force acting between an ion p a i r . The force i s given by the negative d e r i v a t i v e of the p o t e n t i a l and i s generally a more d i r e c t l y interpretable physical quantity. From equation (3.40b) we have that i n the present theory f5wag i s the sum of a coulombic term and a solvent dependent term, ~ r ^ ^ ^ T ^ ' Thus the mean force between two ions, F a g ( r ) , i s given by - 122 --9 3 q a q 8 3 F a 8 ( r ) - 3F ( 3 w a 6 ( r ) ) = + 3 F s ( 5 ' 2 a ) where the f i r s t term i s the coulombic force which i s independent of the solvent and the second term, 6 Fs = il ( T iT ( r ) ) (5-2b) i s the force dependent on the ion-solvent and solvent-solvent structure. We note that by d e f i n i t i o n a negative force i s a t t r a c t i v e . The mean force acting between pairs of l i k e ions i s presented i n Figures 13, 14, and 15. We immediately notice that the t o t a l force i s the sum of two large terms which are opposite i n sign and so tend to cancel each other. The coulombic term i s always repulsive whereas the solvent dependent term i s always a t t r a c t i v e . The r e s u l t i n g suras have both p o s i t i v e and negative segments i n d i c a t i n g that ions w i l l experience forces which prefer s p e c i f i c i n t e r i o n i c separations. In Figures 13 and 14 we can compare P F ^ for T~/I~ and L i + / L i + ions, respectively, i n the water-like solvent. We f i n d that the mean forces have very s i m i l a r structure except near contact where the force between two L i + ions has a much larger r e l a t i v e value. Figures 14 and 15 compare 8F^^ for L i + / L i + i n the water-like and dipolar solvents, r e s p e c t i v e l y . We observe that the force exhibits much more structure for the d i p o l a r solvent. For the water-like solvent, the solvent dependent force term near contact 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 less repulsive - 123 -Figure 13 The force acting between two I - ions i n the water-like solvent. The dash-dot curve represents the mean force between the two ions. The dashed curve represents the repulsive coulombic component and the s o l i d curve represents the negative of the a t t r a c t i v e solvent dependent component of the mean force. - 124 -- 125 -Figure 14 ^ ions : curves are i d e n t i f i e d as i n Figure 13. The force acting between two L i + i n the water-like solvent. The - 126 -- 127 -Figure 15 The force acting between two L i + ions i n the dipolar solvent. The curves are i d e n t i f i e d as i n Figure 13. - 128 -- 129 -mean force near contact. As we would expect for the dipolar solvent, the force changes sign at (r - d-Q)/d s = 1. The mean forces acting between pairs of unlike ions are presented i n Figures 16, 17, and 18. The t o t a l force i s again the sum of two large terms with the coulombic part now always being a t t r a c t i v e and the solvent dependent term always repulsive. If we compare &F±j f ° r L i + / F - and C s + / I ~ i n the water-like solvent ( c f . Figures 16 and 17, r e s p e c t i v e l y ) , we fin d only some s i m i l a r i t y i n r e l a t i v e structure. Not only i s the U + / F ~ fo rce proportionally more a t t r a c t i v e at contact, but the f i r s t repulsive peak i s smaller i n width and r e l a t i v e l y more repu l s i v e . The dipolar structure at (r - dj_j)/d g = 1 i s much more apparent for Cs +/I~. These s t r u c t u r a l changes i n with ion diameter can again be a t t r i b u t e d to the decrease i n r e l a t i v e importance of the quadrupolar i n t e r a c t i o n . Figures 17 and 18 compare 3F^j for C s + / I - i n the water-like and dipolar solvents, r e s p e c t i v e l y . We again observe much more structure for the dipolar solvent. The contact value i s l e s s a t t r a c t i v e in the water-like solvent i n d i c a t i n g that i t i s a "better" solvent. However, the r e l a t i v e structure in the mean forces i s s i m i l a r with both solvents showing a change of sign at ( r - d£j)/d g = 1. Hence, we view the solvation of unlike ions i n the water-like model as becoming more dipolar i n nature as ion size i s increased. 4. The D i e l e c t r i c Constant of 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 constant of s o l u t i o n consists of both dynamical and equilibrium terms. The d i e l e c t r i c - 130 -Figure 16 The force acting between a L i + / F - pair i n the water-like solvent. The dash-dot curve represents the mean force between the oppositely charged ion p a i r . The dashed curve represents the a t t r a c t i v e coulombic component and the s o l i d curve represents the negative of the solvent dependent repulsive term. - 131 -- 132 -Figure 17 L pair curves are i d e n t i f i e d as i n Figure 16. The force acting between a Cs +/I i n the water-like solvent. The - 133 -- 134 -Figure 18 The force acting between a C s + / I ~ pair i n the dipolar solvent. The curves are i d e n t i f i e d as i n Figure 16. - 135 -- 136 -constant of an e l e c t r o l y t e 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 just that of the pure solvent. However, expressions have been developed for c a l c u l a t i n g the contributions of the equilibrium and dynamical terms to the l i m i t i n g slope of the d i e l e c t r i c decrement. Hence, in the present study we calculate the l i m i t i n g slopes and then compare these with experimental data at low concentration. In order to obtain the equilibrium contribution, Ae p, to the t o t a l d i e l e c t r i c decrement, we write equation (4.19) in the reduced form A eP = % - % = i e ( D i p * + • • • ( 5 - 3 a ) where ( D i ( D i ( £ o -9y D ' (I 00110* 00 1_ 11000*. 3 00 ; ' (5.3b) N* - „ ^ A TO0 110* _ T00110,,6 , . ., , , ,11000*. „ p. = p. d , and Inr. = I-,. /d ( s i m i l a r l y f o r I n _ ). We would l i s OU 00 s 00 l i k e to be able to estimate decrements for r e a l e l e c t r o l y t e solutions which can be assumed to be completely di s s o c i a t e d . For a solution of a s a l t of the general form AaB^)Cc... we have N d 3 10 i ' ^ s - 137 -where c s a i ^ t i s the concentration of the s a l t i n moles per l i t e r , i s Avogadro's number, d s i s the solvent diameter expressed i n angstroms and = a, b, c IQQ^^ and IQQ**^ are calculated using the expressions (4.28) and (4.30). The dependence of these quantities on ion diameter i s 00110* i l l u s t r a t e d i n Figure 19. We note that since IQQ depends on h ^ ^ and h ? ^ , i t w i l l be independent of ion charge i n the present SS I S +u AI A T n o o o * , . ,ooo . ,011 , , theory. Also, since depends on h and h, and because we 00 ss i s know h^ g^ scales with ion charge, we would expect IQQ^^ t o scale with z 2 (squared because h^ g^ i s squared). In Figure 19 we see that T11000*, 2 . , , A + * „ i u * +u + T 1 1 0 0 0 * - 1 1 00 i 1 S i n " e P e n " e n t °f z-£* W e again emphasize that IQQ w i l l depend only upon the magnitude but not upon the sign of the i o n i c charge for the present model. From Figure 19 i t can also be seen that IQQ^^ i s r e l a t i v e l y small and p o s i t i v e , although i t does become negative for very small ions and * 11000* becomes larger quickly for d^ > 1.3. IQQ i s r e l a t i v e l y large and 00110* negative and i s more structured than IQQ . It has a l o c a l maximum at * d^ K 0.85 from which i t drops very r a p i d l y with decreasing ion * diameter. Af t e r a shallow l o c a l minimum at d^ » 1.1 i t continues to 11000* become more p o s i t i v e with increasing ion s i z e . The structure i n IQQ may appear s u r p r i s i n g at f i r s t , but i t can be explained by considering two contributing factors which vary with ion s i z e . As the size of an ion i s increased we know that the ion w i l l be surrounded by an - 138 -Figure 19 The terms contributing to e ( 1 ) i f ° r t n e water-like solvent as a function of ion diameter. The s o l i d curve represents ~IQQ /Z^ f o r univalent ions and the s o l i d dots are r e s u l t s for divalent ions. The dashed l i n e 00110* represents IQQ which i s independent of i o n i c charge i n the present theory. Note that two scales are used i n the p l o t . - 140 -increasing number of neighboring solvent molecules, but the strength of the charge-dipole interactions between the ion and these solvent molecules must decrease. These two competing factors w i l l tend to o f f s e t each other to some extent. We tested t h i s explanation by attempting to recalculate values of IQQ^^ for several ions, a. 11000* IQQ for the ion, 3 , with d^ = d s was m u l t i p l i e d by the r a t i o of C.N.a/C.N.g and by 1/s , where s i s an "average" separation for the nearest solvent molecules around the ion. We were able to roughly reproduce the behavior exhibited by *QQ^^* f o r 0.7 < d < 1.3. * * In Figure 20 we have plotted eQ\.|_ a s a function of d^ for * univalent ions i n the water-like solvent. It can be seen that i s negative ( i n d i c a t i n g that e p w i l l decrease with increasing ion concentration) and decreases i n magnitude with increasing ion diameter, a c t u a l l y becoming p o s i t i v e for d^ > 1.9. For a l l but large ions, 11000* * IQQ dominates the value of e ( j ^ > a s I s obvious from the behavior * * of e(i)i« I t : i s i n t e r e s t i n g to note that e ( ^ ) ^ I s nearly constant i n the reduced ion diameter range 0.8 to 1.2 which encompasses many of the * ' ( D i * a l k a l i metal and halide ions. For larger Ions such as I , e,.. begins to show a marked increase because IQQ^^ i s decreasing i n magnitude 00110* and IQQ i s beginning to have a s i g n i f i c a n t a f f e c t . I t i s found [23] that, for ions s i m i l a r i n size to the solvent, * £ ( l ) i * S a-*- m o s t o n e o r < l e r of magnitude more negative for the dipolar - 141 -Figure 20 The dependence of e Q ) ^ o n ion diameter for univalent ions the water-like solvent. - 142 -- 143 -00110* solvent. In the dipolar solvent IQQ i s s t i l l r e l a t i v e l y unimportant 11000* f o r ions of t h i s s i z e , however, IQQ i s much larger (by a factor of 3). The other large contributing factor i s y (proportional to u 2) which appears i n the denominator of equation (5.3b). It i s about 3 times smaller due to the smaller dipole moment of the dipolar s o l v e n t 1 with e Q = 78.5. In order to calculate the k i n e t i c d i e l e c t r i c decrement, Ae^^, we rewrite equation (4.42) i n the form A eKDD • A ( E (1T LC ) V i ^ C s a l t ( 5 ' 5 a ) Ae KDD „ . i D where A e ^ D D ' / T D a i i-s t n e dimensionless r a t i o given by . KDD ST __ In obtaining (5.5a) we have used the fact that at low concentrations we can write °i = c s a l t V i  X°i  ( 5 ' 5 c ) It has been shown [35,38,41] that for equal values of u and p, e Q i s much larger for a purely dipolar system than i f a quadrupole i s included i n the model. - 144 -where X^ i s the molar i o n i c conductance at i n f i n i t e d i l u t i o n . If we express in ST-1 m - 1 then T A = 4 i r P g = 0.7415 ton (5.5d) o where £ 0 i s p e r m i t t i v i t y of free space and we have used [99] — 1 2 T = 8.25 x 10 s. We again note that s l i p boundary conditions have been assumed i n obtaining equations (5.5b). KDD ST In Figure 21 we see that the r a t i o (Ae^ ^ T D a i ^ r i ^ r i ' w h i c h depends only on the i o n i c c r y s t a l radius, r ^ , i s a smooth continuous function of ion diameter with the correct large d^ l i m i t i n g behavior. This r a t i o i s independent of ion charge as would be expected since U^D/N^ scales with charge. Values of A e ^ D / f o r the present study and from the previous r e s u l t s of Hubbard et. a l . [82] are also shown i n Figure 21. We observe that the values of ^^^1°^^ a r e v e r v dependent on the somewhat ar b i t a r y choice of the Stokes r a d i i . In the present c a l c u l a t i o n we have used the Stokes r a d i i given i n Ref. 100 corrected for s l i p boundary conditions by multiplying by 3/2. The comparison between the present and previously calculated values produces f a i r agreement, p a r t i c u l a r l y f or larger ions. The largest discrepancy occurs for the smallest ion compared, L i + . This agreement i s quite encouraging since the values were obtained using approximate methods which are quite d i f f e r e n t i n o r i g i n . - 145 -Figure 21 The dependence on ion diameter of the k i n e t i c d i e l e c t r i c decrement for the water-like solvent. The s o l i d l i n e represents ( A e ^ ^ a ± T Y ) ^ T ± ^ T ± for univalent ions and s o l i d dots are re s u l t s for divalent ions. Results for ^ E^VO^T^ i n the present theory for several p o s i t i v e and negative ions are given by open c i r c l e s and s o l i d squares, respectively, while the stars represent (from righ t to l e f t ) previous r e s u l t s of Hubbard et. a l . [82] for L i + , Na +, C l ~ , Cs +. - 146 -24.0 20.0 16.0 12.0 8.0 o 4.0 / Li+NaVRl> CsT F" CI" Br - ( A e T / o i T D ) ^ '(STR 0 J I I L 0 .0 2.0 _l L 3.0 di - 147 -In the present study we define the t o t a l d i e l e c t r i c decrement Ae = Ae p + A e ^ j j . In Figure 22 we compare for L i C l , KC1, CsCl solutions values of Ae, Ae p as given by equation (5.4), and Ae^Q obtained using l i t e r a t u r e values [4] of X° as given i n Table IV. These three s a l t s give quite s i m i l a r r e s u l t s as i s found to be the case for a l l a l k a l i halides. It i s clear from Figure 22 that the value of Ae for any aqueous a l k a l i halide solution i s dominated by Ae p. This t h e o r e t i c a l p r e d i c t i o n seems consistent with previous experimental data [80,83] although i t contradicts the theories of these authors. Table IV. Molar ion conductances [4] at i n f i n i t e d i l u t i o n at 25°C. Cation Anion X°_ L i + 38.68 F~ 55.4 Na + 50.10 CI" 76.35 K + 73.50 Br" 78.14 Rb+ 77.81 I" 76.84 C s + 77.26 In Figures 23 and 24 we compare t h e o r e t i c a l and experimental r e s u l t s f o r d i l u t e solutions of 1:1 and 2:1 e l e c t r o l y t e s , r e s p e c t i v e l y . The decrement for CuCl2 i s observed to be about four times larger than those obtained for a l k a l i halides at the same s a l t concentration. In 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 have taken the radius of Cu2"*" to be 0.72A. The experimental points from Ref. 80 represent - 148 -Figure 22 A comparison of the equilibrium and dynamical contributions to the t o t a l d i e l e c t r i c decrement. Theoretical r e s u l t s for Ae p, As^p, ari<l Ae f o r solutions of L i C I , KC1, and CsCl i n the water-like solvent are presented. The ion diameters are those given i n Table I. - 149 -I I I I I I I  0 0.02 0.04 0.06 CsALT (moles/litre) - 150 -Figure 23 Comparing experimental and t h e o r e t i c a l d i e l e c t r i c decrements for aqueous 1:1 e l e c t r o l y t e solutions. The open squares, open c i r c l e s , and s o l i d dots represent experimental r e s u l t s , A e e X p t , as obtained from Ref. 80 using equation (5.6a), for KC1, L i C l , and NaCl solutions, r e s p e c t i v e l y . The stars are experimental r e s u l t s obtained from Ref. 81 using equation (5.6b) for a KC1 sol 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), Ae p, and Ae, are for the water-like solvent model. The dashed l i n e represents the equilibrium contribution to the d i e l e c t r i c decrement for a KC1 s o l u t i o n using the present dipolar solvent model. - 151 -0.0 0.05 0.1 CsALT ( m o l e s / l i t r e ) - 152 -Figure 24 Comparing experimental and t h e o r e t i c a l d i e l e c t r i c decrements for aqueous CuCl2 solutions. The s o l i d dots represent experimental r e s u l t s , A e e X p t , as obtained from Ref. 80 using equation (5.6a). The t h e o r e t i c a l r e s u l t s , Ae^nn* Ae p, and Ae, are for the water-like solvent model. 0 0.02 0.04 0.06 0.08 C s A L T ( m ° ' e s / H t r e ) - 154 -A £ e x p t = eSOL ~ £o ~ A £DF ( 5' 6 a> where A e n p i s given by equation (4.37). The experimental points from Ref. 81 represent A £ e x p t = £SOL - £o ( 5 ' 6 b ) where we assume that these data points do not contain a DF contribution since they were obtained using high frequency techniques. As noted i n Chapter IV, we see that i n both Figures 23 and 24 the experimental points of van Beek [80] are a l l sh i f t e d upward. If we assume that his experimental points obey a l i n e a r law (ignoring the two points at lowest concentration for L i C l ) , we find that the experimental and t h e o r e t i c a l slopes agree quite well. However, the experimental l i n e would not extrapolate to zero. It i s obvious from both plots that Ae i s too small to alone account for the d i e l e c t r i c decrement observed. The experimental r e s u l t s of Weiss et. a l . [81] indicate a smaller slope for the d i e l e c t r i c decrement. Although these r e s u l t s (of which only the two lowest concentration points are shown) form a nice smooth curve which extrapolates to zero, we must be conscious of the fact that they were obtained using high frequency techniques with only two frequency measurements made for each point. The nearly i d e a l behavior of these r e s u l t s could e a s i l y be only an a r t i f a c t of f i t t i n g techniques. We would expect r e l a t i v e l y large errors i n these r e s u l t s e s p e c i a l l y f o r the points at lowest concentration. Although van Beek - 155 -[80] predicted r e l a t i v e l y modest errors on his points, the scatter evident i n the d i e l e c t r i c constants for most of the s a l t s he studied appears to exceed his predicted e r r o r s . There also remains the problem of t r y i n g to account for the added d i e l e c t r i c increment which appears i n h i s data for both 1:1 and 2:1 e l e c t r o l y t e solutions. At present there i s no t h e o r e t i c a l basis to support t h i s a d d i t i o n a l increment and i t may well be due to experimental problems. Despite the current lack of "good" experimental data for d i e l e c t r i c constants of aqueous solutions at low concentrations, i t would appear that the present solution model i s i n reasonable agreement with the r e s u l t s which do e x i s t . This i s c l e a r from both Figures 23 and 24. In Figure 24 we have included the d i e l e c t r i c decrement predicted by the dipolar model for a KC1 s o l u t i o n . A purely dipolar solvent model obviously greatly overestimates the d i e l e c t r i c decrement for aqueous so l u t i o n s . - 156 -CHAPTER VT CONCLUSIONS In t h i s thesis we have examined model aqueous e l e c t r o l y t e solutions at i n f i n i t e d i l u t i o n using the LHNC theory. A po l a r i z a b l e hard-sphere f l u i d with embedded point dipoles and tetrahedral quadrupoles with water-like parameters was used to model the solvent into which hard sphe r i c a l ions were immersed. Results obtained with this water-like solvent model were compared with those given by a purely dipolar solvent as well as with the pr i m i t i v e model. The LHNC theory was used to obtain ion-solvent c o r r e l a t i o n functions for e l e c t r o l y t e solutions i n the i n f i n i t e d i l u t i o n l i m i t . These c o r r e l a t i o n functions were then used to determine average ion-solvent i n t e r a c t i o n energies, ion-solvent coordination numbers, and ion-ion potentials of mean force. The mean force acting between ions was also examined. Furthermore, the theory for both equilibrium and dynamical contributions to the apparent d i e l e c t r i c constant of sol u t i o n has been discussed. An expression for the solute dependent dynamical contribution, A eKDD' defined by Hubbard et. a l . [82], was evaluated i n the present theory i n the i n f i n i t e d i l u t i o n l i m i t . Values for A e ^ ^ and Ae p were calculated using the ion-solvent c o r r e l a t i o n functions and both of these terms, as well as the t o t a l d i e l e c t r i c decrement, were compared with experimental r e s u l t s for 1:1 and 2:1 aqueous e l e c t r o l y t e s . The dependence on ion size and charge was of primary concern i n our i n v e s t i g a t i o n of the s t r u c t u r a l and d i e l e c t r i c properties of model - 157 -aqueous e l e c t r o l y t e solutions. The ion-solvent c o r r e l a t i o n functions are found to be quite s e n s i t i v e to ion size and charge. However, i n the LHNC theory these c o r r e l a t i o n functions are found to simply scale with the ion charge, i r r e s p e c t i v e of ion diameter. This behavior indicates that even for univalent ions, the ion-solvent o r i e n t a t i o n a l structure i s e f f e c t i v e l y saturated. We have noted that t h i s charge dependence i s not unexpected for the present model i n the LHNC theory and i t i s also cle a r that t h i s exact " s c a l i n g " could not occur at f i n i t e ion concentrations or i n Oi l 022 "higher order" theories. The s c a l i n g i n h^ g and h^ g leads to sc a l i n g i n other functions and properties of the system. The ion-ion potentials of mean force demonstrate a strong dependence upon diameter and t h e i r charge dependence d i f f e r s for ions of varying s i z e . For small ions the potentials of mean force scale to a f a i r approximation with the product of the ion charges whereas for larger ions t h i s behaviour i s not observed. For ions i n the water-like solvent, 8w^j exhibits highly dampened o s c i l l a t o r y structure, approaching the continuum l i m i t much more rap i d l y than for ions i n a purely d i p o l a r solvent. The water-like f l u i d i s found to be a "better" solvent, reducing to a greater extent the a t t r a c t i o n between two unlike ions at small separation. Quadrupolar in t e r a c t i o n s are found to play a very important role i n reducing the structure i n 8w^j and the po t e n t i a l s of mean force f o r r e l a t i v e l y small ions show d i s t i n c t quadrupolar features. As ion size increases, the average charge-quadrupole energy decreases much more rap i d l y than the average - 158 -charge-dipole energy and the potentials of mean force become more dipo l a r i n nature. However, the quadrupolar in t e r a c t i o n s continue to dampen purely dipolar features. Comparisons of the mean forces lead to s i m i l a r conclusions. We emphasize that i n the p r i m i t i v e model, 3w^j has none of these s t r u c t u r a l features which depend upon ion-solvent c o r r e l a t i o n s . For the present model and theory, Ae p i s found to be negative for small ions, but increases with ion size to a c t u a l l y become p o s i t i v e for larger ions. I n t e r e s t i n g l y , the equilibrium contribution 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 constant for a l k a l i metal and halide ions. The values calculated for the k i n e t i c d i e l e c t r i c decrement, A eKDD' a r e ^ o u n ^ t o b e ^ n reasonable agreement with the previous c a l c u l a t i o n s of Hubbard et. a l . This dynamical contribution to the t o t a l d i e l e c t r i c decrement i s also negative, although i t was found to be only about one-third the size of the equilibrium term for aqueous a l k a l i h a lides. It was possible i n t h i s study to make comparisons between the d i e l e c t r i c decrements predicted by the present theory and those measured experimentally. I t i s found that the t h e o r e t i c a l d i e l e c t r i c decrements calculated for aqueous a l k a l i halide solutions are r e l a t i v e l y i n s e n s i t i v e to ion s i z e . The l i m i t i n g slopes for 1:1 and 2:1 e l e c t r o l y t e s from experimental data [80,81] at low concentrations are not inconsistant with those predicted i n this study. At low concentrations, however, the uncertainties i n the experimental points are s u f f i c i e n t l y large so as to make the trends observed for i n d i v i d u a l a l k a l i halide solutions e s s e n t i a l l y i n d i s t i n g u i s h a b l e . Also, a l i n e a r - 159 -extrapolation of the experimental data of van Beek [80] to zero concentration does not give the d i e l e c t r i c constant of pure water, even though we have attempted to correct for the DF e f f e c t . The magnitude of the DF e f f e c t i s only known in the i n f i n i t e d i l u t i o n l i m i t . Its true behavior at f i n i t e concentrations i s unknown, although i t has i n the 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 be used up to concentrations of at least 10 equivalence per mole. The behavior of the data of van Beek [80] remains a puzzle and deserves further experimental and t h e o r e t i c a l i n v e s t i g a t i o n . More accurate measurements of the d i e l e c t r i c constants of aqueous e l e c t r o l y t e solutions at low concentration are needed not only to confirm or refute the measurements of van Beek [80], but also to further test the t h e o r e t i c a l r e s u l t s presented i n t h i s t h e s i s . The r e s u l t s of the present study may be applied to many other areas. The average ion-solvent i n t e r a c t i o n s can be used i n the c a l c u l a t i o n of solvation energies for ions i n the present water-like solvent. The c a l c u l a t i o n of solvation energies i n a purely dipolar solvent has already been ca r r i e d out [72] with somewhat s u r p r i s i n g r e s u l t s and the a p p l i c a t i o n of t h i s method to the water-like solvent i s c u r r e n t l y taking place. The average ion-solvent energies at i n f i n i t e d i l u t i o n are also presently being used to calculate heat capacities for aqueous e l e c t r o l y t e solutions at high temperatures. The potentials of mean force obtained have been applied to dynamical theory. They have been used i n dynamical c a l c u l a t i o n s [101,102] which predict NMR spin r e l a x a t i o n times for diamagnetic paramagnetic ion p a i r s i n s o l u t i o n . These re l a x a t i o n times can be measured experimentally and are very - 160 -se n s i t i v e to the structure i n the potentials of mean force ( i . e . to the ion-ion c o r r e l a t i o n s ) . These comparisons provide a good test of both dynamical and equilibrium theories. Recent r e s u l t s [102] have shown good agreement between theory and experiment. It has also been found that i f the pri m i t i v e model p o t e n t i a l of mean force i s used, p a r t i c u l a r l y for unlike ions, very poor agreement i s obtained. A further study of e l e c t r o l y t e solutions could be attempted using a "better" theory for the present solution model. In the QHNC or HNC theory we would not expect the ion-solvent c o r r e l a t i o n functions to scale with ion charge since the solvent packing w i l l now depend on the ion-solvent i n t e r a c t i o n s . A f i n i t e concentration c a l c u l a t i o n could be performed using a McMillan-Mayer [103] l e v e l theory. This approach ef f involves using an e f f e c t i v e ion-ion i n t e r a c t i o n p o t e n t i a l , u ^ , which would attempt to include a l l solvent e f f e c t s . For such a c a l c u l a t i o n , ef f t h i s u^j could be taken as a p o t e n t i a l of mean force for the present model at i n f i n i t e d i l u t i o n . The present solution model could also be studied at f i n i t e ion concentration by extending the theory presented i n th i s t h e s i s . This would represent a "true" f i n i t e concentration c a l c u l a t i o n , unlike the McMillan-Mayer approach, and would serve as a test of the McMillan-Mayer theory. Although the theory and c a l c u l a t i o n would be much more complicated, the re s u l t s of such a study could be re a d i l y compared with experimental data. At f i n i t e concentration there i s a wealth of ava i l a b l e experimental data for thermodynamic and d i e l e c t r i c properties with which to compare. The present model could be improved by taking into account the p o l a r i z a t i o n of the solvent by an ion i n s o l u t i o n . We would expect t h i s to increase the degree of - 161 -o r i e n t a t i o n a l saturation of the solvent around an ion. Also, the sol u t i o n model could be altered to allow ions to have dipole and/or quadrupole moments (as i n the case of CN~). F i n a l l y , comparisons should be made with computer simulations to ensure that the theory for the present model has not broken down. Many of the above extensions are currently being examined. In t h i s study i t has been shown that for aqueous e l e c t r o l y t e solutions 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 to describe the general q u a l i t a t i v e behavior of the apparent d i e l e c t r i c constant of solut i o n at low concentrations. Including the solvent as a true molecular species i n the model solutions i s found to greatly influence the short-range ion-ion c o r r e l a t i o n s predicted. 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We fin d i t convenient to use the cartesian representation A A A [104,105] of $ m n ^ where x,, y., z, denote the molecule-fixed unit ' uv i i i vectors of molecule i (defined for the present model in figure 2.2) and r^2 i s t n e unit vector i n the d i r e c t i o n of the segment j o i n i n g molecules 1 and 2. Thus using the shift-up (down) operators given i n [104,105] one obtains * 0 0 0 ( 1 2 ) = 1, $ (12) = z 2 • r 1 2 , $ (12) = - z x • z 2 , 112 $ (12) = 3(zj^ • r 1 2 ) ( z 2 • r 1 2 ) - ( z x • z 2 ) , - 169 -*022 ( 1 2 ) = / 6 [ ( x 2 . ^ ^ 2 - ( y 2 • r 1 2 ) ], A A A A A * 2 1 1(12) =4[xl * z 2 ) ( x l ' r12 } " ( y l ' Z2 } ( y l ' r12 ) ]' * 2 1 3(12) = /6{-5[Xl • r 1 2 ) * - (y x • r 1 2 ) * ] ( z 2 • r 1 2 ) A A A A A A A +2[xx • z 2 ) ( x 1 • r 1 2 ) - (yl • z 2) (y x • r 1 2 ) ] } , A A A A A A A A A A A A A A * 2 2 0(12) = -2[ X l • x 2 ) 2 + ( 7 l • y 2 ) 2 - (x L • y 2 ) 2 - ^  • x 2 ) 2 ] , $ 2 2 2(i2) = - 2 { [ ( x 1 - x 2 ) / + ( y ^ ) * - ( v y 2 ^ ~ ( y r * 2 ) z ] ^ [ ( x j ^ • x 2 ) ( x 1 T 1 2 ) ( x 2 T 1 2 ) + ( y 1 *y 2)(y 1 * rl2^ y2* r12) - ( x 1 « y 2 ) ( x 1 T 1 2 ) ( y 2 T 1 2 ) - ( y 1 •x 2)(y 1 T 1 2 ) ( x 2 » r 1 2 ) ] }, A A A A A A and A A AA AA A A * 2 2 4(12) = | { 3 5 [ ( X l - r 1 2 ) 2 - ( y i T 1 2 ) 2 ] [ ( x 2 - r 1 2 ) 2 - ( y 2 - r 1 2 ) 2 ] -20[(xj * x 2 ) ( x 1 • r 1 2 ) ( x 2 « r 1 2 ) + (y1 *y 2) (y x T 1 2 ) ( y 2 * r 1 2 ) - ( x 1 « y 2 ) ( x 1 « r 1 2 ) ( y 2 T 1 2 ) - (y1 »x2)(y1 « r 1 2 ) ( x 2 * r 1 2 ) ] AA A A AA AA + 2 [ ( X l - x 2 ) 2 + ( y 1 - y 2 ) 2 - ( V y 2 ) 2 " (y !* x 2 ) 2 n. - 170 -APPENDIX B ITERATIVE PROCEDURES Two i t e r a t i v e procedures for obtaining solutions were attempted i n the present study. A modified i t e r a t i v e method, which i s discussed below, was used as the a c t u a l l y working method. We have also examined how one might apply a recently developed method [68] of the Newton-Raphson (NR) type, which had appeared to have promising c h a r a c t e r i s t i c s . 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 [68] to obtain quickly converging solutions i n the study of i o n i c f l u i d s . The method i s a c t u a l l y a hybrid of the t r a d i t i o n a l i t e r a t i v e scheme and of the usual NR technique for finding zeros of equations. This procedure requires that we c a l c u l a t e and invert a jacobian matrix and thus represents an extensive c a l c u l a t i o n . G i l l a n [68] had claimed two advantages to using t h i s method. It was found to greatly reduce the number of i t e r a t i o n s required to reach convergence and the method would v i r t u a l l y 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 point. Unfortunately, the NR method was not e a s i l y applied to the present system. The systems G i l l a n studied were defined using only i s o t r o p i c p o t e n t i a l s and so the generalized convolution of the OZ equation involves only Fourier transforms ( c f . equation (3.15)). For the present multicomponent system, i n which "(12) has been expanded i n terms of r o t a t i o n a l i n v a r i a n t s , the expressions become much more complicated, now inv o l v i n g Hankel transforms. As a r e s u l t each NR cycle represents a - 171 -very laborious numerical c a l c u l a t i o n , requiring about 5 times as many computational steps as i n a normal i t e r a t i v e cycle. For the present system the NR method probably represented a longer computation i n obtaining convergence. There also appeared to possibly be some technical d i f f i c u l t i e s i n the ap p l i c a t i o n of the NR method to the present system. Thus, we did not ac t u a l l y use the NR method i n the present study. A modified i t e r a t i v e method, diagrammatically represented i n Figure 25, was su c c e s s f u l l y used to obtain a l l solutions i n th i s study. The following i s a step-by-step explanation of t h i s procedure. Step (1): Input. The input step involves the r e t r i e v i n g of a previous s o l u t i o n consisting of a set of n^^Cr) which has been stored i s i n a computer f i l e from a previous c a l c u l a t i o n . In the case where no previous s o l u t i o n set 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 point although care i s required. The g ^ s ( r ) must be calculated or retrieved from a f i l e i f previously calculated. At t h i s time we also ca l c u l a t e any c o e f f i c i e n t s or constants, such as the a n a l y t i c a l piece of c ? ^ , which w i l l be needed, i s Step (2): Applying the LHNC Closure. The LHNC closure, as given by equations (3.45), i s applied to the current s o l u t i o n , represented by h m n j , to ca l c u l a t e c ? n A . The long-range term of c ? ^ i s not included old i s i s at t h i s point. Step (3): Hankel Transforms. The Hankel transforms of c m g ^ are performed by f i r s t i n t e g r a l transforming then Fourier transforming the functions. The i n t e g r a l transforms are defined by equations (3.46). - 172 -Figure 25. A diagrammatic representation of the i t e r a t i v e method used to solve the i n t e g r a l equations. - 173 -The integrations are performed using the trapezoidal r u l e . A Fast Fourier transform (FFT) subroutine package [106] which can transform two r e a l functions simultaneously [107,108] was used to evaluate the transforms defined by equations (3.47). Step (4): Applying the Reduced OZ Equation. A new solution set of n m n A i s calculated using (3.42). We note that before applying the 1 s reduced OZ equation we must include the a n a l y t i c a l piece of c ™ ^ . Step (5): Back Hankel Transforms. The back Hankel transforms are performed by f i r s t back Fourier transforming and then reversing the i n t e g r a l transforms. The back Fourier transform uses the same procedure as i n Step (3) where the integ r a t i o n i s now over k. The back i n t e g r a l 022 transforms are defined i n (3.46). When back i n t e g r a l transforming n^ g , the f i r s t ten points of the function are f i t t e d to a polynomial of degree <^  5 and the required integration performed a n a l y t i c a l l y . This improves the numerical accuracy near r = 0, Step (6): Test Convergence. The new solu t i o n set, n m n A , i s f w new compared with the current set, h m n j . The contact values ( r . = d ) are old i s used for comparison since we would expect them to converge most slowly. The convergence condition i s s a t i s f i e d when the old and new values d i f f e r by les s than 0.01% i n a l l cases. Step (7): Mixing. Direct i t e r a t i o n (replacing the current solution by the new solution) i s an unstable method and w i l l diverge unless the i n i t i a l t r i a l s olution i s "close" to the "correct" r e s u l t . It was found [109] that the solution can be forced to converge by mixing successive i t e r a t e s . In order to determine the new value of the current solution - 174 -we have used n m n * - ( l - a) n m n £ + a n m n * (B.l) new old where a = j expH^n^ - n o l d ) / n o l d | } . (B.2) The exponential function was used in (B.2) because i t represented a convenient choice. Also, the constant factor of 4 in equation (B.2) provides an "optimum" mixing parameter, a, for the present system. An upper l i m i t of 0.975 was set on the possible values of oc. The contact values of n ™ g £ were used to calculate ot. We note that a separate mixing parameter was calculated for each n ™ g £ at each i t e r a t i o n . This g r e a t l y improved the speed of convergence because each projection was allowed to converge independently, some projections converging more r a p i d l y than others. This method of mixing also proved very stable. We were always able to converge a solution even a f t e r r e l a t i v e l y large steps, up or down, i n diameter or charge. In a l l cases less than 150 i t e r a t i o n s were required to obtain a convergence of 99.99%. This represented less than 60 seconds of CPU time on an Amdahl 470 V/8 running under an MTS operating system. Step (8): Output. The output step involves the saving of the current solution set 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 - 175 -equilibrium system properties. These values include energies of • - j . . . T00110* T11000* i n t e r a c t i o n , coordination numbers, IQQ , IQQ , as well as functions such as the potentials of mean force. - 176 -APPENDIX C NUMERICAL RESULTS Tables V and VI summarize some of the numerical data obtained i n the present study. Table V contains the data for univalent ions and Table VI contains the data for divalent ions. The reduced average charge-dipole, 3U"cn/Ni» a n d average charge-quadrupole, 3^0/%, energies for a single ion were calculated using equations (3.38b) and 3.38c), r e s p e c t i v e l y . The reduced average ion-solvent i n t e r a c t i o n energies for a single ion, 3Uis/ Ni> were determined using equation ST (3.38a). The reduced k i n e t i c d i e l e c t r i c decrement, (Ae /a.T ) r . / r . , KJJD 1 JJ 1 1 was calculated using equation (5.5b). The reduced terms, IQQ^*^ and IQQ^^ , contributing to a r e given by equations (4.28). The coordination number, C.N., for a given ion was obtained using equation (5.1). Table V. Numerical Results for Univalent Ions. * d. 1 - P UCD N i N i -3 Uis N i " A W r i V T n ' ST i D r ^ 11000* 00 00110* 00 C.N. 0.44 449.4 122.6 572.0 4.14 -100.32 -0.54 7.32 0.68 358.7 73.24 431.9 5.10 -61.50 -0.10 8.97 0.84 316.2 54.09 370.3 5.55 -53.42 0.43 10.38 0.88 307.0 50.32 357.3 5.65 -53.32 0.55 -0.96 290.0 43.76 333.8 5.82 -54.36 0.76 -1.00 282.2 40.91 323.1 5.90 -55.06 0.86 11.55 1.08 267.9 35.92 303.8 6.05 -55.90 1.06 12.33 1.12 261.2 33.73 294.9 6.12 -55.87 1.21 -1.16 254.9 31.71 286.6 6.18 -55.52 1.39 13.15 1.20 248.9 29.85 278.8 6.25 -54.86 1.61 -1.28 237.7 26.52 264.2 6.36 -52.77 2.23 14.4 1.40 222.6 22.38 245.0 6.52 -48.51 3.54 -1.44 218.0 21.19 239.2 6.56 -46.99 4.07 16.25 1.56 205.2 18.09 223.3 6.69 -42.76 5.88 17.6 1.80 183.5 13.47 197.0 6.91 -37.41 10.18 -2.00 168.6 10.75 179.3 7.05 -35.55 14.57 -2.52 139.0 6.42 145.4 7.33 -29.6 — Table VI. Numerical Results f o r Divalent Ions. * d. 1 -*UCD N. l " 3 UCQ N i " 3 U I S N. l " A W r i V T_ ' ST i D r i 11000* 00 00110* 00 C.N. 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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