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The structural, thermodynamic and dielectric properties of electrolyte solutions : a theoretical study Kusalik, Peter Gerard 1987

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THE STRUCTURAL, THERMODYNAMIC AND DIELECTRIC PROPERTIES OF ELECTROLYTE SOLUTIONS: A THEORETICAL STUDY By PETER GERARD KUSALIK B.Sc, The University of Lethbridge, 1981 M.Sc, The University of British Columbia, 1984 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Chemistry) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1987 ® Peter Gerard Kusalik, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) - ii -ABSTRACT In traditional theories for electrolyte solut ions the solvent is treated only as a dielectric continuum. A more complete theoretical picture of electrolyte solut ions can be obtained by including the solvent as a true molecular species. In this thesis we report results for the structural, thermodynamic, and dielectric properties of model electrolyte solut ions which expl ici t ly include a water- l ike molecular solvent. The ions are modelled s imply as charged hard spheres and only univalent ions are considered. The water- l ike solvent is also treated as a hard sphere into which the low-order multipole moments .and polarizabil ity tensor of water are included. The reference hypernetted-chain theory is used to study the model systems. The formal ism of Kirkwood and Buff is employed to obtain general expressions relating the microscopic correlation functions and the thermodynamic properties of electrolyte solut ions without restricting the nature of the solvent. The low concentration l imiting behaviour of these expressions is examined and compared with the macroscopic results determined through Debye-HiJckel theory. The influence of solvent polarizabil i ty is examined at two theoretical levels. The more detailed approach, the R-dependent mean f ield theory, a l lows us to consider the average local electric f ield experienced by a solvent particle as a function of its separation from an ion and is shown to have an effect upon the limiting laws of some thermodynamic properties. Model systems for liquid water are investigated over a large range of temperatures and pressures and are found to have dielectric constants which agree reasonably wel l with experiment. Model aqueous electrolyte solutions are studied both at infinite dilution and at f inite concentration, but only at 25°C. The equilibrium dielectric constants of these solutions are qualitatively consistent with those of experiment. A remarkable diversity of behaviour is obtained for our model solutions by s imply varying the hard-sphere diameters of the ions. In many cases the behaviour observed for thermodynamic quantities is in accord with experiment. The ion - ion , ion-solvent and solvent-solvent correlation functions of the solutions are examined in detai l , revealing a wealth of structural information. Ionic solvat ion is generally found to be very sensit ive to the details of the interactions within the sys tem. - iii -TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iii LIST OF TABLES v LIST OF FIGURES vi ACKNOWLEDGEMENTS ix CHAPTER I. INTRODUCTION 1 CHAPTER II. STATISTICAL MECHANICAL THEORY 6 1. Introduction 6 2. Interaction Potentials 9 3. The Ornstein-Zernike Equation 23 4. The Hypernetted-Chain Approximation 34 5. Method of Numerical Solution 38 6. Averages and Potentials of Mean Force 41 CHAPTER III. THERMODYNAMIC THEORY FOR ELECTROLYTE SOLUTIONS 51 1. Introduction 51 2. General Expressions 52 3. Limiting Behaviour 63 CHAPTER IV. MEAN FIELD THEORIES FOR POLARIZABLE PARTICLES 74 1. Introduction 74 2. The Self-Consistent Mean Field Theory 76 3. The R-Dependent Mean Field Theory 83 - iv -CHAPTER V. RESULTS FOR WATER-LIKE MODELS 108 1. Introduction 108 2. Choice of Basis Set 110 3. Results for Hard-Sphere Models 114 4. Results for Soft Models 139 CHAPTER VI. RESULTS FOR MODEL AQUEOUS ELECTROLYTE SOLUTIONS 150 1. Introduction 150 2. Dielectric Properties 156 3. Thermodynamic Properties 162 4. Structural Properties 202 5. Effects of Including the RDMF 238 6. Results Obtained Employing Different Solvents 254 CHAPTER VII. CONCLUSIONS 272 LIST OF REFERENCES 278 APPENDIX A . TREATMENT OF POTENTIAL TERMS IN c(12) 287 APPENDIX B. REPRESENTATIVE EXAMPLES OF EXPONENTIAL INTEGRALS 291 APPENDIX C. TRANSFORMATION OF THE ROTATIONAL INVARIANT $ 1 2 3 ( 1 2 ) . . 294 - V -LIST OF TABLES I. Reduced ion d iameters , d . * , used in this s tudy 23 II. Exper imental densi t ies of water for the temperatures and pressures examined in this study 109 III. Numbers of unique pro jec t ion terms required in HNC bas is se ts . 110 IV. Pro ject ion te rms included in n = 2 bas is sets 111 1 max V. M a x i m u m numbers of n o n - z e r o te rms fo r any g iven pro ject ion in the HNC binary product 111 VI. C P U t ime required per i teration on an F P S 164 array p r o c e s s o r . 112 VII. Bas is set dependence of e, the average energies and g(r=d) . . . 113 VIII. Parameters for U g R ( r ) 139 IX. M o d e l aqueous e lect ro ly te so lu t ions studied 152 X . A v e r a g e i o n - s o l v e n t energies per ion at infinite di lut ion . 268 XI . Individual ionic partial molar v o l u m e s at infinite di lut ion 270 - vi -LIST OF FIGURES 1. Molecular axis system for the water molecule 18 2. A charge distribution possessing (a) a square quadrupole and (b) a dipole and a square quadrupole 21 3. The angle 6- for (a) a positive ion and (b) a negative ion . . . . 46 4. An illustration of the method used in determining < A E ^ j ( R ) > . . . 87 5. An illustration of the method used in determining < A E ^ p ( R ) > . . . 92 6. An illustration of the method used in determining < A E ^ Q ( R )> . . . 98 7. The mean dipole moment of water-like particles as a function of temperature and pressure 115 8. The dielectric constants of water and of water-like models as functions of temperature and pressure 118 9. Radial distribution functions for water-like fluids at 2 5 ° C 121 10. The projection hQQ 0(r) 124 11. The projection h g Q 2 ( r ) 126 12. The projection h ^ d r ) 128 1 2 3 13. The projection ( r ) 130 14. The projection h 2 Q 4 ( r ) 132 15. The projection ( r ) 134 16. The projection ( r ) 136 17. Soft potentials at 2 5 ° C 140 18. Radial distribution functions for soft water-like models at 2 5 ° C and m *=2.75 142 e 19. Radial distribution functions of water and of water-like fluids at 2 5 ° C and m *=2.75 145 e 20. Structure factors of water and of water-like fluids at 2 5 ° C and m *=2.75 147 e 21. The concentration dependence of Y . . . 154 22. Comparing theoretical and experimental values for the dielectric constant of aqueous KCI solutions 157 2 3 . T h e d i e l e c t r i c c o n s t a n t s o f r e a l a n d o f m o d e l a q u e o u s e l e c t r o l y t e s o l u t i o n s a s f u n c t i o n s o f c o n c e n t r a t i o n 160 2 4 . A v e r a g e t o t a l i o n - i o n e n e r g i e s p e r i o n as f u n c t i o n s o f s q u a r e r o o t c o n c e n t r a t i o n 163 2 5 . A v e r a g e i o n - d i p o i e e n e r g i e s p e r i o n a s f u n c t i o n s o f s q u a r e r o o t c o n c e n t r a t i o n 165 2 6 . A v e r a g e i o n - s o l v e n t e n e r g i e s p e r i o n a s f u n c t i o n s o f s q u a r e r o o t c o n c e n t r a t i o n 168 2 7 . A v e r a g e s o l v e n t - s o l v e n t e n e r g i e s p e r s o l v e n t a s f u n c t i o n s o f c o n c e n t r a t i o n 170 2 8 . T o t a l a v e r a g e e n e r g i e s a s f u n c t i o n s o f c o n c e n t r a t i o n 173 2 9 . T h e s q u a r e o f t he D e b y e s c r e e n i n g p a r a m e t e r a s a f u n c t i o n o f c o n c e n t r a t i o n 176 3 0 . T h e p r o d u c t P 2 ^ + _ a s a f u n c t i o n o f s q u a r e r o o t c o n c e n t r a t i o n 179 3 1 . C j g a s a f u n c t i o n o f s q u a r e r o o t c o n c e n t r a t i o n 182 3 2 . G . a s a f u n c t i o n o f s q u a r e r o o t c o n c e n t r a t i o n + s 184 3 3 . G a s a f u n c t i o n o f s a l t c o n c e n t r a t i o n s s 187 3 4 . I s o t h e r m a l c o m p r e s s i b i l i t y a s a f u n c t i o n o f c o n c e n t r a t i o n 190 3 5 . P a r t i a l m o l a r v o l u m e o f t h e s o l v e n t a s a f u n c t i o n o f c o n c e n t r a t i o n 193 3 6 . P a r t i a l m o l a r v o l u m e o f t h e s o l u t e a s a f u n c t i o n o f s q u a r e r o o t c o n c e n t r a t i o n 195 3 7 . l n y + a s a f u n c t i o n o f s q u a r e r o o t c o n c e n t r a t i o n 199 3 8 . S o l v e n t - s o l v e n t r a d i a l d i s t r i b u t i o n f u n c t i o n s o f t h e p u r e s o l v e n t a n d o f s e v e r a l m o d e l e l e c t r o l y t e s o l u t i o n s 2 0 3 3 9 . < c o s 0 s s ( r ) > f o r t he p u r e s o l v e n t a n d f o r m o d e l e l e c t r o l y t e s o l u t i o n s 2 0 6 4 0 . I o n - s o l v e n t r a d i a l d i s t r i b u t i o n f u n c t i o n s at i n f i n i t e d i l u t i o n . . . . 2 0 8 4 1 . < c o s # . ( r ) > at i n f i n i t e d i l u t i o n 211 4 2 . I o n - s o l v e n t r a d i a l d i s t r i b u t i o n f u n c t i o n f o r C l " 2 1 4 4 3 . < c o s 0 . s ( r ) > o f C h . . 2 1 6 4 4 . P o t e n t i a l s o f m e a n f o r c e at i n f i n i t e d i l u t i o n f o r s e v e r a l p a i r s o f o p p o s i t e l y c h a r g e d i o n s 2 1 9 - viii -4 5 . C o n c e n t r a t i o n d e p e n d e n c e o f g + _(r) f o r KCI 2 2 3 4 6 . C o n c e n t r a t i o n d e p e n d e n c e o f g + _(r) f o r M ' l 2 2 5 4 7 . P o t e n t i a l s o f m e a n f o r c e at i n f i n i t e d i l u t i o n f o r s e v e r a l p a i r s o f l i k e i o n s 2 2 8 4 8 . g^ (^ r ) f o r s e v e r a l i o n s in m o d e l e l e c t r o l y t e s o l u t i o n s at 1 .0M . 2 3 1 4 9 . ^( r ) f o r C h f o r s e v e r a l m o d e l e l e c t r o l y t e s o l u t i o n s 2 3 4 5 0 . C I - / C I " p a r t i a l s t r u c t u r e f a c t o r s f o r m o d e l N a C l s o l u t i o n s 2 3 6 5 1 . A d d i t i o n a l i o n - s o l v e n t i n t e r a c t i o n t e r m d u e t o Ap( r ) f o r a N a + i o n at i n f i n i t e d i l u t i o n 2 4 0 5 2 . E f f e c t o f t h e R D M F u p o n ^( r ) f o r N a C l 2 4 2 5 3 . E f f e c t o f t h e R D M F u p o n w^(r ) f o r M B r 2 4 4 5 4 . E f f e c t o f t h e R D M F u p o n r ) f o r N a + 2 4 7 5 5 . E f f e c t o f t h e R D M F u p o n w. . ( r ) f o r M 4 2 4 9 5 6 . E f f e c t o f t h e R D M F u p o n C I g 2 5 1 5 7 . P o t e n t i a l s o f m e a n f o r c e at i n f i n i t e d i l u t i o n f o r K C I i n p o l a r i z a b l e a n d u n p o l a r i z a b l e t e t r a h e d r a l s o l v e n t s 2 5 5 5 8 . I o n - s o l v e n t r a d i a l d i s t r i b u t i o n f u n c t i o n s at i n f i n i t e d i l u t i o n f o r t h e t e t r a h e d r a l a n d C 2 V o c t u p o l e s o l v e n t s 2 5 8 5 9 . < c o s 0 . s ( r ) > a t i n f i n i t e d i l u t i o n f o r t h e t e t r a h e d r a l a n d C 2 V o c t u p o f e s o l v e n t s ^ 2 6 1 6 0 . P o t e n t i a l s o f m e a n f o r c e at i n f i n i t e d i l u t i o n f o r L i F in t h e C 2 V o c t u p o l e a n d t e t r a h e d r a l s o l v e n t s 2 6 3 6 1 . L i k e - i o n p o t e n t i a l s o f m e a n f o r c e at i n f i n i t e d i l u t i o n f o r L i + a n d F- in t h e C 9 o c t u p o l e a n d t e t r a h e d r a l s o l v e n t s 2 6 6 ACKNOWLEDGEMENTS I wish to thank my academic advisor, Dr. G.N. Patey, for his guidance and support throughout the past six years, and the Chemistry Department of the University of British Columbia and the Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial support. I am also grateful to my family, particularly my parents, for all their help and encouragement. Most of all I would like to thank my wife, Sheila, for the tremendous sacrifices she has made during the preparation of this thesis. - 1 -CHAPTER I INTRODUCTION The study of electrolyte solutions, particularly aqueous electrolyte solutions, has been one of the most active areas of physical chemistry. The considerable attention received by aqueous electrolyte solutions appears to have two principal motivations. The first arises from an innate interest in water, or in this case, in water as a solvent. Of course, the importance of water as a chemical substance cannot be overstated. Not only is water the only naturally occurring inorganic liquid on earth [1], but its unusual physical properties [2-4] and the fact that all biological processes use it as a solvent [2,5] make water essential to all life on this planet. The second arises from the central role charged species play in many chemical reactions and electrochemical processes [5]. It is little wonder then that aqueous electrolyte solutions have received so much attention. An electrolyte solution is a homogeneous liquid consisting of an ionic solid, commonly known as a salt, dissolved in a polar solvent, a liquid usually characterized by a large dielectric constant. For aqueous solutions the solvent is water. An electrolyte solution is often defined as one having a high conductance [3]. In this study we consider only strong electrolytes, i.e., those which can be assumed to fully dissociate (or ionize) in solution. Aqueous electrolyte solutions have been the subjects of numerous experiments and a great deal of experimental data on their macroscopic properties has been accumulated [6-8]. Properties such as density, vapour pressure, and apparent dielectric constant have been measured and their dependence upon salt concentration determined [6-9]. Unfortunately, much of this data has proven difficult to interpret and the details of the underlying microscopic properties remain rather poorly understood [5-12]. We know that the solute exists as a free ionic species in solution, but only recently [13-16] have direct measurements of the microscopic structure in solution (i.e., the ion- ion, ion-solvent, and solvent-solvent distribution functions) been possible. However, these measurements are difficult to perform. Moreover, the results obtained are somewhat ambiguous [13,16] and provide only limited information. Thus, - 2 -the details of the microscopic structure of aqueous electrolyte solutions and how they relate to macroscopic properties remain poorly understood. What is needed then is a more complete microscopic understanding of electrolyte solutions in order to more fully understand the macroscopic observations. It is to this point that this thesis is addressed. t h The study of electrolyte solutions began during the 19 century in that field of chemistry which has now become known as electrochemistry. Investigators [17] were intrigued by the fact that matter was transported in electrolyte conductors but not, of course, in electronic conductors. Clausius [17] noted that ionic solutions obey Ohm's law and concluded that there must be electrically charged particles present. In 1887 van't Hoff [18] published experimental results which clearly indicate that conducting solutions possess colligative properties distinct from those of non-conducting solutions. These results were interpreted by Planck [19] as a possible indication of ionization of the solute. However, the present day theory of electrolyte dissociation in solution is usually credited to Arrhenius [20] who first published his theory in 1887. At the time this seemed like a radical idea, but with the support of a wealth of data it gained general acceptance by the turn of this century. Many attempts were made during the first decades of this century to develop equilibrium theories, and later dynamical theories, to describe electrolyte solutions and their behaviour. The long-range nature of the ion-ion interactions made it possible to derive [21] limiting laws for many of the properties of dilute electrolyte solutions. These theoretical results were found [21] to be in very good agreement with experiment. The theories of Debye and Hiickel [22] and of Onsager [23] stand out today as landmarks. The theory of Debye and Huckel for the equilibrium structure of electrolyte solutions became, and probably remains, the standard approach [3,5,6] used in discussing or describing them. Much of the work done on the equilibrium theory of electrolyte solutions [6,21,24,25] in the 50 years after the advent of Debye-Hiickel theory was concerned with justifying and improving the theory itself. However, the basic approach to electrolyte solutions used by Debye and Huckel [22], in which the solvent is treated simply as a dielectric continuum, remains essentially unchanged. It was actually McMillan and Mayer [26] who formally showed that the solvent need not be explicitly considered as a molecular - 3 -species if an effective solvent averaged ion-ion interaction potential is used. Of course, implicit in the above statement is the assumption that all effective many-ion potentials can be ignored [25], although this assumption is really only valid at very low concentration. Within McMillan-Mayer theory [26], the effective ion-ion potential can be written in the form [13] u. .(r) = u?. ( r ) + , (1 .1) 1 J 1 J e r where r is the distance between the ions i and j, q. and q. are their charges, e is the dielectric constant of the solvent and u^j(r) is the short-range ion-ion interaction. Models for electrolyte solutions which are defined in terms of eq. (1.1), i.e., which treat the solvent as a dielectric continuum, are know as primitive models. If we take U £ j ( r ) = 0 , we obtain the Debye-Huckel primitive model. The restricted and extended primitive models [13,27] result when u^j(r) is a simple hard-sphere potential [27]. The so-called refined primitive models [28,29] attempt to use more realistic short-range ion-ion interactions while also incorporating short-range solvent effects. Primitive model systems have been extensively examined [24,25,27-30] and researchers have been fairly successful at fitting the concentration dependence of many thermodynamic properties of aqueous electrolyte solutions [28] with these simple models. Unfortunately, primitive model studies have given very little insight into the microscopic structure of real aqueous electrolyte solutions because they ignore the molecular nature of the solvent. Therefore, the primitive model is not particularly useful if one wishes to investigate the microscopic properties of electrolyte solutions and determine how they may affect the macroscopic behaviour. Of course, before one can really even begin to consider investigating model aqueous electrolyte solutions which explicitly include the solvent as a molecular species, one must first be able to study and characterize the pure solvent. The first computer simulation studies [31-33] of water-like models took place almost 20 years ago. Since then numerous other computer simulations [33-49] have been carried out on many different water models. Several of these models have been found to reproduce the microscopic structure and many of the thermodynamic properties of liquid water at normal temperature and pressure quite well [34,36,41,47], although almost all models give rather poor results when studied in the gas or solid phase [36]. Also, - 4 -the dielectric properties of two of the more successful models, the MCY [49] and TIP4P [41], have recently been shown to agree quite poorly with those of real water [45,46]. This result has been attributed [46] to the fact that these models neglect molecular polarizability. Computer simulation techniques have also been used fairly extensively to examine model aqueous electrolyte solutions [34,50-59], with alkali halide solutions receiving the most attention. A variety of ion and solvent models have been employed [34,58,59] to study infinitely dilute solutions {i.e., containing only one ion) and those at moderate to high concentration (i.e., >1M). These investigations have concentrated mainly on the determination of the solvent structure around the ions [34,58], for which they obtain reasonable agreement among themselves and with experiment [13,16]. Unfortunately, computer simulation studies of aqueous electrolyte solutions are somewhat limited as to the systems and the properties which can be examined. This is due mainly to the fact that they consider systems of only a small number of particles. Hence, the ion-ion and long-range ion-solvent structure and the many thermodynamic properties which depend upon them (e.g., j+, V^, etc.) are not currently accessible through computer simulation. Moreover, one can not study solutions at low concentration. Integral equation methods, commonly used in liquid state theory [27,33,60], have been used very successfully to investigate primitive model electrolyte solutions [24,25,27,28,30]. They have also been shown to be very useful in the study of multipolar fluids [27,61-72]. Solutions of hard-sphere ions in a dipolar solvent [73-78] were examined extensively with integral equation techniques. More recent work [79-82] has focussed upon the calculation of ion-ion potentials of mean force at infinite dilution in water-like solvents. Unlike computer simulation, integral equation theories consider an infinite system and will, in principle, yield all equilibrium properties of the solution. Furthermore, the entire concentration range can, for the most part, be investigated with integral equation theories. In the present study we will use integral equation methods first to examine a water-like solvent and then to study model aqueous electrolyte solutions, both at infinite dilution and at finite concentration. We stress that we can consider only equilibrium properties of these systems because of our choice of an integral equation approach. The solvent model we shall - 5 -investigate is a simple one which incorporates a set of known (measured) molecular properties of the water molecule with no freely adjustable parameters. When determining the properties of this water-like solvent, particular attention will be paid to its dielectric constant, since e represents the ability of the solvent to screen the coulombic forces between ions that are far apart (cf. eq. (1.1)). Model aqueous electrolyte solutions will then be studied. The structural, thermodynamic and dielectric properties of these systems and their dependence upon salt concentration will be determined. The results obtained will be compared with those of real solutions. These comparisons will be mostly qualitative in nature. Their basic purpose will be to help identify which microscopic properties are important in ionic hydration and in determining the macroscopic properties of aqueous electrolyte solutions. We also hope to demonstrate the usefulness of the current approach. It should also be noted that in the present study a great deal of theoretical formalism is introduced and derived, most of which can be applied to more general models than those examined here. Most of the work presented in this thesis is being prepared for publication [83,84], or has been submitted or accepted for publication [85,86]. In Chapter II we define the models considered in this study and describe the specific integral equation theory (the reference hypernetted-chain [68]) employed. A general formalism which relates certain thermodynamic properties of electrolyte solutions to integrals of radial distribution functions is outlined in Chapter III. The low concentration behaviour of our expressions is examined and the limiting laws obtained. These limiting laws are in terms of microscopic properties and can be compared with the macroscopic (i.e., Debye-Hilckel) results. In Chapter IV we discuss two levels of theory in which the polarizability of the solvent can be taken into account. We find that polarization can have long-range effects which may influence the limiting laws of some thermodynamic quantities. The results obtained for the pure solvent are given in Chapter V, while in Chapter VI we present our findings for model aqueous electrolyte solutions, both at infinite dilution and finite concentration. Finally, Chapter VII will summarize all the results presented in this thesis, pointing out areas which need further investigation and indicating possible extensions of the present study. - 6 -CHAPTER II STATISTICAL MECHANICAL THEORY 1. Introduction Statistical mechanics plays an essential role in the present day study of real systems [27,33,87]. Its principal function is often viewed as being a bridge between the disciplines of thermodynamics and quantum (or classical) mechanics. Thermodynamics is primarily concerned with the measurement and interpretation of the macroscopic, or bulk, properties of materials while quantum mechanics is, at present, restricted to the study of individual (or very small numbers of) atoms or molecules within materials. Like thermodynamics and quantum mechanics, statistical mechanics embodies a very large theoretical framework built upon only a small number of axioms. This development is not given here but may be found in introductory books [87-89] on statistical mechanics. Statistical mechanics provides several different approaches through which to study matter [87,88]. Distribution function language [27,33,87] is frequently used in such studies since it allows a complete but compact description of the microscopic structure. Knowledge of the probability distribution functions is sufficient, in general, to determine all thermodynamic properties of a liquid system. In all statistical mechanical studies of matter, we start by first defining a microscopic model for the system of interest. It is usually sufficient to define such a model by specifying the interaction potential between particles of the system. Then given this interaction potential, statistical mechanical theory provides us with a means of determining the average microscopic structure which in turn specifies the macroscopic properties of the system. This chapter will outline the theoretical approach we have used. In this thesis we are concerned with the study of systems in the liquid state. Most liquids and solutions can be reasonably described using classical statistical mechanics [27,33]. Liquid hydrogen, liquid helium, and solvated electrons are some of the few exceptions. Two basic approaches are currently employed to study classical fluid systems; they are computer simulation and - 7 -approximate methods. Computer simulation [27,33,87] can be regarded as essentially an exact method although it usually requires considerable computational resources. All but the simplest model systems require several hours on a powerful computer to obtain statistically meaningful results, even with present day super-computers. In order to keep such times on a reasonable scale only small systems, usually consisting of 100-1000 particles, are studied. As a result quantities which are very sensitive to boundary conditions, such as dielectric constants, or systems which have long-range forces, such as electrolyte solutions, pose major problems for computer simulation. A great deal of effort has been spent in developing approximate theories [27,33,60,87]. Up until the advent of the modern computer some 25 years ago, they represented the only means through which model systems could be studied. Approximate theories do not suffer from the statistical or boundary condition problems present in computer simulation. Also, they usually require much less computation than do computer simulations. However, being approximate, they can only give estimates for the unique set of properties that exist for a given model system. Integral equation theories are one set of approximate methods which have been used extensively in the study of fluids [27,33,60,61,87]. Most integral equation theories can be written as two coupled equations. One of these, the Ornstein-Zernike (OZ) equation [90], is a basic relationship in the equilibrium theory of fluids. The OZ equation is an exact relationship. A second equation is required to close the system of equations, hence the term closure is given to this expression. An integral equation theory is usually known by the name given the closure equation. At present only approximate closures exist. Therefore, it is the closure approximation which determines the accuracy of the integral equation theory. Also for all but the simplest cases, these theories must be solved numerically. There are several different integral equation theories which have been extensively studied [27,33,60,87]. These include the Mean Spherical Approximation (MSA) [91], the Percus-Yevick (PY) [92] theory, and the Hypernetted-Chain (HNC) [93-97] theory. Further discussions of these theories can be found elsewhere [27,33,60,87]. Of importance here is the fact that the HNC theory is known [27,60,87] to be superior for fluid systems possessing - 8 -long-range interactions (e.g., charged systems). Until recently, it was not possible to solve the HNC theory for systems characterized by non-spherical potentials. As a result further approximations were made to the HNC closure in order to obtain several related theories, including the linearized HNC (LHNC) [62] and quadratic HNC (QHNC) [63] approximations. The LHNC and QHNC theories have been used extensively to study systems with non-spherical interaction potentials and have been shown to give good results for some multipolar fluids [61-65,67,69]. However, recent advances [68] have made the use of the full HNC and the closely related reference HNC (RHNC) [68] possible in the investigation of systems possessing anisotropic potentials. The models studied include dipolar hard spheres [68] and Stockmayer particles [70], as well as dipole linear quadrupole systems [71]. Very recently, the HNC was used to examine liquid crystal models [98] as well as hard ellipsoids and spherocylinders [99,100]. In all cases the HNC has been found to agree reasonably well with computer simulation results. In this chapter we are concerned with the development of the classical statistical mechanical theory necessary to study model water and electrolyte solution systems using integral equation methods, primarily the RHNC theory. Since the water molecule has C2 V symmetry [4], we have restricted ourselves to model systems in which all species have at least C2 V symmetry. We shall consider the simplifications that result from this restriction. Section 2 of this chapter will deal with the interaction potentials used in this study. In section 3 we describe that generalized reduction of the OZ equation for a multi-component system employing the rotational invariant language [68,101-103] outlined in section 2. A discussion of the HNC and RHNC closures and their application is given in section 4. Section 5 of this chapter will outline some of the techniques used to obtain a numerical solution to the equations of sections 3 and 4. Finally, in section 6 we will summarize the relationships used to calculate some of the average properties of polar solvents and electrolyte solutions. In Chapter III we will examine how other thermodynamic properties of electrolyte solutions may be obtained from Kirkwood-Buff [104] theory. - 9 -2. Interaction Potentials In the study of real systems and their physical properties the development of useful models and the potentials that characterize them is an essential step. For a model to be useful it must be simple enough to enable us to produce meaningful results with resources currently available, yet it must have a sufficient degree of sophistication so as to adequately represent the system of interest. The interactions which determine most bulk properties of liquids, and of matter in general, are essentially electrostatic in nature [33]. They arise from the coulombic interactions between nuclei and electrons. At this level we could treat any system exactly by solving the many-body Schrodinger equation describing the motion of all nuclei and electrons. Unfortunately, such a task is several orders of magnitude beyond our present day capabilities. In order to simplify the model, we first use the Born-Oppenheimer approximation [105] in which the heavier nuclei are held fixed while we determine the electronic distributions. The ground electronic states are then used to determine average charge distributions and polarizabilities. A second approximation is made in assuming that all molecules are rigid. We ignore all intramolecular vibrational and rotational modes. (This may not be a good approximation for large polymeric molecules.) A third simplification arises from the fact that the behaviour of the particles within most fluids at ordinary temperatures can be described classically, as was mentioned above. Hence we find it convenient to restrict ourselves to classical mechanics and classical statistical mechanics. Subject to the above assumptions, the total interaction potential, u^, will depend only upon the positions and orientations of all particles within the system. We write [33] u N = u ( X 1 , X 2 , . . . , X N ) , (2 .1) where X^ represents the positional and angular coordinates of particle i, and N is the total number of particles in the system. In general, this N-body interaction potential in very complicated, particularly for liquid systems, and the N-body problem is very difficult to solve. To allow further simplification, - 10 -we express [33] the total interaction potential as an expansion u ( X 1 , . . - , X N ) = .S.u ( 2 >(X. , X . ) + . 5 u ( 3 ) (X . , X . , X . ) + , (2 .2) — i —w i<j —i —j i<j<k ~ 1 ~ J ~ K in which the first term is the sum of all unique pair interactions, the second term is the sum of all unique 3-body interactions, and so on. It is generally agreed that for most liquids the pair interaction potential is the dominant term [27,33]. In most statistical mechanical studies higher order terms are usually neglected or the pair potential is modified in an attempt to take into account higher order terms [27,33]. In Chapter IV we will describe how the many-body problem of polarizability can be reduced to an effective pair potential. It is convenient to write the pair interaction potential term of eq. (2.2) as [106] u ( 2 ) ( X 1 f X 2 ) = u(12) = U*(12) + u e ( l 2 ) , (2 .3) where u*(12) is the interaction due to the overlap and instantaneous e anisotropics of the charge distributions of the particles, and u (12) is the interaction due to permanent anisotropies or net charges associated with the charge distributions of the particles. Thus u*(12) contains the short-range repulsive terms and the long-range dispersion terms of the interaction. It can be approximated by potentials such as the hard-sphere or Lennard-Jones [106] interactions, u (12) is the electrostatic interaction between two non-overlapping charge distributions. It is usually described in one of two ways; either using point charge models or using multipole expansions. the continuous charge distribution of the molecule of interest. Hence u (12) is A point charge model uses a small set of discrete charges in place of >ntir given by •qi(^) qA2) i r ( i 2 ) = z . i<3 (2 .4) t h t h where q. is the i charge on particle 1, cjj is the j charge on particle 2, and r^  is the separation between points i and j. In general, the larger the 11 number of discrete charges, the more closely the real charge distribution can be mimicked. It has been found [39] that for simple molecules relatively few point charges are necessary to give a reasonable description of their electrostatic interaction. u (12) may also be described in terms of multipole expansions. The electric potential produced at a point t by an arbitrary charge distribution can always be expressed as a Taylor series in spherical harmonics known as a multipole expansion [106,107]. In a similar fashion, the electrostatic interaction between two non-overlapping charge distributions is given by [72,102,106] u e ( 1 2 ) = Z mnl av 1 U V n , l | _ 2 m ! 2nlJ fmnl 1+1 v-v ( 2 . 5 ) where 5 is a Kronecker delta function, r is the separation between the centres {i.e., the points of expansion) of particles 1 and 2, and the rotational invariants, ^ " ^ ( f l , , 0 2 , f ) , are defined below. The multipole moments, Q^, are defined for a discrete charge distribution by [72,102,106] (2 .6a) where the coordinates (r .8 . <j> ) of the charge e are given in the molecular v a a a 7 a (rotating) frame of reference. The generalized spherical harmonic [72,108] "So'*-*' L2m+1J m L(m+ hu ) !J m ( 2 . 6 b ) where ^ ( 0 , 0 ) is a spherical harmonic [106] and P^cosfl ) is an associated Legendre polynomial [106]. We note that for a continuous charge distribution the multipole moments will be given by an expression analogous to eq. (2.6a), except now the sum over discrete charges will be replaced by integrals over - 12 -the charge distribution [106]. For an isotropic system we would expect u(12) to be translationally and rotationally invariant. That is to say, the interaction observed between particles 1 and 2 will be invariant to the position and orientation of the frame of reference attached to the vector joining the two particles, with respect to the lab fixed frame. Translational invariance is retained by eq. (2.5) by noting that r is just the interparticle separation. The functions ^1 i&2 f ? ) , which fulfill the requirement of rotational invariance in eq. (2.5), are known as rotational invariants. They are defined by [68,101] ^ > , , a 2 , r ) . t™\zFa J ^ « , ) H g j O , ) ^ » , (2.7) where m , n , l are positive integers, ^J>&) >s again a generalized Wigner spherical harmonic [108], 0= (8, c6,v//) is the set of Euler angles [87,108] for each particle, f is the orientation of the vector from particle 1 to particle 2 and is the usual 3-j symbol [109]. The orientations R 1 f £ 2 2 r f a r e t n e sets of angles of rotation from the lab fixed frame to the molecular fixed . frame [87]. The sum in eq. (2.7) is only over those values of a,P,y for which the 3-j symbol evaluates to a non-zero value. The triangle condition [109] in the 3-j symbol requires that |m-n| <, 1 < m+n . (2 .8a) From the definition of the generalized Wigner spherical harmonics we have \n\ < m and | i>| < n . (2.8b) In eq. (2.7) fmn-'- can.be any non-zero constant. In this thesis we will make use of two different definitions: f ™ 1 = I/ , (2.9a) K0 0 0' f m n l = [(2m+1)(2n+1)J* . (2.9b) We will alternate between eqs. (2.9a) and (2.9b) when it is convenient to do so. - 13 -The rotational invariant functions generated by eq. (2.7) form a basis set of orthogonal polynomials [101] which will span the complete space of orientations of particles 1 and 2. Equation (2.5) is an expansion in this rotational invariant basis set. The expansion is such that the coefficients will only depend upon the interparticle separation r and all angular dependence is in the functions ^ ^ ( f i , , f l 2 > r ). Using f m n ^ as given by eq. (2.9b), we rewrite eq. (2.5) as u e ( 12 ) = L u m" X(r) rt,,fl2,f) , (2 .10a) mnl M»> M P nv where r v 2 i + i H i i Q£ Qri M P v r y v " °m+n,lL(2m+1)!(2n+1) j j 1+1 * (2.10b) r We can expand eq. (2.3) in a similar manner. Later we will find it very convenient to expand other functions in this same basis set. A rotational invariant expansion must also satisfy two other symmetry conditions of isotropic fluids. Since the labels 1 and 2 are totally arbitrary for an isotropic fluid, exchange of these labels should leave ue(12) unchanged. It has been shown [101,110,111] that this condition requires that the invariant expansion coefficients satisfy u m ^ ( r ) = ( - 1 ) m + n u ^ m l ( r ) . (2.11) We also have the requirement that u (12) must remain unchanged under symmetry operations on the individual particles. Conditions for several symmetry groups are given in Blum and Torruella [101]. In this study we will make use of the following requirements: i) for spherical symmetry in both particles m,n,1,MF e = 0 , (2 .12a) ii) for linear symmetry in both particles H,v = 0 , (2.12b) - 14 -iii) for C 2v symmetry in both particles a,v = even (2 .12c) and u mnl ( uv ( r ) = u' mnl ( ±u±v (2.12d) Blum and Torruella [101] also showed that since u (12) must be real then Together eqs. (2.12c), (2.l2d) and (2.13) imply that at least for particles of C symmetry and u ( r ) must be real. The conditions given by eqs. (2.12) and (2.14) serve simply to remove some of the basis functions from the rotational invariant expansion. Let us now return to eq. (2.5). The sum in eq. (2.5) is infinite where m,n, l , / i , j> are subject to the restrictions given by eqs. (2.8), (2.12) and (2.14). We know that the multipole expansion for non-overlapping charge distributions must be convergent [106], in which case u (12) must be given equivalently by eqs. (2.4) and (2.5). Also, as we go to higher moments in the expansion and 1 becomes larger, the terms in eq. (2.5) become shorter ranged as their (1/r) dependence increases. Therefore, we would expect the multipole expansion to converge quickly at large separations while converging more slowly at short-range. This property of multipole expansions will be used in discussing the models used in this study. The multipole expansion is frequently given in Cartesian tensor form [107,112], whereas eq. (2.5) is really a spherical tensor form [113] of the same expansion. In Cartesian notation the first four moments (n = 0,1,2,3) of a discrete charge distribution are [107,112,114] (2.13) m+n+1 = even (2.14) a a (2 .15a) - 15 -u- = E e i a (2.15b) 0:, = As e_ (3r. r . - r 2 5 - .) i a ]a a i ] i ] 2 a 'a (2 .15c) and fl i jk " 2 2 e a [ 5 r i a r j a r k a - r (r, 5... + r . 8. . . + r. 5, (2.15d) • a V i i a " j k ' "ja^k ' "ka^ij'] ' where q is the net charge, are the components of the dipole vector, and 0.j and 0.^ are the components of the quadrupole and octupole tensors, respectively. In eqs. (2.15) the sum is over the discrete charges e Q and r., x. r^  are the i, j , k components, respectively, of the vector x_ given in the molecular axis frame. We again point out that for a continuous charge distribution the sums over charges in eqs. (2.15) become integrals over the charge distribution. A common convention, and one we will use, is to choose this reference frame such that the origin (i.e., the point about which the expansion is made) is at the molecular centre of mass and the axis of highest symmetry is labelled the z-axis. An electric multipole moment has, in general, (2n + 1) independent components. However, this number can be greatly reduced by molecular symmetry [107,114]; of most importance here is the reduction under C2 V symmetry. From Kielich [107] we have that for symmetry, the dipole, quadrupole and octupole moments have 1, 2 and 2 mutually independent components, respectively. Thus, the dipole moment is given by the scalar u and the quadrupole tensor has the form 0 = 0 0 0 X X 0 0 0 yy 0 0 z z (2.16a) where we require [107] that 0 + 0 + 0 xx yy z z (2.16b) To specify the octupole tensor it is sufficient to give & x x z , ^ y y 2 a n d ^ z z z - 16 -where 0 + 0 + 0 =0. xxz yyz zzz The Cartesian representation is the form most often used in the literature to specify the multipole moments of specific molecules. In this study it is convenient to work with multipole moments as defined by eq. (2.6a). To find expressions relating the two representations we use eqs. (2.6) and (2.15) along with explicit forms for the associated Legendre polynomials [106] and the relationships z = r COS0 and x + i y =. r s in© e1^. It is then 1 J K a a a Ja a easy to show that in general Q~0 = Q , (2 .17a) Q° = M2 , (2.17b) Q° = 0 2 2 , (2 .17c) Q J 1 = j ^ U x ± iuy) , (2 .18a) *V - f ( 0 x z + i 0 y z ) ( 2 ' l 8 b ) Q i 2 ' ^ f ( 0 x x - Qyy± 2 i V ' ( 2 - 1 8 c ) and If we restrict ourselves to C„ symmetry where u= u = 0 v = ©„_= 0 = 0 [107], we have immediately from eqs. (2.18) that QJ = Q^ 1 = 0 , (2.19a) Q 2 = = 0 (2.19b) Q2 " Q 2 2 = ^ ( 0 x x " V ' ( 2 ' 1 9 C ) and In a similar fashion we can show that for C 2 v symmetry the components of the octupole moment are given by Q° = fizzz , (2.20a) Q 3 = Q 3 1 = 0 , (2.20b) Q3 = - ^ ( n x x z - ° y y 8 > ( 2 - 2 ° C ) - 17 -and =1 - 3 i 3 = 0 (2.20d) We note that eqs. (2.17) and (2.19) are consistent with the results of Carnie etal. [72]. Price etal. [113] give similar relationships between the Cartesian and spherical multipole moments. It is also obvious from eqs. (2.17), (2.19) and (2.20) that for C2 V symmetry we have the required 1, 2 and 2 independent non-zero components of the dipole, quadrupole and octupole moments, respectively. It is clearly the case that particles of spherical symmetry will possess only an n=0 moment (net charge) as given by eq. (2.15a). Thus the charge distributions of spherical ions, such as the alkali halides, are completely represented by a single point charge at their centre. It follows from eq. (2.5) that the electrostatic interaction between two such ions, i and j , is itself spherically symmetric, so we write where $QQ ( 0 1 , r 2 2 , f ) = 1 is understood and q., q^  are the charges of the ions. For non-spherical particles eq. (2.5) remains, in principle, an infinite sum subject to symmetry conditions, such as those represented by eqs. (2.12). The gas phase dipole and quadrupole moments of water have been measured [115], In this study we will take the permanent dipole moment, u, — 18 as being 1.855D [118] (D = 10~ esu. cm.). The non-zero components of the quadrupole moment [119] are 0 = 2.63B, 0 = -2.50F3 and 0 = -0.13B, where 26 2 X X B = 10~ esu. cm. and the molecular axis system is defined as it appears in Figure 1. This definition for the molecular axis frame for water will be used universally throughout this thesis. For the higher moments of water we must rely on quantum chemistry to give us reasonable estimates. There have been many large scale Cl and SCF calculations done for the water molecule, but only a few of these [115-117] report multipole moments higher than quadrupole order. In this study we will use the octupole moment of Neumann and Moskowitz [116] who reported the values fl = 2.30F, fl =-0.96F and -34 3 x x z ^ z fl = -1.34F, where F = 10 esu. cm. Their calculated values of the dipole zzz ' r and quadrupole moments are in reasonable agreement (within 8%) of the u^.2) = u ? j ( r ) (2.21 ) - 18 -Figure 1. Molecular axis system for the water molecule. The stars indicate the atomic centres. - 19 -measured values. More recent calculations [115,117] for water using larger basis sets show little improvement in their results for the dipole and quadrupole moments. Moreover, their reported octupole moments are in good agreement with Neumann and Moskowitz [116]. The hexadecapole moment of water has also been calculated [115,117] and a very recent publication [117] reports multipole moments up to and including the n=6 moment. In this study we have chosen to ignore the hexadecapole and all higher moments. Hence we have truncated the sum in eq. (2.5) to octupole order {i.e., required m,n<3). We might expect properties such as the dielectric constant and processes such as ion solvation to be dominated by long-range electrostatic interactions and thus not be particularly sensitive to this truncation of the multipole expansion. Several early studies [62,66,73-78] of polar solvents and electrolyte solutions considered electrostatic interactions only up to dipole order. This model was found to be quite unsatisfactory for water, giving very poor results for its dielectric properties and generally a poor description of ion hydration. More recent studies [67,79-81] have indicated that the addition of the quadrupole moment to the solvent model greatly improves the results. The dielectric properties seem to be approaching those of real water [67]. For model aqueous electrolyte solutions at infinite dilution the ion hydration appears more reasonable, much more like what is believed to be the case in real solutions [79-81,120]. Therefore, for most of the electrolyte solutions studied here, all those examined at finite concentration, the electrostatic interaction will contain terms only up to quadrupole order. The influence of the octupole terms will be examined for the pure water-like solvent as well as for electrolyte solutions at infinite dilution. Carnie and co-worker [67,72] have pointed out that to a good approximation the quadrupole tensor of water can be expressed as 0s 0 0 0 (2.22) 0 0 0 by setting © z z to zero in eq. (2.16). Such a quadrupole tensor is totally specified by the single parameter 0 g , which we shall refer to as a square quadrupole moment (it is also known as a tetrahedral quadrupole moment - 20 -[67,72]). In Figure 2(a) we have illustrated the simplest charge distribution that has a square quadrupole as its lowest order moment. It is a charge distribution where two positive and two negative charges of equal magnitude have been placed at opposite corners of a square. In Figure 2(b) we have illustrated what we will refer to as a tetrahedral charge distribution, in which the four charges are now located at the vertices of a regular tetrahedron. This point charge model, which possesses a dipole and square quadrupole as its two lowest order moments, has been used to represent the real charge distribution of water {i.e., the BNS model [44]). Carnie et al. [67] have shown that u (12) is subject to an additional symmetry condition if both particles 1 and 2 have only dipole and square quadrupole moments. In this case, in addition to eqs. (2.12c) and (2.12d), we require that [72] Thus, if we consider moments only up to quadrupole order, restricting ourselves to a square quadrupole moment will result in a smaller number of basis functions. We point out that the tetrahedral charge distribution does not represent a special symmetry group, since if we include the octupole moment we return to general symmetry. The tetrahedral point charge model does have an interesting property in that it will interact with positive or negative charges equivalently. Therefore a solvent with a tetrahedral charge distribution will solvate simple spherical ions of equal size symmetrically. This property will prove useful in the present study. Now let us return to eq. (2.3) to consider the remaining term, u*(12), in the expression for the total pair interaction potential. In this study we take u*(12) to be spherically symmetric, i.e., u*(12) = u*(r). This is frequently done for water-like models [35-39,44,67] since the water molecule is roughly spherical (see Figure 1) and results in a much simpler pair potential. We choose to represent u*(r) with the simplest possible pair interaction, the HS hard-sphere potential. The hard-sphere potential, UFL^ ( r ) , between two particles a and /3 is given by [27,33] (M + v + 21) MOD 4 = 0 (2.23) °°; r<d a/3 (2 .24a) 0; r>d a/5 - 21 -Figure 2. A charge distribution possessing (a) a square quadrupole and (b) a dipole and a square quadrupole. - 22 -where , , , d + d f l da-3 = J L T ~ l > ( 2 ' 2 4 b ) with d f l being the diameter of particle a. It is clear from eq. (2.24a) that the hard-sphere interaction is a purely repulsive potential. In approximating u*(r) HS with u a^( r) we are making two further assumptions; first that the short-range repulsions between the particles can be reasonably represented by the hard-sphere potential, and second that the long-range dispersion forces are small compared to the electrostatic forces in the systems of interest. The validity of these assumptions and their influence upon properties of interest shall be discussed later. The hard-sphere potential requires that we specify hard-sphere diameters for our particles. Although the exact values are somewhat arbitrary, for water a value of d ^ 2.8A is a reasonable choice. It is consistent with 0 -0 s structure as measured by diffraction experiments [121,122] and has been used in previous ° studies [67,79-81]. For ions in solution, the choice is not quite so obvious. We would expect the radius of an ion in solution to be close to its crystal radius. For the alkali halides however, there are several estimates of the crystal radii [123,124]. Recent X-ray electron density measurements of ionic crystals seem to be the most physically realistic method of defining ionic radii. To be consistent with previous work [80,81], we have chosen to use the radii of Morris [124] as determined in this manner. Table I summarizes the values of the ion diameters used in this study. They are expressed as reduced ion diameters (i.e., in terms of solvent diameters), d.*= dj/d where d s = 2 . 8 A . The values have also been rounded to the nearest 0.04 to accommodate the grid width used in the numerical calculations (i.e., 0.02ds), as discussed below. Included in Table I are reduced diameters for the alkali halides, as well as those of four other ions which appear at the bottom of the table. Two of the ions, Eq + and Eq- which have no real counterparts, are the same size as the solvent and will be used simply to test solvation effects. The two other ions, M'+ and M + , are almost twice the size of the solvent. They are similar in size to tetraalkylammonium ions and will be useful in investigating large ion effects. - 23 -TABLE I. Reduced ion diameters, d.*, used in this study. ION Rfcr.CI-Cs+,Br-Na\F-Eq+,Eq-0.68 0.84 1.08 1.16 1.28 1.44 1.80 1.96 1.00 Finally we should point out that we have not ignored terms in the interaction potential due to the polarizability of the particles in our systems. These terms will be treated on a mean field level (i.e., by ignoring fluctuations) and will be included as effective interactions in our pair potentials. Details of how this can be done at two different levels are given in Chapter IV. 3. The Ornstein-Zernike Equation In liquid state theory the pair distribution function, g (0 , ,J22 r£) = 9(12), is of fundamental importance [27,33]. It is a measure of the probability density of finding particle 1 with orientation and particle 2 with orientation fl2 a t t n e separation r_. For a system defined using only pair potentials, knowledge of g(12) is sufficient to completely describe the equilibrium thermodynamics of such a system. The radial distribution function, g(r), is the angle-averaged pair distribution function and is obtained by integrating g(12) over all orientations of particles 1 and 2. More detailed discussions of distribution functions can be found elsewhere [27,81,87]. One important property of g(l2), and of distribution functions in general, is that they are normalized such that g( 1 2) —» 1 as r — (2.25) In the development of integral equation theories [27,33,60] it is convenient to 24 -introduce the pair correlation function h ( !2 ) = g(12) - 1 (2.26) which measures the departure of the pair distribution function from its limiting value. In 1914, Ornstein and Zernike [90] defined a relationship in which h(12) is expressed as a sum of a direct part involving only particles 1 and 2, and an indirect part which takes into account all correlations involving other particles. This expression is known as the OZ equation. When generalized to a mixture [125], it can be written in the form where = N^/V is the number density of species y and the integration is over all positions and orientations of particle 3 of species y. The sum in eq. (2.27) is over all species in the system. The original authors [90] called c(12) the direct correlation function. The second term of eq. (2.27) is a convolution and is often called the indirect part of h(12). As stated earlier, the OZ equation is a basic relationship in liquid state theory and is common to many integral equation theories [27,33,60]. The OZ equation is now regarded as a definition of the direct correlation function, since c(12) has no simple physical interpretation. More detailed discussions of the OZ equation and how it can be derived through diagrammatic expansions or functional differentiation appear in several text books dealing with liquid state theory [27,87]. In order to make the convolution in the OZ equation tractable, one has only to Fourier transform eq. (2.27) with respect to the interparticle position vector jr. We then have [101,110] where the integration is now over all orientations of particle 3 and the ~ denotes the usual Fourier transform [126,127]. If all particles in the system (2.27) (2.28) - 25 have spherical symmetry, then the integration over 0 3 is trivial and eq. (2.28) reduces to a simple algebraic form which is easy to solve numerically [81]. However, this is not the case for systems which may include anisotropic terms in their pair potentials. Blum and Torruella [101] recognized that eq. (2.28) could be reduced for systems with non-spherical pair potentials by expanding h(12) and c(12) in terms of the rotational invariants defined in the previous section. The reduction given below closely follows that of Blum [101-103] but the expressions have been generalized to multi-component systems. The notation and definitions used are those of the more recent literature [68]. Also, the description given here will only summarize the important results. Discussions of the underlying mathematical details can be found elsewhere [110,111]. Analogous to eq. (2.10a), we write the expansions b fl(12) = Z b m n l i r ) ^nl(a, , Q 2 , r ) (2 .29a) aP mnl MP;a/3 juf 1 2 ' and V'2) - mSi^ "Wk>C1(G"n-J) • <2-29b) nv where b „(1 2) can be c „( 1 2) or h „ ( 1 2 ) , b „ (12 ) can be c 0( 1 2) or ap ap ap ap ap h a ^( l2 ) , and tP^Hsi ^ ,Q2 , r) and ^ " 1 ( J 2 1 , f l 2 , k) are defined by eq. (2.7). The coefficients # r) are given by ; ba/3( 12) [^(o, ,n2,f)j* dfi,dn: ^ o . ^ ^ ) [ ^ n l f n 2 f ? ) ] * a o . d o 2 C;a/5(r) - - ' ( 2 ' 3 0 ) and the k-space projections, * 5 ^ l a ^ ' c ^ . a r e t n e Hankel transforms where j-^(kr) is a spherical Bessel function of order 1 [128]. In general, high order {i.e., 1>2) Hankel transforms are very difficult to treat numerically since - 26 -they can not make use of fast Fourier transform techniques [126]. Thus a two-step method of performing these transforms has been introduced [102]. First we define the integral (hat) transforms c m n l (s) CU(r> • #W'> - £ -^-#r/.> as f o r 1 even , (2.32a) and -mnl / „ \ _ mnl , x r c m n l / x £ p £ ^ P ° ( r / s ) ] d s f o r 1 o d d , r S (2.32b) Q O in which the polynomials P^(x) and P^(x) are given by e , , t r -1 fc 1 x 2 1 t + i + 3/2 M p ; . + 1 ( x ) = 2 .1 - — , 2 .33a 2 t + 1 i=0L i ! ( t - i ) ! (i + 1 /2) ! J o , s $ r - 1 x t + i + 5/2) n (x) = 2 I — — , 2 . 3 3 b i = 0 L i ! (t-i)! (i + 3 / 2 ) ! J P 2 t + 3 for t>0 and P ° ( x ) = PQ(X) = 0 , (2 .33c) where we use the general definition of z! [128]. The k-space projections, ~mnl (k) c a n t n e n foe w r jtten [102] as zeroth order (Fourier) and first order uvjaji ' L J v ' Hankel transforms of c a( r ) . Explicitly we have C}a^ k ) = 4 7 r ( r 2 ^ 0 ( k r ) C ; a / J ( R ) D R F O R 1 E V E N ( 2 ' 3 4 A ) and CWk) • 4 d ( r 2 V k r ) w / r ) d r for 1odd' (2-34b) where and j 0 (x ) = (2.35a) - 27 -j,<x> - s i n x cos x ( 2 . 3 5 b ) W e note that the t rans fo rms (2.34) can be c o m p u t e d using fast Fourier t rans form techniques . The expans ions for c f l^( 12 ) and h f l ^( 1 2 ) can then be inserted into e q . (2.28). A f t e r p e r f o r m i n g the n e c e s s a r y angular integrat ions and s i m p l i f y i n g , one obta ins [102] h m n l i k ) uv;ap ~ m n l / , \ v c . A k ) = Z p z [ z 1 ^ 1 l a l . i L m n i i .(-uw hmilc ( k ) c i n l ? J k ) l , ( 2 . 3 6 a ) where 1 2 1 , 1 _ , . x i n + n + i [21 + 1] f m i l 1 f i n l z f l . l , l H l . l i l ) z m n i ' 1 1 ; L 2 i + 1 j f ™ 1 m n 1 0 0 0 ( 2 . 3 6 b ) and is the usual 6-j s y m b o l [109]. F o l l o w i n g Blum [102] we now introduce the X'transform C m n ; x ( k ) = Z ( m n 1 ) c m n l ( k ) (2 3 7 ) in which the sum over 1 is f r o m | m - n | to m + n . It is convenient at this t ime to def ine the funct ion 7) -(12) = h fl(l2) - c -(12) , ( 2 . 3 8 ) 'ap ap ap which unlike h „ ( 1 2 ) or c , , ( 12 ) , wi l l be a s m o o t h cont inuous funct ion of r m n l for hard core m o d e l s . For the cho ice of f g iven by eq . (2.9b), it has been shown [102] that we can rewrite e q . (2.36) as - 28 -N m n ' * X (k) z p z 7 7 1 1 z C J = - I N mi ; x ' nu>; ay (k) + Cm i ? X ( k ) l MCJ; ay J x C i n ; x -uv; 7/3 ,(k) ( 2 . 3 9 ) where N m " j X (k) is the x-transform of T ? m n i „{k) as defined by eq. (2.37). nvjap livjap / - i v / By comparing eqs. (2.36) and (2.39), we see that the x-transform has split the general OZ equation into smaller independent sets of equations which should be easier to invert numerically. A l s o , the numerical constant Z^ 2} 1^" has been greatly simplified. m m Again following Blum [103], we define the matrices N X F I and CX,, whose (i,j) elements are N . o( k) and C ' % ( k ) , respectively, where v , J / uvjap nv;ap ' K '' and i = m(m+1) + u + 1 j = n (n+1) + v + 1 ( 2 . 4 0 a ) ( 2 . 4 0 b ) Equations (2.40) follow from the fact that there are, in general, (2m + 1) values of M allowed for each m. The general OZ equation (2.39) can then be written as N X a/3 Z ( - 1 ) X [ N X + C X |p P C X , ( 2 . 4 1 ) y I ay ayj Hy 7/3 where p^ = p^I is a diagonal matrix. The elements of matrix P are given by ( - 1 ) M f o r ri=m(m+1 1 U f o r [ j=m(m+1 ( 2 . 4 2 ) o t h e r w i s e Now, for an n-component system, we introduce the matrices N A and C —• Y ~~ Y whose ( a , / 3 ) elements are N * ^ and C*^, respectively. We also define £ = d i a g ( p 1 ,p 2, • • • ,p n) ( 2 . 4 3 a ) and - 29 -P = d i a g ( P , P , • • • , P ) . (2.43b) We then rewrite eq. (2.41) as N X = ( - l ) x |g X + C X j j p P C X , (2 .44a) which we can rearrange to obtain N x = C X £ P CX|^(-1 ) X I - £ P C 1 . (2.44b) ~ y Thus, for each value of x we must construct the C matrix and then solve ~ v eq. (2.44b) to determine a N A matrix. Once the values of N m n ' x , , ( k ) are known the projections T j m n ^ i k ) are easily obtained using the inverse x-transform 9 5 m n l (k) = (21+1) I (m n 1 ) N m n ; x ( k ) (2 45) in which the sum over x is from -min(m,n) to min(m,n). Finally, the inn 1 projections a ^ r ^ a r e found by inverting the transforms defined by eqs. (2.34) and (2.32). The inverse Hankel transforms are given by CU(r) " y ( k 2 V k r )CU ( k ) d k for 1even' (2'46a) C;a/3 ( r ) = f7 -C k ^1 ( k r )C ;a /3 ( k ) d k f ° r 1 ° d d ' ( 2 ' 4 6 b ) and the inverse hat transforms by CUM - C U r ) " 3 ^ C^»> PI<^ » d= '<« 1 (2 .47a) (2.47b) where P^(x) and P°(x) are defined by eqs. (2.33). - 30 Hence, given a set of coefficients c „( r) we now have the necessary relationships, as given in eqs. (2.32)-(2.35) and (2.44b)-(2.47), to calculate the set of projections affr) which satisfy the general OZ equation (2.27) for a mixture of non-spherical particles. We should point out ~ V ~ Y that in general C and N A will be complex matrices and complex arithmetic must be used to solve eq. (2.44b). In principle, the expansions in rotational invariants (cf. eq. (2.29)) are infinite sums. For obvious numerical reasons we must truncate the basis set at some point to make it finite. We do this by requiring that [68,70,71] m,n < n m a x . (2.48a) This is not a unique choice for restricting the basis set. However, the finite set of rotational invariants that satisfy eq. (2.48a) does have the special property of being a closed set under the generalized convolution of the OZ equation. That is to say, this particular set will generate only itself when the angular integration in eq. (2.28) is performed. We will also examine the effects of imposing an additional condition 1 * W ' ( 2 ' 4 8 b ) Having truncated the basis set by imposing eqs. (2.48), it follows from eqs. (2.37) and (2.45) that eq. (2.44b) need only be solved for | x | ^ n m a x - A t t h i s point we recall that we have also truncated the electrostatic pair potential (cf. eq. (2.10)) in an equivalent manner to eq. (2.48a). The two truncations need not occur at the same value of n , however, it is obvious that the basis max* sets generated by eqs. (2.48) must always contain all terms we wish to include in the pair potential. Patey and co-workers [68,70,71] have found rapid basis set convergence for several dipolar and dipolar-quadrupolar models. They observed that a value of n of four or five was sufficient to converge max properties such as the average energy and dielectric constant. Furthermore, they found that higher order projections (i.e., n=4,5) only weakly couple to the low order ones (i.e., n=0,1). It is easily shown [103] that the matrices N*^ and C*^ have dimensions DxD where D - ( n m a x + 1 ) 2 . (2.49a) - 31 For |x|>0, it follows from the definition of the x-transform and the properties of the 3-j symbol [109] that these matrices will contain rows and columns of zeros, corresponding to m,n<|x|. The number of such zero rows (or columns) will be { ( X - D + 1}2 = X 2 . (2 .49b) We eliminate these zero rows and columns from the matrices by defining new indices and i = m(m+1) + M - X 2 + 1 (2.50a) A j = n(n+l) + v - x + 1 (2.50b) A to replace those defined in eqs. (2.40). The new dimensionality is D X = { n m a x + 1 ) 2 " * 2 • ( 2 ' 5 1 ) For an n-component system, the matrices N x and C x will then have dimensions (nD )x(nD ). We define the indices A A and g = D ( a - 1 ) + i (2 .52a) A A A h x = D x ( 0 - 1 ) + j x (2 .52b) such that the (g h element of N* will be N m n ' x f l ( k ) , and similarly for C . Also, eq. (2.42) is now replaced by X J X i „ = m ( m + 1 ) - M " X +1 x m(m+1 )+*z-x2 + 1 . (2 .52c) o t h e r w i s e Let us now consider the simplifications that arise if we restrict ourselves to systems in which all the particles have at least symmetry. First we will examine the results of imposing eq. (2.14). Immediately, the properties of the 3-j symbol allow us to rewrite the x-transform, eq. (2.37), as - 32 -pmn;x ( k ) = z ( n m 1 } ~mnl ( k ) (2 53) and then using the relationship given by eq. (2.11) for the exchange of labels, we have It also follows immediately from the properties of 3-j symbols that ~mn;-x/k) = £mn;x ( k ) ( 2 5 4 b ) and similarly for N ^ ' ^ O O . Clearly then, the OZ equation as expressed by eq. (2.39) will be invariant to the sign of x for systems with C 2 y symmetry. Thus we need only solve the OZ equation for x in the range [ ° . n m a x ] - m this case the inverse x-transform, eq. (2.45), becomes CWk> - <21+"^ x<x-; o>CS^k> • < 2-5 5 a ) where the sum over x is from 0 to min(m,n) and r1 f o r x=0 , a v = L r « (2.55b) X |_2 f o r x>0 . Tun * y *wmn * Y We have already pointed out that in general C ' „ (k) and N . „ (k ) are uv, ap uv, ap complex. However, for m+n + l=even we find that there can be no mixing of real and imaginary projections by eq. (2.37). Thus, ^ " ' ^ ( k ) c a n o n | y b e real for m+n=even, or pure imaginary for m+n=odd, and similarly for N m n ' " X , ( k ) . uv,ap Now let us examine simplifications that result from imposing eqs. (2.12c) and (2.12d). We immediately obtain pn;x ( k ) = rmn;x ( k ) (2 56) Then using eq. (2.56) and eq. (2.12c), we can rewrite the general OZ equation, - 33 -as expressed by eq. (2.39), as N m n ; x (k) = 2 p ? uu; a/3 7 7 i i ( - D x a [ > i ; x (k) + C m i ; x ( k ) l co=0 M^;a7 /uco;a7 J (2 .57a) x C i n ; x (k) op; 7/3 where r 1 f o r CJ=0 , a , , = L « rt (2 .57b) w L 2 for CJ>0 . In matrix notation, the general OZ equation (2.44b) wi l l be unchanged but the forms of the matrices wi l l change. If we examine the changes in dimensionality upon applying only eq. (2.12c), it is easily shown that D X = [ ( n m a x + l ) 2 + 1 ] / 2 ' £ X 2 + 1 ] / 2 • (2 .58a) c To determine the dimensional i ty, D under both C 2 v symmetry condit ions, we observe that in eq. (2.57a) we have essential ly decreased the number of terms in the sum over co by a factor of 2, except for the (n +1) terms for ' ' ^ v max which co =0. For x=0 we have D 0 " [ D 0 + nmax + 1 ] / 2 = [ ( n m a x + 2 ) 2 ] / 4 . (2 .58b) We then generalize to obtain D X = [ ( n m a x + 2 ) 2 ] / 4 " Hx+D 2]/4 . (2 .59a) In a similar fashion we determine that and i x = [(m+D 2 ] /4 + [ M /2] - [(x+D 2]/4 + 1 (2.59b) £ = [(n + 1 ) 2 ] / 4 + [v/2] - [ ( x+D 2 ] / 4 + 1 . (2 .59c) We remark that the div is ions in eqs. (2.58) and (2.59) are to be taken as integer d iv is ions. For systems of C 2 y symmetry, eqs. (2.59) can be used to replace eqs. (2.50) and (2.51). However, the indices g and h wil l st i l l be - 34 -C -C given by eqs. (2.52a) and (2.52b) with D l and j replaced by D l and _ A A A A A j respectively. If we examine eq. (2.57), comparing it with eq. (2.39), we find that the matrix P will no longer be given by eq. (2.42). Matrix P will now be diagonal, given by r1 for M=0 , P.c .c = (2.60) 12 for M>0 , • C where the index l is defined by eq. (2.59b). 4. The Hypernetted-Chain Approximation The hypernetted-chain (HNC) approximation was developed simultaneously by several authors [93-97] some 25 years ago. It can be derived from functional Taylor series or cluster series expansions for c(12) which is given exactly by [27,87] c(12) = h(12) - lng ( 1 2 ) - /3u(12) + B(12) , (2.61) where /3=l/kT. The function B(12) represents a class of diagrams known as elementary clusters [33] or bridge diagrams [27,87] which are not easily expressed as simple functions of h(12). The HNC equation is obtained [27,33,87] by setting B(12)=0. The name hypernetted-chain equation reflects the fact that the HNC approximation, c(12) = h ( l 2 ) - l n g ( l 2 ) - /3U(12) , (2.62a) includes contributions to c(12) from classes of diagrams known [33] as simple chains, netted chains and bundles. It is believed [27,87] that B(12) is 2 short-ranged, having a h(12) dependence at large r. Therefore, the HNC is thought to have the correct long-range behaviour, i.e., C(12)—>-J3U(12) as r—>oo. The HNC closure is often rewritten in the form C(12) = exp[-}(12) - 0u (12)] ~ n<12) - 1 , (2.62b) where rj(12) is defined in eq. (2.38). - 35 -The HNC approximation has been used widely [27,33,60,87] to study model systems defined by spherical potentials. For example, the HNC has been found [60,87] to be particularly successful for primitive model electrolyte solutions. Until recently, in order to study systems with angle-dependent interactions, further approximations were made to the HNC closure. The LHNC [62] and QHNC [63] closures were obtained by making a particular expansion of the logarithm in eq. (2.62a) and retaining terms to only linear and quadratic order, respectively. (A further discussion of the LHNC approximation will be included at the end of this section.) Recently however, Fries and Patey [68] have shown how it is possible to analytically expand the full HNC in terms of rotational invariants. The following is essentially a summary of their work. In order to eliminate the logarithmic term from eq. (2.62a), we take the partial derivative with respect to r holding all angular variables fixed. Using the definition W(12) = -17 (1 2) + j3u(12) , (2.63) we obtain 9 c ( 1 2 ) = - h ( l 2 ) 9 W ( l 2 ) - ^ U ( 1 2 ) . (2.64) 9r 9r 9r Later we shall see that W(12) is really a dimensionless angle-dependent potential of mean force (cf. eq. (2.100)). We now re-integrate eq. (2.64), taking advantage of the fact that as r — c ( 1 2 ) — > - / 3 U ( 1 2 ) —>0. One immediately has the result C (12) = f h ( l 2 ) 9 W ( 1 2 ) dr - 0u(12) . (2.65) r 9r In this form the HNC can then be expanded in rotational invariants, where the binary product, h(12)[( 9W( 1 2 ) / 9 r ) ] , can be expressed as a sum over a single invariant. In particular, Fries and Patey [68] have shown that ^1,11 ,1 , ^ t i 2 n 2 l 2 = z pmnl^nnl (2.66a) Mi i 'l Uzv2 mnl " v * v in which the numerical coefficient - 36 -P ™ 1 - f m ' n i l i r 2 l 2 ( 2 ^ l ) ( 2 n + l ) ( 2 h l ) ( - i r l t ^ ^ ^ ^ m n 1 M l M 1 _ M " 1 c , V ( 0 0 O' U . b b b J and { • • • } is the usual 9-j symbol [108]. Thus we have [ 3Wm * n 2 ^" 2 ( r ) pninl » h m 1 n 1 l l { r ) _ J i ^ d r uv J R M 1 P 1 3r M ^ i A 2 P 2 - ^ ^ ( D , (2.67) where the sums are over all allowed projections. For models with hard-sphere potentials (cf. eq. (2.24)), it follows immediately from eq. (2.62b) that c (12) = -1 - T?(12) f o r r<d, (2.68) which is an exact closure. Thus for the models considered in this study, eq. (2.68) will replace eq. (2.67) when r<d. To improve the accuracy of the HNC closure, Fries and Patey [68] employed a well known perturbation technique first suggested by Lado [129]. This technique separates the pair potential and correlation functions into reference and perturbation parts. Explicitly, we write X(12) = AX(12) + X R ( r ) , (2.69) where X(12) can be c(12), h(12), r?(12), or u(12) and XR(r) is the same function for some spherically symmetric reference system. Then applying eq. (2.69) to eq. (2.62a), we obtain the reference HNC (RHNC) closure given by C ( 12 ) = Ah(12) + l n g ^ r ) - l n g ( 1 2 ) - /3AU(12) + c R ( r ) . (2.70) This method assumes that exact results for gR( r ) and cR( r ) can be readily obtained. One can then proceed as before and derive the result - 37 -A c m n l ( r ) = j " av J r L Z 111,0 , 1 , m 2n 2l 2 l/MP M 2 f 2 m r l m n , 9 A W m 2 n 2 l 2 ( r ) -, 3r 9AW m n l(r) . h ^ r ) ^ - A h ^ ^ r ) Bin 3r UV 9r dr - U u P^r) . (2.71) The RHNC closure is easily generalized to mixtures. Clearly, in eq. (2.70) we see that there can be no coupling between different pairs of components in the HNC equation. Hence, for multicomponent systems, we have only to apply eq. (2.71) to each unique component pair, a/3. In the present study the appropriate hard-sphere fluid is the clear choice of reference system. The exact hard-sphere radial distribution functions, HS <3ap( r ) , are determined using the Lee-Levesque [130] generalization of the Verlet-Weis [131] fit to Monte Carlo data. Again, because of the hard-sphere potential, we only apply eq. (2.71) for rxi^^ and use eq. (2.68) when r < ^ a ^ . Expanding eq. (2.68) in rotational invariants, we have that for r<&ap c O O ; a 0 ( r ) " 1 T J 0 0 ; a / 3 ( r ) (2.72a) and for mnl^OOO. Equation (2.71) need not be used for spherically symmetric components. In such cases it is numerically expedient to directly apply eq. (2.70), which we can rewrite as W r ) = 9 a f ( r ) e x p [ A V r ) " **W r ) ] ' W r ) " 1 * ( 2 ' 7 3 ) In this thesis we will also report a few results obtained using the reference LHNC (RLHNC) theory [62,74,81]. For hard core models, the RLHNC closure is equivalent to the RHNC closure for v<da^ (cf. eqs. (2.72)). When r>dfl0, we can show [81] that the RLHNC is given by - 38 000 / x HS/ x [A 000 / x O A 000 / c 0 0 ; a / 3 ( r ) = %(f r ) e x P [ ^ 0 0 ; a ( S { r ) " ^ u 0 0 ; a / 3 ( r ) J rP^Jr) - 1 (2.74a) and A m n l _ ,000 / x f . m n l , v _ flAi]mnl , »"] " ^ C ; a ^ r ) ( 2 ' 7 4 b ) 000 HS for mnl*000. We point out that in the RLHNC theory, h^n j r ) = h I r ) for U U ; a p ap a single component system, but this is not true for a multi-component system. Examination of eq. (2.74a) reveals that there is no coupling of the anisotropic projections into c o o ^ a / / r ' ' n t ' i e ^HNC closure approximation. Therefore, the angle-dependent terms of the pair interaction potential between components a/3 can have no effect on 9 ( )0^a / / r ^ ' " ^ ' s deficiency and its consequences will be discussed later. However, comparing eqs. (2.74a) and (2.73) we see that the RLHNC approximation is equivalent to the RHNC approximation for spherically symmetric components. 5. Method of Numerical Solution In the previous two sections we have described how the two equations which compose the RHNC theory can be solved by expanding them in terms of rotational invariants. Hence we have two set of equations, the OZ and the RHNC equations, and two sets of unknowns, the T ? m n ^ fl(r) and c m n } ,,(r) uv;ap u^jap coefficients. The equations must be solved numerically and so the projections, mnl / j a n cj mn (r-), must be represented with a set of discrete points 'uv,ap uv;ap on a numerical grid of width Ar . In this study we will use a value of Ar=0.02ds, which is consistent with previous work [68,70,71,79-81]. It represents a reasonable compromise between data storage, computational requirements and numerical accuracy. The necessary Hankel transforms are performed using fast Fourier transform techniques [126,127] and thus we require 2 n grid points. For the calculations done on pure solvent systems, we find that 512 points are sufficient. For ionic solutions we find that more - 39 -points are usually necessary to accommodate the longer ranged correlations. The number of points varies with concentration as the screening length changes. At 1.0 molar, 1024 points are required, while concentrations between 0.1 and 0.02 molar need 4096 points. The RHNC theory is solved by iterating the RHNC and OZ equations in a manner similar to that used in earlier work [68,81]. The iterative cycle begins with an initial guess for one of the functions. In this study we have chosen . r ) . This guess is usually a converged result from a previous calculation at slightly different conditions (e.g., concentration, total density, temperature, etc.). We then solve the OZ equation to determine a set of coefficients n m n ^ J. r ) . This set along with our initial guess for c m n ^" J. r ) 'uv;a(l 3 3 uv;ap is used as input into the RHNC closure which returns a new estimate for m n 1 c „( r ) . Direct substitution of the new estimate for the initial guess will uvjaf not provide a stable solution unless the initial guess is already very close to the correct result. However, convergence can usually be obtained by mixing successive approximations. The (i + 1) approximation is given by mnl / ^ \ ( i + 1) _ _ mnl , mnl / , - \ ( i ) , mnl mnl , i n e w ) , _ K x + V;a /3 c M p ; a 0 ( r ) ' ( 2 * 7 5 ) where the mixing parameter, a m n ^ „, determines how much of the previous and new estimates are taken and it satisfies 0<a o<1. Separate mixing uv; ap ^ 3 parameters are used for each projection to speed convergence and their values are allowed to increase as convergence is approached. The iteration continues in this manner until a desired state of convergence is attained. However, unlike the mixing procedures used in previous studies in which C ; a 0 ( r ) I81!- o r b o t h V ; a 0 ( r ) a n d cZ)aP{r) [ 6 *J°' 7 1 ] - W e r e m i X e d ' , n this study we have chosen to mix only the functions c . „ ( r ) . For the ' uv;ap model systems being considered here, this is a better procedure. The functions which are actually mixed are short-range c's; they are the projections c m n ^ Q( r ) with the potential terms subtracted, as described in K ' uv;a(S K Appendix A. This method has several obvious advantages. These short-range c's contain no long-range tails due to the pair potential. As a result, they can be readily truncated to reduce storage requirements. The long-range tail - 40 -of Cq^( 12) will always be given exactly, and hence it is relatively easy to obtain solutions after making adjustments to the potential. For electrolyte solutions, particularly at low concentration, 0^ ( 1 2 ) will change very little with a small shift in the concentration, whereas r? J, 12) will show much 'ap larger changes because of its dependence upon the screening length. Experience with the present systems would also indicate that, in general, the convergence is faster and larger changes in parameters are tolerated when c m n 0( r) is being mixed. The one obvious disadvantage is that c a( 12) is uv,afS 3 3 aj3 discontinuous for hard-sphere models. Thus we would expect it to be more difficult to change diameters in a multi-component system. The iterative procedure described above is a task well suited for an automated program. Such a program has been written and was used to generate all the results, both for one and three component systems, presented in this thesis. The program uses the general forms of the multipole potential (cf. eq. (2.10a)) and the OZ and RHNC equations as described in the previous sections of this chapter. This same program is also being used to study several different systems, including liquid crystal models [98], as well as models for pure ammonia [132] and ions dissolved in ammonia. It has been extended in order to study four component systems, in particular systems consisting of a colloidal particle in an electrolyte solution [133]. With only slight modifications, the program is being used to investigate systems of hard ellipsoids [99] and spherocylinders [100]. All one dimensional integrals required by the Hankel transforms are calculated using the trapezoidal rule. However, for the integration in the HNC equation (cf. eq. (2.71)), the trapezoidal rule was found to be inadequate near contact (i.e., r—&an) f ° r some of the systems studied here. Hence, a higher order rule [134] (n=6) was used in this region for these systems. The numerical derivatives needed in the HNC closure were computed using a standard 4**1 order central difference formula [134]. Comparison with results nd obtained using only a 2 order formula showed almost no change. Care must also be taken in computing the binary product in the HNC closure equation. The number of terms in the double sum for a given projection will grow as the square of the total number of projections. Fortunately, for as many as 90% of these terms, the is zero. Even so, for large basis sets several hours on a large computer are required in order - 41 -to calculate a complete set of coefficients. Therefore it becomes uv important to compute the coefficients only once for a given model, storing them in a file in such a fashion so as to avoid storing zero values. A further reduction in storage can be achieved by storing only unique values. It was also found that for very large basis sets, many of the non-zero terms of the double sum could be ignored because they were very small. Another means of saving substantial amounts of time when computing the binary product in the HNC equation is to limit the range in r over which the calculations are done. For most projections, particularly those with larger m or n, the contribution from the binary product is relatively short-range. For the model systems considered here the contribution to most projections is essentially zero after the first 200-300 points. By automating a truncation procedure, the binary product is computed only over that range in r where its contribution to c m n ' ' " „( r) is significantly different from zero. Of course, this range will vary with the projection being considered. A further discussion of computational details and their relationships to basis set will included in Chapter V. When studying multipolar models using integral equation methods, care mn X must be taken in treating the long-range tails in c a(r) due to the electrostatic potential. This is particularly true here, where the ion-ion and some ion-solvent c m n ^" „( k)'s will have divergent behaviour at small k [74,135] due to the long-range nature of the charge-charge and charge-dipole interactions. A further discussion of how we treat these and other long-range tails can be found in Appendix A. 6. A v e r a g e s a n d P o t e n t i a l s o f M e a n F o r c e In section 2 of this chapter we have defined the models we will investigate. The RHNC theory has been described in sections 3 and 4, and the scheme for numerical solution is outlined in section 5. In this section we will examine in detail how average properties of our systems can be calculated once we have solved the RHNC theory. We should again point out that the RHNC is an approximate theory and will give only approximate results for the correlation functions of our system. Hence, any properties determined using these correlation functions will only be estimates of the true values for the - 42 -model system. The RHNC approximation, or any related integral equation theory, will provide us with numerical solutions for h a ^( l2) and 12) which satisfy the OZ and RHNC equations (to within numerical accuracy). General statistical mechanical theory [30,33] tells us that for a model system defined by only a pair potential, knowledge of the pair distribution function, <3ap( 12), is sufficient to completely describe the thermodynamic properties of that system. It can be shown [27,33] from the definition of g f l ^(l2) that the average value, of any mechanical quantity, m a^(l2), associated with the pair a/3 is given by the general expression Ma/3 = ( 8 7 r 2 ) 2 v J g ^ J m ^ J d O ^ a r , (2.76) where V is the volume of the system. Now the total interaction potential for a multi-component system (cf. eq. (2.2)) is given by 1 N a ty u t o t = \ i u i j ( l 2 ) > <2'77> where N f l and N^ are the numbers of particles of species a and /3 in the system. We have, of course, assumed that the system is completely characterized by a pair potential. Using eqs. (2.76) and (2.77) one obtains [27,33,74] an expression for the total average configurational energy i * & W j ^ r f r ' V 1 2 * V ^ d ^ d P ^ d r , (2.78) where N is the total number of particles in the system, p ,^= N/V is the total number density and X f l = N f l/N is the mole fraction of species a. Expanding <3ap( 1 2) and ua^( 1 2) in terms of rotational invariants, and using the orthogonality condition [110,111] /^ l(fi 1,n 2,f)[#^; lMn 1,n 2,f)]*dn 1dn (_1jm+n+l 2 mnl x 2 , „ 2,2 5 6 5,, 6 5 [  {- } ( 8 7 r } 1 (2.79a) mm, nn , 11, MMi f e , |_(2m+1 ) (2n+1 ) (21 + 1 )J - 43 -and we have 0 0 UpGT N — E Ap a/3 a 0 mnl 47rr d r ^ jinn 1 ) 2 (2.79b) (2m+1)(2n+1)(21+1) x f r 2 g m n l Iv) u m n 1 ' ( r ) d r l J0 *uv;ap uv;ap J (2.80) Here we note that g m n l J r ) = h m n l „( r ) for mnl^OOO. We will use eq. (2.80) to calculate all average energies reported in this thesis. Care must be taken in computing some terms such as the ion-dipole energy at infinite dilution and the ion-ion energies. The contribution to the energy from each individual ion-ion pair is divergent. However, for a charge neutral system these divergences cancel and the total ion-ion energy, UJJ, is a meaningful quantity. Also of interest in this study will be the total average energy, U f l ^, of single component pairs within the system. It is usually convenient to express these energies per N q rather than N . After eq. (2.80) we .write U & = 2*a N a *P PP mnl ^^mn1)2 (2m+1)(2n+1)(21+1) f5 2 mnl / \ mnl , \ , x f r g A r ) u A r ) d r (2 .81a) where 1 i f a=0 2 i f (2.81b) Another average quantity we can calculate for our systems is the average pressure as given by the compressibility factor, Z=PV/NkT. From the virial expression for the equation of state (cf. eq. (2.28) of Ref. 27), together with eq. (2.76) we have that Z = 1 -6kT h Xfl^ 1 ( 8 T T 2 ) 2 / r g a / 3 ( l 2 ) 9u „ ( 1 2 ) ap  9r d n , d f i 2 c l r (2.82) - 44 -Equation (2.82) is often referred to as the pressure or virial equation. We determine Z by again expanding 9a^( 12) and U q ^( 12) in rotational invariants. In evaluating eq. (2.82), the required derivative of the multipolar potential is easy to perform analytically, while for the hard-sphere potential we must use the identity [27] 9u A 12) — ^ = - k T 5 ( r = d a j 3 ) , (2.83) where 8 is the Dirac delta function. We must again treat the ion-ion terms and the ion-solvent terms carefully. In general, we can also determine average quantities as functions of particle separation. Let us define Ma/3 ( r ) = \ ^ maV 1 2 )) ( 2 ' 8 4 ) as the average total value of m°^( 12) at r, where <•••> denotes the ensemble average. m°^( 12) is some property of the particles of species /3 to be evaluated at a distance r from a particle of species a, and Na^( r) is the number of particles of species /3 at the separation r. Using eqs. (2.76) and (2.79b), and recalling that [13,27] 4 « S 9 o o ? a ^ r ) d r = < V r ) > ' ( 2 - 8 5 a ) we have the expression M a 0 ( r ) " T T T ^ W - T T S 9 a / / 1 2 ) n £ (12) d^dO, . (2.85b) ( 8 7 r ) 9 00;a /3 ( r ) If we assume there is very little correlation between M i r ) and <N „(r) > ap ap then the average r) per particle is given by < m*(r)> ~ — a/3 <N i r ) > ap l a 2 ,2 1 000 , , J 9 a ^ 2 ) m ° ( 1 2 ) d n l d Q 2 . (2.86) - 45 -In the study of liquids, the average orientations of the molecules as functions of separation are usually of interest. We define <$^ u . a ^ r ) > a s being the average orientation of a particle of species /3 at a distance r from a particle of species a. Expanding ga^( 12) in eq. (2.86) and taking advantage of the orthogonality of rotational invariants as given in eq. (2.79a), we obtain « i jnn l ( r ) > = ( f m n 1 ) 2  M^-a/J (2m+1)(2n+1)(21+1) ,mnl / \ 0 0 6 , * 9 0 0 ; a / 3 ( r ) (2 .87) where we have assumed that m+n + l=even. The average orientation of solvent molecules around an ion is an important property of electrolyte solutions and will be examined in this study. The averages of most interest are and <P1 (cos0£ )> = <cos0^ s> < P 2 ( c o s 0 i s ) > = ! < c o s 2 0 i s > - 1 , (2 .88a) (2.88b) where & s is the angle between the dipole vector and the vector joining the ion, i, and the solvent, s. In Figure 3 we have illustrated our convention in choosing 0.g for positive and negative ions. This convention guarantees that cos0. g will be positive for favourable dipole orientations for both positive and negative ions. For the choice of f m n l given by eq. (2.9a), and using explicit forms for the rotational invariants [61,81], it is easy to show that eqs. (2.88) can be written as <P1 (cos0 + £ . )> ( r ) h 0 0 ; + s ( r ) o „ 0 0 0 T T 3 - 0 0 ; + s ( r ) (2 .89a) and <P1 (cos0_ s )> ( r ) < P 2 ( c o s 0 I £ J > ( r ) h ° 1 1 (r- \ " h 0 0 ; - s ( r ) 2 ^ i s ( r > (2.89b) (2 .89c) where + and - denote the positively and negatively charged ionic species. It - 47 -is also easy to demonstrate that for a random distribution of dipole orientations <P1 ( c o s 0 i s ) > ( r ) = < P 2 ( c o s 0 i s ) > ( r ) = 0 . (2.90) Clearly then, eq. (2.90) must represent the large r limits of these averages. From any introductory textbook in statistics [136], we have that the standard deviation, a, of a distribution, y, is given by a = (<y 2 >-<y> 2 ) 2 , (2.91) where a is a measure of the width of the distribution. Therefore we can easily compute the standard deviation of (cos0.g) using eqs. (2.88) and (2.89). The equilibrium, or static, dielectric constant is another quantity we will calculate since it is an important property of polar solvents and of electrolyte solutions [8,10,81,137]. For a pure solvent, the static dielectric constant, e, is well defined [81,137-140] and is readily measured. For electrolyte solutions the equilibrium dielectric constant, e g , is theoretically well defined [61,81], but the measured dielectric constant, e*(co), diverges at low frequencies, co, of the applied field [10,81]. This is due to the conducting properties of ionic solutions. Hence, one defines [10,81] an apparent dielectric constant, e , for electrolyte solutions given by e a = l i m i t \e*(co) - * £ ] , (2.92) co—> 0 L J where X is the zero-frequency conductivity. However, e is not a true ct equilibrium quantity since several authors [10,21,141] have shown that it contains dynamical contributions. These dynamical contributions are not well understood and e g can not, at present, be unambiguously determined experimentally for electrolyte solutions. Thus, for electrolyte solutions we would not expect exact agreement between the dielectric constants calculated for model systems and those determined experimentally using eq. (2.92). However, the agreement that is obtained should give a further indication of how large the dynamical contributions to e may be. cl In this study we will make use of three different expressions for determining the dielectric constant of a pure solvent. The first is the Kirkwood [138,142] relationship - 48 -( e - l ) ( 2 e + l ) = yg , (2 .93a) 9e where . 2 4irp M y = * , (2 .93b) 9kT in which u is the dipole moment of the solvent. The Kirkwood g-factor can be expressed [61,62] as 4tfp oo 7 . . n  9 = 1 + W h O O ; s s ( r ) d r * ( 2 ' 9 3 c ) It has been shown [143] that the dielectric constant can be obtained through the limit h O O ; s s ( r ) - * A { € ' ] ) 2 3 a s r - * ° ° • ( 2 ' 9 4 ) 47rp syer We will also determine e using the relationship [74] 1 " %KJ?ss(k=0) + 2£JJ;ss<k=°>] ' • jry 7: 7T T . (2.95a) 1 - T [ ? i i ? s s < k - ° > " Soo?s. ( k-°>] where 1 1 9 - 4 7 T M 2 ? o o 2 s s ( k = 0 ) - u s " ( 2 ' 9 5 b ) and it follows from eq. (2.31) that 5 0 0 ? s s ( k = 0 ) " 4 f l r - C r 2 c 0 0 ? s s ( r ) d r ' ( 2 ' 9 5 c ) For electrolyte solutions, eq. (2.95a) is still a valid route [74] to the dielectric constant. However, eqs. (2.93a) and (2.94) are no longer valid because of the Debye screening of ^ Q Q ^ S S ^ ^ a n c ' b O O ^ s s ^ r ^ ' ' - e v e s c 1 u e et al. [74] have shown that for a screened ionic system ^ S - l = yg ' (2.96) where y and g are still given by eqs. (2.93b) and (2.93c). Chan etal. [135] - 4 9 -have also found that e 4TT s Fr Z p- p. Q. a. Iv • i j " i "3 y i *3 1 ] ( 2 . 9 7 a ) where the sums are over ionic species and / h . . ( r ) r 4 d r , 0 1 J ( 2 . 9 7 b ) in which n ^ j ( r ) is understood to be the spherically symmetric ion-ion pair correlation function. Equations (2.97) are known as the Stillinger-Lovett [144] second moment condition. It is important to point out that for the HNC (and LHNC) theory, the three formulas valid for a pure solvent (eqs. (2.93), (2.94) and (2.95)) and the three formulas valid for electrolyte solutions (eqs. (2.95), (2.96) and (2.97)) must, in principle, all yield e consistently for their respective systems. In this study we examine many electrolyte solutions at infinite dilution. In discussing such systems it is convenient to introduce the ion-ion potential of mean force, w- •( r ) , defined by [27] where 9^j( r ) ' s t n e ion-ion radial distribution function and |3=l/kT. At infinite dilution, j( r ) is the potential associated with the solvent averaged force acting between the two ions, i and j . It includes all solvent effects that influence the ion-ion correlations. The ion-ion potential of mean force at infinite dilution is a measure of the free energy change of the system in taking the two ions from infinite separation to some separation r. Pettitt and Rossky [82] have exploited this relationship to determine the entropic and energetic contributions to w^j(r). We point out that w^(r ) at infinite dilution is the solvent averaged ion-ion potential required by McMillan-Mayer theory [26] and it clearly follows from eq. (1.1) that ( 2 . 9 8 ) e r as r —> 0 0 ( 2 . 9 9 ) Thus this effective ion-ion potential could be used to perform McMillan-Mayer level theory for model electrolyte solutions at finite concentration, as was - 50 -done by Pettitt and Rossky [82]. In the HNC theory, we rewrite eq. (2.98) using the HNC equation (2.62a) to obtain However, since all our calculations are done using the RHNC approximation, we will report the ion-ion potentials of mean force as given by which is the correct expression for hard-sphere ions in the RHNC theory. Finally, we note that whenever a dielectric constant, energy, or other average quantity is computed, the required integrations are usually performed using both trapezoidal and Simpson's rule [134]. This is done, in part, to check our numerical accuracy. In most cases it is the result obtained using Simpson's rule which is reported. /5w .j(r) = P u . j ( r ) - T ? . j ( r ) (2.100) (2.101) - 51 -CHAPTER III THERMODYNAMIC THEORY FOR ELECTROLYTE SOLUTIONS 1. Introduction In the statistical mechanical theory of multi-component systems the formalism of Kirkwood and Buff [104] often provides a convenient route to the thermodynamic properties. The Kirkwood-Buff approach is well known [87-89]. It uses grand canonical concentration fluctuation relationships in order to relate certain thermodynamic functions to integrals of the type Ga/3 = 4 H r 2 V r ) d r ' ( 3 * 1 a ) where for notational convenience we use and 9"o0^ ap^ r^ i s t h e r a d i a l distribution function defined in Chapter II. This makes the Kirkwood-Buff theory particularly useful in extracting thermodynamic properties from the integral equation theories discussed in the previous chapter. In this study we would like to be able to apply the Kirkwood-Buff method to model electrolyte solutions in which the solvent has been included as a discrete molecular species. For mixtures of uncharged particles, each species is an independently variable component and the expressions given by Kirkwood and Buff [104] can be directly applied. However, for electrolyte solutions where one has correlation functions between dependent constituents rather than independent components (i.e., the concentrations of individual ions cannot be varied independently), the computational application of the Kirkwood-Buff theory is not immediately obvious. The ambiguity stems from the fact that when charge neutrality conditions are applied, all Kirkwood-Buff expressions [104] for the thermodynamic properties (e.g., the partial molecular volume of the salt, the compressibility of solution, etc.) are indeterminate. This problem has been previously recognized and dealt with by Friedman and Ramanathan [145] for model electrolyte solutions which treat the solvent at the continuum level. - 52 -Several other authors [13,146,147] have employed the Kirkwood-Buff formalism in order to relate the structure and thermodynamics of real electrolyte solutions. However, the results reported are either limited to particular systems and thermodynamic properties or are not applicable to the present study. In this chapter we will use the Kirkwood-Buff expressions to derive more general results for electrolyte solutions. The model we will consider incorporates the solvent as a true molecular species, and hence the relationships obtained are directly applicable to real systems. Although we will only give expressions explicitly for a two component salt/solvent system, the method we will outline in this chapter is totally general and can be readily applied to solutions of more than one salt. We will also examine the low concentration limiting behaviours of our expressions and compare these with macroscopic results obtained through Debye-Hiickel theory [6]. 2. General Expressions The exact formulation of Kirkwood and Buff [104] expresses the thermodynamic properties of a multi-component system in terms of a matrix B. The elements of B are defined by ap 'a a/3 'a'p a/3 ' (3.2) where is given by eq. (3.1a) and p^. N Q / V is again the number density of species a. If we consider a mixture of m species and denote the chemical potential of species a by M q , the partial molecular volume by V and the isothermal compressibility of the system by Xip, then the relevant relationships given by Kirkwood and Buff [104] can be expressed as follows: _V_ kT 3N al = 1 r 9 M a ] | B | (3.3a) xr. kT L 9 N « J T , P , N kTL 3 Pf l jT , a /3 kTXrp (3.3b) - 53 -V~ = P s S - l = i " P « I B I a ' (3 .3C) y L 9 N J T , P f N S 0=1 ^ pi-lye k T x T = ^ |B| , (3 .3d) where S = . J . i '.'/llSLd ' < 3 - 3 e ) |B| is the determinant of B, and \0L\ap indicates the cofactor of the element Bap. Also if we label the solvent as component 1 and the remaining species by integers ranging from 2«««m, the derivative of the osmotic pressure, II, with respect to pa is given by 1 pn 1 m | i r k T L 9 P A J T , M L , P _ M 0 = 2 "P | B 8 P a l ± 2 * L , (3 .4a) where the elements of B' are defined by Bdp = ?a8ae + pappGaP ' 1 ' ( 3 ' 4 b ) It should be emphasized that eqs. (3.3) and (3.4) apply to ionic solutions in only a formal sense. This is because single ion properties can not be evaluated by thermodynamic methods [6,7]. However, the physically meaningful quantities that apply to the electrically neutral salt can be obtained from the single ion expressions. Although the method described below can be applied to any electrolyte solution, we shall write explicit results only for a two component system consisting of a solvent and a salt of the general type . Throughout this chapter, the solute (salt) will be referred to as component 2 and the subscripts s, + and - will denote the solvent and the positively and negatively charged ionic species. Also, it is convenient to introduce the parameter v = v+ + v_, as well as the relationships p + = f + P 2 , P_ = V~P2 and p + =(v+/v_)p_. Also, since the salt molecule is electrically neutral, we have that q + = -(v_/u+)q_. For electrolyte solutions, we have charge neutrality conditions which can be expressed as [74,135,148,149] - 54 -and f p i q i G i j = ~ q j (3. 5a) ? Pi G i s = 0 . (3.5b) For the systems we will consider, eqs. (3.5a) and (3.5b) can be rewritten in the form G+_ = G + + + 1 = G__+ 1 (3.5c) and G + s = G_s . (3.5d) As mentioned earlier, the charge neutrality conditions render indeterminate all thermodynamic quantities obtained by direct substitution into the Kirkwood-Buff equations. Therefore, in order to proceed it is necessary to employ a formalism which allows the charge neutral limit to be taken analytically in such a way that useful determinate expressions are obtained for the thermodynamic properties. One way of doing this in a general systematic manner is described below. We begin by realizing that Gafi = V k = 0 ) E ' ( 3 * 6 a ) where ~ Air 0 0 h a 0 ( k ) = i f S0 r ha/3 ( r ) s i n ( k r ) d r (3.6b) follows immediately from eqs. (2.34a) and (2.35a). At finite ion concentration hfl^( r) is screened and decays exponentially at large r (as will be discussed below). Hence at small k, h ^ k ) can be expanded in the form [73,74,135] V k ) = *a¥  + * 2*aV  + '  ( 3 ' 6 C ) ~( 2) where the second moment, t r^ , is given by eq. (2.97b). Thus we can introduce the matrix B(k) whose elements are B / , ( k ) = p 8 r t + p p „ h / J ( k ) . (3.7) - 55 -Th is then a i l o w s de te rm ina te e x p r e s s i o n s f o r the t h e r m o d y n a m i c p roper t i es t o be ob ta ined by tak ing the k—>0 l imi t o f the appropr ia te k -dependen t quan t i t i es . For the / s o l v e n t s y s t e m w e c o n s i d e r , B(k) has the exp l i c i t f o r m B(k) = p + 2 h * + ( k ) p+p_h+_(k) p + p h (k) P + P_ h +_( k) p_ 2 h!_( k) p_ p s h_ s( k) P + P s h + S ( k ) p _ p s h _ ( k ) p 2 h + ( k ) ( 3 . 8 a ) where h1* (k) = h (k) + 1 aa aa ( 3 . 8 b ) and w e have made use o f the requ i rement that h „{ r ) = ( r ) , in wh i ch ap pa a,/3 = +,-,s. In order to take the requi red k—> 0 l i m i t s , it is n e c e s s a r y to k n o w the s m a l l k behav iour o f the determinant |B(k)| and of the sum SU) = Z P a o ^ \ B ^ ) \ a p ( 3 . 9 ) where again | B ( k ) | a ^ d e n o t e s the c o f a c t o r o f the e lement B f l ^ ( k ) . U s i n g e q . (3.7) together w i t h the charge neut ra l i ty c o n d i t i o n s g i ven by e q s . (3.5c) and (3.5d), one f i nds that at s m a l l k | B ( k ) | ++ = p_ 2p 2 »r(0)c;t(0) r ^ ° h 2 h + _ h s s - [ h + s ] + k^  ["ht<°>h(2) + h<0)h(2) - 2h<° )h ( 2 )l ss +- ss +s -s J ( 3 . 1 O a ) | B ( k ) | + _ = | B ( k ) | _ + = p +p_p c [h<°>] 2 - h ^ h t J O ) + k 2 h! 0 )[hi 2 ) +h ( 2 )] - h<0)h(2) - ht ( 0)K | 2 ) l + +s L +s -s ( 3 . 1 Ob) - 56 -|B(k)|_. = p + 2 p 2 K ( 0 ) K t ( 0 ) . [ K ( 0 ) , 2 (3.10c) |B(k>| = |B(k)| = P + p / p • K ^ ' t h l ^ - h l 2 ' ) (3.10d) |B(k)l - | B ( k ) L . = p.Vp, k2[h<°>[h<2>-h<2> • e't '^-hl2')] • and l < k > l S s - > + 2 <>- 2 k2 [ K i ? ) [ h i 2 ) + h ! ? > - 2 h i ? ) ] ] -( 3 . I 0 e ) ( 3 . 1 O f ) We remark that in the k—»0 limit |B(k)| = |B(k)| = |B(k)| - 0. The ~\~ S s s s explicit forms for the cofactors can then be used to show that as k—> 0 |S(k)| - P + 2 p J V [ h t < ° > h < 0 > - [h<°»]2]Dk: , ( 3 . 1 1 a ) S(k) 2 2 2 P + P- Pc h t ( 0 ) + K ( 0 ) _ 2 K ( 0 ) l D k 2 s s +- +s J ( 3 . 1 1 b ) where D = h<J> + h ! ? > - 2h<?) ( 3 . 1 1 c ) It is also obvious from eqs. (3.11a) and (3.11b) that as k—>0, both S(k) and | B ( k ) | - » 0 . - 57 -First w e sha l l de r i ve exp l i c i t e x p r e s s i o n s f o r the v o l u m e t r i c p rope r t i es . For the present s y s t e m the part ia l m o l e c u l a r v o l u m e of the sa l t , V ^ , is de f i ned by [6] % = = ^ V . + v V , ( 3 . 1 2 ) where V + and V_ are g i ven by e q . (3.3c). It then f o l l o w s that the appropr ia te k -dependen t quant i ty is V 2 ( k ) = v+V+(k) + v_V_(k) , ( 3 . 1 3 a ) where V . ( k ) = L \pn | B ( k ) | . 1 . ( 3 . 1 3 b ) 1 S ( k ) a=+,-,sL ' i a j S ( k ) o.= +,-,s N o w us ing e q s . (3.10) and (3.11b) and c o l l e c t i n g t e r m s , w e f ind that the 2 resu l t ing e x p r e s s i o n con ta ins the fac to r Dk in both i ts numera tor and i ts denomina to r . W e then s i m p l y cance l th is c o m m o n fac to r to ob ta in Jjt(O) _ £ ( 0 ) l imit V ? ( k ) = s s ——L§ _ _ , ( 3 . 1 4 ) o 2 p 2 [ h t ( ° > -5(0) . ^ 0)3 wh ich in the conven ien t no ta t ion g i v e s V 2 = P 2 [ 1 • , S ( G S S + G + _ - 2 G + S ) ] • < 3 - ' 5 > Clea r l y e q . (3.15) is the des i red de te rm ina te e x p r e s s i o n f o r V^ . In a s i m i l a r f a s h i o n w e de f ine the k -dependen t quant i ty V ( k ) . Then inser t ing e q s . (3.10) and (3.11b) and tak ing the k—>0 l im i t , w e ob ta in the part ia l mo lecu la r v o l u m e of the s o l v e n t , G+- " G + s V c = - — — r . ( 3 . 1 6 ) s 1 + p ( G + G ^ - 2 G ^ ) s ss +- +s It is e a s y to see that e q s . (3.15) and (3.16) s a t i s f y the requi red re la t i onsh ip " s V s + P2V2 = 1 ' ( 3 - 1 ? a ) - 58 -From eq. (3.15) we immediately have that l i m i t V 0 = ( 3 . 1 7 b ) Also, since in eq. (3.16) only G + _ is divergent in the limit P 2 — > 0 (as shown below), it follows that Obviously eqs. (3.17b) and (3.17c) represent the correct single component results for the partial molecular volumes. Finally, we note that for the particular case when f + = 1, eq. (3.16) can easily be shown to be equivalent to the expression given by Enderby and Neilson [13]. In general, the isothermal compressibility [27] of a system is given by It should be pointed out that when k appears in the combination kT, as in eq. (3.18), it refers to the Boltzmann constant and is not to be confused with k in the Fourier transform. By analogy with eq. (3.3d), we define the k-dependent isothermal compressibility The k—> 0 limit follows immediately from eqs. (3.11a) and (3.11b), and yields the relationship G + _ + P S ( G + . G - G 2 ) k T X t r = — S + s s ±2-r , ( 3 . 2 C 1 + < > s ( G s s + G + - " 2 CW which agrees with the result previously given by Levesque etal. [74]. Equations (3.15), (3.16) and (3.20) will be used to determine the volumetric properties of the model electrolyte solutions being considered in this study. We also point out that these expressions are totally general and can be applied directly to those solutions which contain only a single salt. ( 3 . 1 7 c ) ( 3 . 1 8 ) k T x T ( k ) = ^ — | B ( k ) | . T S ( k ) " ( 3 . 1 9 ) - 59 -The chemical potential is a fundamental quantity in thermodynamics [6,150] and is particularly useful in describing non-ideal behaviour in solutions [1,5,7]. Using notation consistent with Harned and Owen [6], we express the chemical potential of species a as M a = u°a + k T l n a a = M ° . m + k T l n ( 7 a m a ) = M ° ; c + k T l n ( y a c a ) , ( 3 . 2 1 ) where a f l is the activity of species a , ma and c f l are concentrations expressed as molality and molarity, 7 and y are activity coefficients and u° u° a J a ' a ' arm 0 and u are the chemical potentials of the standard states. Since u must be independent of the concentration scale in which a f l is expressed, the chemical potential of the standard state will contain a term dependent on the choice of scale. Thus we find u° m^ p° and it follows that 7 * y . ' a r m M a ; c 'a 2 a Expressions relating the logarithms of the various activity coefficients can be easily obtained (cf. eqs. (1-8—13)-(15) of Ref. 6) and it can be shown that the differences between the logarithms of the activity coefficients always have a linear dependence on concentration at low solute concentration. We point out that the molarity activity coefficient, y remains unchanged if the concentration scale is expressed as a number density. For the electrolyte solutions being considered here, the chemical potential of the solute (salt), J L ^ , is given by M2 = v+u+ + v - u - > ( 3 . 2 2 a ) where the single ion quantities are defined by eq. (3.21). If we introduce the mean activity coefficient of the salt defined [6] such that y ± = y++y!" , ( 3 . 2 2 b ) then it follows from eqs. (3.21) and (3.22a) that M2 = + k T l n ( j ^ + / - ) + ^ k T l n ( y ± p 2 ) . ( 3 . 2 2 c ) We take the partial derivative of eq. (3.22c) with respect to p 2 holding T and p ox 1 and P, fixed to obtain s 60 -r91ny ± p s o r P TTTf ^ 2 bp. p s o r P ( 3 . 2 3 ) Expressions for the right hand side of eq. (3.23) can now be found by applying the Kirkwood-Buff equations. Using eq. (3.22a) together with the mathematical relationship [ 9 p 2 J j = + f _ ] L 9 P j J p k 5 4 J I = + o r -one immediately finds that ( 3 . 2 4 a ) & - *+2feL* & + + feL * * - 2 & + • ( 3 - 2 4 b ) where in addition to the variables specifically indicated T and p g or P are also held constant. The partial derivatives, (da^/b p^), required in the constant volume case are given by eq. (3.3a). Again, in order to obtain determinate results it is necessary to define 1 kT « | ( k ) = ^ [a£. | B ( k ) | ( 3 . 2 5 ) Substituting eq. (3.25) into the right hand side of eq. (3.24b) and using eqs. (3.10) and (3.11a), then simplifying and taking the k—>0 limit yields the determinate result 1 r 9 M 2 i k T | _ 9 POJT, 1 + p G SS p / [ G + _ + P s ( G s s G + _ - G + J ) ] ( 3 . 2 6 ) From eqs. (3.3b), (3.12) and (3.24b) we can show that the constant pressure derivative can be expressed as _V_ kT ^ 2 ] 9 N 2 J T , P , N kT L 9 P 2 J T , p, kTx^ ( 3 . 2 7 a ) Inserting results from eqs. (3.15), (3.20) and (3.26) we obtain - 61 -k T L 9 N 2 . l T , P , N c p2 [ 1 + ' s ( G s s + G + - " 2 G + s ) ] ( 3 . 2 7 b ) Clearly, eq. (3.27b) does not represent the constant pressure derivative required in eq. (3.23). In order to proceed, we introduce the relationship J _ r ^ l = J L f i ^ l / vl"-^ 2! (3 28a) k T [ 9 p 2 J T f P kTL 9 N2JT,P,N L3N2-rT,P,N ' where N is being implicitly held fixed on the left-hand side of the equation. However, p g can not be held fixed because the volume is allowed to vary. It can easily be shown that Then combining eqs. (3.27b) and (3.28), and using eq. (3.16) we obtain [ w H = i < 1 " P ? v ? > = ^ P e v c • (3.28b) L 9 N2JT, P, N v 2 2 V s s 1 r 9 M 2 ] k T L 9 p 2 j T , : 1 P 2 ^(G + _ - G + s ) ( 3 . 2 9 ) Expressions for the mean activity coefficients now follow immediately from eqs. (3.23), (3.26) and (3.29). Explicitly, we have r-ainy+n _ ] and r 9 1 n y ± i L 9^2 J T , I 1 + P G ^s ss v p 2 [ G + _ + P s ( G s s G + _ - G+p] - 1 - 1 vp2(G+_ - G + s ) (3.30a) (3.30b) Equations (3.30) will be used in this study to calculate derivatives of the mean activity coefficient. Again, these are general expressions for two component salt/solvent systems but results for more complex mixtures could be found with relative ease. - 62 -In a similar manner, we can also derive relationships involving the solvent chemical potential. In particular, we can show that and L9PSJT, _Lf_^2 kT +-kT ° s p 2 p [G, + p (G GA - G. *) ] * ps + - s ss +- +s L 9P sJT,p 0 kTL 9P2- l T ' - G + s P 2[G +_ + p_<G, ss G+- " G + s ) ] (3.31a) (3.31b) Finally, we will consider the osmotic pressure. The derivative of the osmotic pressure with respect to p 2 is given by '3IT I = Z v \— -9P2-JT,MC i=+," " i L gPi. (3.32) where (3II/3p^) is defined by eq. (3.4a). Again direct substitution into eq. (3.32) leads to an indeterminate result when the charge neutrality condition (3.5c) is applied. Therefore, proceeding as before we define the matrix B'(k) (cf. eq. (3.4b)) and the k-dependent derivatives analogous to eq. (3.4a). Substituting the k-dependent quantities for (3II/3p^) into eq. (3.32) and taking the k—>0 limit yields the expression j_rin_i k T|_3p 2J T / P 2 G+- • (3.33) For 1:1 electrolytes this result is equivalent to that given in Ref. 74, although eq. (3.33) is a more general relationship. - 63 -3. Limiting Behaviour In order to determine the limiting behaviour as p^—>u ° f t n e expressions given in the previous section, it is first necessary to deduce the low concentration limiting laws for , G, and G . For continuum level theories of electrolyte solutions only G + _ is relevant and this function has been previously considered by Rasaiah and Friedman [151]. The ion-ion distribution function, g + _ ( r ) , can be written in the form g + _ ( r ) = exp [ - /3w + _(r ) ] , ( 3 . 3 4 ) where w + _ ( r ) is the ion-ion potential of mean force (cf. eq. (2.98)). For both continuum and molecular solvents it is possible to show [61] that as r—>a> and K—>0, w + _ ( r ) -H> 12±_SL e " K r , ( 3 . 3 5 a ) where * - [T&IW]* (3-35B) is the usual Debye screening parameter and e is the dielectric constant of the pure solvent. For a solution containing only a single salt we have K = [jZT l q + q - l  vplY ' ( 3 . 3 5 c ) If we now expand the exponential in eq. (3.34) and keep terms to order 2 [/3w+_(r)] , eqs. (3.1), (3.35a) and (3.35c) yield (see Appendix B) the limiting law G + _ = — + - — = • + • • • , ( 3 . 3 6 a ) where up2 STpZ We emphasize that eq. (3.36a) holds for both continuum and molecular level theories of electrolyte solutions. In order to obtain limiting expressions for G + g and G g s , it is necessary to reintroduce the direct correlation function, 0 ^ ( 1 2 ) , and to apply the OZ - 64 -e q u a t i o n , both o f w h i c h are d e s c r i b e d in the p rev ious chapter . Our a n a l y s i s w i l l requi re that w e k n o w the l ong - range behav iour o f 0 ^ ( 1 2 ) . T h e r e f o r e , w e must res t r ic t o u r s e l v e s to s y s t e m s wh i ch can be desc r i bed by pa i rw i se or e f f e c t i v e pa i rw i se add i t i ve po ten t i a l s . For the present p u r p o s e s , the on l y re levant p ro jec t i ons are hgo^a//1"^" From the O Z equa t i on , as g i ven by (2.36a), w e have that y jmmO 5 z00m m L .u=-m ( - D w h ° m m ( k ) c m 0 m „ ( k ) l O C J ; ay -uOjyP J ( 3 . 3 7 ) where Z ^ m ^ is a nonze ro c o e f f i c i e n t g i ven by e q . (2.36b). O b v i o u s l y , it is the sma l l k dependence o f th is e x p r e s s i o n w h i c h is requi red here. In order to reduce e q . (3.37) fur ther , w e take advantage o f a proper ty of the Hankel t r a n f o r m s , b . „( k) (cf. e q . (2.31)). It can be s h o w n (by us ing exp l i c i t ix v r ap f o r m s f o r the spher i ca l B e s s e l f unc t i ons and expand ing the s i n and c o s t e r m s ) that if b m n l „ ( r ) d e c a y s fas te r than 1 / r 3 , then £ M " ; a / 3 ( k = 0 ) " 0 f o r 1 > 0 ' ( 3 . 3 8 a ) Of c o u r s e , fo r al l e l e c t r o l y t e so l u t i ons at f in i te c o n c e n t r a t i o n s , s c reen ing ensures that hg^a/^ r ) d e c a y s exponen t i a l l y , and hence h S m m „(k = 0) = 0 fo r m>0. U P ; ap ( 3 . 3 8 b ) Thus , if we c o n s i d e r the k—> 0 l imi t o f e q . (3.37), w e can then app ly e q . (3.38b) to ob ta in the e x p r e s s i o n a s - C = I p G C a s 7 a7 7s ' ( 3 . 3 9 a ) where f o r no ta t iona l c o n v e n i e n c e w e have in t roduced -000 2 000 ( 3 . 3 9 b ) Equa t ion (3.39a) is an exact re la t i onsh ip sub jec t on l y to the res t r i c t i on that the po ten t i a l s desc r i b i ng our s y s t e m be pa i rw i se add i t i ve . - 65 -For the s y s t e m s be ing c o n s i d e r e d here w e can exp l i c i t l y wr i te out the t e r m s in the s u m o f e q . (3.39a) and then rearrange to ob ta in G + s ( 1 " p s C s s ) = ( 1 + P + G + + ) c + s + P - G + - c - s ( 3 . 4 0 a ) and G _ s ( 1 - P S C S S ) = (1 + p _ G _ _ ) C _ s + P + G + _ C + S , ( 3 . 4 0 b ) w h i c h , w h e n c o m b i n e d w i t h the charge neut ra l i t y c o n d i t i o n (3.5c), y i e l d the resu l t r " + C + + v_C_ -, ° + S = G " S = 1 ' C P 2 G + - ' ( 3 ' 4 1 ) S SS Equa t ion (3.41) is in fac t the or ig in o f the charge neutra l i ty c o n d i t i o n (3.5d). A l s o , s i n c e e q . (3.41) is ob ta ined w i th the a id o f the charge neut ra l i ty c o n d i t i o n (3.5c), it ho lds on l y fo r p2>0. A t th is po in t it is in te res t ing to note that if w e examine e q . (3.37) at s m a l l k, w e have a re la t ionsh ip ana logous to e q . (3.39a), n a m e l y h0 0 0 (k) = Tc000 (k) + o h0 0 0 (k)c000 (k) n 0 0 ; a s { K ) [ c 0 0 ; a s l K ; p + n 0 0 ; a+K K ; c 0 0 ; + s l K ' trOOO x - 0 0 0 / , xl T 1 0 0 0 s p s c 0 0 ; s s ( k ) + p - b O O ; a - ( k ) g O O ; - s ( k ) ] [ 1 _ y 0 0 - -I • ( 3 ' 4 2 a ) The s m a l l k behav iour o f hnn . .(k) (an apparent 1/k d i v e r g e n c e ) w i l l ' J ~ 0 0 0 d o m i n a t e th is e x p r e s s i o n , and s o it f o l l o w s that n o o « a s ^ k ^ m u s t a ' s o n a v e the s a m e s m a l l k dependence . T h e r e f o r e , at large r we can wr i te that 0 0 0 h! and 0 0 ; i s ( r ) " * a i s p 2 h + - ( r > ( 3 . 4 2 b ) C ? s s ( r ) a s s p 2 2 h + - ( ^ . ( 3 . 4 2 c ) whe re a is a cons tan t dependent on the va lues o f c?5u „ ( k ) at s m a l l k. a s „ „ „ . „ U u j a s 0 0 0 0 0 0 ' W e remark that if c n n . . ( r ) = c n n . _ ( r ) (i.e., the ions are s o l v a t e d UU;TS UU, S equ i va len t l y ) , then a a s = °- T n e l o n g - r a n g e ta i l s in ^ 0 0 * a s ^ r ^ t h a t a r e a resu l t o f e q s . (3.42b) and (3.42c) w i l l be d i s c u s s e d in more deta i l in later chap te rs . 6 6 Inspection of eqs. (3.41) and (3.36a) shows that at low concentration G + s = G _ s will have a term due to P 2 G + _ which varies like \fp^. We might also expect C ^ s (i = + or -) to have a \/p~^ dependence at low concentration, so we write ' I S = " + C + S + ^ _ C _ S = (v+cls+ p _ c f s ) + + . . . , (3 .43) where the superscript o indicates the infinite dilution result. It is not possible, at present, to obtain an exact expression for the slope S However, the HNC equation (2.62a) can be used to find an approximate form for S c . We start by expanding the logarithm in eq. (2.62a) for large r and using eq. (2.25), which immediately yields the result ap ^ [ h a / 3 ( 1 2 ) ]2 - 0u a / 3 (12) as r — ( 3 . 4 4 ) We then expand c(12), h(12), and u(12) in rotational invariant as in Chapter II. It is possible to deduce, with that aid of eq. (3.42b), that as p2—> 0 and ,000 -oo 1 r > 0 l 1 , . .2 fll,000 ( r . , 6 [ h 0 0 ; i s ( r ) ] " ^ u 0 0 ; i s ( r ) (3.45) Thus for nonpolarizable particles the first term in eq. (3.45) is the leading concentration dependent term and it is sufficient to determine the limiting HNC slope for C ^ g . It is known from the work of Hcfye and Stell [61,152] that in the limits p 2 — > 0 and r — > t » t h 0 0 ; i s ( r ) + /cr -/cr (3.46) where y has been defined in eq. (2.93b). Now substituting eqs. (3.45) and (3.46) into eq. (3.39b), it is possible to show, after considerable manipulation (see Appendix B), that as p2—>0, i s C? is «2 [e-\' „ 2 2 2 ? i M 9 / K + (3.47) - 67 -C o m b i n i n g e q s . (3.43) and (3.47) a long w i t h e q . (3.35c), and then rearranging g i v e s - Kv* ( e - 1 ) 2 Sc = oye • (3 .48 ) A g a i n w e note that e q . (3.48) is a p p r o x i m a t e , and fu r thermore ho lds on l y fo r f l u ids o f nonpo la r i zab le p a r t i c l e s . The accu racy of the HNC e s t i m a t e fo r S c w i l l be d i s c u s s e d b e l o w . F rom e q s . (3.36a), (3.41) and (3.43) w e have the l im i t i ng l aw G+s - G-s • < + [^  + ^ <"+C°s+ "-c-s'K ' (3-«a> where G° = G° = l imit G + = X _ ± s ^ . ( 3 . 4 9 b ) W e po in t out that p2—>0+ is the appropr ia te l imi t here s i n c e e q . (3.41) ho lds on l y f o r P 2 > 0 . It is in te res t ing to app ly the in f in i te d i lu t ion l imi t (i.e., p+ = p_ = 0) to e q s . (3.40) to ob ta in 1 — P L S S S and G ° e = 2— . ( 3 . 5 0 b ) 1 P L ^S SS It is o b v i o u s that these e x p r e s s i o n s do not agree w i th e q . (3.49b), and hence Gj_ and G must be d i s c o n t i n u o u s at p 0 = 0. M o r e o v e r , it f o l l o w s f r o m eqs . + S _ s 2 ' ^ (3.49b) and (3.50) that < = G ° s = ( , + G : s + , _ G f S ) A . ( 3 . 5 ! ) C l e a r l y , G?_ and G°_ are just the w e i g h t e d ave rages o f G°_ and G°_ In ~ 0 0 0 te rms o f the Four ier t r a n s f o r m s , ^ o O ' * i s ^ k ^ ^ = + o r ~)> t ' 1 ' s d i scon t i nuous behav iour can be e x p r e s s e d in the f o r m - 68 -l i m i t l i m i t h ° S ? . ( k ) * l i m i t l i m i t h ° ° ° . (k) . (3.52) p 2 ^ 0 k - ^ 0 U U ' 1 S k - ^ 0 P 2 - ^ 0 U U ' 1 S The left and right hand sides of eq. (3.52) give eqs. (3.49a) and (3.50), respectively. We note that this discontinuous behaviour in (for a charged-uncharged pair) was also indicated in earlier considerations [149] of this function at the second virial coefficient level. The limiting form for G can also be obtained by considering the OZ equation as expressed by eq. (3.39a). It can be shown that (p.C,c + p C )G. e + C G s s = + + S - + s SS- , (3.53) S S 1 " p s C s s from which it follows that at low concentration C° G s s = , S S r o + 0 ( " 2 > ' ( 3 ' 5 4 ) 1 - p C ^S s s We also point out that 1 1 ' o = 1 + » s G s s • p s k T 4 ' ( 3 ' 5 5 ) 1 — P ^S s s where Xip* is the isothermal compressibility of the pure solvent. Equation (3.55) is a well known result [27], which can be obtained from the one component limit of eq. (3.3d) or from the p2—>0 limit of eq. (3.20). We can now examine the limiting behaviours of the thermodynamic functions discussed in the previous section. First let us consider the mean activity coefficient. Substituting eqs. (3.36a), (3.49a) and (3.54) into eqs. (3.30), it can be shown that in the limit p2—>0, « ] _ « ] _ . ( 3 . B 6 ) Equation (3.56) agrees with the derivative of the usual Debye-Huckel limiting law [6] for l n 7 + , where it follows from the discussion in the previous section that the limiting law for the mean activity coefficients must be independent of the concentration scale. - 6 9 -Now let us consider the limiting behaviour of the derivative of the osmotic pressure. Combining eqs. (3.33) and (3.36a) we immediately obtain T-\¥l-\ ~* V-F= ~> v(\-h}/VpZ) . (3.57) We recall [150] that the osmotic pressure is a measure of the change in chemical potential of the solvent due to the presence of the salt. A more frequently used measure of the solvent chemical potential is the osmotic coefficient [6,7], <t>, defined by where M is the molecular weight of the solvent m is the concentration in s 3 molality and a g is the solvent activity. If we assume the solvent is incompressible over the pressure change of II, then the osmotic pressure is related [6,7] to the osmotic coefficient by j>kTM n = I r - 4>m . (3.59) i o o o v s The limiting law for <p (cf. eq. (3-5-12) of Ref. [6]) can be written as <p —» 1 - |A i / pp 2 as p 2 —»0 . (3.60) At very low concentration eq. (3.59) can be rearranged to give n = i > k T 0 p 2 . (3.61 ) Now substituting eq. (3.60) into (3.61) and taking the derivative with respect to P 2 , we again obtain eq. (3.57). Clearly, the limiting behaviours of the osmotic pressure and of the osmotic coefficient are simply related. The limiting behaviour of the partial molecular volume, V 2 , is of particular importance and requires careful attention. From eq. (3.15) and the limiting expressions (3.36a), (3.49a) and (3.54), it is possible to show that as - 70 -V„ 1 + p G ^s s s p s p 2 G + - • ] " +s P2G+_ or using eq. (3.41) f 1 + P S G S 5 l _ P V j H S LP SP 2G +_J L 1 -v, C.B + v C n + s - - s 1  p C Hs s s (3. 6 2 a ) ( 3 . 6 2 b ) Now applying eqs. (3.36a), (3.43) and (3.55) we obtain the limiting law expression where V 2 = V 2 + \^~2 V° = * + k T X J ( l - p_CJ ) + r . k T x J O - P.C° ) ( 3 . 6 4 a ) s +s s - s (3.63 ) p k T x ^ " pkTx°?{p+C0++ v_Q°_ ) and S„ = AkTp 2 • kV2 ( 3 . 6 4 b ) ( 3 . 6 5 ) We remark that as one would expect, splits into two independent terms which depend upon the interaction of the positive and negative ions with the solvent (cf. eq. (3.64a)). It is also interesting to note that V?, can be written as the sum of two terms, only one of which depends upon the ion-solvent interactions, as in eq. (3.64b). It is very instructive to compare eq. (3.65) with the exact macroscopic (i.e., Debye-Hiickel) result for S which can be expressed in the form [6] S v = AkTV 2 •x£ + 3 L 9 P JT ( 3 . 6 6 ) where again e is the pure solvent dielectric constant. Clearly, our microscopic result for S as given by eq. (3.65), is functionally equivalent to eq. (3.66). Comparing eqs. (3.65) and (3.66) we obtain the differential equation - 71 -f a ine l L 9 P JT ( 3 . 6 7 ) If we introduce the identity [oVlne] = 1 [3e 1 L 9P J T = e[dps] T P S X T ' ( 3 . 6 8 ) which follows from eq. (3.18), we can then rewrite eq. (3.67) as ( 3 . 6 9 ) We note that at least for systems characterized by pairwise additive potentials, eqs. (3.67) and (3.69) are exact expressions. If the HNC result for S c (i.e., eq. (3.48)) is substituted into eq. (3.69), then we obtain the differential equation This is exactly the equation obtained by Rasaiah era/. [153] in their consideration of electrostriction in polar fluids at the HNC level. Equation (3.70) integrates to give [153] which is the Debye approximation for the dielectric constant [61] of the pure solvent. Of course, eq. (3.71) is not a very accurate theory and overestimates e. This means that one cannot expect the HNC theory to give very accurate values for S y since the HNC approximation appears to overestimate the effect of electrostriction. Rasaiah [154] has shown that when bridge diagrams missing in the HNC approximation are included in the closure, improved results are In the LHNC theory, it clearly follows from eq. (2.74a) and eqs. (3.39b) and (3.43) that S c =0 . As discussed earlier, this is a result of the lack of coupling between the anisotropic potential terms and the radial distribution function in the LHNC closure equation. This is consistent with the observation of Rasaiah era/. [153] that the LHNC approximation does not predict ( 3 . 7 0 ) obtained. - 72 electrostriction in polar fluids. We now examine the low concentration behaviour of the partial molecular volume of the solvent, V First we rewrite eq. (3.17a) as Vs = " o2V • ( 3 - 7 2 ) Then inserting eq. (3.63) into eq. (3.72) one immediately has the limiting law Vs = 7 T ( 1 " ^ * { 3 ' 7 3 ) s Clearly, V g has a linear dependence on p 2 at low concentration and its limiting slope will be almost totally determined by unless p g has a strong P 2 dependence. Finally, from eqs. (3.20), (3.36a), (3.49a) and (3.54) we can show that at low concentration X T —> xj + 0 ( p 2 ) . (3.74) Thus we find that the compressibility also has a p 2 dependence in the limit p2 It also possible to deduce limiting laws for some of the average energy terms. In order to simplify the expressions slightly, we will consider only symmetric electrolytes (i.e., v+= v_= 1). Using eq. (2.81a), eq. (2.10b), and eqs. (3.34), (3.35) and (3.46) we can show that at low concentration ^ = " 2 p f f | g + g j ^ (3.75, and Nj N. e -€ i ] C T l q + q . l * ^ . ( 3 . 7 6 ) where U J J / N ^ is the total average ion-ion energy per ion and U ^ / N ^ is the average ion-dipole energy per ion. In a similar manner, we can obtain an expression for the dipole-dipole energy by inserting the long-range low 1 1 2 concentration form for hfjO'SS^ 1^ (cf- eci- (2.35a) and Ref. 61) into eq. (2.81a). Integrating and then simplifying yields the result - 73 -X - X + - 'MM i W ^ • «3 .77> It is interesting to note that for systems in which the pure solvent has a large dielectric constant, the limiting slopes of the first two energy terms, i.e., eqs. (3.75) and (3.76), are almost equal in magnitude but opposite in sign. The limiting slope of the dipole-dipole energy (the last term in eq. (3.77)) is also very similar in form, and since there are two ionic species present, it will be almost cancelled by the two ion-dipole terms. In subsequent chapters, in particular Chapter VI, we will use the expressions derived in section 2 of this chapter to compute the various thermodynamic properties of the electrolyte solutions being studied. We will also test the validity of the limiting laws given in this section and examine the ranges over which they hold. - 74 -CHAPTER IV MEAN FIELD THEORIES FOR POLARIZABLE PARTICLES 1. Introduction In Chapter II we have developed a theoretical approach and the necessary methodology with which to study liquid systems of several components. One requirement of this theory was that the total interaction potential of the system contain only pairwise additive terms (cf. eq. (2.2)). As mentioned earlier, most statistical mechanical studies of fluid systems [33] use only pair potentials to describe the interactions within the systems being investigated. However, recent studies of polar-polarizable fluids using both approximate theories [61,67,155-158] and computer simulations [158-162] have shown that the many-body interactions due to molecular polarizability are important in determining the equilibrium properties of the systems. The importance of polarization effects in water [35-38,67] and in electrolyte solutions [38,54] is now well known. Thus, in this study we have chosen to include polarizability in the models we will consider. In general, the many-body problem of polarizability is difficult to treat. In the current theoretical framework (i.e., the RHNC theory as described in Chapter II) it is not possible to treat it exactly. Fortunately, recent work [61,67,156] has demonstrated that it is possible to take into account the influence of many-body interactions due to polarizability through effective pair potentials. The self-consistent mean field (SCMF) approximation [67] has been shown [158,163] to be an accurate means of reducing the many-body potential when applied to a fluid of polarizable particles with dipole and square quadrupole moments. For purely dipolar fluids, the SCMF approximation is equivalent to the 1-R theory of Wertheim [156]. Unlike other methods, however, the SCMF theory uses approximations that are distinctly physical in nature. It is also easily generalized to include contributions to the average local electric field from higher order multipole moments. In section 2 of this chapter, we will extend the SCMF theory of Carnie and Patey [67] to include ion and octupole terms in order to facilitate its application to the systems being studied here. - 75 -in the SCMF theory the pairwise additive potentials, which result from the reduction of the many-body interactions of a polarizable system, are written in terms of an effective permanent dipole moment, m as described below. This effective dipole moment is an average molecular property of the system, that is, it is the same for all molecules. It will depend upon the polarizability and permanent multipole moments of the model, as well as upon other properties of the model and the state parameters of the system. Within the SCMF appproximation, the systems characterized by this effective dipole moment will have the same structural and dielectric properties as the true polarizable fluid. The SCMF theory has been previously used [79-81] in the study of model electrolyte solutions at infinite dilution. There the effective moment, m must simply be that of the pure solvent. At finite concentrations we might expect the effective dipole moment to vary due to the presence of the ions and the resulting changes in the solvent structure. We know that at small separations (~3A) an ion is surrounded by an intense electric field g (~10 V/cm.) which will greatly alter the local solvent structure. Hence, the average local solvent electric field might be expected to change appreciably. Moreover, we would also expect an ion itself to significantly alter the local field in its immediate vicinity. Thus it would be very interesting to be able to examine the average local electric field experienced by a solvent molecule in solution as a function of its separation, R, from an ion, and thereby determine the R-dependence of the average dipole moment of a solvent molecule. If we consider an ion and only one polarizable solvent molecule, it can be shown [139,140] that the dipole moment, p, induced in the molecule will be given by 2 p = aq/r , (4.1a) where a = - jTr a (4.1b) is the isotropic polarizability of the molecule, a being its polarizability tensor, and q is the charge on the ion. It immediately follows that the interaction between the charge and the induced moment is - 76 -u q p(r) = - - j . (4.1c) H o w e v e r , in s o l u t i o n an ion is sur rounded by more than one so l ven t m o l e c u l e . These m o l e c u l e s w i l l order in s o m e f a s h i o n a round the i o n , and this must g i ve r ise to l oca l changes in the e lec t r i c f i e l d around the i on . A t f i n i te concen t ra t i on the ion w i l l a l s o be sur rounded by other i ons w h i c h w i l l tend to s c r e e n its charge and again alter the loca l e l ec t r i c f i e l d . T h u s , w e w o u l d expect e q s . (4.1a) and (4.1c) to be poo r a p p r o x i m a t i o n s in a dense s y s t e m such as an e l e c t r o l y t e s o l u t i o n . In th is chapter w e w i l l desc r i be t w o d i f fe ren t l e v e l s o f theory in wh i ch a po la r i zab le s o l v e n t may be s tud ied . The f i rs t is the S C M F app rox ima t i on [67] wh i ch w e w i l l ou t l ine in s e c t i o n 2. In s e c t i o n 3 o f th is chapter w e w i l l d e v e l o p a s e c o n d and more de ta i led f o r m a l i s m through w h i c h w e can e s t i m a t e the average l oca l f i e l d expe r i enced by a s o l v e n t at a d i s t ance R f r o m an i on . A s in the S C M F theo ry , w e w i l l f o l l o w a mean f i e l d app roach {i.e., w e w i l l ignore f l uc tua t i ons ) . Th is R -dependen t mean f i e l d (RDMF) theory g i ves r i se to an e f f e c t i v e spher i ca l po ten t ia l be tween the ion and the s o l v e n t at R. M o r e o v e r , th is spher i ca l po ten t ia l is f ound to have an e f f e c t on the l im i t i ng l aws fo r t h e r m o d y n a m i c quan t i t i es , such as V ^ , w h i c h depend upon the i o n - s o l v e n t c o r r e l a t i o n s . 2 . The Se l f -Cons i s ten t Mean Field Theory The S C M F theory of Carn ie and Pa tey [67] reduces the m a n y - b o d y p r o b l e m o f po la r i za t i on into a p rob lem i n v o l v i n g an e f f e c t i v e pa i rw i se add i t i ve po ten t i a l . It d o e s s o by ignor ing f l uc tua t ions in the l oca l e lec t r i c f i e l d . The f o l l o w i n g is s i m p l y an e x t e n s i o n of the S C M F a p p r o x i m a t i o n [67] to inc lude oc tupo le and ion f i e l d con t r i bu t i ons thus mak ing it app l i cab le to the s y s t e m s o f in terest in th is s tudy . In gene ra l , w e w i l l c o n s i d e r a s y s t e m w h i c h c o n t a i n s three mo lecu la r s p e c i e s , one o f these be ing a p o l a r - p o l a r i z a b l e s o l v e n t . The t w o ion ic s p e c i e s , des igna ted + and - , are a s s u m e d to be s i m p l e sphe r i ca l ions p o s s e s s i n g o n l y charges and no higher order mu l t i po le m o m e n t s . The po la r i zab i l i t y o f t hese - 77 -ions must also be spherically symmetric. When solvated, these ions will experience no net average polarization since the average electric field generated at the centre of such an ion by the surrounding solvent must be zero. Therefore, we can ignore the polarizability of simple spherical ions at a mean field level. (Of course, we have already chosen to ignore all dispersion terms for the models we will consider.) For the solvent, we start by writing the total instantaneous dipole t h moment of the j solvent molecule as mj = Mj + 2j > (4.2a) where p. is the permanent dipole moment of the solvent, Pj = a - f B ^ j (4.2b) is the instantaneous induced dipole moment and (E^)j is the total instantaneous electric field felt by the solvent j. If we let <E-^> be the average electric field experienced by a solvent molecule, then the average total dipole moment, <m>=m', (measured in the molecular frame) of each molecule is given by m' = M + a- < E 1 > . (4.3) For molecules of C2y (or higher) symmetry in an isotropic fluid, we know that <E-^> will be non-zero only in the direction of This immediately implies that fx, <E^> and m' are all in the same direction. As a result, we can write < E 1 > = C(m')m' , (4.4) where the scalar C(m') will depend upon the properties of the system. If one inserts eq. (4.4) into eq. (4.3) and then iterates the result with itself, one obtains m' = M + C (m' )a '«M , (4 .5a) where we define a? = a + C (m')a-a' (4.5b) as being a renormalized polarizability. This renormalized polarizability of a - 78 -molecule in a fluid of polarizable particles plays the same role as a does for an isolated molecule. It will describe the fluctuations of the total dipole moment of a polarizable molecule in solution about its mean value and thus one can write <m2> = m'2 + (<p 2> - <2> 2 ) = m'2 + 3a'kT , (4 .6a) where a' = j T r a ' (4.6b) 2 and <m > is just the mean square dipole moment of the solvent. In order to determine an expression for the scalar C(m'), we now examine the configurational energy of the polarizable system. For the system we are considering, the instantaneous conf igurational energy is given by UN = "HS + U I I + UIQ + "lO + + MgO + MDO iffflj-(£lD>j " 5 m j -(E 1 Q)j - L n . . ( E 1 0 ) . - ^ y ( E n ) . + - ^ ' ( E ^ , (4.7) in which the sums over j are over the number of solvent molecules N and s' where { * l > j = ( = I D > J + ( E-lQ>j + (5l0>j + % i > j ' <4'8> —ID' — 1Q' —10 a n C ' —II a r e t ' i e ^'P 0 ' 3 1 "- puadrupolar, octupolar and total ionic field contributions, respectively. In eq. (4.7) u^g is the instantaneous hard-sphere energy and Uj j , U J Q , ^ Q ' ^ 0 a n C' HDO a r e t' i e t o t a ' instantaneous ion-ion, ion-quadrupole, ion-octupole, quadrupole-quadrupole, quadrupole-octupole and octupole-octupole energies, respectively. The first term of the second line of eq. (4.7) is the total dipole-dipole contribution to the energy, followed by the dipole-quadrupole, dipole-octupole and total ion-dipole terms. The last term in eq. (4.7) is the energy of polarization. Since all the solvent molecules present are equivalent, the total average energy can be expressed as - 7 9 -^TOT = U t " K < ^ 1 D > " N s < ^ ^ l Q > " N s < i ^ l O > " Ns<if5ll> " l N S < 2 - E l Q > " K < 2 - E 1 0 > " i N s < e - E n > , (4 .9a) where ° t = + U I I + U I Q + U I 0 + UQQ + UQO + U 0 0 • ( 4 ' 9 B ) In the SCMF theory one ignores fluctuations in the electric field and assumes that <P_*E^>= < £ > • <E^>. Using this assumption and the fact that for molecules of C 2 v symmetry the average electric field must always be directed along the permanent dipole moment, we can rewrite eq. (4.9a) in the form ^ O T = U t " i N s " < E l D > " ^ S ^ ' ^ ^ ^ I Q ^ ^ I O ^ ^ I I ^ • ( 4 ' 1 0 ) If we now let C(m') = ( y m ' ) + CQ(m') + CQ(m') + Cjdn') , (4.11) then it clearly follows from eqs. (4.4) and (4.8) that <E_ 1 D> = CpdnOm' , (4.12a) < E 1 Q > = CQ(m')m' , (4.12b) < E 1 Q > = C0(m')ni' (4. 1 2c) and < E 1 : > = CjdnOm' . (4. 12d) Inserting eqs. (4.12) into eq. (4.10) yields - ^Nsm'(m'+M) [cQ(m')+C0(m') +Cj(m')J . (4.13) - 80 -To obtain estimates for the average local fields, or for C(m'), we now define an effective system characterized by an effective pair potential. This system should have very similar structure to the polarizable system (the structures must become identical if fluctuations are unimportant), and hence one assumes that they will have the same local fields. For this effective system the instantaneous conf igurational energy will be given by P i 2 where the sum over j and k is over all particles of all species present. The second term in eq. (4.14) represents the polarization energy which will be non-zero only for solvent molecules. The pair potential, U J ^ ( 1 2 ) , is described in detail in Chapter II. Of importance here are the terms (as given by eq. (2.10b) and eqs. (2.17), (2.19) and (2.20)) which involve the dipole moment, including u00;ss (r) = -/nJ73[-ij^] r ( 4 . 1 5 a ) "oiUr) - -^KH ' U- I 5 B ) and n 1 1 q- i m k where m. and are the total instantaneous dipole moments of solvents j and k. To facilitate the simplification of the pair potential, one again ignores 2 2 fluctuations in the local field. In eq. (4.14) we replace p. by <p >. Consequently, the polarization energy becomes a constant term which can be dropped without altering the physics of the effective system. In eq. (4.15a) we use 2 2 m. m, = <m > = m , (4.16) J K 6 where m g is an effective permanent dipole moment. In all other potential - 81 -terms involving the dipole moment (e.g., eqs. (4.15b)-(d)), one could replace m. by m'. However, the effective potential would then involve two parameters, m' and m and considerable effort would be required to determine the self-consistent average local field for the polarizable system. Therefore it is convenient, from a computational viewpoint, to ignore the difference between m' and m g (for the systems we will consider this difference is only 2-4%) and to use m g in place of m' in the effective pair potential. Thus the pair potential of the effective system becomes equivalent to that of a nonpolarizable system in which we have replaced the permanent dipole moment by m . 7 e We have yet to determine an explicit form for C(m'). For the effective system we know that = - i N s m e < E l D > e ' { 4 ' 1 7 a ) "DQ = - N s m e < E l Q > e ' ( 4 ' 1 7 b ) and "DO = - N s m e < E 1 0 > E ( 4 ' 1 7 C ) °ID = - N s m e < E l I > e ' ( 4 ' 1 7 d ) where the superscript e indicates the effective system. Since the effective and polarizable systems are assumed to have the same structure, it immediately follows that < E 1 Q > E = < E 1 Q > ( 4 . 1 8 a ) and similarity for <E^Q> and < E ^ j > , but m < E 1 D > " S T < B 1 D > * ( 4 - L 8 B ) Combining eqs. (4.18) and eqs. (4.17) and then using eqs. (4.12), one simply rearranges to obtain the desired expressions CJm') = ' - ^ 2 - , (4 .19a) s e - 82 -and C(m') = _ ^ Q _ , (4.19b) Q N m m' s e Cn(m') = D O (4.19c) s e - u e CT(m') = . (4.l9d) 1 N m m' s e We note that although the effective and polarizable systems have the same structure, their total average energies must be different since in the effective system we have ignored the polarization energy. Substituting eqs. (4.19) into eq. (4.13) we find that the energy of the polarizable fluid is given by «TOT - " t * • ^ . [ ^ • l £ • 0« D ] . (4.20, e e Patey etal. [163] have also shown that the dielectric constant, e, of the polarizable system is simply that of the effective system and does not depend upon the method used to obtain the properties of the effective system. In order to solve the SCMF theory, we must first determine the average energies of the effective system at several values of m g while all other parameters are held fixed. Of course, in this study we employ the RHNC theory, as described in Chapter II, to perform this task. The energies must then be accurately fit to interpolating curves (here we have used cubic splines [134]) so that C(m') is known as a function of m For given values of u, a and x s = N s / N , eqs. (4.5), (4.6) and (4.19) are solved iteratively, to give values of m', a' and m g that are consistent with the given molecular and state parameters. - 8 3 -3 . The R-Dependent Mean Field Theory The R D M F theory , as ou t l i ned b e l o w , is d i rec t l y app l i cab le on l y to s o l u t i o n s o f sphe r i ca l l y s y m m e t r i c i o n s , a l though e x t e n s i o n s to more genera l s y s t e m s m a y be p o s s i b l e . In the f o l l o w i n g de r i va t i on w e res t r ic t o u r s e l v e s to a te t rahedra l s o l v e n t m o d e l , that is a so l ven t mode l w i t h o n l y a d i p o l e and a square quad rupo le , as d e s c r i b e d in Chapter II. For s i m p l i c i t y , we w i l l aga in c o n s i d e r s o l u t i o n s of on l y a s i ng le sal t in wh i ch the ions are s p h e r i c a l l y s y m m e t r i c . A l s o , al l pa r t i c les w i l l be t reated as hard sphe res where the h a r d - s p h e r e con tac t d i s t a n c e s , &ap, are g i ven by e q . (2.24b). The p r o b l e m w e w i l l be a d d r e s s i n g w i l l be that of an ion be ing i m m e r s e d in to a m u l t i p o l a r - p o l a r i z a b l e so l ven t wh i ch may or may not con ta in other i ons . The R D M F theory w i l l examine the changes in the average loca l f i e l d w h i c h are a resul t o f the p resence o f an i on . C o n s i d e r i n g on l y the c a s e of a s p h e r i c a l l y s y m m e t r i c ion a l l o w s us to take advantage of the fac t that the f lu id su r round ing the ion must be i s o t r o p i c f r o m the v i e w p o i n t o f the i on . Thus , w e need o n l y examine the dependence o f the average loca l f i e l d upon R, the d i s tance f r o m the i on . Fu r the rmore , al l add i t iona l average f i e l d s genera ted in the su r round ing f l u i d , due to the p resence o f the i on , must be d i rec ted rad ia l l y at or a w a y f r o m the i o n , and hence appear as though they are be ing p roduced by add i t i ona l (sc reen ing) charges p laced at the cent re o f the i on . W e w i l l de te rm ine the average l oca l f i e l d s at R by c o n s i d e r i n g the average in te rac t ion b e t w e e n the d ipo le o f a so l ven t par t i c le f i xed at a d i s tance R f r o m the ion and al l o ther pa r t i c les in the s y s t e m . The average in te rac t ion can then be e a s i l y re la ted to the average l oca l e lec t r i c f i e l d s . The e l ec t r i c f i e l d expe r i enced by a so l ven t m o l e c u l e at a d i s tance R f r o m an ion w i l l , in genera l , have both ion and s o l v e n t c o m p o n e n t s . For the e l e c t r o l y t e s o l u t i o n s be ing c o n s i d e r e d here , the average loca l e lec t r i c f i e l d at R w i l l be g i ven by ^ ( R ^ = <E 1 D(R)> + <E 1 Q(R)> + <E n(R)> . ( 4 . 2 1 ) W e poin t out that th is e x p r e s s i o n is just an R -dependen t ana logue of e q . (4.8). W e n o w rewr i t e e q . (4.21) in the f o r m <E^(R)> = <E X> + <AE1(R)> , ( 4 . 2 2 a ) - 84 -w h e r e < E ^ > is just the average loca l f i e l d of the bulk s o l u t i o n as g i ven by the S C M F theo ry . < A E - ^ ( R ) > is the c o r r e c t i o n to the bulk f i e l d w h e n a s o l v e n t par t i c le is at a d i s tance R f r o m an i o n , that is to s a y , it g i v e s the change in the average loca l f i e l d at R due to the p resence of the i on . For the present s y s t e m we can exp ress th is c o r r e c t i o n te rm as < A E 1 ( R ) > = < A E 1 D ( R ) > + < A E 1 Q ( R ) > + < A E N ( R ) > . ( 4 . 2 2 b ) A s in the S C M F t h e o r y , w e w i s h to de te rm ine the average to ta l m o l e c u l a r d i po le m o m e n t . In the present con tex t , w e have m ' ( R ) = JX + a - < E 1 ( R ) > , ( 4 . 2 3 ) whe re m ' ( R ) is the appropr ia te R-dependen t quant i t y . U s i n g e q s . (4.3) and (4.22a) it c l ea r l y f o l l o w s that m ' ( R ) = m' + a - < A E 1 ( R ) > , ( 4 . 2 4 a ) w h i c h w e then wr i t e as m ' ( R ) = m' + A 2 ( R ) , ( 4 . 2 4 b ) whe re Ap_(R) is the average e x c e s s induced m o m e n t g i ven by A £ ( R ) = a . < A E 1 ( R ) > . ( 4 . 2 5 ) In the present theory w e w i l l f i nd it conven ien t to exp ress < A E ^ ( R ) > in an i n te rmo lecu la r r e fe rence f r ame in w h i c h the z - a x i s is a long the i o n - s o l v e n t v e c t o r . Hence , i ns tead of e q . (4.25) w e use A E ( R ) = a < A E 1 ( R ) > . ( 4 . 2 6 ) whe re the average p o l a r i z a b i l i t y , a , o f a s o l v e n t m o l e c u l e is de f i ned by e q . (4.1b). W e po in t out that fo r the wa te r m o l e c u l e a is near ly sphe r i ca l l y s y m m e t r i c [164], and the re fo re e q . (4.1b) w i l l be a ve ry g o o d a p p r o x i m a t i o n . A l s o , s i nce the present theory w i l l be used in con junc t i on w i t h the S C M F a p p r o x i m a t i o n , m ' shou ld be rep laced e v e r y w h e r e w i th m e , the e f f e c t i v e permanent d i p o l e m o m e n t . - 85 -We now define the additional ion-solvent interaction term due to Ap_(R) as U i s ( l 2 ) = -^e(R)-<AI 1 I(R)> - Ap_(R) • |^<AE1D(R)> + <AE 1 Q(R )>J + ^Ap_(R) • <AE1(R)> . (4.27) The first term in eq. (4.27) is the interaction between the excess induced moment and the excess ion field felt by a solvent at a distance R. The second term takes into account the interaction between the excess moment and the surrounding solvents, while the last term in eq. (4.27) is simply the polarization term. Since eq. (4.26) ensures that Ap(R) and <AE^(R)> will always be in the same direction, eq. (4.27) can be written in the form U i s ( R ) = " I A P ( R ) <AE1(R)> . (4.28) Finally, combining eqs. (4.26) and (4.28) yields the expression uff(R) = -[Ap(R ) ] 2 / 2 a . (4.29) It is interesting to note that u^g(R) is a spherically symmetric interaction which will always be attractive relative to infinite separation. We now have only to determine expressions for each term of eq. (4.22b) contributing to <AE^(R)>. However, even for the current simplified model, this is a non-trivial task. In the present theory we will consider only those contributions which can be more easily characterized and- which require knowledge of only the pair correlation functions. These terms should exactly (at the mean' field level) determine the long-range behaviour of <AE^(R)>. Hence the RDMF theory, as presented here, will be most accurate at large R. Moreover, since we would expect the long-range behaviour of <AE^(R)> to be more important for electrolyte solutions at infinite dilution or low concentration, the RDMF theory should provide the best results for these systems. We also point out that our approximation for <AE^(R)> is not unique and that others may be possible, particularly if 3-body correlation functions were available. - 86 -Let us first examine the simplest case, that of <AE^j (R)> , the average excess local ion field at a distance R from an ion. We will identify contributions from three terms, the first term being the direct term, <AE^g(R)>, due to the charge on the ion. The other two terms, < A E ^ + ( R ) > and <AE^_(R)>, are essentially screening terms and are due to all other positive and negative ions in the system. Therefore we write < A E 1 I ( R ) > = < A E l g ( R ) > + < A E 1 + ( R ) > + < A E 1 _ ( R ) > . (4 .30a) It is obvious that < A E l g ( R ) > = q i / r 2 , (4.30b) where q. is the charge on the ion, which we have labelled i, and < A E ^ g ( R ) > will always be directed along the vector joining the ion i and the solvent at R. In order to determine the two other terms of eq. (4.30a), we will first examine the average interaction between the dipole moment of a solvent particle at R, which we will call the reference particle (see Figure 4), and all other ions (i.e., excluding q.). Now we know that because the solvent molecule has symmetry, the average orientation of the total dipole moment of the reference solvent will be in the direction of <AE^g(R)> (i.e., along the ion-solvent vector) and all other orientations will average to zero. Thus, if we ignore fluctuations we can define m f(R) = m'(R) < ^ u ! i s ( R ) > = m'(R) <cos0 i £ . (R ) > (4.31) as the average projection of m'(R) onto the ion-solvent vector, where <cos0. (R)> is given by eqs. (2.89a) and (2.89b) and the angle 6. is IS IS illustrated in Figure 3. Again, we stress that in the intermolecular reference frame this will be the only non-zero projection of m ' (R). If we can obtain an expression for the interaction between m'(R) and a spherical shell of ions at a distance r from q., then we have only to integrate (i.e., sum over all such shells) to obtain an expression for the total interaction. In Figure 4 we have illustrated the problem being considered and have indicated all the variables used in the derivation outlined below. - 8 7 -Figure 4. An illustration of the method used in determining < A E , T ( R ) > . The t case where q^  is a negative ion is shown. The dipole moment, nr , located at a distance R from the ion, q., is that of the reference solvent particle. - 88 -From e q s . (2.5) and (2.17) it is e a s y to s h o w [79,81] that the in te rac t ion b e t w e e n m ' ( R ) and q. is g i ven by J U m j ( l 2 ) = 2 ^ * 0 0 ( 1 2 ) ' ( 4 ' 3 2 ) r m j mn 1 w h e n w e take f as g i ven by e q . (2.9a). F o l l o w i n g the c o n v e n t i o n s g i ven in F igures 3 and 4 , and us ing an exp l i c i t f o r m fo r the ro ta t iona l invar iant (see A p p e n d i x A o f Ref . 81 or A p p e n d i x B o f Re f . 61), w e rewr i te e q . (4.32) in the f o r m r m T ( R ) q . -, u m j ( l 2 ) = - s ign(q i ) | ^ I c o s f l , (4 .33) r m j where the s i g n f unc t i on equa ls 1 if q. is p o s i t i v e and -1 if q. is nega t i ve . F r o m the law of c o s i n e s [165] we have the re la t i onsh ips r m 2 = r 2 + R 2 - 2 r R c o s t f » (4 .34a) and cose = R " r c o s * , (4 .34b) m j w h i c h , when subs t i tu ted into e q . (4.33), y i e l d u .(R,r,tf,) = - s i g n ( q i ) m T ( R ) q . R ~ r c o s ^ i . (4 .35) [r + R - 2 r R c o s « ] 2 Then the average in te rac t ion ene rgy , U ^ ^ ( R , r ) , be tween rn^(R) and the spher i ca l she l l o f ions (pos i t i ve or nega t i ve ) at a d i s tance r f r o m q. is g iven by UJj !?(R ,r) = P j J g s j ( R , r R 0 , i f O i ^ ( R , r , * ) dAdr , (4 .36a) in wh i ch the e lemen t o f area dA = r 2 s i n 0 d 0 d i / / . (4 .36b) The l im i t s o f in tegra t ion fo r d^j are 0 to 27T (a fu l l r evo lu t i on ) , wh i l e f o r d0 they are 0„ t o ir, where m - 89 -0, i f | r - R | > d j s ^ r W - d , 2 - , (4.37) cos 1 -L?- , i f | r - R | < d . L 2rR J 1 J s guarantees that the ion q. and the re fe rence so l ven t do not in te rpenet ra te . N o w , in p r i nc i p l e , we do not k n o w g -(R, r , 0, \ / / ) . H o w e v e r , s i nce w e s J k n o w that the d i s t r i bu t i on o f i ons must be un i f o rm in the spher i ca l she l l (val id on l y w h e n q. is a sphe r i ca l l y s y m m e t r i c ion) and if w e a s s u m e it to be independent of R, the p o s i t i o n the re fe rence s o l v e n t , then w e can w r i t e gsj(R,r,4>,<//) = g i : j ( r ) , (4.38) where 9 j j ( r ) ' s J u s t t h e i o n - i o n radial d i s t r i bu t ion f u n c t i o n . Th is app rox ima t i on s h o u l d b e c o m e exact at large r. Insert ing e q s . (4.35) and (4.38) in to e q . (4.36a) and in tegrat ing ove r d\p, one ob ta ins ^ ( R , r ) = - 2 7 r r 2 p j g i j ( r ) s i g n f q ^ q^m^R) dr x J* (R-rcos»)sin^ d 0 ^ ( 4 > 3 9 ) [ r 2 + R 2 - 2 r R c o s 0 ] 2 Then in tegrat ing e q . (4.39) ove r d# (using s tandard f o r m s fo r the t r i g o n o m e t r i c in tegra ls w h i c h m a y be f o u n d in tab les [165]) and s i m p l i f y i n g y i e l d s the e x p r e s s i o n Tjjj!?(R,r) = - 2 7 r r 2 P : j g i : J ( r ) s i g n ( q i ) q j m t ( R ) L 2 r " R c o s * r f ^ • 4r[r^+R - 2 r R c o s ^ ] 2 j ^ m (4.40) If one ca re fu l l y eva lua tes e q . (4.40) at i ts l i m i t s , one f i nds that u S(R,r) = 0 f o r r^R+d. , (4.41a) m j j s - 90 --V, 9 3-i m ^ R ) U ^ ( R , r ) = - 2 7 r r ^ p j g i j ( r ) s i g n ( q i ) - l _ R 2 - r 2 - d , 2 1 + 2rd. - ^ I d r j s f o r R - d j s ^ r < R + d j s , (4.41b) u P ^ ( R , r ) = - 4 7 r r ^ p j g i j ( r ) s i g n ( q i ) ^ - L _ J d r f o r r < R - d DS (4 .41c) W e remark that e q s . (4.41) are c o n s i s t e n t w i t h b a s i c e l e c t r o s t a t i c theory [140] w h i c h s ta tes that f r o m an internal po in t o f v i e w a spher i ca l she l l o f charge is e l e c t r i c a l l y nonex is ten t (cf. e q . (4.41a)); f r o m an external po in t o f v i e w it is e l e c t r i c a l l y equ iva len t to a point charge w h o s e charge is equal to that 2 con ta ined in the she l l (cf. e q . (4.41c) where N j = 47rr p ^ g ^ _j( r ) d r ) . The to ta l average in te rac t ion ene rgy , U m j (R) , b e t w e e n m ' (R) and al l other i o n s , q . , in the s y s t e m is found by s i m p l y in tegrat ing e q s . (4.41) ove r al l va lues o f r. The ions are not a l l o w e d to in terpenet ra te s o we wr i t e and inser t ing e q s . (4.41) one ob ta ins the n e c e s s a r y resul t [ r 2 9 i j ( r ) [ q_; mT(R) U m j ( R ) = - 4 1 r p j s i g n ( q i ) ^ L _ R _ d i s d i j dr 1 R + d j s R ' d j s R r 2 - d 2 1 + 2rd DS ( 4 . 4 2 ) ( 4 . 4 3 ) N o w tak ing advantage o f the fac t that a l l o ther ions w i l l be u n i f o r m l y d is t r ibu ted in spher i ca l she l l s about q . , it c l ea r l y f o l l o w s that < A E , . ( R ) > ' t (j = + or - ) w i l l be n o n - z e r o on l y a l ong m ' ( R ) , and c o n s e q u e n t l y w e have that U m j ( R ) = - m 1 " ( R ) < A E l j ( R ) > . ( 4 . 4 4 ) U s i n g e q s . (4.43) and (4.44) one i m m e d i a t e l y ob ta ins - 91 R d. • dr R+d. R-d. ^ p 9 i j ( r ) [ 1 + R' dr ( 4 . 4 5 ) where j = + o r - . To ensure that <AE-> (R)> has the s a m e d i rec t i ona l sense where <AE^j(R)> w i l l a l w a y s be d i rec ted a long the i o n - s o l v e n t v e c t o r . W e n o w turn our a t ten t ion to the e x c e s s loca l d i p o l e f i e l d , <AE^^(R)>. Let us p ic ture an ion in a po la r so l ven t o f C 2 v s y m m e t r y . W e f i nd that on ave rage al l the d ipo le m o m e n t s po in t e i ther d i r ec t l y at or d i rec t l y a w a y f r o m the i o n , depend ing upon the s ign of the charge on the i on . C l e a r l y , al l d i p o l e s in the s a m e spher ica l she l l (we sha l l re fer to them as be ing la tera l ) w i l l mu tua l l y repel one another . It is th is con t r i bu t i on to the l oca l d i p o l e f i e l d due to the lateral d i po le m o m e n t s wh i ch we w i l l e x a m i n e . W e de te rm ine the lateral d i po le f i e l d , and hence <AE-^(R)> in the present t heo ry , in a manner qui te s i m i l a r to that e m p l o y e d fo r <AE^j(R)>. A g a i n , it is on l y the average p r o j e c t i o n s , m^ ( r), o f the to ta l d ipo le m o m e n t s that need to be c o n s i d e r e d . In order to de r i ve an e x p r e s s i o n fo r <AE^^(R)> w e f i r s t examine the in te rac t ion b e t w e e n our re fe rence d i p o l e , rn^(R), and al l sphe r i ca l she l l s of d i po le m o m e n t s , rn'(r), at a d i s tance r f r o m the i o n . In Figure 5 w e have i l l us t ra ted the s i t ua t i on be ing e x a m i n e d a long w i t h al l the n e c e s s a r y v a r i a b l e s . F rom eq . (2.5) and (2.17b) w e can s h o w [67] that the in te rac t ion t f be tween rn'(r) and m'(R) is g i ven by as <AE^j(R)>, w e must mu l t i p l y e q . (4.30b) by s ign(q.) . Equa t ions (4.30) and (4.45) are c o m b i n e d to g i ve the des i red re la t i onsh ip ( 4 . 4 6 ) Figure 5. A n i l l us t ra t ion o f the m e t h o d used in de te rm in ing <AE^_(R)>. The c a s e whe re q. is a nega t i ve ion is s h o w n . - 93 -u m m ( 1 2 ) = - m T ( r ) m t ( R ) ^ i i 2 ( l 2 ) f ( 4 > 4 7 )  rmm where w e again take f m n ^ a s g iven by e q . (2.9a). The ro ta t iona l invar iant can be wr i t t en [61,81] in the f o r m 4>JJ2( 1 2 ) = 2005^00502 - s i n 0 1 sin0 2 cos(</> 1-0 2) , ( 4 . 4 8 ) where 0^ and 0 2 are the ang les ind ica ted in F igure 5 and <j>^ and 0 2 are the az imutha l ang les . It i s e a s y to see f r o m Figure 5 that <j>^ and # 2 must a l w a y s be equa l . A n a l o g o u s to e q s . (4.34) w e have the re la t i onsh ips rmm = r 2 + r 2 " 2 r R c o s ^ ' ( 4 . 4 9 a ) COS0, = R ~ r r C O S ^ ( 4 . 4 9 b ) mm and c o s « = RCOS0- r m ( 4 . 4 9 c ) rmm F r o m the law of s i n e s [165] it is easy to s h o w that s i n 0 = 2 L | i M ( 4 . 4 9 d ) mm and s i n « = R | i H i . ( 4 . 4 9 e ) mm Subs t i t u t i ng e q s . (4.49) into e q . (4.48) and c o m b i n i n g th is resul t w i t h e q . (4.47) y i e l d s Unn/R 'T'*) = m ( r ) ™ ( R ) | ^3 rR - 2 ( r 2 + R 2 ) c o s 0 + r R c o s V ] . ( 4 . 5 0 ) rmm 4. W e note that th is e x p r e s s i o n is invar iant to s ign(q. ) because both m ( r) and t ' m'(R) w i l l r eve rse d i rec t i on if the charge on the ion i is r e v e r s e d (i.e., 0 1 — » 1 8 O ° + 0 1 , 0 2 — > 180° + 0 2 ) . The average lateral d i p o l e - d i p o l e in te rac t ion ene rgy , U^^R,!:), be tween . mm m'(R) and the spher i ca l she l l o f d i p o l e s at a d i s tance r f r o m the s a m e ion w i l l be g i ven by - 94 -U m m t ( R ' r ) = ps ^  9 s s ( R ' r ' ^ ) u m m ( R ' r ' 0 ) d A d r ' (4.51) where d A is e x p r e s s e d in e q . (4.36b). The l im i t s o f in tegra t ion are s t i l l 0 to 27T fo r d\p and <pm to it f o r d0, as w a s the c a s e fo r <AE^j(R)>, but n o w <t> = 0, i f |r-R|>ds _ l p r 2 + R 2 _ d 2 . (4.52) cos '| : s_ f i f | r-R|<ds 2rR A g a i n , as w a s the case fo r <AE^j(R)>, w e k n o w that g s s(R,r,0,^) is sphe r i ca l l y s y m m e t r i c (i.e., the s o l v e n t s are u n i f o r m i l y d i s t r i bu ted in the she l l ) . W e w i l l a l so a s s u m e it is independent o f R and take 9 s s ( R , r , ^ ) = g. s(r) = ggjj°is<r> . ( 4 . 5 3 ) N o w subs t i tu t i ng e q s . (4.50) and (4.53) into e q . (4.51) and in tegrat ing ove r d0, one ob ta ins U ^ a t ( R , r ) = 2 7 r r 2 p s g i s ( r ) m t ( r ) m t ( R ) d r x f [3rR - 2(r 2+R 2)cos0 + rRcos 2fl] s i n ^ ^ (4 54) [r 2+R 2- 2rRcos0] 2 The in tegra t ion o f e q . (4.54) is n o n - t r i v i a l to p e r f o r m , but af ter c o n s i d e r a b l e man ipu la t i on (again mak ing use o f s tandard t ab les o f t r i g o n o m e t r i c in tegra ls [165]) it can be s h o w n that T J ^ V r ) = 2 ^ g. ( r ) m t ( r ) m t ( R ) [ — ° ° S * m J d r . ( 4 . 5 5 ) m L [ r 2 + R 2 - 2rRcos4>PJ it then f o l l o w s f r o m e q s . (4.52) and (4.55) that I J ^ R ^ ) = 0 for | r-R| >ds (4.56a) and - 95 -S 3 1 S 2 r R for |r-R|<d (4.56b) F i n a l l y , the to ta l average lateral d i p o l e in te rac t ion energy U ( R ) b e t w e e n m T ( R ) and al l o ther d i po le m o m e n t s is f ound by in tegra t ing e q s . (4.56) o v e r al l va lues o f r, a l though on l y t hose she l l s f o r wh i ch | r - R | < d g w i l l con t r i bu te . W e ob ta in the e x p r e s s i o n U ( R ) = mm ,p .«t(H) R ; + d s r m t ( r ) ( r ) [ ( 2 r R ) 2 _ ( r 2 + R 2 2 ) 2 , 1 d r _ R - d L 1 3 s J 2 R 2 d 3 R 0 s s ( 4 . 5 7 ) N o w tak ing advantage of the fac t that the s o l v e n t d i s t r i bu t ion about the ion is independent o f the ang le , and b e c a u s e al l the average p ro jec ted m o m e n t s , m'(r), are d i rec ted a long the i o n - s o l v e n t v e c t o r , it must f o l l o w that the to ta l average lateral d ipo le f i e l d w i l l be in the d i r ec t i on o f m ( R ) . A l l other c o m p o n e n t s to the average f i e l d must average to z e r o . Thus , ana logous to e q . (4.44) w e wr i t e that U M M ( R ) = - m T ( R ) < A E L D ( R ) > , ( 4 . 5 8 ) w h i c h w h e n c o m b i n e d w i th e q . (4.57) y i e l d s R+d <AE 1 D ( R ) > = Y S 3 * S U f ( r ) g i s ( r ) [ ( r 2 + R 2 - d s 2 ) 2 - ( 2 r R ) 2 ] | d r . 2R d s R-d s L (4 .59) t W e po in t out that in e q . (4.59) m ( R ) w i l l con ta in con t r i bu t i ons due to bo th the bulk average d ipo le m o m e n t , m ' , and the average e x c e s s induced m o m e n t , Ap_(R). F r o m e q s . (4.24b) and (4.31) w e f i nd that m T ( R ) = m ' < c o s 0 . ( R ) > + Ap (R ) , ( 4 . 6 0 ) where w e have taken advantage of the fac t that Arj ( R ) , and hence <AE-,(R)>, t w i l l be n o n - z e r o on l y a long m ' ( R ) . Th is has a l ready been s h o w n to be the c a s e f o r < A E - ^ J ( R ) > and < A E ^ ^ ( R ) > in the present theory and b e l o w w e f ind th is to be a l s o true fo r < A E - ^ Q ( R ) > . Insert ing e q . (4.60) into e q . (4.59) and us ing e q s . (2.89) w e obta in the re la t i onsh ip - 96 -< A E 1 D ( R ) > r s s s m ' h 0 0 1 - i s ( r ) , °°', 1 S , + Ap(r)g._(r) [ ( r 2 + R 2 - d s 2 ) 2 - ( 2 r R ) 2 dr (4.61) w h i c h is the des i red resul t . Inspec t ion of e q . (4.61) revea ls that t w o d is t inc t con t r i bu t i ons to < A E ^ D ( R ) > can be i d e n t i f i e d , one due on l y to m ' , and the other due to A p ( R ) . Th is sepa ra t i on w i l l p rove use fu l in d i s c u s s i o n s b e l o w . The e x c e s s l oca l quadrupo le f i e l d , < A E ^ Q ( R ) > , is de te rm ined in a ve ry s i m i l a r f a s h i o n to that used fo r < A E ^ D ( R ) > . W e w i l l again cons i de r the con t r i bu t i on to < A E - ^ Q (R ) > due to lateral f i e l d s . These lateral f i e l d s are a c o n s e q u e n c e of the average p r o j e c t i o n s , ©£(r), o f the quadrupo le m o m e n t s of the s o l v e n t pa r t i c l es around the ion i. The p ro jec t i ons © ' ( r ) are ana logous to t S the p ro jec t i ons m'(r) and w i l l be de f i ned b e l o w . W e e m p h a s i z e that th is de r i va t i on app l i es s t r i c t l y to square quadrupo le m o m e n t s as d e f i n e d by e q . (2.22) and i l lus t ra ted in Figure 2(a). F i rs t let us de f ine [67,72] the f u n c t i o n s and $ 0 2 2 ( 12) = $°22(n1,n2,f) + *°?|(0 1 fO 2 ff) (4.62a) * 1 2 3 ( 1 2 ) = *J23(n! , 0 2 r f ) + # J ^ ( f l 1 f B 2 l r ) , (4.62b) inn 1 w h i c h , if f is g i ven by e q . (2.9a), can be wr i t ten [81,166] in the exp l i c i t f o r m s and $ 0 2 2 ( 1 2 ) $ 1 2 3 ( 1 2 ) - ^ [ ( x 2 . f 1 2 ) 2 - ( y 2 . f 1 2 ) 2 ] (4.63a) ^ 6 [ 5 [ ( x 2 . f 1 2 ) 2 - ( y 2 . f 1 2 ) 2 ] ( z 1 -z 2) - 2[ ( x 2 - z 1 ) ( x 2 . f 1 2 ) - ( y 2 . z 1 ) ( y 2 - r 1 2 ) ]j . (4.63b) Then us ing the d e f i n i t i o n o f the Euler ang les a, p\ 7 as w e l l as the ro ta t ion mat r ix (see e q . (39)) o f Re f . 166, it can be s h o w n that - 97 -$ 0 2 2 ( 1 2 ) = »/6 s i n 2 p \ COS 2 T 7 ( 4 . 6 4 a ) and $ 1 2 3 0 2 ) = Ve . 2 . . 3 c o s / 3 1 s i n /3 2 c o s 2 7 2 + 2 s i n 0 1 s i n / 3 2 x [ c o s / 3 2 c o s 2 7 2 cosia^-a^) + s i n 2 7 2 s i n f c ^ - c ^ ) ]J . ( 4 . 6 4 b ) A s in the case o f < A E ^ D ( R ) > , w e w i l l de te rmine the average in te rac t ion be tween al l average p r o j e c t i o n s , 0^( r ) , o f the quadrupo le m o m e n t s in a spher i ca l she l l at a d i s tance r f r o m the ion q. and the d ipo le m o m e n t m ^ R ) . In Figure 6 w e have t r ied to represent th is geome t r i ca l p r o b l e m , ind ica t ing the va r i ab les used in the f o l l o w i n g d e r i v a t i o n . H o w e v e r , b e f o r e w e can examine the d i p o l e - q u a d r u p o l e i n te rac t i on , w e must f i r s t de f i ne @ g ( r ) . F r o m e q s . (2.5), (2.19c) and (4.64a) it f o l l o w s that the i o n - s q u a r e quadrupo le in te rac t ion [79,81] is g i ven by q i % 2 u i Q ( l 2 ) = ^ y ^ s i n z / 3 2 c o s 2 7 2 . ( 4 . 6 5 ) It is the m o s t ene rge t i ca l l y f avou rab le quadrupo le o r ien ta t ion w i th respec t to the ion i w h i c h is p ic tu red in Figure 6(a). Th is o r ien ta t ion of the quadrupo le momen t is the o n l y one that w i l l not average to zero fo r m o l e c u l e s of C 2 v s y m m e t r y and it is on to th is o r i en ta t i on that w e de te rm ine the average p ro jec t i on of the quadrupo le m o m e n t . It is c lear f r o m e q . (4.65) that fo r 0 g p o s i t i v e , w h i c h is the case fo r a w a t e r - l i k e s o l v e n t [72], th is p ro jec t i on c o r r e s p o n d s to and p*2 = 9 0 ° ( 4 . 6 6 a ) c o s 2 7 2 = - s i g n ( q ^ ) , ( 4 . 6 6 b ) where the supersc r ip t I i nd ica tes the ang les as de f i ned fo r the ion re fe rence f rame (see Figure 6(b)). W e po in t out that U ^ Q ( 1 2 ) is independent o f and the re fo re © g ( r ) is a l l o w e d to sp in f r e e l y about the ax i s . W e can then wr i te that Figure 6. A n i l l us t ra t ion o f the me thod used in de te rmin ing < A E ^ g ( R ) > . The c a s e w h e r e q. is a p o s i t i v e ion is s h o w n . For c la r i t y the va r i ous re fe rence f r a m e s and their r o ta t i ons are separa te ly g i ven in (b). 99 -© I ( r ) = s i g n ( q i ) < * 0 2 2 ( r ) > , ( 4 . 6 7 a ) w h i c h , w h e n c o m b i n e d w i th e q s . (4.62a) and (2.87) b e c o m e s 0 2 2 ej(r) - 0 8 h 5 2 T i s ( r ) WO s i g n ( g i ) 5 g ™ ? u ( r > ( 4 . 6 7 b ) N o w us ing e q s . (2.5), (2.17b), (2.19c) and (4.62b) w e can s h o w that the d i p o l e - s q u a r e quadrupo le i n te rac t ion , ^ 0 ( 1 2 ) , can be e x p r e s s e d [67,72] in the f o r m u m © ( 1 2 ) = ^ 4 * ( 1 2 ) i/6 r, ( 4 . 6 8 ) m© , 1 2 3 , H o w e v e r , the f unc t i on $ ( 1 2 ) , as g i ven by e q . (4.64b), is not e x p r e s s e d in t e r m s o f ang les s h o w n in Figure 6. T h u s , b e f o r e p roceed ing w e must f i rs t 1 2 3 I I w r i t e $ ( 1 2 ) in t e rms of these ang les , i.e., p^ , 7 2 , P^ and co. W e beg in b y exp ress i ng the unit vec to r s x\~, y 9 and z ~ in te rms of the Euler ang les I I I cu, , $2 and 7 2 a s s o c i a t e d w i t h the ion re fe rence f rame (see A p p e n d i x C) . N o w in order to go f r o m the ion re fe rence f r ame to the f r ame ( x , y , z ) , w e must rotate about the y^ axis b y an angle co (see Figure 6(b)). The re fo re w e app l y the ro ta t i on matr ix [166] R = C O S C J 0 .sincj 0 1 0 - s i x\co' 0 C O S U . ( 4 . 6 9 ) t o the unit v e c t o r s xV^, y 2 and ^ e c a n a ' s o e x P r e s s * 2 > Y 2 a r , d z 2 in t e r m s of the Euler ang les a ^ , /3 2 and 7 2 , as done in A p p e n d i x C . Equat ing the c o m p o n e n t s o f the t w o f o r m s fo r xV^, y 2 and ^ y i e l d s seve ra l r e l a t i onsh ips (g iven in A p p e n d i x C ) b e t w e e n the t w o se t s o f Euler ang les a s s o c i a t e d w i t h the t w o d i f fe ren t re fe rence f r a m e s . W e subs t i tu te these e x p r e s s i o n s in to e q . (4.64b), fo r wh i ch w e take =0 (this f o l l o w s f r o m our c h o i c e of r e f e rence f rame) , and af ter c o n s i d e r a b l e man ipu la t i on and s i m p l i f i c a t i o n (see A p p e n d i x C) w e ob ta in - 100 -123 T 2 2 1 2 $ (12) = v/6 3cos/? 1 (cos u> - s i n s i n u) 2 I "I I + 2sin/3 1 sinco C O S C J ( 1 + s i n 02 ) c o s 2 7 2 . (4.70) F i n a l l y , w e rep lace by 0^  and e l im ina te the c o s 2 7 2 dependence in e q . (4.70) by no t ing that the p roduc t s cos/31 c o s 2 7 2 = cos0 1 (4 .71a) and sin/? 1 co s27 2 = s i n 0 1 (4.71b) are independent o f the s ign o f q . . C o m b i n i n g e q s . (4.68), (4.70) and (4.71) y i e l d s the n e c e s s a r y resul t mT(R) e t ( r ) u J 1 2 ) = s Tn0v ' 4 r m0 2 2 1 2 3cos0 1 (cos GD - s i n s i n w) + 2 s in0 1 sina> cosco( 1 +s in 2 a 2 I ) j . (4.72) W e have a l ready p o i n t e d out that the average p ro jec t i on o f the quadrupo le m o m e n t is a l l o w e d to sp in f r e e l y about the z T a x i s , and hence w e I I cannot s p e c i f y the ang le . H o w e v e r , w e can r e m o v e the dependence f r o m e q . (4.72) by tak ing advantage of the fac t that a l l a n g l e s , , are equa l l y p robab le (at the m e a n f i e l d leve l ) and s i m p l y a n g l e - a v e r a g i n g ( integrat ing) ove r t h e m . A f t e r pe r f o rm ing th is in tegra t ion one has the resul t 3 m T ( R ) © J r ) r 2 , um0 ^ 1 2 ^ = 4 cos0 1 (cos (J - 2 s i n w) rm0 1 + s i n 0 1 sincucoscj . (4.73) A s in the c a s e o f < A E ^ j - j ( R ) > , w e e m p l o y re la t i onsh ips ana logous to e q s . (4.49) in order to exp ress f unc t i ons o f 0^  and co as func t i ons of r, R and 0 , and c o n s e q u e n t l y e q . (4.73) b e c o m e s - 101 -<*> 31 (^11) rf(r) r 2 R3 5 2 3 um© ( R ' r ' 0 ) = =T |_ " 1 " <f rR Z+r 3 ) c o s 0 rm0 2 + (r2R+|R3)cos20 - |^-cos30 . (4.74) The average lateral q u a d r u p o l e - d i p o l e in te rac t ion ene rgy , ^ 0 ^ R ^ r K ' s aga in f o u n d by in tegrat ing ove r d A . U s i n g the a p p r o x i m a t i o n (4.53) and in tegrat ing ove r d ^ one has ^ ( R r r ) = 6*r2psq-is{r)ds(r)J{R)ar v 2 r 2R - - (f rR 2+r 3 ) c o s 0 + (r2R+f R3 )cos20 - 5^-cosV x j £ 1 £—_ ± s i n 0 d 0 , *m [ r 2 +R2- 2 r R c o s 0 ] 2 (4.75) where </>m is g i ven by e q . (4.52). Eva lua t ing the integral o v e r d<p in e q . (4.75) requ i res a great dea l o f e f f o r t , but w i th the a id of t ab les o f s tandard t r i g o n o m e t r i c in tegra ls [165] and af ter much s i m p l i f i c a t i o n the resul t can be wr i t t en in the f o r m U ^ ^ r ) - 3*r 2p sg i s(r)0t(r)mt(R) 2 r( r-Rcostf>) (1 -cos 0 )-x | 55 ! ^ | d r . ( 4 . 7 6 ) L [ r 2 + " R2- 2 r R c o s 0 ] 2 A p p l y i n g e q . (4.52) i m m e d i a t e l y y i e l d s LT I^^ R, r) = 0 for |r-R|£ds (4.77a) and . . 7 m T ( R ) 0 j r ) r r r 2 + R 2 - d 2 1 2 l I & W ) . 3 , r 2 p s g i s ( r ) - - ^ - [ l - ] ds dr for |r-R|<d . (4.77b) W e remark that these e x p r e s s i o n s bear s t r i k ing s i m i l a r i t y to e q s . (4.55), the lateral d ipo le resu l t . - 102 The to ta l average lateral q u a d r u p o l e - d i p o l e in te rac t ion energy is again f o u n d by in tegrat ing e q s . (4.77) ove r all va lues o f r (i.e., s u m m i n g o v e r s h e l l s ) . E x p l i c i t l y , we ob ta in 3rrp_mT(R) R + d r ©t ( r ) U m 0 ( R ) = V - 5 - J " H H ^ i s ^ 8 R ' d ^ R - d ^ s s x [ ( 2 r R ) 2 - ( r 2 + R 2 - d s 2 ) 2 ] ( r 2 - R 2 + d s 2 ) ] d r . ( 4 . 7 8 ) One can use arguments ve ry s im i l a r to those used fo r < A E ^ Q ( R ) > to s h o w that < A E - ^ Q ( R ) > w i l l a l s o on l y be n o n - z e r o in the d i rec t i on o f m ^ ( R ) , s o w e wr i te U M Q ( R ) = - m T ( R ) < A E 1 Q ( R ) > . ( 4 . 7 9 ) F i n a l l y , the c o m b i n a t i o n o f (4.67b), (4.78) and (4.79) g i ves the d e s i r e d resul t 3 T T P 0 R + d s r h 2 2 2 i s ( r ) < A E , ( R ) > = S_s ; s 0 2 , i s i Q 5 s i g n ( q . ) R ^ d c b R - d L r • L o o x [ ( 2 r R ) 2 - ( r 2 + R 2 - d s 2 ) 2 ] ( r 2 - R 2 + d s 2 ) ] d r . ( 4 . 8 0 ) It is in te res t ing to note that bo th < A E ^ Q ( R ) > and < A E ^ Q ( R ) > , as g i ven by e q s . (4.61) and (4.80), r e s p e c t i v e l y , can be p e r c e i v e d as hav ing an apparent 1 / R 2 d e p e n d e n c e . Thus , they can be v i e w e d as add i t iona l s c reen ing t e r m s , appear ing to be the resul t o f e f f e c t i v e c h a r g e s , ana logous to the c a s e fo r < A E 1 + ( R ) > and < A E 1 _ ( R ) > . W e n o w have e x p r e s s i o n s fo r al l three te rms con t r ibu t ing to < A E ^ ( R ) > . G i v e n the n e c e s s a r y p ro j ec t i ons o f the co r re l a t i on f u n c t i o n s , < A E ^ j ( R ) > and < A E ^ Q ( R ) > can be eva lua ted d i rec t l y us ing e q s . (4.46) and (4.80), r e s p e c t i v e l y . H o w e v e r , < A E ^ ^ ( R ) > , as g i ven by e q . (4.61), must be for s o l v e d i te ra t i ve ly s ince it depends on the va lue o f A p ( R ) , w h i c h in turn depends upon < A E ^ D ( R ) > . O n c e the to ta l e x c e s s loca l f i e l d , < A E ^ ( R ) > , has been d e t e r m i n e d , e q s . (4.26) and (4.29) w i l l i m m e d i a t e l y g ive us the e f f e c t i v e i o n - s o l v e n t po ten t i a l , u f ^ ( R ) , f o r the R D M F theo ry . In the present s tudy w e e m p l o y the RHNC theo ry , as d e s c r i b e d in Chapter II, to de termine the co r re la t i on f unc t i ons fo r a s o l u t i o n . These - 103 -co r re l a t i on f u n c t i o n s can then be used in the R D M F theory to eva lua te wh i ch in turn m o d i f i e s the i o n - s o l v e n t pair po ten t i a l , and hence the co r re l a t i on f u n c t i o n s . A s m e n t i o n e d a b o v e , the R D M F theory must be s o l v e d in con junc t i on w i t h the S C M F a p p r o x i m a t i o n . T h e r e f o r e , f o r a g i ven s y s t e m w e s o l v e the S C M F / R D M F / R H N C theory in the f o l l o w i n g manner . For a f i x e d set o f s y s t e m p a r a m e t e r s , inc lud ing m g and a , the RHNC theory is s o l v e d numer i ca l l y s t i l l us ing the i terat ive m e t h o d ou t l i ned in s e c t i o n 5 of Chapter II, but n o w at each i te ra t ive c y c l e w e update the current es t ima te fo r Ap ( R ) . In p r i nc ip le , th is c a l c u l a t i o n must then be repeated f o r seve ra l va lues o f the e f f e c t i v e permanent d i p o l e , m to a l l o w the s e l f - c o n s i s t e n t va lues o f m ' , a' and m g to be eva lua ted us ing the S C M F a p p r o x i m a t i o n . W e poin t out that at in f in i te d i lu t ion the va lue of m g is just that o f the pure s o l v e n t . It f o l l o w s f r o m e q . (3.45) that b e c a u s e u ^ g ( R ) ' s a sphe r i ca l l y s y m m e t r i c po ten t ia l t e rm it may make a con t r i bu t i on to the l o w concen t ra t i on l im i t i ng behav iour o f C- In order to i nves t i ga te th is p o s s i b i l i t y w e need to examine the large R and l o w concen t ra t i on d e p e n d e n c e s of the te rms cont r ibu t ing to < A E ^ ( R ) > . Firs t w e w i l l examine the large R behav iour o f < A E ^ Q ( R ) > . It can be s h o w n [61] that at p 2 =0 h 0 2 ; i s ( r ) a / r 3 a s r - > ™ ' ( 4 . 8 1 ) 0 2 2 where a is s o m e cons tan t . If w e then insert th is f o r m fo r • ( r ) into U 2 ; l s e q . (4.80) and in tegra te , one f i nds that the integral eva lua tes to an ident i ty 0. C l e a r l y , w e then have that at in f in i te d i l u t i on < A E 1 Q ( R ) > = 0 a s R — . ( 4 . 8 2 ) T h e r e f o r e , < A E , N ( R ) > has no l o n g - r a n g e t a i l , b e c o m i n g zero as s o o n as 0 2 2 h Q 2 # ^ g ( r ) a t ta ins i ts large r behav iou r , and c o n s e q u e n t l y it cannot con t r ibu te to the l o w c o n c e n t r a t i o n l im i t i ng behav iour o f U ^ ( R ) . W e n o w turn our a t ten t ion to < A E ^ j ( R ) > . F r o m e q s . (3.34) and (3.35a) w e have that as r—>°> and K—>0, - 104 -where K is g iven by e q . (3.35b). W e inser t e q . (4.83) into e q . (4.46) and expand and integrate (see A p p e n d i x B). Then c o l l e c t i n g t e r m s and app l y i ng the s m a l l K l imi t y i e l d s < A E 1 X ( R ) > = s i g n ^ ) D = + , " 4 * p j q j ^ ( R 3 - d , 3 ) + 2e k T 2i-5l[e k R ( R + 1 / / C ) - 1/ K]1 K e k T J ( 4 . 8 4 ) If w e n o w use e q . (3.35b) and charge neut ra l i t y , d ropp ing any te rms l inear in P 2 , w e ob ta in the resul t I*' <AE 1 I(R)> - - 2 1 1 eKR{^+KR) , ( 4 . 8 5 ) w h i c h rep resen ts the R — a n d K—>0 l im i t i ng behav iour o f <AE^j(R)>. W e note that in the in f in i te d i lu t ion l imi t {i.e., /c=0) eq . (4.85) b e c o m e s <AE 1 I(R)> - ( 4 . 8 6 ) w h i c h is cons i s ten t w i t h e q . (4.30). In cons ide r i ng the l o w concen t ra t i on l im i t ing behav iour o f < A E ^ Q ( R ) > w e f i nd it conven ien t t o sp l i t <AE^^(R)>, and hence e q . (4.61), into t w o p a r t s ; <AE^ M ( R ) > due o n l y to m ' , and <AE^p(R)> d i r ec t l y dependent on l y upon Ap ( R ) . F i rs t w e w i l l examine <AE-^M(R)>. F rom e q . (3.47) it f o l l o w s that as p 2 — ^ - 0 and R—>°>, iQfc I r _ n R + c L r ( 4 . 8 7 ) If one p e r f o r m s the in tegra t ion in e q . (4.87) and c o l l e c t s t e r m s (see A p p e n d i x B) , then it can be s h o w n that in the l imi t K—>0 and R — <AE l m(R)> = - 2 | q i l 3R" e k R ( 1 + / C R ) . ( 4 . 8 8 ) - 105 -W e remark that e q s . (4.85) and (4.88) are ve ry s i m i l a r . It is c l ea r l y the c a s e that at l o w concen t ra t i on and long range near ly t w o th i rds o f the ion f i e l d w i l l be c a n c e l l e d by the lateral d ipo la r f i e l d due to m g w h e n the pure s o l v e n t d ie lec t r i c cons tan t , e, is large (as is the c a s e fo r wa te r ) . Th is is an o b v i o u s ind ica t ion o f h o w poor an a p p r o x i m a t i o n e q . (4.1a) is f o r aqueous e l e c t r o l y t e s o l u t i o n s . In order t o de te rm ine the l o n g - r a n g e l o w c o n c e n t r a t i o n behav iou r o f <AE^p(R)> w e must k n o w Ap(R), but Ap(R) depends upon <AE^(R)>, wh ich in turn depends upon <AE, (R)>. T o ob ta in a f i rs t a p p r o x i m a t i o n fo r ™ ( 1 ) Ap(R), wh i ch w e w i l l des igna te as Ap (R), w e use e q s . (4.22b) and (4.26), c o m b i n e d w i t h the resu l ts f r o m e q s . (4.82), (4.85) and (4.88), w h i c h y i e l d A p ( i ) ( R , = _ l L ^ i r i i 2 ] e - « R ( 1 + K R ) . ( 4 . 8 9 ) 3R^  L - I Th is resul t can then be subs t i t u ted into the e x p r e s s i o n R+d <AE, (R)> = * PS_ J s[Ap(r)[(r 2+R 2-d 2 ) 2 - (2rR) 2]ldr , ( 4 . i p 2R^dJ* R-d L S J s s 90) w h i c h f o l l o w s f r o m e q . (4.61) w h e n g . (r) = 1 (va l id at large r). A f t e r p e r f o r m i n g the n e c e s s a r y in tegra t ion and s i m p l i f y i n g (see A p p e n d i x B) , one has that < A E l p ( R ) > ( l ) = ^ [ i l 2 ] ( - j ) e - K R < 1 + i c R ) , (4.91a) where 8 7T p a. t i = — . ( 4 . 9 1 b ) 3 W e n o w c o m b i n e th is resul t w i t h e q s . (4.82), (4.85), (4.88), (4.22b) and (4.26) to ob ta in a s e c o n d es t ima te fo r Ap (R), name ly Ap ( 2 ) ( R ) = ^ I f l l ^ l e ' ^ d + z c R ) d-£) . ( 4 . 9 2 ) 3R^  L e J By repeat ing the above p rocedure to ob ta in higher and higher o rder e s t i m a t e s of Ap(R), it is p o s s i b l e to s h o w that - 106 -Ap ( o 5 )(R) = -I^ir.£l2]e- K R( 1 + KR)( 1^^ 2-^ 3 +...) . (4.93) 3RZ L e J For | £ | < 1 (i.e., fo r ap <3/87r), w e r e c o g n i z e the last f a c t o r in e q . (4.93) as the T a y l o r se r i es e x p a n s i o n fo r 1 / ( 1 + £ ) . W e note that fo r wa te r at 2 5 ° C , 1=0.403. T h e r e f o r e , w e have the s e l f - c o n s i s t e n t resul t Ap(R) = i ! f ! L ! [ l l l l e - ^ d + K R ) , (4.94) (3+87rpsa)R^ L e J . w h i c h w i l l be va l i d o n l y at long range and l o w c o n c e n t r a t i o n . Let us n o w c o n s i d e r the average e x c e s s loca l f i e l d , <AE^(R)>, at in f in i te d i lu t ion (i.e., p^ =0) f o r the s p e c i a l c a s e when the s o l v e n t is polarizable but non-polar. In such a c a s e m'=0, and fu r the rmore <E_1> = <AE^m(R)>=0. S ta r t i ng w i th an in i t ia l guess f o r <AE^(R)> as g i ven by e q . (4.86), and us ing e q . (4.90) in the s a m e i terat ive s c h e m e ou t l ined a b o v e , w e can s h o w that fo r th is s y s t e m at large R |q . | <AE,(R)> = —4- . (4.95) 1 R 2 3 + 87rp sa Subs t i t u t i ng the C l a u s i u s - M o s o t t i r e la t i onsh ip [139], °° „ = ^irp_a , (4.96a) e +2 J s OO in to e q . (4.95) i m m e d i a t e l y y i e l d s the re la t i onsh ip <AE,(R)> = l i - i ^ l 2 . f (4.96b) 1 R 3 e oo where is the high f requency d ie l ec t r i c cons tan t due o n l y to mo lecu la r p o l a r i z a b i l i t y . P o l l o c k etal. [167] have a l s o s tud ied po la r i za t i on e f f e c t s in th is s y s t e m . They ob ta ined exac t l y the s a m e e x p r e s s i o n fo r the large R dependence o f the average loca l f i e l d . M o r e o v e r , P o l l o c k etal. [167] f o u n d that con t i nuum theory p red i c ted a d i f fe ren t resu l t . W h e n the t w o e x p r e s s i o n s w e r e c o m p a r e d w i t h repor ted va lues f o r the average loca l f i e l d at large R ob ta ined f r o m compu te r s i m u l a t i o n [167], e q . (4.96b) w a s f o u n d to g ive e s s e n t i a l l y exact r esu l t s , wh i l e the con t i nuum e x p r e s s i o n w a s on l y accurate f o r - 107 -s m a l l va lues o f p g a . It is a l so in te res t ing to examine the behav iour o f <AE^(R)> at in f in i te d i l u t i on w h e n the so l ven t is polar but not polarizable. F r o m e q s . (4.86) and (4.88) it is e a s i l y s h o w n that at large R <AE,(R)> = iSi . ( 4 . 9 7 ) 1 R 3e C u r i o u s l y , eqs . (4.96b) and (4.97) are equ iva len t in f o r m ind ica t ing that the f i e l d due to a charge w i l l be screened to the s a m e extent in ei ther s y s t e m p r o v i d e d that they have the same d ie l ec t r i c cons tan t . W e point out that Ap(R), as g iven by e q . (4.94), has that s a m e l a r g e - R l o w concen t ra t i on l im i t i ng behav iour as Q ] i s ^ r ^ ' anc* T , 1 U S W E W O U L C L E X P E C T it to a f f e c t the l im i t i ng behav iou r o f C- U s i n g e q s . (3.39b), (3.45), (4.29) and (4.94) it is again p o s s i b l e to s h o w , af ter c o n s i d e r a b l e man ipu la t i on (see A p p e n d i x B) , that as p^—>0, 2 1 S + n)2l € J 2 ( 3 + 8 7 r p _ a ) L  i s where is the con t r i bu t i on to C- due on l y to u^?(R) and fl = 1 A T . If w e I S - I S I S n o w de f i ne S c " as be ing the con t r i bu t i on to the to ta l l im i t i ng s l o p e , S o f C j g (as g iven in e q . (3.43)) due on l y to u^P(R), then c o m b i n i n g e q s . (3.43) and (4.98) a long w i th e q . (3.35c) y i e l d s A P = - 1 2 ^ ^ ^ t ( 4 ; g g ) Fur the rmore , it f o l l o w s f r o m e q . (3.41) that if u^P(R) make a con t r i bu t i on to the l im i t i ng s l o p e of C j g , then it must a l so in f luence the l im i t i ng behav iour of G.„=G „. T h e r e f o r e , u^?(R) w i l l a f f ec t the l im i t ing l a w s of al l + S - s i s t h e r m o d y n a m i c p roper t ies o f e l e c t r o l y t e s o l u t i o n s wh i ch have a dependence on i o n - s o l v e n t co r re l a t i ons at l o w concen t ra t i on (e.g., N^). W e again e m p h a s i z e that the va lue of S A p shou ld be accura te l y g i ven by e q . (4.99) s i nce the l o n g - r a n g e l o w concen t ra t i on behav iour o f Ap(R) p red i c ted by the R D M F theo ry , i.e., e q . (4.94), shou ld be an exact result at the mean f i e l d l e v e l . - 108 -CHAPTER V RESULTS FOR WATER-LIKE MODELS 1. Introduction In Chapter II w e have de f i ned the w a t e r - l i k e m o d e l s w e w i l l i n ves t i ga te . W e w i l l repor t resu l t s ob ta ined us ing the R H N C / S C M F theory w h i c h w a s d e s c r i b e d in Chapte rs II and IV. For the m o s t par t , th is s tudy w i l l f o c u s upon the fu l l C 2 v wa te r m o d e l , w i th and w i thou t the oc tupo le m o m e n t . W e w i l l f i nd it conven ien t to re fer to the C 2 y m o d e l w h i c h inc ludes on l y d ipo le and quadrupo le m o m e n t s as the quadrupole m o d e l , whe reas the te rm C ^ ^ octupole m o d e l w i l l be used w h e n the oc tupo le has a l s o been i nc luded . H o w e v e r , the tetrahedral m o d e l (i.e., a m o d e l con ta in ing on l y a d i p o l e and a square quadrupo le ) w a s a l so exam ined to a l l o w c o m p a r i s o n w i t h p rev ious wo rk [67,168]. M o r e o v e r , th is s i m p l i f i e d m o d e l w a s e m p l o y e d e x t e n s i v e l y as the s o l v e n t in our s tudy o f m o d e l aqueous e l e c t r o l y t e s o l u t i o n s (as w i l l be d i s c u s s e d in Chapte r VI) . It a l s o p r o v e d use fu l in exp lo r i ng the b a s i s set dependence o f the RHNC theory fo r m o d e l s o f n o n - l i n e a r s y m m e t r y . W e w i l l d i s c u s s bas i s se t s and bas i s set dependence in s e c t i o n 2 of th is chapter . W e f ind that the b a s i s set c o r r e s p o n d i n g to n m a x = 4 rep resen ts a reasonab le c o m p r o m i s e b e t w e e n c o n v e r g e n c e and c o m p u t a t i o n a l r equ i remen ts . Hence , th is bas i s set w a s used to ob ta in v i r tua l l y al l resu l t s repor ted in s e c t i o n s 3 and 4. In s e c t i o n 3 w e w i l l p resent resu l ts ob ta ined f o r the mu l t i po la r h a r d - s p h e r e m o d e l s . A m o d i f i e d v e r s i o n of th is m o d e l , w h i c h i nco rpo ra tes a so f t spher i ca l po ten t i a l , w i l l be e x a m i n e d in s e c t i o n 4 . Our d i s c u s s i o n s w i l l f o c u s ma in l y on the st ructura l and d ie lec t r i c p rope r t i es o f the m o d e l s y s t e m s be ing i n v e s t i g a t e d . In th is s tudy w e have examined w a t e r - l i k e m o d e l s at seve ra l d i f fe ren t tempera tu res and p r e s s u r e s . The temperature and pressure po in t s at w h i c h ca l cu l a t i ons w e r e done are g iven in Tab le II, a l ong w i th the c o r r e s p o n d i n g exper imen ta l d e n s i t i e s . The po in t s s a m p l e a re l a t i ve l y large po r t i on o f the phase d iag ram o f l iqu id wa te r and w e r e c h o s e n to c o r r e s p o n d w i t h t hose exam ined in p r e v i o u s w o r k [67,168]. Of impo r tance here is the fac t that the d ie lec t r i c cons tan t has been measu red [48,169,170] at a l l these p o i n t s . W e shal l re fer to po in t s at 1 a tm. or at vapour p ressure as be ing at normal - 109 -TABLE II. Exper imen ta l dens i t i es of water f o r the tempera tu res and p ressu res e x a m i n e d in this s tudy . Tempera tu re ( ° C ) P ressu re Dens i t y (9 /m I) 25 1 a t m . 0.99707 [169] 90 1 a t m . 0.9653 [48] 200 Vapour P r e s s . 0.865 [48] 300 Vapour P r e s s . 0.710 [48] 370 Vapour P r e s s . 0.452 [170] 100 5000 bars 1.106 [170] 200 5000 bars 1.051 [170] 300 5000 bars 0.993 [170] 400 5000 bars 0.931 [170] p r e s s u r e , wh i l e those at 5000 bars shal l be re fe r red to as high p ressure p o i n t s . In Chapter IV w e have ou t l ined h o w po la r i zab i l i t y m a y be inc luded in the present m o d e l s through the S C M F a p p r o x i m a t i o n . A g a i n , it is an exper imen ta l va lue of the po la r i zab i l i t y tensor w h i c h w e have used in the m o d e l s . The ind iv idua l c o m p o n e n t s of the tenso r were d e t e r m i n e d us ing the - 2 4 3 average po la r i zab i l i t y , a = 1.444X10 c m , repor ted by E i s e n b e r g and Kauzmann [171] and the exper imen ta l resu l t s of Mu rphy [164] to f i nd their re la t ive v a l u e s . The f o l l o w i n g va lues o f the c o m p o n e n t s of the po la r i zab i l i t y tensor we re o b t a i n e d : a = 1 . 5 0 1 X 1 0 ~ 2 4 c m 3 , X A. -24 3 a y y = 1.390X10 c m , a = 1 . 4 4 2 X 1 0 " 2 4 c m 3 . z z Th i s f o r m f o r the tenso r w a s e m p l o y e d in al l S C M F c a l c u l a t i o n s . W e a l so note that the S C M F resu l ts w e w i l l repor t are those d e t e r m i n e d when the energ ies g i ven by S i m p s o n ' s rule in tegra t ion o f e q . (2.80) are used in the c a l c u l a t i o n . The d ie lec t r i c c o n s t a n t s w e w i l l g i ve were de te rm ined through e q s . (2.93), again e m p l o y i n g S i m p s o n ' s rule to p e r f o r m the requ i red in tegra t ion . In our c a l c u l a t i o n s , and in s o m e of the f o l l o w i n g d i s c u s s i o n , it is conven ien t to exp ress al l pa ramete rs in reduced uni ts . The w a t e r - l i k e f l u i ds w e sha l l c o n s i d e r can be t o ta l l y charac te r i zed by the f o l l o w i n g reduced - 110 -p a r a m e t e r s : * p = p d s 3 ' ( 5 . 1 a ) d * " d / d s ' ( 5 . 1 b ) * M = <0M 2 / d s 3 ) * , ( 5 . 1 c ) 0 * = ( / 3 © 2 / d s 5 ) 2 , ( 5 . 1 d ) n * = ( ^ / d g 7 ) ^ , ( 5 . 1 e ) * a = a / d s3 , ( 5 . 1 f ) where /3 = 1 /kT and d =2 .8A is the h a r d - s p h e r e d iameter o f the w a t e r - l i k e s * m o d e l . It c l e a r l y f o l l o w s f r o m e q . (5.1b) that d =1 . 2. Cho ice of Bas is Set In the present s tudy it is impor tant to f i rs t de te rm ine an HNC b a s i s set wh i ch is c o m p u t a t i o n a l l y p r a c t i c a l , ye t g i v e s reasonab le c o n v e r g e n c e f o r the p rope r t i es w e sha l l c o n s i d e r . In Tab le III w e have p resen ted the number o f unique p ro j ec t i on te rms w h i c h must be inc luded in the HNC b a s i s se ts f o r both C» and tet rahedra l m o d e l s fo r n =2,3,4,5,6. In Tab le IV w e have 2v max ' ' ' ' exp l i c i t l y g i ven the unique p r o j e c t i o n s f o r n = 2 for bo th m o d e l s . A set o f TABLE III. Numbe rs o f unique p ro jec t i on t e rms requi red in HNC b a s i s s e t s . Bo th te t rahedra l and C „ m o d e l s are c o n s i d e r e d . max Tet rahedra l M o d e l C 2 v M o d e l 2 3 4 5 6 19 49 130 262 532 T A B L E IV. P r o j e c t i o n te rms inc luded in n = 2 bas i s se t s ' max M o d e l # o f Un ique P r o j e c t i o n s P r o j e c t i o n s Inc luded (mn\;uv) Tet rahedra l 12 (000;00),(022 (121;02),(123 (222;00),(222 00),(110 02),(220 22),(224 00),(112;00), 00),(220;22), 00),(224;22) ' 2 v 19 al l t hose a b o v e p lus (011 ;00),(022 ;02),(121 ;00),(123 ;00), (220;02),(222;02),(224,02) T A B L E V . M a x i m u m numbers o f n o n - z e r o te rms for any g i ven p ro jec t i on in the HNC b inary p roduc t . V a l u e s are g i ven f o r each b a s i s set inc luded in Tab le III. max Tet rahedra l M o d e l C 2 v M o d e l 2 3 4 5 6 28 150 1000 3200 12000 48 225 1800 6200 unique p ro j ec t i ons c o n s i s t s of al l t e rms w h i c h cannot be re la ted by s o m e s y m m e t r y requ i rement o f the m o d e l , i.e., e q s . (2.11), (2.12), (2.14), or (2.23). It is o b v i o u s f r o m Tab le III that the number of unique p ro jec t i ons g r o w s very rap id l y as n is i n c r e a s e d , w i t h the number of te rms requ i red by the max ' te t rahedra l m o d e l be ing s l i gh t l y more than half the number needed by a genera l m o d e l o f C 2 v s y m m e t r y . A s one might expec t , the HNC bas i s se t s fo r s y s t e m s o f l inear s y m m e t r y c o n s i s t o f far f ewe r p ro j ec t i ons when n m a x is large (e.g., 84 fo r n m a x = 6 [71,110]). In Tab le V w e have reco rded the m a x i m u m number of n o n - z e r o t e rms in the HNC b ina ry p roduc t , or doub le sum (cf. e q . (2.67)), that any g i ven p ro jec t i on in a s p e c i f i c bas i s set can ever have . A s w e w o u l d expec t , th is number g r o w s ve ry rap id l y as n is i nc reased s ince the to ta l number o f m 3 x t e rms in the doub le sum w i l l i nc rease as the square of the number of - 112 -T A B L E VI . C P U t ime requ i red per i te ra t ion on an F P S 164 array p r o c e s s o r . T i m e s are g i ven fo r each b a s i s set inc luded in Tab le III and are in C P U s e c o n d s . The va lues in pa ren theses are fo r ca l cu l a t i ons w h i c h l im i t the number of te rms c o n s i d e r e d in the b inary product (as d i s c u s s e d in the text). max Tet rahedra l M o d e l C 2 v M o d e l 2 3 4 5 6 2.5 7.0 30 90(80) 470(330) 3.0 9.5 50 (230) not a t tempted p r o j e c t i o n s in the bas i s se t . H o w e v e r , i nspec t i on of the va lues g i ven in T a b l e s III and V i m m e d i a t e l y leads us to conc lude that at least 8 5 % o f these t e rms are zero fo r a te t rahedra l m o d e l , and th is f igure r i ses to 90% fo r a genera l m o d e l o f C 2 y s y m m e t r y . F i n a l l y , in Tab le VI w e report the t i m e s requ i red fo r an F P S 164 array p r o c e s s o r (i.e., the C P U t ime requ i red) to c o m p l e t e one fu l l i te ra t ion o f the RHNC theory (see s e c t i o n 5 o f Chapter II). Va lues are g i ven fo r each bas i s set c o n s i d e r e d in th is s tudy . A l s o inc luded in the tab le are the t imes wh ich resul t w h e n small t e rms in the b inary product are i gno red . W e see that fo r the largest b a s i s set e x a m i n e d , the n = 6 tet rahedral there is a cons ide rab le ' max t ime s a v i n g s (~30%), a l though the amount of t ime s a v e d b e c o m e s much sma l l e r fo r s m a l l e r b a s i s s e t s . C o n s e q u e n t l y , th is t runcat ion p rocedure w a s e x t e n s i v e l y used fo r ca l cu l a t i ons i n v o l v i n g on l y the t w o largest bas i s se t s (i.e., the n = 6 te t rahedra l and the n = 5 C» ). It shou ld a l so be no ted that there max max 2 v y w a s no de tec tab le change o b s e r v e d in the so l u t i on se t s ob ta ined as a resul t o f th is add i t i ona l t runca t ion o f the HNC b inary p roduc t . The b a s i s set dependence of the RHNC resu l ts f o r the d ie lec t r i c c o n s t a n t , e , the average energ ies and the con tac t va lue of the rad ia l d i s t r i bu t i on f u n c t i o n , g ( r=d) , f o r a s p e c i f i c te t rahedra l f l u id is g i ven in Tab le VII. The paramete rs o f the m o d e l be ing exam ined w e r e c h o s e n to be s im i l a r to t hose o f the w a t e r - l i k e m o d e l s at 2 5 ° C . It is o b v i o u s f r o m Tab le VII that there is s t r ong bas i s set dependence when go ing f r o m n = 2 to n = 3 to m sx m sx nm a x = 4 . Fo r tuna te l y , w e f i nd that there is on l y s l ight bas i s set dependence - 113 -T A B L E VII. B a s i s set dependence o f e, the average energ ies and g( r=d) . A tet rahedra l f l u id fo r w h i c h p*=0.7317, M *=2 .50 and 0 g * = O.94 is c o n s i d e r e d . n I max 1 2 3 4 5 6 6 | 89.1 83.7 66.5 66.4 66.2 - I L ^ / N k T | 9.19 9.42 9.17 9.15 9.14 - U D Q / N k T | 5.93 6.49 7.01 7.06 7.07 - U Q Q / N k T | 1.49 1.50 1.76 1.76 1.79 9 ( r=d ) | 10.14 11.24 11.95 12.07 12.16 w i t h the n = 5 and n = 6 s y s t e m s . For al l the p rope r t i es c o n s i d e r e d in max max 1 r r . Tab le VI I . there is l ess than 2% d i f f e r e n c e b e t w e e n the n = 4 and n = 6 max max resu l t s . The b a s i s set dependence w e o b s e r v e fo r this tet rahedral m o d e l is s i m i l a r to that repor ted f o r m o d e l s w i t h d i p o l e s and l inear quadrupo les [71,110]. C o n s i d e r i n g the fac t that 100, and s o m e t i m e s m o r e , i te ra t ions are requ i red t o c o n v e r g e a s o l u t i o n set fo r a g i ven m o d e l s y s t e m , the n m a x = 4 bas i s set w o u l d s e e m to be a reasonab le c o m p r o m i s e b e t w e e n compu ta t i ona l requ i rements and accu racy . T h e r e f o r e , the n = 4 b a s i s se t w a s used M 7 ' max e x c l u s i v e l y to ob ta in a l l the resu l ts p resen ted in s e c t i o n s 3 and 4 . One o b s e r v a t i o n wh i ch can be made f o r the larger b a s i s se t s is that when g i ven in Car tes ian rep resen ta t ion (i.e., w h e n f m n ^ js g i ven by e q . (2.9a)) mn 1 h (r) b e c o m e s ve ry s m a l l f o r large 1, e.g., fo r 1>7 al l con tac t va lues are less than 1 0 ~ 3 . Thus one might expect that these b a s i s se t s c o u l d reasonab l y be t runcated on 1. Equat ion (2.48b) rep resen ts the requ i red c o n d i t i o n . T w o c a l c u l a t i o n s w e r e done us ing the same te t rahedra l m o d e l ; one w a s n „ = 4 , max 1 = 6 . and the other w a s n = 4 , 1 = 4 . In both c a s e s the s o l u t i o n max ' max ' max se t s ob ta ined we re marked ly d i f fe ren t (in an u n s y s t e m a t i c manner ) f r o m the s o l u t i o n w i t h no t runcat ion on 1. C l e a r l y , t runca t ion o f the bas i s set on 1 d o e s not represent a v iab le app roach f o r the m o d e l s w e are c o n s i d e r i n g . U p o n r e - e x a m i n a t i o n o f the con tac t va lues o f h ( r ) , th is t ime in B lum 's [101-103] rep resen ta t i on (i.e., w h e n fm n- '- j s g i ven by e q . (2.9b)), w e f i nd that the va lues do not dec rease f o r large 1. O b v i o u s l y , it is the 1/1! f ac to r in - 114 -the Ca r tes i an rep resen ta t i on o f h ( r ) wh ich c a u s e s the large 1 te rms fo r these s y s t e m s to appear t o b e c o m e s m a l l . F i n a l l y , w e po in t out that part ia l bas i s se t dependence (i.e., fo r nm a x = 2,3,4) w a s de te rm ined fo r a C 2 v quadrupo le f lu id at the s a m e reduced d e n s i t y and w i t h the s a m e d ipo le moment as the above tet rahedra l m o d e l . The reduced quadrupo le momen t w a s taken as be ing that o f wa te r at 2 5 ° C . A s one w o u l d expec t , w e found behav iour v e r y s im i l a r to that d e m o n s t r a t e d by the tetrahedral m o d e l , as g i ven in Tab le VI I . The bas i s set dependence o f a C 0 oc tupo le f l u id f o r n „ — A and n = 5 w a s a l so e x a m i n e d . W e found 2v r max max a s l i gh t l y s t ronger dependence (e.g., about a 4% drop in e and a 2% inc rease in the magni tude o f the tota l average ene rgy ) than w a s the c a s e fo r the tetrahedral s y s t e m . Th is is not a surp r i s ing resu l t s i nce we w o u l d expect the oc tupo le momen t to inc rease the magn i tudes o f the higher order p r o j e c t i o n s , and hence inc rease their impor tance . 3. Resul ts fo r Hard -Sphere Mode ls Ca l cu l a t i ons us ing the n = 4 bas i s set we re car r ied out at al l the max tempera tu res and p ressu res l i s ted in Tab le II. The C 2 v quadrupo le m o d e l w a s s tud ied at al l p o i n t s , wh i l e the C 2 v o c t upo le f l u i d w a s examined on l y at norma l p ressure at 2 5 ° C and 300°C . The c a l c u l a t i o n at 2 5 ° C w a s repeated w i th a tet rahedral m o d e l where 0 *=0.94 , th is va lue be ing c o n s i s t e n t w i th p r e v i o u s work [67,72,168]. The average d i p o l e m o m e n t s , m ' , as de te rm ined w i t h the S C M F a p p r o x i m a t i o n , are s h o w n in Figure 7. Not su rp r i s i ng l y , we f ind that m ' d e c r e a s e s w i t h i nc reas ing tempera tu re , but i n c reases w i t h inc reas ing p ressu re . Y e t , even at high tempera ture the value o f m ' is s t i l l w e l l a b o v e that o f the permanent d i po le m o m e n t o f wate r . The add i t i on of the oc tupo le m o m e n t is found to cause o n l y a s l ight inc rease in the average d ipo le m o m e n t . The resu l t s ob ta ined here are in s t r i k ing agreement w i t h p rev ious w o r k [67,168] in w h i c h the tet rahedra l m o d e l w a s examined o v e r the s a m e tempera ture and p ressure ranges . H o w e v e r , these ear l ier s tud ies [67,168] e m p l o y e d the RLHNC ins tead o f the RHNC theo ry . In the present s t u d y , w e found that the tet rahedra l f l u id at 2 5 ° C gave va lues f o r m ' a l m o s t ident ica l to those - 115 -Figure 7 . The mean d i p o l e m o m e n t of w a t e r - l i k e pa r t i c l es as a f u n c t i o n of temperature and p ressu re . The va lues of the m o m e n t s are in D e b y e s . The s o l i d and dashed l ines are the S C M F resu l ts f o r the C 2 y quadrupo le m o d e l at no rma l and high p r e s s u r e , r e s p e c t i v e l y . The do t t ed l ine rep resen ts the permanent d i po le m o m e n t o f wa te r . The open squares are va lues ob ta i ned fo r C 2 y o c t upo le f l u ids at no rma l p r e s s u r e , wh i l e the s ta rs represent S C M F resu l ts at the same po in ts for the s o f t C „ m o d e l d i s c u s s e d in the next s e c t i o n . - 116 -117 ob ta ined for the C 2 v quadrupo le s y s t e m . The average d ipo le m o m e n t s w e de te rm ine at 2 5 ° C c o m p a r e ve ry w e l l w i th va lues that have been ca lcu la ted fo r ice [172] or repor ted fo r another po la r i zab le w a t e r - l i k e m o d e l [37]. The d i e l ec t r i c cons tan t s ob ta ined fo r the po la r i zab le s y s t e m s be ing i nves t i ga ted here are s h o w n in Figure 8. W e f i nd that at higher t empera tu res , i.e., > 100°C , the s i m p l e h a r d - s p h e r e m o d e l s w e are c o n s i d e r i n g g i ve resu l ts wh i ch are in g o o d agreement w i th exper imen ta l da ta , both at no rma l and high p ressu re . A t l ower t empera tu res , h o w e v e r , w e f i nd that the h a r d - s p h e r e m o d e l s c o n s i s t e n t l y o v e r e s t i m a t e e. W e remark that th is w a s not the c a s e in ear l ier RLHNC s tud ies o f the te t rahedra l m o d e l [67], where g o o d agreement w i t h exper iment w a s f ound even at 2 5 ° C . U s i n g the RHNC theo ry , w e ob ta in e = 105 fo r the tet rahedral f l u id c o n s i d e r e d in Ref . 67 (we remark that th is value is larger than the v a l u e , e =97.4, f ound fo r the C 2 v quadrupo le f lu id) , even though the e f f e c t i v e d i po le momen t w a s e s s e n t i a l l y the s a m e . The RHNC is usua l l y the more accurate t heo ry , there fo re w e w o u l d expect it to g ive the better es t ima te of the d ie lec t r i c cons tan t fo r th is mode l s y s t e m . Later in this s e c t i o n w e w i l l s h o w that the pack ing s t ructure (i.e., the radia l d i s t r i bu t i on func t i on ) p red i c ted by the RHNC theory f o r the present h a r d - s p h e r e m o d e l s at 2 5 ° C is qui te d i f fe ren t f r o m that o f real wa te r . A t higher t empera tu res , both real wa te r and our mode l f l u ids b e c o m e less s t ruc tu red , and hence the st ructura l d i f f e r e n c e s b e c o m e less s i gn i f i can t . The fac t that w e ob ta in g o o d agreement w i t h exper iment fo r e at higher tempera tu res but not at l o w e r tempera tu res s t r o n g l y s u g g e s t s that the unique pack ing st ructure o f wa te r at l o w tempera tu re , i.e., 2 5 ° C , a f f e c t s its d ie lec t r i c p rope r t i es . Th is h y p o t h e s i s w i l l be e x a m i n e d in deta i l in s e c t i o n 4 . The impor tance o f po la r i zab i l i t y in the present w a t e r - l i k e f l u i ds can be e a s i l y d e m o n s t r a t e d . If w e ignore po la r i zab i l i t y and take the e f f e c t i v e d ipo le momen t to be s i m p l y the gas phase va lue , then w e ob ta in e =28.4 f o r the C 2 v quadrupo le f l u id at 2 5 ° C . It is a l so in te res t ing to point out that the M C Y [49] and T IP4P [41] m o d e l s , t w o popular w a t e r - l i k e m o d e l s , have recen t l y been s h o w n [45,46] to g ive d ie lec t r i c cons tan t s of 34 and 53 , r e s p e c t i v e l y , at 2 0 ° C . Bo th m o d e l s are n o n - p o l a r i z a b l e and have d i p o l e m o m e n t s of about 2.2D. In c o m p a r i s o n w i t h these resu l t s , t hose ob ta ined here at 25 °C fo r our po la r i zab le h a r d - s p h e r e f l u i ds appear more respec tab le . - 118 -Figure 8. The d ie lec t r i c cons tan t s o f wa te r and o f w a t e r - l i k e m o d e l s as f u n c t i o n s of tempera ture and p ressu re . The do ts and t r i ang les are S C M F resu l t s fo r C 2 V quadrupo le f l u i ds at no rma l and high p r e s s u r e , r e s p e c t i v e l y . The open squares and s tars re fer to the s a m e m o d e l s as in F igure 7. The s o l i d and dashed l ines represent exper imen ta l va lues [48,169,170] at no rma l and high p r e s s u r e , r e s p e c t i v e l y . - 119 -- 120 -In genera l , w e f i nd that add i t i on of the oc tupo le momen t to the C 2 v quadrupo le m o d e l c a u s e s a no t i cab le drop in e fo r a f i x e d d ipo le m o m e n t . W e have a l so f ound (see Figure 7) that the add i t i on of the oc tupo le momen t to the mode l resu l t s in a larger average d ipo le m o m e n t , w h i c h shou ld g ive r i se to a larger d ie lec t r i c cons tan t . W e f i nd that the t w o e f f e c t s tend to c a n c e l one another . C o n s e q u e n t l y , in Figure 8 w e see that the C 2 v quadrupo le and C 2 v oc tupo le f l u i ds g ive qui te s im i l a r d ie lec t r i c c o n s t a n t s . The to ta l average con f i gu ra t i ona l ene rg ies , U-j-Q-p/NkT, we re c o m p u t e d f o r these po la r i zab le s y s t e m s us ing e q . (4.20). A t 2 5 ° C the te t rahedra l , the C 2 v quadrupo le and the C 2 v o c t upo le f l u i ds g ive va lues of - 1 6 . 4 , - 16 .8 and - 1 8 . 1 , r e s p e c t i v e l y . These c o m p a r e w e l l w i t h the exper imen ta l value [41] o f - 1 6 . 7 . Th i s agreement is perhaps qui te fo r tu i t ous s i nce we might expect s o m e o f the te rms wh ich w e have ignored in the poten t ia l (e.g., d i s p e r s i o n and s h o r t - r a n g e repu l s i ve t e r m s ) to make fa i r l y large con t r i bu t i ons to the energy . W e remark that the RLHNC resul t [67] o f - 16 .9 f o r the tet rahedra l f l u i d at 2 5 ° C is again c l o s e to the value g i ven by the RHNC theory . The radia l d i s t r i bu t i on f u n c t i o n , g(r)=gQQ^( r ) , ob ta ined f o r the C 2 v quadrupo le and the C 2 y o c t upo le f l u i ds at 2 5 ° C are s h o w n in Figure 9. Bo th s y s t e m s have the s a m e e f f e c t i v e d i po le m o m e n t , m e * = 2 . 7 5 , and hence any d i f f e r e n c e s in s t ructure are due s o l e l y to the in f luence of the oc tupo le m o m e n t . W e point out . that th is va lue of m g * is very c l o s e to the S C M F resu l t s o f 2.74 and 2.77 fo r the po la r i zab le C 2 v quadrupo le and C 2 y oc tupo le f l u i d s , r e s p e c t i v e l y . A l s o s h o w n in Figure 9 is the radia l d i s t r i bu t ion func t i on o f the ha rd -sphe re re fe rence s y s t e m (which w o u l d a l so be the RLHNC resul t f o r g(r)). The e f f e c t s of the s t rong mu l t i po la r in te rac t ions are c lea r l y ev iden t . For the t w o mu l t i po la r f l u i d s , the con tac t va lue o f g(r) has inc reased d r a m a t i c a l l y f r o m the h a r d - s p h e r e v a l u e , wh i l e the p o s i t i o n of the f i rs t m i n i m u m has m o v e d inward . The resul t is a ve ry sharp f i r s t peak in g(r), as is ev iden t in F igure 9. The s e c o n d peak has a l s o sharpened and sh i f t ed inward fo r bo th mu l t i po la r s y s t e m s . H o w e v e r , i ts m a x i m u m s t i l l o c c u r s at a sepa ra t i on of about 2 d s > w h e r e a s fo r real wa te r the max imum in the s e c o n d peak appears at about 1.65d c o r r e s p o n d i n g to the tetrahedral d i s tance [121,122]. If w e c o m p u t e the c o o r d i n a t i o n numbe rs , - 1 2 1 -Figure 9 . Radia l d i s t r i bu t ion f unc t i ons fo r w a t e r - l i k e f l u i d s at 2 5 ° C . The s o l i d and dashed l ines are RHNC resu l t s f o r the C 2 v quadrupo le and the C 2 y oc tupo le m o d e l s , r e s p e c t i v e l y , w h e n m e * = 2 . 7 5 . The do t t ed l ine represen ts the h a r d - s p h e r e radia l d i s t r i bu t i on f unc t i on fo r that dens i t y . 1.45-. 1.30H 1.15H g(r) 1.00-^  0 .85H 0.70-0.0 16.-i 1 — I — r — I 0.0 0.08 0.16 i — r 0.4 n — T 0.8 1—r 1.2 n—r 1.6 ~i—r 2.0 n — i 2.4 to (r-ds)/ds - 123 -R 2 C N = 47r p f r g ( r ) d r , ( 5 . 2 ) d s where R rep resen ts the sepa ra t i on c o r r e s p o n d i n g to the f i rs t m i n i m u m o f the in tegrand , w e f ind that both the m o d e l s g i ve va lues of about 5.7 at 2 5 ° C . A g a i n , these do not c o m p a r e w e l l w i t h the resul t fo r water at 2 5 ° C , where C N ^ 4 . 5 [122]. Thus , even accoun t i ng fo r the s t ructura l e f f e c t s o f the unrea l i s t i c r epu l s i ve co res o f the C 2 v m o d e l s , the RHNC resu l t s fo r g(r) fo r these t w o f l u i ds are s t i l l qu i te d i f fe ren t f r o m that of real wa te r . The e f f e c t s o f the add i t i on o f the o c t u p o l e momen t to the C 2 V quadrupo le m o d e l can a l s o be seen in Figure 9. A s might be e x p e c t e d , the con tac t peak o f g(r) b e c o m e s s o m e w h a t s teeper due to the ext ra te rms in the mu l t i po le po ten t i a l . The m a x i m u m in the s e c o n d peak s h o w s v i r tua l l y no change ; h o w e v e r , a s m a l l shou lder on the s e c o n d peak has d e v e l o p e d at the te t rahedra l d i s t a n c e . M o r e o v e r , the th i rd and four th peaks appear to be sh i f t ed s l i gh t l y i nwa rd . C l e a r l y , the oc tupo le m o m e n t d o e s in f luence the pack ing s t ructure in a des i rab le w a y , but the magn i tude of i ts e f f e c t s are s t i l l r e l a t i ve l y s m a l l . In an a t tempt to t ry and i m p r o v e our resu l ts f o r the rad ia l d i s t r i bu t ion f u n c t i o n , w e a l s o exam ined the e f f e c t s of i nc reas ing the va lues o f the quadrupo le and oc tupo le m o m e n t s . W e found that inc reas ing the quadrupo le and o c t u p o l e by 15% and 50%, r e s p e c t i v e l y , p r o d u c e s ve ry l i t t le change in the RHNC resul t fo r g(r) except to generate an even s teeper con tac t peak. T h e r e f o r e , at least w i th in the RHNC theo ry , h a r d - s p h e r e m o d e l s con ta in ing on l y the l o w order mu l t i po le m o m e n t s o f wa te r appear unable to g i ve a tet rahedral s t ructure s i m i l a r to that of wa te r . In F igu res 10 -16 w e have s h o w n s o m e of the p ro j ec t i ons of the pair co r re l a t i on f u n c t i o n s of the C 2 V quadrupo le and C 2 V o c t upo le f l u i d s . The p ro j ec t i ons w h i c h have been p lo t ted are al l t h o s e wh i ch con ta i n po ten t ia l t e rms fo r bo th m o d e l s , as w e l l as h g Q ^ ( r ) . W e remark that these p ro jec t i ons represent o n l y a s m a l l subse t o f the to ta l number of unique p ro jec t i ons in the HNC b a s i s se t used in the c a l c u l a t i o n s . M o s t o f the co r re l a t i on f unc t i ons are at least m o d e r a t e l y a f f e c t e d by the add i t i on o f the oc tupo le m o m e n t to the 1 2 3 2 2 4 1 2 3 C 2 v quadrupo le m o d e l ; b.QQ ( r ) and YIQQ ( r ) change m a r k e d l y , wh i l e ( r ) - 124 -Figure 10. The p ro j ec t i on h Q Q ^ ( r ) . The s o l i d and dashed l ines represent RHNC resu l ts f o r the C 2 v quadrupo le and the C 2 v o c t u p o l e m o d e l s , r e s p e c t i v e l y , at 2 5 ° C and m *=2 .75 . e - 126 -Figure 11. 1 1 2 The p r o j e c t i o n h n n ( r ) . The cu rves are d e f i n e d as in F igure - 128 -Figure 12. '00 1 2 3 The p ro jec t i on h n n ( r ) . The cu rves are de f i ned as in F igure 10. 0.08-1 0.06H 0 .04H .123 '00 0.02H 0.00--0.02 1.0-0.5H 0.0-0.0 1 1 1 0.08 0. I 1 1 1 1 1 1 1 1 1 1 1 1 0.0 0.4 0.8 1.2 1.6 2.0 2.4 (r-ds)/d5 - 130 -Figure 13. J02 1 2 3 The p r o j e c t i o n hno ( r ) . The cu rves are de f i ned as in Figure 0 . 0 5 - 1 o.ooH - Q . Q 5 - J 123 - 0 . 1 0 -- 0 . 1 5 H - 0 . 2 0 -0 . 0 - 132 Figure 14. 2 2 4 The p ro j ec t i on h.QQ ( r ) . The cu rves are de f i ned as in Figure 0.06-. 0.04H 0 . 0 2 ^ u224 "oo o.oo-- 0 . 0 2 H -0 .04-0 . - 134 -Figure 15. '02 224 The p ro jec t i on hno ( r ) . The cu rves are de f i ned as in Figure 0 . 0 8 - n 0 . 0 6 H 0.04-^ u224 n 02 0 . 0 2 H 0 . 0 0 -- 0 . 0 2 -A 0 . 0 0.30-1 —\ \ 0 . 1 5 - \ \ \ \ 0 . 0 0 0 . 0 0 . 0 8 0 . 1 6 CO ~ i — i — i — r n — i 1.6 2.0 2 .4 - 136 -Figure 16. 224 The p r o j e c t i o n ( r ) . The cu rves are de f i ned as in Figure 10. 0.05-1 0.02H 0.0 0.4 0.8 1.2 1.6 2.0 2.4 (r-ds)/ds - 138 -and Y122 ( r ) appear to be the least a f f e c t e d . H o w e v e r , w e f i nd that there is no s y s t e m a t i c va r i a t i on in the co r re l a t i on f unc t i ons s ince each rep resen ts a d i f f e ren t angle dependence . The d i p o l e - d i p o l e c o r r e l a t i o n s , as g i ven by h.QQ^(r) and h^^it), are of par t icu lar in teres t . In both c a s e s w e f ind that the co r re la t i on f unc t i ons b e c o m e l ess s t ruc tured w i th the add i t i on o f the o c t u p o l e m o m e n t . Th is is c o n s i s t e n t w i t h the o b s e r v e d d rop in the d ie lec t r i c cons tan t and the dec rease in the average d i p o l e - d i p o l e ene rgy . C l e a r l y , the o c t u p o l e f o r c e s act to d isrupt the d ipo la r s t ructure w i th in the f l u i d . Th is i s , o f c o u r s e , a l s o true of the quadrupo le [67]. B e f o r e c o n c l u d i n g this d i s c u s s i o n of resu l t s fo r h a r d - s p h e r e w a t e r - l i k e f l u i d s , w e po in t out that because o f i ts s i m p l i c i t y , the te t rahedra l m o d e l w a s used e x t e n s i v e l y in our s tudy of m o d e l aqueous e l e c t r o l y t e s o l u t i o n s (as w i l l be d i s c u s s e d in Chapter VI). It is o b v i o u s f r o m Tab les III and VI that th is s o l v e n t m o d e l reduces the c o m p u t a t i o n a l r e s o u r c e s requ i red to s o l v e the RHNC theo ry . Th is b e c o m e s an impor tan t c o n s i d e r a t i o n in the c a s e o f e l ec t ro l y te s o l u t i o n s , pa r t i cu la r l y at l o w concen t ra t i on where a much larger number of po in t s is requ i red in the numer ica l g r i d . In p rev ious s tud ies [67,72,79-81] in w h i c h the te t rahedra l mode l w a s e m p l o y e d , the parameter 0 w a s s o m e w h a t —26 2 ^ arb i t ra r i l y set to the value of 2 . 5 0 X 1 0 - esu c m . In the p resen t s tudy , w e have found that th is va lue of the square quadrupo le m o m e n t unde res t ima tes the e f f e c t o f the fu l l quadrupole tenso r o f wa te r . H o w e v e r , w e do f i nd that —26 2 © s =2 .57X10 esu c m (which is just s l i gh t l y larger than hal f the sum of the magn i tudes of 0 and © ) w o r k s ve ry w e l l as an effective square quadrupo le xx y y m o m e n t . In th is c a s e the te t rahedra l and C 2 y quadrupole m o d e l s g i ve a l m o s t iden t i ca l resu l ts f o r ai l average p r o p e r t i e s , inc lud ing the d i e l ec t r i c cons tan t and average ene rg ies . The radial d i s t r i bu t i on f u n c t i o n s appear i nd i s t i ngu i shab le . T h u s , th is va lue o f the square quadrupo le m o m e n t w a s used in the te t rahedra l s o l v e n t m o d e l e m p l o y e d in our s tudy o f aqueous e l e c t r o l y t e s o l u t i o n s . - 139 -4. Resul ts f o r So f t Mode ls In the p rev ious s e c t i o n w e have found that fo r ha rd - sphe re w a t e r - l i k e f l u i d s , the RHNC resu l ts fo r e do not agree w e l l w i t h exper iment at l o w e r t empe ra tu res , i.e., < 100°C. W e have specu la ted that th is d i s c r e p a n c y might be due to the fac t that the s t ruc ture , i.e., the rad ia l d i s t r i bu t ion f u n c t i o n , o f these m o d e l s y s t e m s is quite d i f fe ren t f r o m that o f real wa te r . In th is s e c t i o n w e w i l l s h o w h o w the correct s t ructure can be ob ta ined by m o d i f y i n g on l y the spher ica l po ten t ia l (i.e., by mak ing the m o d e l so f t ) . W e w i l l then examine the d ie lec t r i c p roper t i es o f th is new m o d e l . F i r s t , let us de f ine an emp i r i ca l s h o r t - r a n g e poten t ia l U g R ( r ) = r[hear + B e b r + C e c x 2 + D e d y " ] , ( 5 . 3 a ) where x = r - X q ( 5 . 3 b ) and y = r - y Q . ( 5 . 3 c ) Th is po ten t ia l w a s added to the C 2 v o c t upo le m o d e l as a sphe r i ca l l y s y m m e t r i c te rm and the h a r d - s p h e r e d iamete r w a s reduced to 0.92d . The -1 -1 paramete rs a , b, c and d w e r e e m p i r i c a l l y a s s i g n e d the va lues - l O d , - 4 0 d - 2 - 4 _ 1 3 S S - 3 5 d and - 7 0 0 d , r e s p e c t i v e l y , T w a s taken as 1.34X10 ergs per m o l e c u l e , wh i l e x = 1 . 1 6 d „ and y = 1 . 6 5 d „ . In Tab le VIII the va lues of A B C o s o s ' and D used to generate three d i f fe ren t f o r m s of U g p ^ 1 " ) a r e g i v e n . T h e s e w i l l p rove use fu l in examin ing the e f f e c t s o f turn ing o n , or o f f , cer ta in parts o f the p o t e n t i a l . In Figure 17 w e have s h o w n the three f o r m s of the po ten t ia l T A B L E VIII. Pa ramete rs fo r U c 0 ( r ) . Po ten t ia l | A I B | c I D 4>> i 40000 | 1 X 1 0 1 5 | 0 I o 40000 | 1 X 1 0 1 5 | 0.18 I o I 40000 | 1 X 1 01 5 | 0.18 | - 0 .08 - 140 -Figure 17. S o f t po ten t ia l s at 2 5 ° C . The s o l i d , do t t ed and dashed l ines represent the po ten t i a l s / 3 u ^ j ^ ( r ) , j 3 u ^ R \ r ) and / 3 u ^ R ' \ r ) , r e s p e c t i v e l y , where the f o r m s o f these po ten t i a l s are de f i ned by the pa ramete rs g i ven in Tab le VIII. 9 . 0 7 . 0 5 . 0 -(r) -3 . 0 -1 . 0 -- 1 . 0 0 - 142 -Figure 18. Radia l d i s t r i bu t i on f unc t i ons fo r s o f t w a t e r - l i k e m o d e l s at 2 5 ° C and m *=2 .75 . e The s o l i d , dashed and do t ted l ines are RHNC resu l ts fo r the m o d e l s e m p l o y i n g ^ U S R ^ r ^ ' ^ ^ R ^ 1 ^ a n d ^ ^ R ^ 1 ^ ' r e s P e c t ' v e l v - T n e d a s h - d o t l ine rep resen ts g(r) fo r the m o d e l s y s t e m us ing | 3 u ^ 3 \ r ) but w i th no oc tupo le m o m e n t . - 144 -(2) w h i c h w e w i l l i nves t iga te at 2 5 ° C . W e po in t out that j 3 u u D ( r ) m a y be hard ( 3 ) to de tec t because it is i nd is t ingu ishab le f r o m / J u u - ( r ) fo r r<1 3 5 d and ( 1 ) S / 3 U g R ( r ) f o r r > 1 . 4 5 d g . It is c lear f r o m Figure 17 that these po ten t i a l s are s i m p l e s m o o t h f u n c t i o n s . In F igure 18 w e have s h o w n RHNC resu l ts f o r the radia l d i s t r i bu t i on f unc t i on fo r 4 d i f fe ren t so f t w a t e r - l i k e m o d e l s 2 5 ° C . The dependence upon the va r i ous te rms in the so f t p o t e n t i a l , as w e l l as upon the o c t u p o l e m o m e n t can be s e e n . W e f i nd that when a s i m p l e so f t po ten t ia l l ike | 3 u ^ R \ r ) is e m p l o y e d , the f i rs t peak in g(r) b e c o m e s qui te b r o a d , i ts m a x i m u m be ing less than 2.5. Th is is c o n s i s t e n t w i th resu l ts wh i ch have been repor ted [121] fo r wa te r at 2 5 ° C . H o w e v e r , the f i rs t m i n i m u m in g(r) appears a p p r o x i m a t e l y where the s e c o n d peak shou ld be . M o r e o v e r , the c o o r d i n a t i o n number is s o m e w h e r e w i th in the range 6 - 8 , depend ing upon where w e s top the in tegra t ion in e q . (5.2). In order to co r rec t the p o s i t i o n of the f i rs t m i n i m u m and to reduce the C N , another repu l s i ve te rm is added to the spher i ca l ( 2 ) p o t e n t i a l , i.e., w e use 0Ug R ( r ) . The f i rs t m i n i m u m in g(r) n o w appears in about the co r rec t p o s i t i o n and C N = 4 . 7 w h i c h is c l o s e to that of real wa te r . H o w e v e r , the radia l d i s t r i bu t ion f unc t i on s t i l l has a m i n i m u m at about r = 1.7d s wi th s m a l l peaks on ei ther s i d e . These t w o s m a l l peaks are d rawn into a s ing le peak cen t red at r = 1.65d w i t h the add i t iona l a t t rac t ive te rm con ta ined ( 3 ) in / 3 U g R ( r ) . W e a l s o see f r o m Figure 18 that if the oc tupo le m o m e n t is not inc luded in the m o d e l , then g(r) is c l ea r l y a f f e c t e d . The m o s t s i gn i f i can t o f the e f f e c t s is the drop in the s e c o n d peak at the te t rahedra l d i s t a n c e . Th is w o u l d s e e m to ind ica te that the o c t u p o l e momen t is impor tan t in s tab i l i z i ng the te t rahedra l s t ruc ture . In the d i s c u s s i o n b e l o w w e sha l l re fer to the ( 3 ) w a t e r - l i k e m o d e l w h i c h u t i l i zes UgR ( r ) and inc ludes the o c t u p o l e m o m e n t as the soft C 2 v m o d e l . The rad ia l d i s t r i bu t i on f unc t i ons o f the so f t C 2 v and o f the C 2 y o c t u p o l e f l u i d s at 2 5 ° C are c o m p a r e d w i t h the exper imen ta l g(r) o f Nar ten and L e v y [121] in Figure 19. The RHNC resul t fo r the so f t C 2 y m o d e l is in g o o d agreement w i t h the exper imen ta l cu r ve , w h i l e that of the C 2 y o c t u p o l e f l u i d is c l ea r l y in v e r y poor ag reemen t . In F igure 20 the s t ructure f a c t o r s , S(k) , o f the t w o m o d e l s y s t e m s are aga in c o m p a r e d w i th the exper imen ta l resul t o f Nar ten and L e v y [121]. In gene ra l , the par t ia l s t ructure fac to r is de f i ned [13] by - 145 -Figure 19. Radial d i s t r i bu t i on f unc t i ons o f wa te r and o f w a t e r - l i k e f l u ids at 2 5 ° C and m *=2 .75 . The s o l i d and dashed l ines are RHNC resu l t s fo r the so f t C - and e 2v the C 2 v o c t u p o l e m o d e l s y s t e m s , r e s p e c t i v e l y . The do t ted l ine is the exper imenta l resu l t o f Nar ten and L e v y [121]. W e no te that m o s t of the contac t peak f o r the C - o c t u p o l e m o d e l d o e s not appear on the p lo t . 2 . 4 r/ds- 0.92 - 147 -Figure 20. Struc ture f a c t o r s of wa te r and of w a t e r - l i k e f l u i ds at 25 °C and m *=2 75 The e cu rves are as in Figure 19. - 149 S a 0 ( k ) = 1 + J" ^ ( g a ^ ( r ) - l ) sinkr dr , (5.4) C l e a r l y , the s o f t C 2 v mode l is aga in in g o o d agreement w i t h exper imen t . The S(k) o f the C 2 v o c t upo le f l u id d o e s not s h o w the charac te r i s t i c sp l i t peak ev ident in the exper imen ta l curve and its amp l i tude of o s c i l l a t i o n is t o o large. W e s t r e s s that the on l y d i f f e r e n c e b e t w e e n the C 0 oc tupo le and the s o f t ( 3 ) C 0 m o d e l s is the add i t i on o f the spher i ca l po ten t i a l , uu^ , ( r ) . H o w e v e r , what ( 3 ) is not en t i re ly c lear is whether U g R ( r ) is s i m p l y co r rec t i ng fo r i nadequac ies in the ha rd -sphe re m o d e l , or whether it is a l s o c o m p e n s a t i n g fo r a d e f i c i e n c y in the RHNC theory . Ve ry recent w o r k [173] sugges t s that part o f the p rob lem may ac tua l l y l ie w i th the RHNC theory . Ful l R H N C / S C M F ca l cu l a t i ons we re then done at 2 5 ° C and 300°C us ing the s o f t C 2 v m o d e l . The resu l t ing average d ipo le m o m e n t s and d ie lec t r i c cons tan t s are s h o w n in F igures 7 and 8, r e s p e c t i v e l y . A t 2 5 ° C the average d ipo le m o m e n t is s l i gh t l y larger than the C 2 y oc tupo le resu l t , wh i l e at 300°C the o p p o s i t e is f o u n d . O f m o s t impo r tance here is t he . fac t that the d ie lec t r i c cons tan t s ob ta ined at both tempera tu res are in ve ry g o o d agreement w i t h exper imen t . Thus , it w o u l d appear that the unique pack ing s t ructure of wa te r at l o w tempera ture d o e s in f luence i ts d i e l ec t r i c p rope r t i es . A g a i n , w e e m p h a s i z e that there is no th ing unique about our c h o i c e o f a ( 3 ) so f t p o t e n t i a l , i.e., u ^ p ( r ) . Our pu rpose w a s m e r e l y to s h o w that the exper imenta l s t ructure for wa te r at 2 5 ° C c o u l d be f i t w i t h a s i m p l e mu l t ipo la r mode l by ad jus t ing the s o f t spher i ca l p o t e n t i a l . - 150 -CHAPTER VI RESULTS FOR MODEL AQUEOUS ELECTROLYTE SOLUTIONS 1. Introduction In th is s tudy w e have i nves t i ga ted m o d e l e l ec t ro l y te s o l u t i o n s c o n s i s t i n g of a s ing le sal t d i s s o l v e d in one of the w a t e r - l i k e s o l v e n t s d e s c r i b e d in Chapter V. The vas t ma jo r i t y of the s o l u t i o n s exam ined (v i r tua l ly al l t hose at f in i te concen t ra t i on ) u t i l i ze the tet rahedral s o l v e n t m o d e l because o f i ts s i m p l i c i t y , as d e s c r i b e d in Chapters II and V . A l s o , in Chapter V w e have s h o w n that when the fu l l quadrupo le tenso r o f wa te r is rep laced by an e f f e c t i v e square quadrupo le m o m e n t , the p rope r t i es o f the pure w a t e r - l i k e s o l v e n t rema in e s s e n t i a l l y unchanged. The ions are t reated s i m p l y as charged hard s p h e r e s , as d i s c u s s e d in Chapter II. O n l y un iva lent ions w e r e c o n s i d e r e d ; their h a r d - s p h e r e d i ame te r s are g i ven in Tab le I. A l l the m o d e l s o l u t i o n s i nves t i ga ted we re at 2 5 ° C . M o d e l aqueous e l e c t r o l y t e s o l u t i o n s o f th is t ype had been p r e v i o u s l y s tud ied at in f in i te d i lu t ion w i t h the RLHNC theo ry [79-81 ] . A s an e x t e n s i o n of th is ear l ier wo rk [ 79 -81 ] , w e began the p resen t s tudy by f i r s t a t tempt ing to examine the s a m e m o d e l s o l u t i o n s , aga in us ing the R L H N C , but n o w at f in i te c o n c e n t r a t i o n . W e found that a l though the RLHNC theory appears to g i ve reasonab le resu l ts fo r i o n - i o n po ten t ia l s o f mean f o r c e at in f in i te d i l u t i on , it d o e s not w o r k w e l l f o r m o d e l aqueous e l e c t r o l y t e s o l u t i o n s at f in i te c o n c e n t r a t i o n . S m a l l i ons appear to be qui te i nso lub le (e.g., f o r KCI w e w e r e ab le to ob ta in numer ica l s o l u t i o n s on l y up to a concen t ra t i on of 0 .125M), wh i l e larger ions appear to be quite s o l u b l e (e.g., f o r C s l w e w e r e ab le to reach a concen t ra t i on o f 2 M w i th no apparent d i f f i cu l t i es ) . C l e a r l y , th is behav iour is not c o n s i s t e n t w i th what is o b s e r v e d fo r real aqueous s o l u t i o n s of the a lka l i ha l ides [169]. Th is unrea l i s t i c behav iour o f the R L H N C theo ry is la rge ly due to the fac t that the RLHNC c l o s u r e a l l o w s no d i rec t c o u p l i n g b e t w e e n a n i s o t r o p i c t e r m s in the pair po ten t i a l and the radial d i s t r i bu t i on f u n c t i o n , as d i s c u s s e d in Chapter II. C o n s e q u e n t l y , w e f ind that in the RLHNC theory the degree of s o l v a t i o n of an ion (i.e., the pack ing of the s o l v e n t around the ion) is de te rm ined a lmos t e x c l u s i v e l y by the h a r d - s p h e r e pack ing s t ruc ture . O b v i o u s l y t hen , w e w o u l d not expec t the RLHNC theory to g i ve - 151 -g o o d resu l ts where there are s t rong in te rac t ions b e t w e e n the ion and the s o l v e n t , as is the case fo r aqueous e l ec t r o l y t e s o l u t i o n s . The re fe rence Q H N C theory [63] w a s a l so b r i e f l y e x a m i n e d fo r the s a m e m o d e l fo r aqueous e l ec t r o l y t e s o l u t i o n s . The ion s o l v a t i o n , par t i cu la r ly f o r s m a l l e r i o n s , w a s f o u n d to great ly imp rove (i.e., the con tac t va lues o f 9 i s ( r ) i nc reased sharp ly ) . Fur thermore , th is change in the degree o f s o l v a t i o n had a s t rong in f luence upon not on l y the t h e r m o d y n a m i c p rope r t i es o f the s o l u t i o n , but a l so the i o n - i o n s t ructure (i.e., g^jCr)). These resu l t s c l ea r l y s h o w e d the s e n s i t i v i t y o f these s y s t e m s and ind ica ted that a ve ry accura te theory w o u l d be requ i red in order to ob ta in accurate resu l ts fo r the present m o d e l s . T h e r e f o r e , in the remainder o f this Chapter e s s e n t i a l l y a l l the resu l ts repor ted fo r mode l e l e c t r o l y t e s o l u t i o n s we re ob ta ined us ing the RHNC theo ry , as d e s c r i b e d in Chapter II. The HNC b a s i s set dependence o f the p rope r t i es o f in terest in th is s tudy w a s e x a m i n e d . The par t icu lar c a s e o f a m o d e l KCI s o l u t i o n at 0.5M w a s c o n s i d e r e d . A s w a s the case fo r the pure s o l v e n t , s t r ong b a s i s set dependence w a s o b s e r v e d fo r al l p roper t i es when go ing f r o m n m a x = 2 to n = 3 to n = 4 . For tuna te l y , w e again f ound on l y s l igh t b a s i s set IT13X IT13X dependence fo r a l m o s t al l resu l ts when go ing to n = 5 (e.g., the cons tan t v o l u m e de r i va t i ve o f the ac t i v i t y c o e f f i c i e n t i nc reased by about 1%). Of the proper t ies e x a m i n e d , s h o w e d the greatest s e n s i t i v i t y , i nc reas ing 2.6 c c / m o l e . W e remark that the n m a x = 5 ca l cu la t i on w a s push ing our compu ta t i ona l resou rces to their l im i t , and hence the n = 6 ca l cu la t i on w a s not even max a t t emp ted . Thus , the n m a x = 4 HNC bas i s set (now con ta in i ng 95 unique p ro j ec t i ons ) w a s used e x c l u s i v e l y to ob ta in al l other resu l ts repor ted in this chapter . In th is s tudy w e have exam ined seve ra l m o d e l aqueous e lec t ro l y te s o l u t i o n s ( e m p l o y i n g the tet rahedra l w a t e r - l i k e s o l v e n t ) ove r a range of c o n c e n t r a t i o n s , inc lud ing in f in i te d i l u t i on , as g iven in Tab le IX . (We remark that the m o l a r i t y concen t ra t i on sca le has been used un i ve rsa l l y throughout th is chapter.) A q u e o u s s o l u t i o n s o f a lka l i ha l ides have r e c e i v e d the m o s t a t ten t ion , a l though three other sa l t s have a l so been c o n s i d e r e d . In order to m i m i c the cons tan t p ressu re c o n d i t i o n s under w h i c h m o s t real e l e c t r o l y t e so l u t i ons are s t u d i e d , w e have used the exper imen ta l dens i t i es [174,175] o f the real s y s t e m s wheneve r p o s s i b l e in our m o d e l c a l c u l a t i o n s . For bo th M B r and M' l w e have - 152 TABLE IX. M o d e l aqueous e l e c t r o l y t e s o l u t i o n s s tud ied . T h o s e concen t ra t i ons g i ven in paren theses are fo r s o l u t i o n s wh ich are b e y o n d the so lub i l i t y l imi t o f their real coun te rpa r t s . The concen t ra t i ons ind ica ted w i th a star represent t hose b e y o n d w h i c h numer ica l s o l u t i o n s cou ld not be ob ta i ned . Sa l t | C o n c e n t r a t i o n ( m o l e s / l i t r e ) L iF I 0 LiCI NaC l KCI M B r M ' I o-0, 0.025, 0.05, 0.075, 0.1, 0.25, 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 4.0, (6.0), (8.0), (12.0), (16.0) 0, 0.025, 0.05, 0.075, 0.1, 0.15, 0.25, 0.5, 0.75. 1.0. 1.5. 2.0. 3.0. 4.0 C s l 0, 0.025, 0.05, 0.075, 0.1, 0.15, 0.25, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5, (3.0), (4.0), (6.0), (9.0), (9.1)* 0, 0.025, 0.05, 0.075, 0.1, 0.25, 0.5, 0.6, 0.7, 0.75, 0.8, 0.85, 0.9, 1.0 0, 0.025, 0.05, 0.075, 0.1, 0.25, 0.5, 0.6, 0.65. 0.7. 0.725. 0.74* EqEq | 0, 0.025, 0.05, 0.075, 0.1, 0.25, 0.5 e m p l o y e d the dens i t y data o f a ^ H ^ N B r s o l u t i o n [175]. W e point out that M B r and M'l have the s a m e va lue of d + _ (as g i ven by e q . (2.24b)). In the d i c u s s i o n s b e l o w w e w i l l use d + _ as measure of the ion size o f a sa l t . For the EqEq s o l u t i o n s w e have e m p l o y e d the dens i t i e s of aqueous NaC l s i nce bo th sa l t s aga in have the s a m e va lue of d + _ . Thus , resu l ts f r o m m o d e l EqEq and N a C l s o l u t i o n s , as w e l l as f r o m M B r and M'l s o l u t i o n s , can be c o m p a r e d in order to exam ine the e f f e c t s of ion a s y m m e t r y . In the present s tudy w e found that the RHNC theory cou ld be s o l v e d fo r our m o d e l N a C l and C s l s o l u t i o n s at concen t ra t i ons above the s o l u b i l i t y l im i t s of their real coun te rpa r t s , as ind ica ted in Tab le IX. For e x a m p l e , w e w e r e s t i l l ab le to s tudy our m o d e l NaC l s o l u t i o n at a concen t ra t i on o f 1 6 M , w h i c h c o r r e s p o n d s to the m o l e f r ac t i ons X ; = 0 . 2 5 and x_ = 0.50! D e n s i t i e s fo r these s y s t e m s w e r e ob ta ined by ex t rapo la t ing the exper imen ta l resu l t s . W e po in t out that fo r the sa l t s M' l and C s l there we re concen t ra t i ons b e y o n d - 153 -w h i c h we w e r e numer i ca l l y unable to s o l v e the RHNC theory (a lso ind ica ted in Tab le IX). The behav iour o f these s o l u t i o n s near these p o i n t s w i l l be d i s c u s s e d b e l o w . It is o b v i o u s f r o m Tab le IX that f o r the m o s t part the a lka l i ha l ides are very so l ub le in the te t rahedra l s o l v e n t . A n apparent excep t i on to th is rule appears to be L i C l . H o w e v e r , the d i f f i c u l t i e s encoun te red w i t h L i C l m a y qui te p o s s i b l y be numer ica l in o r i g i n . In o rder to reach a f in i te (but l o w ) concen t ra t i on of a g i ven sa l t , a numer i ca l s o l u t i o n set fo r that s a m e sa l t at in f in i te d i lu t ion a l w a y s s e r v e d as the in i t ia l guess ( input) into the RHNC theory . For al l the other sa l t s i nves t i ga ted {i.e., t hose g i ven in Tab le IX) th is s c h e m e w o r k e d w e l l . Un fo r t una te l y , because of the ex t reme behav iour d e m o n s t r a t e d by L i + at in f in i te d i lu t ion (as d e s c r i b e d b e l o w ) , th is me thod may not be app l i cab le to mode l s o l u t i o n s of L i + s a l t s . The s o l v e n t m o d e l s w e have c o n s i d e r e d in th is s tudy are po la r i zab le . Hence , a fu l l S C M F ca l cu l a t i on (as ou t l i ned in Chapter IV) w o u l d be requ i red , in p r i nc i p l e , at e v e r y concen t ra t i on fo r each s o l u t i o n w e w i s h to i nves t i ga te . Of c o u r s e , f o r a so l u t i on at in f in i te d i lu t ion al l p roper t i es o f the s o l v e n t , inc lud ing i ts e f f e c t i v e d i po le m o m e n t , rema in unchanged f r o m those o f the pure s o l v e n t . A fu l l S C M F ca l cu l a t i on w a s car r ied out on a KCI so l u t i on at 2 . 0 M . W e found that even at th is re la t i ve l y high c o n c e n t r a t i o n the average l oca l e lec t r i c f i e l d in the bulk (as g i ven by the S C M F t h e o r y ) changed by l ess than 1% f r o m the pure s o l v e n t va lue . C o n s e q u e n t l y , the e f f e c t i v e d ipo le m o m e n t o b t a i n e d , m *=2 .734 , d i f f e r s f r o m the pure te t rahedra l so l ven t resul t ' e by less than 0.25%. If w e de f i ne the quant i ty 2 N k T N k T N k T s s s then it c l ea r l y f o l l o w s f r o m e q s . (4.12) and (4.19) that w i t h i n the S C M F theory Y w i l l be re la ted to the average loca l f i e l d in the bulk. In Figure 21 w e have s h o w n Y fo r seve ra l s o l u t i o n s w h i c h we re e x a m i n e d us ing m e * = 2 . 7 4 , the pure so l ven t va lue . The va lue o f Y ob ta ined fo r a 2.0M KCI s o l u t i o n has a l s o been c lea r l y i nd i ca ted . It is o b v i o u s f r o m Figure 21 that even at concen t ra t i ons o f 4 . 0 M , w e w o u l d s t i l l expec t the average l oca l f i e l d in the bulk to remain ve ry c l o s e to the pure so l ven t resu l t . Thus , at least fo r the current m o d e l s w i t h i n the S C M F a p p r o x i m a t i o n , it w o u l d appear that as the concen t ra t i on of an e l e c t r o l y t e s o l u t i o n is i n c r e a s e d , the dec rease in the s o l v e n t con t r i bu t ion to - 154 -Figure 21. The concen t ra t i on dependence of Y . Resu l t s are g i ven fo r fou r sa l t s in te t rahedra l s o l v e n t f o r wh i ch m e *=2 .74 . The point ind ica ted w i t h a c i r c l e rep resen ts a 2.0M KCI s o l u t i o n . Its s i g n i f i c a n c e is d i s c u s s e d in the text . - 156 -the average l oca l e lec t r i c f i e l d is a l m o s t exac t l y c o m p e n s a t e d by an i nc reased con t r i bu t i on f r o m the i ons . T h e r e f o r e , to a ve ry g o o d a p p r o x i m a t i o n , the e f f e c t i v e d i p o l e momen t of the pure te t rahedra l so l ven t can be taken as be ing independent of sa l t c o n c e n t r a t i o n . Th is is wha t w a s done in the present s t u d y , and hence al l f i n i te concen t ra t i on resu l ts p resen ted in th is chapter w e r e de te rm ined us ing m g *=2 .74 . M o r e o v e r , the s o l v e n t m o d e l then b e c o m e s equ iva len t to a n o n - p o l a r i z a b l e m o d e l w i th a permanent d i p o l e m o m e n t equal to m It shou ld a l so be no ted that not p e r f o r m i n g a fu l l S C M F ca l cu l a t i on at each concen t ra t i on for e v e r y so l u t i on rep resen ts a subs tan t ia l reduc t ion (by at least a f ac to r of three) in the to ta l number o f c o m p u t a t i o n s requ i red . In Chapter IV w e have in t roduced the R D M F theory w h i c h a l l o w s us to examine the average loca l f i e l d expe r ienced by a so l ven t par t i c le as a f u n c t i o n o f i ts sepa ra t i on f r o m an i on . The e f f e c t s of t reat ing the po la r i za t i on of the so l ven t at th is leve l w i l l be exam ined in s e c t i o n 5, where w e w i l l p resent resu l ts ob ta ined at in f in i te d i l u t i on and at l o w c o n c e n t r a t i o n . A l l the resu l ts fo r m o d e l aqueous e l e c t r o l y t e s o l u t i o n s repor ted in s e c t i o n s 2 through 5 w e r e de te rm ined us ing the te t rahedra l s o l v e n t m o d e l w i t h m e = 2 . 6 0 5 D . In s e c t i o n 6 w e w i l l cons i de r m o d e l e l e c t r o l y t e s o l u t i o n s at in f in i te d i lu t ion w h i c h e m p l o y seve ra l d i f f e ren t so l ven t m o d e l s , inc lud ing the nonpo la r i zab le te t rahedra l s o l v e n t (i.e., m g =u = 1.855D), as w e l l as the C 2 y quadrupo le and C 2 v o c t u p o l e s o l v e n t s d e s c r i b e d in Chapter V . Par t icu lar a t ten t ion w i l l be pa id to the e f f e c t s the d i f fe ren t w a t e r - l i k e s o l v e n t s have upon i o n - s o l v e n t and i o n - i o n s t ruc ture . 2. Dielectric Properties The equ i l i b r ium d ie l ec t r i c c o n s t a n t s , e of our m o d e l e l e c t r o l y t e s o l u t i o n s we re de te rm ined us ing e q s . (2.95), (2.96) and (2.97). In p r i nc i p l e , these three f o r m u l a s shou ld al l y i e l d the s a m e value fo r e H o w e v e r , in Figure 22 w e see that n u m e r i c a l l y the agreement is not exac t . W e f i nd that e q s . (2.95) and (2.96) g i ve resu l ts w h i c h are e s s e n t i a l l y in mutual agreement and w h i c h ex t rapo la te to an in f in i te d i lu t ion va lue that is c o n s i s t e n t w i th t hose ca l cu la ted fo r the pure s o l v e n t . The d ie l ec t r i c cons tan t s de te rm ined f r o m e q s . (2.97) are c o n s i s t e n t l y s m a l l e r (by about 5%) than those ob ta ined f r o m the - 157 -Figure 22. C o m p a r i n g theore t i ca l and exper imen ta l v a l u e s f o r the d ie lec t r i c cons tan t o f aqueous KCI s o l u t i o n s . The l ines labe l led w i t h ^ and represent RHNC resu l ts fo r m o d e l s o l u t i o n s ca l cu la ted f r o m e q s . (2.96) and (2.97), r e s p e c t i v e l y , e m p l o y i n g S i m p s o n ' s rule f o r the requi red i n teg ra t i ons . A t f i n i te c o n c e n t r a t i o n e g as g i v e n b y e q s . (2.95) fa l l ve ry c l o s e to the ^ ^  cu rve . The star and the s o l i d t r iang le are va lues fo r the pure te t rahedra l s o l v e n t (m g *=2 .74 ) ob ta i ned f r o m e q s . (2.93) us ing S i m p s o n ' s rule and f r o m e q s . (2.95) us ing t rapezo ida l ru le , r e s p e c t i v e l y , w h i l e the dot is the pure so l ven t d ie lec t r i c cons tan t ob ta ined f r o m the l im i t i ng s l o p e o f P2^+_. The exper imen ta l l ine rep resen ts the resu l ts o f Behret etal. [176] fo r aqueous KCI , wh i l e the open square , c i r c l e and t r iangle are measu remen ts of Harr is and O ' K o n s k i [177], G i e s e etal. [178] and Hagg is etal. [179], r e s p e c t i v e l y . The dashed l ine is e s fo r a m o d e l C s l so l u t i on s tud ied w i th the R L H N C theo ry . - 159 -other t w o rou tes and ex t rapo la te to a va lue lower than t hose found fo r the pure s o l v e n t . Th i s ex t rapo la ted va lue d o e s , h o w e v e r , agree very w e l l w i th the pure s o l v e n t d ie lec t r i c cons tan t , e = 8 8 . 3 , w h i c h is c o n s i s t e n t w i t h al_[ l im i t ing law behav iour repor ted in s e c t i o n 3. Un fo r t una te l y , th is d i s c r e p a n c y appears to be due to the lack of numer ica l accuracy in the Hankel t r a n s f o r m s , w i th the l im i t ing f a c t o r s be ing the FFT and the numer i ca l gr id w id th e m p l o y e d , A r=0 .02 . (We remark that fo r so l u t i on s y s t e m s w i t h s m a l l e r va lues of the e l ec t r os ta t i c pa ramete rs the agreement b e t w e e n the va r i ous rou tes to e s is much better.) In Figure 22 w e see that the d i s c r e p a n c y b e t w e e n the va r i ous rou tes to the d ie lec t r i c cons tan t is c o m p a r a b l e to the va r i a t i on f ound b e t w e e n t y p i c a l exper imenta l r esu l t s . M o r e o v e r , the qua l i ta t i ve behav iour o f e f o r our mode l s so lu t i ons is independent of the route used to de te rm ine it. The d ie lec t r i c cons tan t s w e w i l l report b e l o w w e r e al l ob ta ined f r o m e q . (2.96). A l s o inc luded in Figure 22 are RLHNC resu l ts fo r e g fo r a m o d e l C s l s o l u t i o n . Th is theore t i ca l cu rve is ve ry s i m i l a r to the exper imen ta l curve for KCI . W e note that the d ie lec t r i c p roper t i es o f aqueous KCI and C s l so l u t i ons are qui te s i m i l a r at concen t ra t i ons b e l o w 1M [176,180]. T h u s , as w i t h the pure s o l v e n t , the RLHNC va lues (when they can be ob ta ined ) f o r the d ie lec t r i c cons tan t s o f the m o d e l aqueous e l ec t r o l y t e s o l u t i o n s be ing c o n s i d e r e d here are in su rp r i s i ng l y g o o d agreement w i t h exper imen t . Un fo r t una te l y , the RLHNC resu l ts are p robab l y less accura te that those fo r the RHNC fo r the present m o d e l s . In Figure 23 w e have c o m p a r e d exper imen ta l va lues o f the d ie lec t r i c cons tan t s w i th t hose de te rm ined in the present s tudy for m o d e l aqueous e l e c t r o l y t e s o l u t i o n s . W e f i n d that at l o w concen t ra t i on the RHNC resu l ts fo r e g are c o n s i s t e n t l y larger than t hose o f expe r imen t . Th is is s i m p l y a c o n s e q u e n c e of the fac t that the pure te t rahedra l so l ven t has a larger d ie lec t r i c cons tan t than d o e s pure wa te r . W e a l s o o b s e r v e in Figure 23 that fo r our m o d e l s o l u t i o n s the l im i t i ng s l o p e s fo r e g are s teeper than those found fo r their real coun te rpar ts . A g a i n , the larger va lue o f e f o r the pure so l ven t can pa r t i a l l y accoun t fo r th is d i s c r e p a n c y s ince the l im i t i ng s l o p e depends upon e [79]. A t higher concen t ra t i ons {i.e., > 1 M ) the resu l ts fo r our m o d e l e l e c t r o l y t e s o l u t i o n s appear to be in qua l i ta t i ve agreement w i t h exper imen t . E v e n at ve ry high concen t ra t i on (i.e., 8 .0M) our m o d e l NaC l s o l u t i o n has ve ry s im i l a r behav iour to that o f a real L iC l s o l u t i o n . For our - 160 -Figure 2 3 . The d ie l ec t r i c c o n s t a n t s of real and of m o d e l aqueous e l e c t r o l y t e s o l u t i o n s f unc t i ons of c o n c e n t r a t i o n . The s o l i d l ines are RHNC resu l t s f o r m o d e l s o l u t i o n s of N a C l , KCI and C s l , wh i l e the dashed l ines represent the exper imen ta l va lues [176] fo r L iC I , NaC l and KCI . The do t ted po r t i on of the m o d e l NaC l curve ind ica tes the concen t ra t i ons for wh i ch ex t rapo la ted va lues the dens i t y w e r e u s e d . - 162 -m o d e l s o l u t i o n s e g dec reases more s l o w l y w i t h c fo r larger i o n s , w h i c h is a l s o c o n s i s t e n t w i t h what is o b s e r v e d expe r imen ta l l y . Thus , the d ie l ec t r i c p rope r t i es o f our m o d e l aqueous e l e c t r o l y t e s o l u t i o n s are in reasonab le qua l i ta t i ve agreement w i th exper imen t . 3. Thermodynamic Proper t ies F i rs t w e w i l l examine the average energ ies o f the m o d e l e l e c t r o l y t e s o l u t i o n s be ing i nves t i ga ted in th is s tudy . The average energ ies o f the e f f e c t i v e s y s t e m s were ca l cu la ted us ing e q s . (2.80) and (2.81). In p r i nc i p l e , in order to de te rm ine the to ta l average energ ies o f the po la r i zab le s y s t e m s w i th in the S C M F a p p r o x i m a t i o n , w e must e m p l o y eq . (4.20) wh i ch w e can rewr i te as UpO-r = ^roT + U POL ' (6.2) where U p Q ^ is the to ta l po la r i za t i on ene rgy . H o w e v e r , w e have a l ready found that to a ve ry g o o d app rox ima t i on < E ^ > e (and hence U p Q ^ ) is independent of sa l t c o n c e n t r a t i o n and w e have taken m to be cons tan t . The re fo re in the e ' present s t udy the energ ies w e w i l l repor t are those of the e f f e c t i v e s y s t e m (when app l i cab le ) , s i nce those o f the po la r i zab le s y s t e m are very c l o s e l y re la ted , never d i f f e r i ng by more than an add i t i ve cons tan t or mu l t i p l i ca t i ve f ac to r . In Figure 24 we have s h o w n the average to ta l i o n - i o n energ ies per ion fo r m o d e l N a C l , KCI and M B r s o l u t i o n s a long w i t h the l im i t i ng s l ope g i ven by e q . (3.75). The i o n - i o n energ ies o f al l three s o l u t i o n s inc rease in magni tude w i th i nc reas ing concen t ra t i on and approach their l im i t i ng behav iour at l o w c o n c e n t r a t i o n {i.e., <0 .1M) . A t higher concen t ra t i on w e f i nd that KCI has the m o s t nega t i ve v a l u e s , a l though fo r a_H the sa l ts e x a m i n e d the average i o n - i o n energ ies are a l w a y s sma l l e r in magn i tude than those p red ic ted by the l im i t i ng law re la t i onsh ip (i.e., e q . (3.75)). W e remark that the C s l l i ne , wh i ch has not been inc luded in Figure 24, l ies be tween the KCL and NaC l cu r ves . For s o l u t i o n s o f both larger and sma l l e r ions than KCI (e.g., NaC l and MBr ) , it can be s e e n that the average i o n - i o n energ ies dev ia te more qu i ck l y f r o m the l im i t i ng l aw . - 163 -Figure 24. A v e r a g e to ta l i o n - i o n energ ies per ion as f u n c t i o n s of square root c o n c e n t r a t i o n . The s o l i d l ines are RHNC resu l t s f o r three o f the m o d e l e l e c t r o l y t e s o l u t i o n s s t u d i e d , wh i l e the dashed l ine rep resen ts the l im i t ing s l o p e de te rm ined f r o m e q . (3.75) us ing e =88.3. W e point out that it is the negat ive energ ies wh i ch have been p l o t t e d . - 164 -165 -F i g u r e 25. A v e r a g e i o n - d i p o l e energ ies per ion as f unc t i ons o f square root c o n c e n t r a t i o n . The s o l i d l ines are RHNC resu l ts fo r four of the ions be ing i nves t i ga ted , N a * and C h o f N a C l , C s + of C s l and M + o f M B r . The dashed l ine represents the l im i t ing s l o p e de te rm ined f r o m e q . (3.76) e m p l o y i n g e =88.3. For ease of c o m p a r i s o n the in f in i te d i lu t ion va lues (which are nega t i ve ) have been sub t rac ted f r o m all the ene rg i es . W e a l s o point out that it is the energ ies o f the e f f e c t i v e s y s t e m s wh i ch have been p lo t t ed . - 167 The average i o n - d i p o l e energ ies p_er j on fo r four of the i ons c o n s i d e r e d in the present s tudy have been p lo t ted in Figure 25. O n l y the l o w concen t ra t i on behav iour has been i l l us t ra ted . It can be c l e a r l y s e e n f r o m F igure 25 that at ve r y l o w concen t ra t i on {i.e., <0 .025M) the i o n - d i p o l e energ ies do take on the l im i t i ng l aw dependence g iven by e q . (3.76). W e f ind that as the concen t ra t i on is i nc reased the energ ies b e c o m e more nega t i ve than t hose p red i c ted by the l im i t i ng l aw , a l though fo r C s + w e o b s e r v e that f o r a s m a l l range o f c o n c e n t r a t i o n s the va lues are m o r e p o s i t i v e than the va lues g i ven by e q . (3.76). A g a i n , it is the s m a l l e s t and the largest ions c o n s i d e r e d in F igure 25 w h i c h dev ia te m o s t rap id l y f r o m the l im i t i ng s l o p e . It shou ld be p o i n t e d out that un l ike the i o n - d i p o l e ene rg ies , the average i o n - q u a d r u p o l e energ ies per ion s h o w a n o n - u n i v e r s a l (i.e., a d i f fe ren t s l o p e for ions of d i f fe ren t s i z e ) l inear dependence upon c at l o w sal t c o n c e n t r a t i o n . In Figure 26 w e have p lo t ted the average i o n - s o l v e n t ene rg i es , aga in per  i o n , fo r four o f the m o d e l aqueous e l e c t r o l y t e s o l u t i o n s w e have e x a m i n e d . A s w e w o u l d expec t , the energ ies b e c o m e more nega t i ve (i.e., i nc rease in magn i tude) as ion s i ze is d e c r e a s e d , but dec rease in magn i tude qui te rap id l y as the concen t ra t i on is i n c r e a s e d . A t l o w concen t ra t i on the i o n - s o l v e n t energ ies b e c o m e l inear in ]/c because of the dominan t i o n - d i p o l e t e r m . C o u n t e r - i o n e f f e c t s can a l s o be seen in F igure 26. For C h at modera te concen t ra t i on (i.e., 0 .25M to 0 .75M) w e f i nd that the i o n - s o l v e n t energ ies are mo re negat ive w h e n N a + , rather than K + , is the c o u n t e r - i o n . Th is w o u l d s e e m to ind icate that N a + is more e f f e c t i v e than K + at d i s rup t ing the s o l v e n t s t ruc tu re , thus a l l o w i n g the so l ven t to interact more s t r o n g l y w i th a C l * ion at these c o n c e n t r a t i o n s . A t higher concen t ra t i ons (i.e., > 1 . 0 M ) the c o n v e r s e is t rue. In these s o l u t i o n s the ions w i l l on average be c l o s e r toge ther , and hence the p re fe ren t ia l s o l v a t i o n o f the N a + ion (as w i l l be d i s c u s s e d in mo re de ta i l in s e c t i o n 4) is at the expense o f the C l " i on . C o u n t e r - i o n e f f e c t s can a l s o be e x a m i n e d by c o m p a r i n g the C s + and B r - cu rves in Figure 26. T h e s e t w o ions w i l l be s o l v a t e d s y m m e t r i c a l l y by the te t rahedra l s o l v e n t . In th is c a s e w e f i nd that at mode ra te concen t ra t i ons the Br- i o n , wh i ch is pa i red w i t h the larger IVh i o n , has the more nega t i ve ene rgy . The average s o l v e n t - s o l v e n t energ ies p_er so l ven t f o r m o d e l N a C l , KCI and C s l s o l u t i o n s are s h o w n in Figure 27. It can be s e e n that w i t h inc reas ing c o n c e n t r a t i o n the s o l v e n t - s o l v e n t energy d e c r e a s e s in magn i tude . The m o s t 168 -Figure 26. A v e r a g e i o n - s o l v e n t energ ies per ion as f u n c t i o n s of square root c o n c e n t r a t i o n . The s o l i d , d a s h e d , d a s h - d o t and do t ted l ines are RHNC resu l ts f o r m o d e l s o l u t i o n s of N a C l , KC I , C s l and M B r , r e s p e c t i v e l y . Each l ine has been labe l l ed w i t h i ts appropr ia te i on . It is the energ ies o f the e f f e c t i v e s y s t e m s w h i c h have been i l l us t ra ted . - 170 -Figure 27. A v e r a g e s o l v e n t - s o l v e n t ene rg ies per s o l v e n t as f unc t i ons o f concen t ra t i on . RHNC resu l ts f o r three of the m o d e l aqueous e l ec t r o l y t e s o l u t i o n s c o n s i d e r e d in th is s tudy have been i l l us t ra ted . The ene rg ies s h o w n are t hose o f the e f f e c t i v e s y s t e m s . - 172 -rap id d e c r e a s e is o b s e r v e d fo r N a C l , the s l o w e s t fo r C s l . Th is c l ea r l y i nd i ca tes that s m a l l i ons are more e f f e c t i v e at d is rup t ing the so l ven t s t ruc ture . W e no te that the average energ ies fo r a M B r s o l u t i o n , w h i c h have not been inc luded in Figure 27 , are a l m o s t i nd i s t i ngu ishab le f r o m those of C s l , be ing on l y s l i gh t l y more nega t i ve at l ower c o n c e n t r a t i o n s . The to ta l average energ ies fo r four o f the m o d e l s o l u t i o n s i nves t i ga ted in th is s tudy are s h o w n in Figure 28. W e f i nd that fo r al l the s o l u t i o n s e x a m i n e d , the to ta l ene rgy b e c o m e s mo re nega t i ve {i.e., i n c reases in magn i tude) as the concen t ra t i on is i nc reased . Fu r the rmore , w i t h s m a l l e r ions the rate o f inc rease in magni tude o f the to ta l energy is mo re rap id . W e a l so o b s e r v e f r o m Figure 28 that to a g o o d a p p r o x i m a t i o n the tota l average energ ies have a l inear dependence upon c , even at h igher c o n c e n t r a t i o n s . Th is is a s o m e w h a t su rp r i s ing resul t s ince F igures 24 , 26 and 27 w o u l d ind ica te that the average i o n - i o n , i o n - s o l v e n t and s o l v e n t - s o l v e n t con t r i bu t i ons to the to ta l average energy are not l inear in c o n c e n t r a t i o n . C lea r l y then , the n o n - l i n e a r con t r i bu t i ons to the to ta l energ ies must cance l one another to a large degree . In Chapter III (cf. e q s . (3.76) and (3.77)) w e have o b s e r v e d s o m e c a n c e l l a t i o n o f th is t y p e fo r the to ta l average i o n - d i p o l e energy and average d i p o l e - d i p o l e energy in the l o w concen t ra t i on l imi t (on ly w h e n e is large) . What is not c lear is w h y the apparent l inear i ty in the to ta l average energy shou ld pe rs i s t ove r such a large concen t ra t i on range (ev ident to at least 4 M ) . H o w e v e r , th is l inear i ty is c o n s i s t e n t w i t h what is o b s e r v e d expe r imen ta l l y f o r the heats of d i l u t i ons o f at least s o m e s t rong e l e c t r o l y t e s [174]. The p ressu res of our m o d e l s o l u t i o n s w e r e ca l cu la ted to RHNC leve l accu racy us ing the p ressu re equat ion (2.82). W e f i nd that even at 1.0M the ca l cu la ted p ressu res have changed by l e s s than 10% f r o m the pure s o l v e n t va lue . Hence , cons tan t p ressure c o n d i t i o n s appear to have been a p p r o x i m a t e l y ma in ta ined for our m o d e l e l ec t r o l y t e s o l u t i o n s . W e again po in t out that w e have used the expe r imen ta l dens i t i es in our m o d e l ca l cu l a t i ons in order to m i m i c real cons tan t p ressure c o n d i t i o n s . F r o m our d i s c u s s i o n in Chapter III w e k n o w that at f in i te c o n c e n t r a t i o n h f l ^ ( r ) (a,P* = +,-,s) are al l sc reened at large r and so w e can wr i t e h J r ) = ^S.e~Kt > ( 6 . 3 ) ap r - 173 -Figure 2 8 . To ta l average energ ies as f unc t i ons o f c o n c e n t r a t i o n . The four cu rves s h o w n represent RHNC resu l t s fo r m o d e l N a C l , KCI , C s l and M B r s o l u t i o n s . W e no te that it is the energ ies of the e f f e c t i v e s y s t e m s w h i c h have been p l o t t e d . - 175 -where is a cons tan t dependent upon the parameters o f the s y s t e m . The sc reen ing paramete r , K , is g i ven by e q . (3.35c) at ve ry l o w c o n c e n t r a t i o n . B y f i t t i ng the l o n g - r a n g e ta i l s o f h Q ^ ( r ) to the func t iona l f o r m g i ven by e q . (6.3), w e have ob ta ined numer i ca l va lues f o r K f o r the m o d e l aqueous e l e c t r o l y t e s o l u t i o n s w e have e x a m i n e d . Va lues f o r K * = K6s we re de te rm ined f r o m each o f the func t i ons h + + ( r ) , h ( r ) , h + _ ( r ) , h ( r ) and h ( r ) , as 1 1 0 w e l l as f r o m h f j O'SS^ 1^" m o s t ° * t n e s o l u t i o n s i nves t i ga ted the l o n g - r a n g e ta i l o f h g s ( r ) w a s v e r y s m a l l because o f i ts p£ dependence (cf. e q . (3.42c)), and hence w a s not used to ca l cu la te an add i t i ona l va lue o f K * . The s i x va lues that w e r e ob ta ined we re a l w a y s in very g o o d agreement at concen t ra t i ons l ess than 0 .5M. A t concen t ra t i ons higher than 0.5M the numer i ca l f i t s usua l l y b e c a m e t o o d i f f i cu l t to p e r f o r m as the l o n g - r a n g e ta i l s b e c a m e shor ter ranged (i.e., t hey w e r e d e c a y i n g fas te r because K w a s inc reas ing) . In Figure 29 w e have s h o w n resu l ts f o r K* de te rm ined in the present s tudy . A t ve ry l o w concen t ra t i on w e f i nd that the ca l cu la ted sc reen ing pa ramete rs app roach the l im i t i ng l aw g i ven by e q . (3.35c). Bo th p o s i t i v e (tend t o w a r d larger va l ues ) and nega t i ve (tend t o w a r d sma l l e r v a l u e s ) d e v i a t i o n s f r o m the l im i t i ng law can be seen in Figure 29 . Of the s o l u t i o n s s t u d i e d , K* i n c reases m o s t rap id l y w i t h concen t ra t i on f o r N a C l . For both N a C l and KCI * 2 s o l u t i o n s the va lues o f K * are c o n s i s t e n t l y larger than those g i ven by e q . (3.35c), ind ica t ing that fo r these s o l u t i o n s the l o n g - r a n g e ta i l s o f h ^ C r ) are shor ter ranged (i.e., they are sc reened more q u i c k l y ) than those p red i c ted by * 2 the l im i t i ng l aw . The s m a l l e s t va lues o f K * w e r e ob ta ined fo r M ' l , these va lues a l w a y s be ing s m a l l e r than those g i ven by e q . (3.35c). Thus the l o n g - r a n g e ta i l s o f h a ^ ( r ) fo r the M' l s o l u t i o n are longer ranged (i.e., t hey are sc reened mo re s l o w l y ) than those p red i c ted by the l im i t i ng l aw . W e remark that fo r M' l r ) w e r e s u f f i c i e n t l y l ong ranged s o as to enab le us to de te rm ine va lues f o r the sc reen ing parameter up to 0.74M (which is the highest c o n c e n t r a t i o n s tud ied f o r th is so lu t i on ) . Over the concen t ra t i on range • 2 0 .6M to 0.7M w e d i s c o v e r that K * s t o p s inc reas ing and rema ins a p p r o x i m a t e l y • 2 c o n s t a n t . For concen t ra t i ons a b o v e 0.7M K ac tua l l y beg ins to dec rease w i t h i nc reas ing c. W e a l so f ind that at these concen t ra t i ons the l o n g - r a n g e ta i l s o f h + + ( r ) and h _ _ ( r ) ac tua l l y change s ign and b e c o m e p o s i t i v e , i nd ica t ing an apparent l o n g - r a n g e a t t rac t ion b e t w e e n l ike i o n s . Th is behav iou r w o u l d s e e m to con t rad ic t the usual no t i ons o f ion ic sc reen ing and o f ion ic in te rac t ion - 176 -Figure 29 . The square of the D e b y e sc reen ing parameter as a f unc t i on of c o n c e n t r a t i o n . The s o l i d l ines are resu l t s f o r four of the m o d e l aqueous e l e c t r o l y t e s o l u t i o n s s tud ied . The dashed l ine rep resen ts the l im i t ing law as g i ven b y e q . (3.35c) when e =88.3 is used . - 178 wi th in s o l u t i o n . C l e a r l y , s o l v e n t e f f e c t s (in add i t i on to those d e s c r i b e d by the d ie lec t r i c cons tan t ) are mak ing s ign i f i can t con t r i bu t i ons to the l ong - range i o n - i o n co r re l a t i ons in the M'l s o l u t i o n . W e w i l l d i s c u s s these so l ven t e f f e c t s in greater de ta i l b e l o w . F r o m Figure 29 it can be seen that f o r the C s l s o l u t i o n the sc reen ing pa ramete rs are just s l i gh t l y sma l l e r than those g i ven by e q . (3.35c). It shou ld a l s o be po in ted out that the va lues o f K * fo r M B r and EqEq s o l u t i o n s , wh i ch have not been inc luded in Figure 29 , l ie ve ry c l o s e to the KCI curve . T h e r e f o r e , w e f i nd that the behav iour o f K depends not o n l y upon the ion s i ze {i.e., the value o f d + _ ) , but a l so upon the s i ze a s y m m e t r y of the t w o ions . A t a g i ven concen t ra t i on the va lue o f K d e c r e a s e s w i th in i nc reas ing ion s i z e , but inc reases w i th i nc reas ing a s y m m e t r y . In order to make use of the t h e r m o d y n a m i c e x p r e s s i o n s g i ven in Chapter III, w e must f i r s t de te rmine va lues f o r by eva lua t ing e q . (3.1a). W e w i l l a l s o examine C j g as de f i ned by e q s . (3.43) and (3.39b). The requ i red in tegra t ions we re p e r f o r m e d us ing the t r apezo ida l rule because al l the Four ier t r a n s f o r m s in our ca l cu l a t i ons we re eva lua ted w i th t rapezo ida l rule and G f l ^ and represent the k—>0 l im i t s o f these t r a n s f o r m s . Care w a s taken in c o m p u t i n g the con t r i bu t i ons to G f l ^ due to the l o n g - r a n g e ta i l s o f n a ^ ( r ) , par t i cu la r l y at l o w c o n c e n t r a t i o n . These con t r i bu t i ons w e r e de te rm ined a n a l y t i c a l l y wheneve r the l o n g - r a n g e ta i l o f h f l ^ ( r ) w a s s u c c e s s f u l l y f i t to e q . (6.3). F i n a l l y , w e remark that numer i ca l l y the charge neut ra l i ty c o n d i t i o n s g i ven by e q s . (3.5c) and (3.5d) w e r e a l w a y s s a t i s f i e d to a reasonab le leve l o f accu racy (e.g., at 0.1M to w i th in 0.01%, w h i l e at 4 .0M to w i t h i n 0.1%). F i rs t w e w i l l examine our resu l ts f o r G + _ . It is c lear f r o m e q . (3.36a) that G + _ w i l l d i ve rge as p2—>0. Thus , in Figure 30 w e have p lo t ted the produc t P 2 ^ + _ wh i ch rema ins f in i te in the P2—>0 l imi t and has a l im i t ing va lue o f 1 / f = 0 . 5 fo r a l l the mode l s o l u t i o n s w e have i nves t i ga ted . A t l o w c o n c e n t r a t i o n the va lues o f P 2 G + _ a l w a y s approach the l im i t i ng l aw , a l though the resu l t s f o r M'l dev ia te ve ry rap id ly f r o m l im i t i ng behav iou r . A t l o w concen t ra t i on the M ' l , M B r and NaCl s o l u t i o n s al l d e m o n s t r a t e d what w e sha l l re fer to as super limiting-law behav iou r , that is P 2 G + _ i nc reases more rap id l y w i th than p red ic ted by the l im i t ing l aw . For NaC l the P 2 G + _ curve d o e s even tua l l y c r o s s the l ine represen t ing the l im i t i ng s l o p e at a concen t ra t i on o f about 2 . 0 M . For M'l P 2 G + _ appears to s h o w d ivergent behav iou r . It i nc reases ve ry rap id l y w i th j / c , reach ing a va lue o f 2.55 (not s h o w n in Figure 30) at - 179 -Figure 3 0 . The produc t P 2 G + _ as a f unc t i on o f square root concen t ra t i on . The s o l i d l ines represent resu l ts f o r f i v e of t h e ' m o d e l aqueous e l e c t r o l y t e s o l u t i o n s c o n s i d e r e d in th is s tudy . The dashed l ine is the l im i t i ng s l o p e de te rm ined f r o m e q s . (3.36) us ing e =88.3. 181 -0 .74M. The va lues of P 2 G + _ a l s o inc rease rap id l y w i th i nc reas ing concen t ra t i on fo r the M B r s o l u t i o n , h o w e v e r , they do not appear to be d i v e r g i n g . W e f i nd that at the h ighest concen t ra t i ons s tud ied fo r M B r the rate o f inc rease appears to have b e c o m e cons tan t . For KCI the va lues of P 2 G + _ are a l w a y s sma l le r than t hose g i ven by e q . (3.36). W e s e e in F igure 30 that the C s l curve in i t i a l l y f o l l o w s the l im i t i ng law quite c l o s e l y , then turns d o w n quite sharp ly at a c o n c e n t r a t i o n o f about 1 M . W e note that the resu l ts f o r the EqEq s o l u t i o n (not inc luded in Figure 30) are on l y just s l i gh t l y larger than those fo r N a C l . Thus , the behav iour of P 2 G + _ appears to have no s i m p l e dependence upon ion s i ze or a s y m m e t r y . Fu r the rmore , w e o b s e r v e super l i m i t i n g - l a w behav iour f o r both large and s m a l l i ons . It is in te res t ing to point out that at l o w (but s t i l l f i n i t e ) concen t ra t i on the va lues of P 2 G + _ are not en t i re ly de te rm ined by the l o n g - r a n g e behav iour of h + _ ( r ) , at least fo r s o m e o f the s o l u t i o n s inves t iga ted here . W e f ind that a l though both NaC l and M B r so l u t i ons s h o w super l i m i t i n g - l a w behav iour fo r P 2 G + _ , their va lues of K w o u l d pred ic t the o p p o s i t e to be true (al though fo r M B r the l o n g - r a n g e ta i l s o f h f l ^ ( r ) do s h o w s o m e o f the s a m e pecu l ia r behav iour found for an M' l so lu t i on ) . The apparent d ivergent behav iour o f P 2 G + _ fo r M' l i s , h o w e v e r , c o n s i s t e n t w i t h the fact that i ts va lue fo r K i nc reases re la t i ve l y s l o w l y and then ac tua l l y beg ins to dec rease w i th inc reas ing c o n c e n t r a t i o n . C o n s e q u e n t l y , even at r e l a t i ve l y l o w concen t ra t i on the s h o r t - r a n g e i o n - i o n st ructure can p lay an impor tant ro le in de te rm in ing the behav iour of P 2 G + _ . M o r e o v e r , th is s h o r t - r a n g e i o n - i o n s t ruc ture w i l l depend s t r o n g l y upon the de ta i l s o f the ion s o l v a t i o n , and hence upon the nature of the s o l v e n t i t se l f . In Figure 31 w e have c o n s i d e r e d the quant i ty C ^ g as d e f i n e d by e q . (3.43). A t ve r y l o w concen t ra t i on w e f i nd that our numer ica l resu l t s do agree w i t h the l im i t i ng s l o p e , S c , de te rm ined f r o m e q . (3.48). There d o e s , h o w e v e r , appear to be a s l ight d i s c r e p a n c y be tween C ^ g , the in f in i te d i l u t i on va lue , and the va lue ob ta in f r o m the ex t rapo la t i on o f C T g to in f in i te d i l u t i on . Th is d i s c r e p a n c y is e a s i l y accoun ted fo r by the d i f f e rence in numer i ca l accu racy o f the d i f fe ren t c a l c u l a t i o n s i n v o l v e d . W e a l s o o b s e r v e f r o m F igure 31 that, fo r the m o s t par t , the larger the ions the more rap id ly C j g d e v i a t e s f r o m i ts l im i t i ng behav iou r . - 182 -Figure 31. C j g as a f unc t i on o f square roo t c o n c e n t r a t i o n . For ease o f c o m p a r i s o n w e have p lo t t ed the d i f f e rence b e t w e e n the in f in i te d i lu t ion va lue , C j g , and C j g i t se l f . The s o l i d l ines are RHNC resu l ts f o r f i v e o f the s o l u t i o n s exam ined in the present s t u d y . The dashed l ine represen ts the l im i t i ng s l o p e , S de te rm ined us ing e =88.3 in e q . (3.48). - 183 -- 184 -Figure 3 2 . G + s as a f unc t i on o f square roo t concen t ra t i on . The d i f f e rence be tween the l im i t i ng va lue G°s (as de f i ned by eq . (3.49b)) and G + g has been p lo t ted fo r ease of c o m p a r i s o n . RHNC resu l ts fo r m o d e l N a C l , KCI , C s l , M B r and M'l s o l u t i o n s have been i nc luded . - 186 -The dependence of G + g = G _ s upon v/c f o r f i v e m o d e l e l e c t r o l y t e s o l u t i o n s has been i l lus t ra ted in Figure 32. A t ve ry l o w concen t ra t i on G + g appears to be a l inear f unc t i on o f y/c, a l though the l im i t i ng s l o p e fo r each s o l u t i o n is c l e a r l y d i f f e ren t . Bo th of these o b s e r v a t i o n s are c o n s i s t e n t w i th e q . (3.49a). A t h igher concen t ra t i ons we f i nd G + g d e m o n s t r a t e s a va r i e t y of behav iou r s , w i t h no s i m p l e re la t i onsh ip to ion s i ze or a s y m m e t r y be ing i nd i ca ted . For e x a m p l e , G + g is a m o n o t o n i c d e c r e a s i n g func t i on fo r N a C l , wh i l e fo r C s l it ac tua l l y beg ins to inc rease and c r o s s e s the NaC l curve at a concen t ra t i on of about 1.5M. For the M' l and M B r s o l u t i o n s G . d e c r e a s e s + s rap id l y , and in fac t appears to be d i ve rg ing fo r M ' l . The apparent d i ve rgen t behav iour of G . f o r the M' l so l u t i on is an e x p e c t e d resu l t . Th is is because +s in Figure 30 w e have a l ready found that P 2 G + _ a P P e a r s t o d i ve rge fo r M'l and f r o m e q . (3.41) w e k n o w that G + g depends upon the produc t P 2 G + _ . In F igure 33 w e have s h o w n resu l ts fo r G„ ob ta ined for m o d e l N a C l , ° SS KCI C s l M B r and M'l s o l u t i o n s . A t ve r y l o w concen t ra t i on w e f i nd that G „ „ SS has a l inear dependence on c , wh i ch is cons i s t en t w i t h e q . (3.54). W e see f r o m Figure 33 that G s g is a m o n o t o n i c d e c r e a s i n g f unc t i on fo r the N a C l and KCI s o l u t i o n s , wh i l e f o r C s l , M B r and M' l it s t r i c t l y i nc reases w i th i nc reas ing concen t ra t i on . The s l o p e o f the cu rves for G appears to inc rease w i t h 3 O i nc reas ing ion s i z e , b e c o m i n g ve ry large fo r M B r and M ' l . A g a i n , w e o b s e r v e that G appears to d i ve rge f o r the M'l s o l u t i o n as the concen t ra t i on 3 app roaches 0.74M (where G has a va lue o f 4 .08d ). W e remark that this OS o behav iour is c o n s i s t e n t w i th e q . (3.53) and the apparent d ivergent nature o f G + s o b s e r v e d fo r M' l in F igure 32. B e f o r e p r o c e e d i n g , it shou ld be po in ted out that d ivergent behav iour in p_G^ G ^ and G w a s a l s o d e m o n s t r a t e d b y our C s l so l u t i on at a r2 +-' +s s s concen t ra t i on o f about 9 M . For th is s o l u t i o n the magn i tudes of P 2 G + _ , G+Q and G s s w e r e f ound to inc rease sharp ly w i th c f o r concen t ra t i ons a b o v e 8 .5M. A s ind ica ted in Tab le IX , w e we re o n l y able to reach a concen t ra t i on of 9 .1M, above w h i c h w e c o u l d not ob ta in numer ica l s o l u t i o n s f o r the RHNC theo ry . A l t h o u g h the va lues o f P 2 G + _ , G + g and G g s a l s o inc rease very rap id l y w i th c for the M B r s o l u t i o n , they do not appear to d i ve rge (at least f o r the concen t ra t i ons exam ined ) l ike t hose o f the M' l s o l u t i o n . C o n s e q u e n t l y , f o r M B r w e w e r e ab le to reach a concen t ra t i on o f 1.0M w i t h no apparent d i f f i c u l t i e s . It is i n te res t ing to note that fo r the t w o s a l t s M B r and M ' l , d + _ (i.e., the ion - 187 -Figure 33. G as a f u n c t i o n o f sa l t c o n c e n t r a t i o n . W e have s h o w n RHNC resu l ts f o r f i ve of the m o d e l aqueous e l e c t r o l y t e s o l u t i o n s exam ined in th is s tudy . - 189 -s i z e ) is the s a m e . These t w o sa l t s d i f f e r on l y in the s i ze a s y m m e t r y o f their r e s p e c t i v e ions w h i c h c lea r l y has a large e f f e c t . The d ive rgen t behav iour d e m o n s t r a t e d by the C s l and M' l s o l u t i o n s appears to c o m e through the O Z equa t ion (see e q . (3.37)) but the actua l s o u r c e s remain unc lear . The p h y s i c a l s i g n i f i c a n c e o f th is behav iour is a l s o not en t i re ly c lear . H o w e v e r , if w e examine the l o n g - r a n g e ta i l s o f h f l ^ ( r) w e f ind that the va lues are p o s i t i v e f o r aj3 = + + , — , H — , s s and nega t i ve f o r a/3 = +s , - s {i.e., at long range l ike s p e c i e s a t t rac t , unl ike repe l ) . T h u s , it appears as though these s o l u t i o n s m a y be prepar ing to undergo a phase separa t ion (e.g., sa l t p rec ip i t a t i on ) as P2G+_, G + g and G g s beg in to d i v e r g e . For M B r s o m e w h a t s im i l a r but more exo t i c behav iour can be seen in the l ong - range ta i l s o f h ^ ^ r ) . There w e f i nd that the magn i tude of h + + ( r ) is c o n s i s t e n t l y must larger than that of h__(r) at long range. M o r e o v e r , the l ong - range ta i l s o f h f l ^ ( r ) are not e x c l u s i v e l y p o s i t i v e or nega t i ve (i.e., repu ls i ve or a t t rac t i ve ) , but rather s h o w long reg ions (the order of 5 to 10d ) s over wh ich the f u n c t i o n s appear m o n o t o n i c , separa ted by in te rva ls of rap id changes in s i g n . A g a i n , w e s t r e s s that the s o l v e n t must be p l ay i ng a major ro le in de te rm in ing the behav iour of the C s l , M' l and M B r s o l u t i o n s . C l e a r l y , these s y s t e m s and their behav iour requi re mo re de ta i l ed s tudy . N o w hav ing de te rm ined the G Q ^ fo r our m o d e l aqueous e l e c t r o l y t e s o l u t i o n s w e can use the e x p r e s s i o n s g i ven in Chapter III to ca lcu la te s o m e of their t h e r m o d y n a m i c p rope r t i es . F i rs t w e sha l l cons ide r the i so the rma l c o m p r e s s i b i l i t y , Xip. In Figure 34 w e have p lo t ted kTXrp as ob ta ined f r o m e q . (3.20) fo r the N a C l , KCI , and C s l s o l u t i o n s . Resu l t s fo r the three rema in ing s o l u t i o n s w h i c h w e r e s tud ied at f in i te concen t ra t i on were de te rm ined but are not inc luded in F igure 34. W e f i nd that the i so the rma l c o m p r e s s i b i l i t y o f the pure tet rahedral s o l v e n t d o e s not c o m p a r e w e l l w i t h the resul t fo r real wa te r at 2 5 ° C , kTx T= 45 .7X10 Bar [169]. A t l o w concen t ra t i on the va lues of x T s h o w a l inear dependence upon c in a c c o r d a n c e w i t h e q . (3.74). A s is o b s e r v e d e x p e r i m e n t a l l y [6,181], the c o m p r e s s i b i l i t i e s o f our m o d e l s o l u t i o n s dec rease w i t h i nc reas ing c o n c e n t r a t i o n , a l though the rates of dec rease are seve ra l t i m e s larger than those of the real s o l u t i o n s . F rom Figure 34 w e see that x.p d e c r e a s e s more rap id ly fo r N a C l than fo r KCI wh ich is c o n s i s t e n t w i t h exper iment [181]. H o w e v e r , con t ra ry to what is found expe r imen ta l l y [6], w e f ind that Xm d e c r e a s e s more rap id l y f o r C s l than fo r KCI . F i n a l l y , w e po in t - 190 -Figure 34. I so thermal c o m p r e s s i b i l i t y as a f unc t i on of c o n c e n t r a t i o n . Resu l t s fo r the three m o d e l a lka l i ha l ide s o l u t i o n s c o n s i d e r e d in th is s tudy are s h o w n . The va lues w e r e ob ta ined f r o m e q . (3.20). - 191 -- 192 -out that the i so the rma l c o m p r e s s i b i l i t y o f the M' l s o l u t i o n (not inc luded in Figure 34) s h o w s none o f the d ivergent behav iour d e m o n s t r a t e d by G ^ , but rather f o r m s a s m o o t h curve wh i ch w o u l d l ie just above the KCI resu l t . C l e a r l y , the apparent d i v e r g e n c e s in the fo r M' l must cance l in the e x p r e s s i o n f o r Xrp (cf. e q . (3.20)). The par t ia l mo la r v o l u m e of the s o l v e n t w a s de te rm ined us ing e q . (3.16) and resu l ts f o r seve ra l m o d e l s o l u t i o n s are s h o w n in Figure 35 . A t l o w c o n c e n t r a t i o n w e f i nd that V g b e c o m e s l inear in c o n c e n t r a t i o n , as p red i c ted by e q . (3.73). W e o b s e r v e that V g i nc reases w i t h i nc reas ing concen t ra t i on f o r NaC l and KCI , but d e c r e a s e s w i t h inc reas ing concen t ra t i on f o r C s l and M B r . In Chapter III w e have c o n c l u d e d f r o m e q . (3.73) that the behav iour o f V g (i.e., whether it i nc reases or d e c r e a s e s w i t h c ) , at least at l o w c o n c e n t r a t i o n , w i l l be de te rm ined by V ^ . For al l the s o l u t i o n s exam ined here w e f i nd th is to be true (values fo r can be e a s i l y c o m p u t e d f r o m the resu l ts f o r V.° g i ven b e l o w in Tab le X I ) . The par t ia l mo la r v o l u m e s of four of the sa l t s exam ined in the present s tudy are s h o w n in Figure 36. A t ve ry l o w concen t ra t i on w e f i nd that the par t ia l mo la r v o l u m e s of al l the sa l t s s tud ied o b e y the HNC l im i t i ng l aw g iven by e q . (3.63). In F igure 36 w e see that w i t h ' i nc reas ing ion s i z e dev ia tes more rap id ly f r o m l im i t i ng l aw behav iour . For the C s l and M B r s o l u t i o n s the s l o p e s o f the cu rves fo r ac tua l ly change s ign and beg ins to dec rease w i t h inc reas ing c o n c e n t r a t i o n . W e a l so o b s e r v e that once the cu rves fo r fo r these t w o s o l u t i o n s have turned o v e r , they again appear to b e c o m e l inear in v/c. Th is behav iour (i.e., the negat ive s l o p e together w i th an apparent l inear i ty in \/c) fo r V2 has been found expe r imen ta l l y fo r s o m e s o l u t i o n s , par t i cu la r l y those o f t e t r a a l k y l a m m o n i u m sa l t s [182]. In the past there has been cons ide rab le debate in the l i terature [183] as to whether or not the par t ia l mo la r v o l u m e s of these t e t r a a l k y l a m m o n i u m sa l t s do in fac t o b e y the un ive rsa l l im i t i ng l aw p red ic ted by D e b y e - H i i c k e l t heo ry [6]. For many o f the t e t r a a l k y l a m m o n i u m sa l t s V2 appears to turn ove r at ve ry l o w c o n c e n t r a t i o n , as is the c a s e here f o r M B r . C o n s e q u e n t l y , it b e c o m e s ve ry d i f f i cu l t to ob ta in exper imen ta l resu l ts [184] o f su f f i c i en t accu racy at l o w enough concen t ra t i on in o rder to demons t ra te that the l im i t i ng l aw d o e s s t i l l ho ld f o r these s a l t s . Fo r tuna te l y , fo r our m o d e l s o l u t i o n s w e are able to pe r f o rm ca l cu l a t i ons at in f in i te d i l u t i on . T h u s , un l ike exper imen t , we k n o w the value o f \i° fo r a - 193 -Figure 3 5 . Par t ia l mo la r v o l u m e of the s o l v e n t as a f unc t i on of c o n c e n t r a t i o n . Resu l t s ob ta ined f r o m e q . (3.16) fo r m o d e l N a C l , KCI , C s l and M B r s o l u t i o n s have been i nc luded . - 195 -Figure 36. Par t ia l mo la r v o l u m e of the s o l u t e as a f unc t i on of square roo t concen t ra t i on . The s o l i d l ines represent resu l ts ob ta ined f r o m e q . (3.15) fo r mode l N a C l , KCI , C s l and M B r s o l u t i o n s . For ease o f c o m p a r i s o n w e repor t the d i f f e r e n c e s be tween V2 and its in f in i te d i l u t i on va lue , V ^ . t h e dashed l ine is the l im i t ing s l o p e , S de te rm ined f r o m e q . (3.65) us ing e =88.3 and S c as g i ven by e q . (3.48). The do t ted l ine is the l im i t i ng s l o p e , s j , f o r real 1:1 aqueous e l e c t r o l y t e s o l u t i o n s at 25 °C [6], wh i le the d a s h - d o t l ine is the corrected l im i t i ng s l o p e wh i ch resu l ts f r o m the mu l t i p l i ca t i on o f by the ra t io Xm( tetrahedral s o l v e n t ) / Xm(real water ) . - 197 -g iven sa l t , and hence fo r sa l t s such as M B r there can be no amb igu i t y about the l o w concen t ra t i on behav iour o f N^. It is o b v i o u s f r o m Figure 36 that V 2 s h o w s much larger va r i a t i on w i th c o n c e n t r a t i o n fo r our m o d e l aqueous e l e c t r o l y t e s o l u t i o n s than fo r thei r real coun te rpa r t s . The l im i t i ng s l o p e de te rm ined fo r our m o d e l s o l u t i o n s is a p p r o x i m a t e l y 20 t i m e s larger than that o f real aqueous s o l u t i o n s at 2 5 ° C . A f a c t o r o f about 5 can be i m m e d i a t e l y accoun ted fo r by the d i f f e r e n c e in the i so the rma l c o m p r e s s i b i l i t i e s of the t w o s o l v e n t s . A corrected l im i t i ng s l o p e fo r the real s o l u t i o n s w h i c h takes th is d i f f e r e n c e into accoun t has been inc luded in Figure 36. H o w e v e r , th is co r rec ted s l o p e is s t i l l much sma l l e r than that of our m o d e l s o l u t i o n s . In Chapter III it w a s s h o w n that fo r a g i ven s o l v e n t m o d e l the HNC theory o v e r e s t i m a t e s S c > and hence d o e s not g ive accura te resu l ts f o r S . W e a l so reca l l that t h e r m o d y n a m i c s [6] p r o v i d e s us w i t h an a l te rna t i ve route fo r de te rm in ing S cf. e q . (3.66). N u m e r i c a l l y w e can ob ta in an app rox ima te va lue fo r ( 3 1 n e / 3 P ) T fo r the te t rahedra l so l ven t by repeat ing the pure s o l v e n t ca l cu la t i on at one or t w o s l i gh t l y higher d e n s i t i e s . The i so the rma l c o m p r e s s i b i l i t y can then be used to de te rm ine the c o r r e s p o n d i n g p ressure changes . Th is p rocedure w a s car r ied out and the resu l ts ob ta ined ind ica te that the HNC theory o v e r e s t i m a t e s S c fo r the te t rahedra l s o l v e n t by about an order o f magn i t ude ! Th is error is more than su f f i c i en t to m o v e the l im i t i ng s l o p e repor ted in Figure 36 f o r the te t rahedra l s o l v e n t to a p o s i t i o n b e l o w that o f the co r rec ted l im i t ing s l o p e f o r wa te r at 2 5 ° C . Th is reduc t i on o f S fo r the te t rahedra l so l ven t is a des i rab le resul t s i nce in Chapter IV w e v have s h o w n that the R D M F makes an add i t i ona l con t r i bu t i on to S w h i c h w i l l c i nc rease S . W e w i l l d i s c u s s the e f f e c t s of the R D M F in deta i l in s e c t i o n 5. v The cons tan t p ressure d e r i v a t i v e s o f the loga r i t hms of the mean m o l a r i t y ac t i v i t y c o e f f i c i e n t s w e r e de te rm ined in the present s tudy us ing e q . (3.30b). Integrat ing these va lues d i r ec t l y in o rder to ob ta in l n y + (we note that here the cons tan t p ressure de r i va t i ves are the appropr ia te va lues to in tegrate s i nce w e have c h o s e n to m i m i c cons tan t p ressure c o n d i t i o n s ) w o u l d p rove ve ry d i f f i cu l t , h o w e v e r , because o f the s ingu la r i t y in the de r i va t i ve at p 2 =0. W e deal w i t h th is p rob lem in a manner v e r y s i m i l a r to that used by Rasa iah and F r i edman [151]. The s ingu la r i t y is c o n v e n i e n t l y r e m o v e d by sub t rac t i ng the - 198 -l im i t i ng l aw fo r the de r i va t i ve of l n y + (as g iven by e q . (3.56)) f r o m ( 9 1 n y + / 9 p 2 )p. The resu l t ing d i f f e rence func t i on is s m o o t h and equa ls zero at P 2 = 0 , and hence can be e a s i l y i n teg ra ted . W e then s i m p l y add on the va lue o f l n y + g i ven by the D e b y e - H u c k e l l im i t ing law. A l l the va lues f o r l n y + w h i c h w e shal l repor t we re ca l cu la ted in th is f a s h i o n . In F igure 37 w e have c o m p a r e d the va lues o f l n y + ob ta ined f o r our m o d e l aqueous e l e c t r o l y t e s o l u t i o n s w i t h t hose of real s o l u t i o n s . W e po in t out that the l im i t i ng law s l o p e s o f the m o d e l and o f the real s o l u t i o n s d i f f e r s l i gh t l y because o f the d i f f e rence in the d ie lec t r i c c o n s t a n t s o f their r e s p e c t i v e s o l v e n t s . F r o m Figure 37 w e o b s e r v e that in general l n y + dev ia tes much mo re s l o w l y f r o m l im i t i ng l aw behav iour f o r the m o d e l e l ec t r o l y t e s o l u t i o n s be ing c o n s i d e r e d here than it d o e s fo r thei r real coun te rpa r t s . If w e c o m p a r e resu l ts fo r the real and m o d e l a lka l i h a l i d e s , w e f ind that l n y + turns up much mo re q u i c k l y fo r the real s o l u t i o n s . L ike the exper imen ta l c u r v e s , l n y + f o r our m o d e l KCI s o l u t i o n turns up more rap id l y than does the m o d e l C s l cu rve . H o w e v e r , un l ike expe r imen t , w e f i nd that our resu l ts fo r l n y + f o r N a C l are c o n s i s t e n t l y l ess than t hose of both C s l and KCI . A l s o inc luded in Figure 37 are the measu red v a l u e s o f l n y + f o r an aqueous s o l u t i o n o f E t ^ N B r wh i ch w o u l d s e e m to be a reasonab le counterpar t fo r the current M B r s y s t e m . W e see that f o r M B r l n y + is a l w a y s s m a l l e r than fo r the m o d e l a lka l i ha l i des . S i m i l a r behav iour can be o b s e r v e d fo r E t ^NBr w i t h respec t to the real a lka l i ha l i des . Un l i ke E t ^ N B r , h o w e v e r , the va lues of l n y + f o r M B r are a l w a y s less than those g i ven by the D e b y e - H u c k e l l im i t i ng law. For l n y + we shal l re fer to th is as super Debye-Huckel behav iou r . W e remark that fo r M' l ( resul ts fo r it have not been inc luded in F igure 37) l n y + d e c r e a s e s even more rap id l y than for M B r . Super D e b y e - H u c k e l behav iour has been o b s e r v e d expe r imen ta l l y f o r s o m e t e t r a a l k y l a m m o n i u m sa l t s (e.g., Pr^NI and Et^NI [186]) and has been in terpre ted [5,186] as be ing a c o n s e q u e n c e o f the hyd rophob i c natures of these re l a t i ve l y large i o n s . Fu r the rmore , it has been h y p o t h e s i z e d [5,186] that in aqueous s o l u t i o n there w i l l be an a t t rac t i ve f o r c e (due to the s o l v e n t ) be tween t w o such large h y d r o p h o b i c i o n s . In the present s tudy w e have seen s t rong e v i d e n c e f o r l o n g - r a n g e a t t rac t i ve f o r c e s be tween bo th l ike and unl ike ions in our m o d e l M' l s o l u t i o n (and to a l esse r degree f o r M B r ) , as w a s d i s c u s s e d ear l ie r . C l e a r l y , many of the p rope r t i es exh ib i ted by the M'l and M B r s o l u t i o n s - 199 -Figure 37. l n y + as a f unc t i on o f square root c o n c e n t r a t i o n . The s o l i d l ines represent RHNC resu l t s f o r four o f the m o d e l s o l u t i o n s i nves t i ga ted in th is s tudy . The do t ted l ines are exper imen ta l resu l ts [185,186] f o r seve ra l 1:1 aqueous e l e c t r o l y t e s o l u t i o n s at 2 5 ° C . The l im i t i ng law s l o p e s ( labe l led w i th L.L.) have a l so been inc luded fo r bo th the m o d e l and real s o l u t i o n s . - 201 -are c o n s i s t e n t w i t h what is o b s e r v e d expe r imen ta l l y fo r aqueous s o l u t i o n s o f t e t r a a l k y l a m m o n i u m s a l t s . It is o b v i o u s f r o m e q . (3.30b) that the cons tan t p ressure d e r i v a t i v e s of l n y + depend upon both G + _ and G + g . If w e ignore G + g in e q . (3.30b), then it i m m e d i a t e l y f o l l o w s that super l i m i t i n g - l a w behav iou r f o r P 2 ^ + - ' m P M e s super D e b y e - H u c k e l behav iour fo r l n y + . In Figure 30 w e have found that N a C l , M B r and M' l al l s h o w super l i m i t i n g - l a w behav iou r , a l though in Figure 37 w e see that on l y the M B r and M'l cu r ves are super D e b y e - H u c k e l . For our m o d e l M B r and M' l s o l u t i o n s G + g is a l w a y s nega t i ve . W e no te that this nega t i ve va lue fo r G + s w i l l tend to dec rease the va lue o f ( 9 1 n y + / 3 p 2 )p, and hence w i l l e m p h a s i z e super D e b y e - H u c k e l behav iou r . For our m o d e l NaCl s o l u t i o n G + g is p o s i t i v e (up to a concen t ra t i on o f 3.0M), and c o n s e q u e n t l y w i l l tend to inc rease l n y + . W e f ind that even though G + g is genera l l y much s m a l l e r than GL. f o r N a C l (about 40 t i m e s s m a l l e r at 1.0M), the e f f e c t of G ^ is s u f f i c i e n t l y large to cause l n y + to turn up f r o m the D e b y e - H u c k e l l im i t i ng l aw . Un fo r t una te l y , the va lues o f G . are not large enough to m o v e the NaC l curve fo r l n y + a b o v e those o f C s l and KCI . C l e a r l y , h o w e v e r , the va lues of G + s (which depend upon the s o l v e n t st ructure around an ion) can have a s t r ong in f luence upon resu l ts f o r l n y + even at ve r y l o w c o n c e n t r a t i o n . Let us n o w return again to Figure 30 . Wha t w e w o u l d l ike to de te rm ine is whether or not a real aqueous so l u t i on o f N a C l (or any a lka l i ha l ide fo r that mat te r ) s h o w s super l i m i t i n g - l a w behav iour f o r P 2 G + _ . O b v i o u s l y , the mean ac t i v i t y c o e f f i c i e n t d o e s not represent a conven ien t means of i nves t i ga t i ng th i s . H o w e v e r , if w e examine e q . (3.33), w e i m m e d i a t e l y see that the de r i va t i ve o f the o s m o t i c p ressure depends o n l y upon the rec ip roca l o f P 2 G + _ . U n f o r t u n a t e l y , o s m o t i c p ressure m e a s u r e m e n t s fo r aqueous e l e c t r o l y t e s o l u t i o n s are d i f f i cu l t to p e r f o r m and have r e c e i v e d re la t i ve ly l i t t le a t ten t ion [7]. On the other hand m e a s u r e m e n t s o f the o s m o t i c c o e f f i c i e n t s (as de f i ned in e q . (3.58)) f o r aqueous e l e c t r o l y t e so l u t i ons have r e c e i v e d a great deal o f a t ten t ion [6,7] and numerous tab les of va lues are ava i lab le [6,7,185]. W e note that a genera l t h e r m o d y n a m i c re la t i onsh ip b e t w e e n the ac t i v i t y and o s m o t i c c o e f f i c i e n t s [6,7] d o e s ex i s t . Un fo r t una te l y , the s i m p l e re la t ionsh ip b e t w e e n II and <f> g i ven in e q . (3.61) ho lds on l y in the l imi t p 2 —>0, and c o n s e q u e n t l y canno t be used to deduce the behav iour o f II f r o m that of <f> at f i n i te c o n c e n t r a t i o n . Equa t ion (3.58) can be e m p l o y e d at f in i te concen t ra t i on if the - 202 -behav iour o f V g is k n o w n , a l though th is e x p r e s s i o n is s t i l l on l y an approx ima te one . W e po in t out that the de r i va t i ve o f the o s m o t i c p ressure in eq . (3.33) is not o n l y at cons tan t tempera tu re , but a l so at cons tan t s o l v e n t chem ica l po ten t i a l . In our mode l ca l cu l a t i ons M s has been a l l o w e d to va ry w i th concen t ra t i on (exper imen ta l l y , M g w o u l d n o r m a l l y be he ld f i x e d at the pure so l ven t va lue) , and hence our de r i va t i ve of II w i l l re f lec t th is fac t . T h e r e f o r e , no de f i n i t i ve answer can cur rent ly be g i ven as to whether or not the present resu l t s fo r P 2 G + _ f o r a m o d e l NaC l s o l u t i o n are c o n s i s t e n t w i t h those o f i ts real counterpar t . Exper imenta l va lues fo r 0 fo r aqueous s o l u t i o n s of NaC l at 2 5 ° C w o u l d sugges t that th is is not the c a s e un less s o m e o f the d i s c r e p a n c i e s men t i oned above make large con t r i bu t i ons to (bU/bp2^-H o w e v e r , w e po in t out that the present resu l ts f o r NaC l m a y p rov ide s o m e exp lana t ion fo r the unconven t i ona l behav iour d e m o n s t r a t e d by other aqueous s o l u t i o n s o f s m a l l i o n s , e.g., the ca t i on order reve rsa l f o r y+ and <t> fo r f l uo r ide sa l t s [6]. W e again e m p h a s i z e that o s m o t i c p ressure measu remen ts w o u l d p rov ide a ve ry s i m p l e and d i rec t route for ob ta in ing i n f o rma t i on about i o n - i o n s t ructure in e l ec t r o l y t e s o l u t i o n s . 4. Structural Proper t ies Fi rst w e sha l l cons ide r the s o l v e n t - s o l v e n t s t ructure o f our mode l aqueous e l e c t r o l y t e s o l u t i o n s . In F igure 38 we have p lo t t ed the s o l v e n t - s o l v e n t rad ia l d i s t r i bu t ion f unc t i ons of mode l KCI and NaC l so l u t i ons at 4 .0M and 4 .0M and 1 2 M , r e s p e c t i v e l y , a long w i th g c c ( r ) f o r the pure tet rahedral s o l v e n t at 2 5 ° C . W e f i nd that the s o l v e n t - s o l v e n t pack ing s t ructure in both s o l u t i o n s at 4 .0M s t i l l r e s e m b l e s that of the pure s o l v e n t , a l though the p resence o f the i ons at th is concen t ra t i on is hav ing an o b v i o u s in f l uence . The so l ven t s t ructure genera l l y appears to be d a m p e n e d , that is to s a y , the peaks in g (r) have b e c o m e s m a l l e r and the w e l l s are not as deep . It is a l so ev ident f r o m Figure 38 that NaC l is more e f f e c t i v e than KCI at d is rup t ing the s o l v e n t - s o l v e n t pack ing s t ruc ture . If w e n o w examine g c_(r) f o r NaC l at 12M w e see that the so l ven t s t ructure has changed d rama t i ca l l y f r o m that of the pure s o l v e n t . The con tac t va lue o f g s s ( r ) n a s d e c r e a s e d subs tan t i a l l y and the p o s i t i o n o f the f i r s t m in imum has sh i f t ed o u t w a r d . The s e c o n d ne ighbor peak, w h i c h w a s o rg ina l l y cent red at r = 2 d fo r the pure s o l v e n t , n o w appears as - 203 -Figure 3 8 . S o l v e n t - s o l v e n t radia l d i s t r i bu t ion f unc t i ons o f the pure s o l v e n t and of seve ra l m o d e l e l e c t r o l y t e s o l u t i o n s . The s o l i d l ine is the radia l d i s t r i bu t i on func t i on of the pure te t rahedra l s o l v e n t at 2 5 ° C , wh i l e the dashed l ine rep resen ts 9 s s ( r ) f o r a m o d e l KCI s o l u t i o n at a concen t ra t i on of 4 . 0 M . The do t ted and d a s h - d o t l ines are resu l ts f o r mode l NaC l s o l u t i o n s at 4 .0M and 1 2 M , r e s p e c t i v e l y . 1.45-1 1.30H l A 5 - \ 1.00—i 0.85H 0.70 16-n i — i — i — r ~ 0.0 0.4 0.8 "1 I—I—I—I—I—I—I 1.2 1.6 2.0 2.4 ( r - d J / d - 205 -three d i s t i nc t peaks at r=1.8d 2 d and 2.15d fo r 12M N a C l . The s m a l l e s t o f 5 *D o t hese p e a k s (the one at r = 2 d ) is what rema ins o f the s e c o n d ne ighbor peak o f the pure s o l v e n t . The t w o rema in ing peaks appear at sepa ra t i ons w h i c h c o r r e s p o n d to so l ven t pa r t i c l es sepa ra ted by either a N a + or C l " i on . The N a + peak is the sharper o f the t w o w h i c h w o u l d sugges t that the s o l v e n t s su r round ing a N a + ion are more r i g id l y he ld in p o s i t i o n . The present resu l t s f o r 9 s s ( r ) c lea r l y ind ica te that f o r 12M N a C l a vast ma jo r i t y of the s o l v e n t pa r t i c l es in the mode l s o l u t i o n are d i rec t l y i n v o l v e d in ion s o l v a t i o n {i.e., c o m p r i s e the f i rs t s o l v a t i o n she l l ) . F i n a l l y , w e point out that s m a l l peaks at r = 1 .85d g and 2 . 1 5 d g can a l ready be seen in 9 s s ( r ) f o r a m o d e l NaC l s o l u t i o n at 4 . 0 M . In the present s tudy the quant i ty < c o s 0 _ _ ( r ) > = r ) , where 6 5 S UUfSS 5 5 is just the angle b e t w e e n the t w o d ipo le v e c t o r s o f t w o s o l v e n t p a r t i c l e s , can be de te rm ined us ing e q . (2.87). Resu l t s f o r < c o s 0 g s ( r ) > f o r the pure te t rahedra l so l ven t and f o r m o d e l NaC l and C s l s o l u t i o n s at 4 .0M are s h o w n in F igure 39 . W e f ind that < c o s 0 g s ( r ) > s h o w s o s c i l l a t o r y behav iour s i m i l a r to that of 9 s s ( r ) - A g a i n , the p resence of the ions in s o l u t i o n is found to disrupt s o l v e n t - s o l v e n t c o r r e l a t i o n s , in th is c a s e d i p o l e - d i p o l e c o r r e l a t i o n s . F r o m F igure 39 w e o b s e r v e that NaC l is much more e f f e c t i v e than C s l at d is rup t ing the d ipo la r co r re l a t i ons be tween s o l v e n t pa r t i c l es . If w e examine < c o s 0 ( r )> for NaC l and C s l mo re c l o s e l y , d rops in the va lues o f the ID o f unc t i ons can be seen at sepa ra t i ons c o r r e s p o n d i n g to the d iame te rs o f the ions p resen t . Hence , these fea tu res o f < c o s 0 ( r )> appear to be due to the o p p o s i n g (i.e., 0 s s = l 8 O ° ) d ipo le m o m e n t s o f t w o so l ven t pa r t i c l es separa ted by a s i n g l e i on . Next w e sha l l examine the i o n - s o l v e n t s t ructure o f our m o d e l aqueous e l e c t r o l y t e s o l u t i o n s . The i o n - s o l v e n t rad ia l d i s t r i bu t ion f unc t i ons at in f in i te d i l u t i on f o r three o f the ions c o n s i d e r e d in th is s tudy (spanning a large range o f ion s i z e ) have been c o m p a r e d in Figure 40 . The dependence o f 9 i g ( r ) upon i on s i ze is much as w e might expec t . For s m a l l i ons 9 i s ( r ) ' s mo re s t ruc tured (i.e., the con tac t and al l subsequen t peaks b e c o m e larger and the f i rs t and a l l subsequent m i n i m a g r o w deeper ) . A s s h o w n in Figure 4 0 , g;_(r) fo r N a + has a ve ry large con tac t va lue w h i c h then ve ry qu i ck l y d rops to a deep m i n i m u m at a reduced sepa ra t i on (i.e., r - d - ) o f 0.2d Th is w o u l d ind ica te that the f i rs t she l l o f s o l v e n t pa r t i c l es around a N a + ion is he ld in - 206 -Figure 39. < c o s 0 s s ( r ) > fo r the pure so l ven t and fo r m o d e l e l e c t r o l y t e s o l u t i o n s . The s o l i d l ine is < c o s 0 (r)> de te rm ined fo r the pure te t rahedra l so l ven t at 25 wh i l e the dashed and do t ted l ines represent resu l ts f o r m o d e l NaC l and C s l s o l u t i o n s , r e s p e c t i v e l y , at a concen t ra t i on of 4 . 0 M . - 208 Figure 4 0 . I o n - s o l v e n t radial d i s t r i bu t i on f unc t i ons at in f in i te d i l u t i on . The s o l i d , dashed and do t ted l ines are resu l ts ob ta ined in the present s tudy fo r N a + , Ch and M + , r e s p e c t i v e l y . - 2 1 0 -ve ry t i gh t l y . For large ions such as M + , the con tac t va lue o f 9 i g ( r ) I S much s m a l l e r . The f i rs t m i n i m u m is a l so much s h a l l o w e r and has m o v e d out to r - d ^ s = 0 . 5 d s (which is a l so the p o s i t i o n o f the h a r d - s p h e r e m i n i m u m ) . Thus , the f i r s t s o l v a t i o n shel l o f a M + ion appears to be held r e l a t i ve l y l o o s e l y . A s in the c a s e o f the pure te t rahedra l s o v e n t , the p o s i t i o n s o f the peaks in g. (r) are found to be s t rong l y d i c ta ted by the h a r d - s p h e r e pack ing o f the s o l v e n t around the ions . The s e c o n d peak a l w a y s has i ts m a x i m u m at a reduced separa t i on o f one s o l v e n t d iame te r . H o w e v e r , w e see in Figure 40 that at least fo r the sma l l e r N a + and C l - ions th is s e c o n d peak is d i s to r t ed t o w a r d s sma l l e r sepa ra t i ons s o that the average reduced sepa ra t i on o f the s e c o n d s o l v a t i o n she l l is s l i gh t l y l ess than one so l ven t d iame te r . The coo rd i na t i on numbers o f the ions (i.e., the numbers o f s o l v e n t s in the f i rs t s o l v a t i o n she l l ) can be de te rm ined us ing e q . (5.2). W e remark that f o r s m a l l ions such as N a * the C N is r e a s o n a b l y w e l l d e f i n e d because the f i rs t m i n i m u m of g;_(r) is s o deep . H o w e v e r , f o r larger ions such as M + it b e c o m e s much less o b v i o u s at what po in t to s top in tegra t ing in e q . (5.2) (i.e., the f i r s t s o l v a t i o n shel l b e c o m e s more p o o r l y de f ined) . For both the N a * and C h w e f i nd C N ^ 7 . 5 , wh i le fo r M + the C N is s o m e w h e r e in the range o f 10 to 15. For M + the va lue o f i ts C N appears reasonab le because o f i ts large s i z e . The C N fo r N a * is s o m e w h a t larger than the va lues usua l l y repor ted by compu te r s imu la t i on s tud ies [50,51,53,54,58], C N =5 .4 -7 .3 , a l though fo r C h the agreement is bet ter , C N =5.6-8 .4 . Neut ron d i f f r a c t i o n s tud ies [13,16] a high concen t ra t i on repor t a c o o r d i n a t i o n number o f about 5.8 f o r C h . In the present s tudy w e have a l s o exam ined the va lues o f < c o s # . _( r )> ob ta ined f r o m e q . (2.89a), where the ang le is de f i ned in Figure 3. Inf in i te d i lu t ion resu l ts f o r th is f unc t i on are g i v e n in Figure 41 f o r the N a * , C h and M * i o n s . (We again note that p o s i t i v e va lues o f < c o s 0 . _( r )> represent f avou rab le d ipo le o r ien ta t i ons w i t h respec t to the i on , w h i l e nega t i ve va lues ind ica te un favourab le ones. ) The o s c i l l a t o r y behav iour d e m o n s t r a t e d by < c o s 0 . ( r )> is s im i l a r to that o f g. (r), a l though the p o s i t i o n s o f the f i rs t 1 S 15 m i n i m a and s e c o n d m a x i m a do not c o i n c i d e exac t l y w i th t hose o f g. (r). A s in the c a s e o f g-_(r), w e f i nd that the structura l fea tu res o f < c o s 0 . ( r )> 1 3 1 b are genera l l y more d is t inc t f o r s m a l l e r i o n s . F r o m Figure 41 we see that f o r both the C h and M * i o n s , the average d i p o l e o r ien ta t ion o f the so l ven t is f a v o u r a b l e at al l sepa ra t i ons f r o m the i on . H o w e v e r , f o r N a * the average - 211 -Figure 41. < c o s 0 . _(r )> at in f in i te d i l u t i on . The l ines are as de f i ned in Figure - 213 -d ipo le o r i en ta t i on ac tua l l y b e c o m e s un favou rab le fo r a s m a l l range o f sepa ra t i ons around r - d - =0.35d W e o b s e r v e that the con tac t va lue of < c o s # . _(r)> d o e s not s h o w s t rong ion s i ze dependence . N e v e r t h e l e s s , if w e we re to compu te < c o s 0 ^ s > fo r the f i rs t s o l v a t i o n s h e l l , N a + g i ves a much larger resul t than d o e s M + s i nce far more o f the s o l v e n t s in the f i rs t s o l v a t i o n shel l o f N a + are at or ve ry near con tac t . It is in te res t ing to po in t out that if w e use the con tac t va lues o f < c o s t 9 ^ s ( r ) > in order to c o m p u t e an <c9. > at con tac t (i.e., a s s u m i n g the d i s t r i bu t i ons are ve ry nar row) , w e ob ta in an angle o f a p p r o x i m a t e l y 5 4 ° , w h i c h is e s s e n t i a l l y half o f the te t rahedra l ang le . Th is va lue c o m p a r e s f a v o u r a b l y w i t h resu l ts f r o m compu te r s i m u l a t i o n s [50,53,58] and f r o m d i f f r ac t i on expe r imen ts [13]. In Figure 42 w e have p lo t ted 9£s(r) f ° r a C h ion at in f in i te d i lu t ion and fo r seve ra l m o d e l e l ec t r o l y t e s o l u t i o n s at f in i te c o n c e n t r a t i o n . W e i m m e d i a t e l y o b s e r v e that at f in i te c o n c e n t r a t i o n the pack ing of the so l ven t around a C h ion is a f f e c t e d by the p resence of other ions in the s y s t e m , par t i cu la r ly at high concen t ra t i on . A s in the case o f 9 s s( r). w e s e e f r o m Figure 42 that the st ructura l fea tu res of 9 ^ s ( r ) b e c o m e d a m p e n e d at f in i te concen t ra t i on (e.g., the con tac t peak d rops and the f i rs t m i n i m u m b e c o m e s sha l l ower ) . C o m p a r i n g g . ( r ) f o r N a C l and KCI at 4 .0M w e f i nd that g . ( r ) X 5 1 3 has changed m o s t marked l y f r o m i ts in f in i te d i lu t ion resul t f o r N a C l . Th is w o u l d again sugges t that sma l l e r i o n s , in th is case N a + , are more e f f e c t i v e than ions of mode ra te s i z e , such as K + , at d i s rup t ing the s o l v e n t s t ruc ture . For NaC l at 1 2 M , g - ( r ) fo r C h bears su rp r i s i ng l y l i t t le r e s e m b l a n c e to the X s in f in i te d i lu t ion resu l t . The f i rs t m i n i m u m has been d i s p l a c e d o u t w a r d , wh i l e the s e c o n d ne ighbor peak has sp l i t into t w o d is t inc t peaks . The s m a l l e r o f the t w o is what rema ins o f the peak due to the s e c o n d s o l v a t i o n she l l o f a C h i on . C l e a r l y , at th is concen t ra t i on the C h ion appears to have o n l y a s ing le s o l v a t i o n s h e l l . The larger peak at a reduced sepa ra t i on of about 0 . 8 d g c o r r e s p o n d s to the ar rangement whe re the c o u n t e r - i o n is s i tua ted in be tween the C h ion and the s o l v e n t . Th i s feature is a l so ev iden t , though much s m a l l e r , fo r both the N a C l and KCI s o l u t i o n s at 4 . 0 M . If w e c o m p u t e the C N of a C h ion in our 12M N a C l s o l u t i o n , w e ob ta in a va lue of 5.5 w h i c h is in g o o d agreement w i t h exper imen ta l e s t i m a t e s [13]. M o r e o v e r , f o r th is s a m e s o l u t i o n w e f i nd that the C N of a N a + ion is app rox ima te l y 4 , w h i c h rep resen ts a c o n s i d e r a b l e d rop f r o m its in f in i te d i l u t i on va lue . T h e r e f o r e , in the present m o d e l o f a 12M NaC l so l u t i on a N a * ion has f e w e r s o l v e n t s in i ts f i rs t - 214 -Figure 42. I o n - s o l v e n t radia l d i s t r i bu t i on func t i on fo r C h . The s o l i d l ine is the in f in i te d i l u t i on resu l t , wh i l e the dashed l ine rep resen ts 9 jg ( r ) f ° r C h in a m o d e l KCI s o l u t i o n at a c o n c e n t r a t i o n o f 4 .0M. The do t ted and d a s h - d o t l ines are resu l t s fo r m o d e l NaC l s o l u t i o n s at 4.0M and 1 2 M , r e s p e c t i v e l y . - 216 -Figure 43. <cosc9. ( r )> o f C l " . The l ines are as d e f i n e d in F igure - 218 -s o l v a t i o n she l l than d o e s a C h i o n , even though the N a + ion in teracts more s t r o n g l y w i th each o f the s o l v e n t pa r t i c l es than does the C h i o n . W e have a l s o ca l cu la ted < c o s 0 ^ s ( r ) > f o r the s o l v e n t pa r t i c l es su r round ing a C h ion . In Figure 43 w e have s h o w n < c o s 0 . _(r)> at in f in i te d i l u t i on a long w i t h resu l ts f o r N a C l and KCI s o l u t i o n s at 4 .0M and a NaC l s o l u t i o n at 1 2 M . W e o b s e r v e that the con tac t va lue o f < c o s 0 ^ s ( r ) > d e m o n s t r a t e s o n l y s l ight concen t ra t i on dependence and a l m o s t no c o u n t e r - i o n dependence (even at 4 .0M) , w h i c h is aga in cons i s t en t w i t h exper imen t [13,16]. A t in f in i te d i l u t i on < c o s # ^ s ( r ) > f o r a C h ion is a l w a y s p o s i t i v e , ind ica t ing that the average d ipo le o r ien ta t ion is a l w a y s favou rab le w i t h respec t to the i o n , a l though the degree o f d ipo le a l ignment va r i es w i t h sepa ra t i on as s h o w n by the o s c i l l a t o r y behav iour o f < c o s # ^ s ( r ) > . A t a c o n c e n t r a t i o n o f 4 .0M the i o n - d i p o l e co r re l a t i ons b e c o m e h igh ly s c r e e n e d . W e see in Figure 43 that at 4 .0M < c o s c 9 ^ s ( r ) > n o w o s c i l l a t e s about zero (except near c o n t a c t ) ind ica t ing ranges of favou rab le as w e l l as ranges o f un favourab le d i p o l e o r ien ta t i on . C o m p a r i n g the t w o cu rves at 4 .0M w e o b s e r v e that < c o s 0 ^ s ( r ) > is genera l l y more negat ive fo r N a C l than f o r KCI . Th is aga in i m p l i e s that sma l l e r ions such as Na* are more e f f e c t i v e at d is rup t ing the order ing o f the so lven t a round a C h ion . A t 12M the shape o f the curve fo r <cos t9 . ( r )> has changed marked l y f r o m the in f in i te d i lu t ion resu l t . S t r o n g d i p o l e c o u n t e r -a l ignment is ev iden t ove r a large range o f s e p a r a t i o n s , s ta r t i ng at r - d - — 0.2d and pe rs i s t i ng unti l 0.9d A sharp drop in the value o f <cost9 . _(r)> can be s e e n at a reduced sepa ra t i on o f about 0.85d c o r r e s p o n d i n g to the Na* peak i den t i f i ed in 9^J-r^ • The NaC l and KCI s o l u t i o n s at 4 .0M s h o w s im i la r but s m a l l e r d rops in < c o s 0 ^ s ( r ) > at sepa ra t i ons c o r r e s p o n d i n g to the arrangement where the c o u n t e r - i o n is s i tua ted in b e t w e e n the C h ion and the so l ven t . W e w i l l n o w turn our a t ten t ion to the i o n - i o n s t ruc ture o f our m o d e l aqueous e l e c t r o l y t e s o l u t i o n s . F i rs t w e sha l l cons i de r the po ten t i a l s of mean f o r c e at in f in i te d i l u t i on f o r pa i rs o f o p p o s i t e l y charged i o n s . In Figure 44 w e have p lo t ted resu l ts f o r / 3 w ^ j ( r ) f o r L iF , N a C l , EqEq and M B r . For sma l l ion pai rs (e.g., L iF ) w e f ind that w ^ j ( r ) i s v e r Y s t ruc tured at shor t range, o b v i o u s l y depend ing very s t r o n g l y upon the mo lecu la r nature o f the s o l v e n t . For larger pa i rs o f ions (e.g., M B r ) w ^ j ( r ) b e c o m e s a much l e s s s t ructured f u n c t i o n w i th much sma l l e r o s c i l l a t i o n s . T h u s , fo r large ion pa i rs w ^ j ( r ) takes on i ts l o n g - r a n g e a s y m p t o t i c behav iou r , as g i ven by e q . (2.99), ve ry qu ick l y - 219 -Figure 44. P o t e n t i a l s o f mean f o r c e at in f in i te d i l u t i on fo r seve ra l pa i rs o f o p p o s i t e l y charged i o n s . The s o l i d , d a s h e d , do t ted and d a s h - d o t l ines represent resu l t s L i F , N a C l , EqEq and M B r , r e s p e c t i v e l y , in the te t rahedra l s o l v e n t . - 221 -{i.e., w i t h i n one or t w o s o l v e n t d i a m e t e r s ) , wh i l e f o r s m a l l e r pa i rs o f ions j ( r ) t akes three or four so l ven t d i ame te r s be fo re app roach ing i ts a s y m p t o t i c f o r m . F r o m Figure 44 we see that at con tac t the magn i tude of w^j(r) i nc reases s l igh t l y fo r s m a l l e r ion pa i r s , a l though w i j ( r ) r i ses much mo re q u i c k l y f r o m its con tac t va lue f o r pa i rs o f sma l l i o n s . If w e compu te the numbers o f contac t or near con tac t ion pa i rs f o r these s a l t s (exc lud ing L iF ) at s o m e l o w but f in i te c o n c e n t r a t i o n (e.g., 0.1M), w e f i nd that M B r has by far the largest number (~0.26) and N a C l the least (~0.04). In Figure 44 w e o b s e r v e that the f i rs t m a x i m u m o f j ( r ) i nc reases w i t h d e c r e a s i n g ion s i ze but appears at app rox ima te l y the s a m e reduced separa t i on f o r a l l ion pai rs except M B r . For both L iF and N a C l th is f i rs t max imum ac tua l l y b e c o m e s p o s i t i v e (i.e., w i j ( r ) is repu l s i ve w i t h respec t to in f in i te separa t ion ) . W e note that th is behav iour is c o n s i s t e n t w i th that o b s e r v e d by Pet t i t t and R o s s k y [82] fo r w i j ( r ) f ° r sma l l ion pa i r s . In Figure 44 w e see that the s e c o n d m i n i m u m in w i j ( r ) f ° r L iF is ve ry deep and broad and is cent red at a reduced sepa ra t i on of on l y 0.8d For N a C l and EqEq the s e c o n d m i n i m u m in w j j ( r ) ' s v e r Y s i m i l a r to that o f LiF o n l y not as deep , wh i l e fo r M B r it has b e c o m e very s h a l l o w and has i ts m i n i m u m value at one s o l v e n t d iamete r . The s i tua t ion where a so l ven t par t i c le sepa ra tes t w o o p p o s i t e l y charged ions is a ve ry favou rab le d ipo la r con f i gu ra t i on [79,81], and hence w o u l d be expec ted to b e c o m e more impor tant fo r larger ions w i th the present s o l v e n t m o d e l (because o f the sho r t - r ange nature o f the i o n - q u a d r u p o l e in te rac t ion w i t h respec t to the i o n - d i p o l e in te rac t ion) . The s o l v e n t - s e p a r a t e d ion pair is a l so f a v o u r e d by the h a r d - s p h e r e pack ing o f the s o l v e n t . The fac t that the s e c o n d m i n i m a in w. . (r) f o r L i F , NaCl and EqEq al l appear at r - d - • < d i nd i ca tes that a s o l v e n t b r idg ing structure (where the ions might be l oca ted near the tet rahedral co rne rs o f the so l ven t pa r t i c l e ) is m o r e s tab le than the s o l v e n t - s e p a r a t e d g e o m e t r y f o r sma l l e r pa i rs o f ions in the tet rahedral s o l v e n t . C l e a r l y , the quadrupo le m o m e n t must be p lay ing an c ruc ia l ro le in s t a b i l i z i n g th is b r idg ing s t ruc ture . S i m i l a r resu l ts we re again repo r ted by Pett i t t and R o s s k y [82]. The e f f e c t s o f ion s i ze a s y m m e t r y upon ion s o l v a t i o n can a l s o be seen in F igure 44 by c o m p a r i n g w i j ( r ) f ° r NaC l and E q E q . W e have p r e v i o u s l y no ted that d + _ is the s a m e fo r both o f these s a l t s . W e f i nd that w i j ( r ) i s a l w a y s mo re p o s i t i v e (i.e., l ess a t t rac t i ve ) f o r NaC l than f o r E q E q . O b v i o u s l y - 222 -i nc reased ion a s y m m e t r y i m p r o v e s s o l v a t i o n fo r the present m o d e l . Th is o b s e r v a t i o n d i f f e r s f r o m p rev ious RLHNC resu l ts [81] f o r the s a m e s o l u t i o n m o d e l w h i c h s h o w e d e s s e n t i a l l y no dependence upon ion s i ze a s y m m e t r y . If w e c o m p a r e the present RHNC resu l ts fo r w j j ( r ) w ' t n t hose ob ta ined f r o m the RLHNC theory [80,81] us ing the s a m e m o d e l s , w e f i nd rather poo r ag reemen t , as might be expec ted f r o m our ear l ier d i s c u s s i o n s . For both t heo r i es w i j ( r ) s h o w s the s a m e bas i c o s c i l l a t o r y behav iour and p o s i t i o n i n g of the p e a k s . H o w e v e r , fo r s m a l l ion pa i rs such as L iF , the RHNC resul t f o r w^j(r) is genera l l y more s t ruc tured {i.e., has larger o s c i l l a t i o n s ) than that of the R L H N C . The RLHNC a l so p red i c t s much more nega t i ve con tac t va lues fo r w^j(r) ( for L iF the RLHNC g i v e s about - 1 2 k T ) c lear l y ind ica t ing that the RHNC p r o v i d e s much bet ter s o l v a t i o n o f s m a l l i o n s . For larger ions the agreement i m p r o v e s on l y s l i gh t l y , w i th the RHNC resul t fo r w^j(r) n o w be ing less s t ruc tured than that of the R L H N C . W e note that th is rather poo r qua l i ta t i ve agreement b e t w e e n the RHNC and RLHNC resu l ts fo r w i j ( r ) c a n n o t be exp la ined s i m p l y in t e rms of the d i f f e r e n c e in the pure s o l v e n t d ie lec t r i c cons tan t s ob ta ined f r o m the t w o t heo r i es . The concen t ra t i on dependence of g + _ ( r ) has been s h o w n in Figure 45 . The par t icu lar case of a mode l KCI s o l u t i o n has been c o n s i d e r e d . W e o b s e r v e that g + _ ( r ) f o r KCI has a f a i r l y s i m p l e concen t ra t i on d e p e n d e n c e ; as the concen t ra t i on is i nc reased (at least up to 4 .0M) g + _ ( r ) e s s e n t i a l l y s h o w s on ly sc reen ing e f f e c t s . W i th i nc reas ing concen t ra t i on g + _ ( r ) is sh i f t ed d o w n w a r d and the magn i tude of the o s c i l l a t i o n s d e c r e a s e s , a l though the shape is e s s e n t i a l l y re ta ined . M o s t of the other s o l u t i o n s i nves t i ga ted in th is s tudy s h o w the s a m e s i m p l e concen t ra t i on dependence . The one major excep t i on is the M' l s o l u t i o n (and to a much lesse r degree the M B r c a s e ) . In the p rev ious s e c t i o n w e have d i s c u s s e d h o w the l o n g - r a n g e tai l o f g + _ ( r ) g r o w s w i t h i nc reas ing concen t ra t i on fo r M ' l . In F igure 46 w e have p lo t t ed g + _ ( r ) fo r the M' l s o l u t i o n at three o f the concen t ra t i ons s t u d i e d , inc lud ing the h ighest concen t ra t i on w e we re able to r each , s p e c i f i c a l l y 0 .74M. A t shor t range w e see the s a m e concen t ra t i on dependence found fo r KCI . A t long range the tai l o f 9+_(r) d rops w h e n go ing f r o m 0.1M to 0 .5M, but then c l ea r l y i nc reases aga in w h e n g o i n g to 0 .74M. B e f o r e p r o c e e d i n g , let us return to Figure 30 and the super l i m i t i n g - l a w behav iour d e m o n s t r a t e d by N a C l fo r P 2 G + _ . If we examine g + _ ( r ) fo r N a C l 223 -Figure C o n c e n t r a t i o n dependence o f g + _ ( r ) f o r l ines are resu l ts f o r m o d e l KCI s o l u t i o n s 45. KCI. The s o l i d , dashed and do t ted at 0 . 1 M , 1.0M and 4 . 0 M , r e s p e c t i v e l y . - 225 -Figure C o n c e n t r a t i o n dependence of g + _ ( r ) f o r l ines are resu l ts f o r m o d e l M' l s o l u t i o n s 46. M ' l . The s o l i d , dashed and do t ted at 0 . 1 M , 0.5M and 0 . 7 4 M , r e s p e c t i v e l y . - 227 -we f i nd that it s h o w s very s i m i l a r behav iour and concen t ra t i on dependence to that o b s e r v e d f o r m o d e l KCI s o l u t i o n s (see Figure 45). N o o b v i o u s a n o m a l y appears in g + _ ( r ) fo r our m o d e l NaC l s o l u t i o n s wh ich might exp la in i ts super l i m i t i n g - l a w behav iou r . H o w e v e r , we sha l l see b e l o w that g + + ( r ) d o e s s h o w s o m e unexpec ted behav iour and w e k n o w f r o m the charge neut ra l i t y c o n d i t i o n s (cf. e q . (3.5c)) that G + _ and G + + are inex t r i cab ly l inked to one another . T h e r e f o r e , the super l i m i t i n g - l a w dependence d e m o n s t r a t e d by N a C l fo r P 2 G + _ may in fac t be more a m a n i f e s t a t i o n o f the unconven t i ona l behav iour o f i ts N a + / N a + radia l d i s t r i bu t ion f u n c t i o n . N o w let us turn our a t ten t ion to l i k e - i o n co r re l a t i ons in our mode l e l e c t r o l y t e s o l u t i o n s . W e have s h o w n | 3 w ^ ( r ) fo r N a + , K + , |- and M + i ons at in f in i te d i lu t ion in Figure 47 . W e again remark that fo r the present ion and s o l v e n t m o d e l s , ions equal in s i ze but o p p o s i t e in charge w i l l be s o l v a t e d equ i va len t l y (i.e., w + + ( r ) =w__( r ) ) . The ion s i ze dependence o f w j i ( r ) i s f ound to be not as s i m p l e as that o f w ^ j ( r ) . For the four ions c o n s i d e r e d in F igure 47 w e o b s e r v e that the con tac t va lue o f A r ) i nc reases w i th d e c r e a s i n g ion s i ze (although th is is not the c a s e fo r L i + , see Figure 61). For ioris o n l y s l i gh t l y larger than the s o l v e n t (e.g., K + and h ) w ^ 1 " ) i s a su rp r i s i ng l y f ea tu re less f u n c t i o n w h i c h d e c r e a s e s rap id ly f r o m i ts con tac t va lue to a l m o s t i m m e d i a t e l y take on i ts l o n g - r a n g e a s y m p t o t i c behav iour (as g iven by e q . (2.99)). A s ion s i ze is inc reased w £ £ ( r ) c lea r l y b e c o m e s less r e p u l s i v e , par t i cu la r l y near con tac t , due in part to the dec reas ing c o u l o m b i c r e p u l s i o n . For M + , ^( r ) ac tua l l y has a loca l m i n i m u m ve ry near con tac t (at a reduced sepa ra t i on o f about 0.256",) w h i c h must be a resul t o f the s o l v e n t f o r c i n g t w o such large and s o m e w h a t h y d r o p h o b i c ions c l o s e r together . H o w e v e r , it is the behav iour o f ^( r ) fo r ions sma l l e r than the so l ven t w h i c h appears the mos t s t r i k i ng . W e see in Figure 47 that f o r N a + A r ) d e c r e a s e s ve ry rap id ly f r o m i ts con tac t va lue to a ve ry b road w e l l w i t h i ts m i n i m u m at r - d ^ — 0 . 3 d g . Th is m i n i m u m in w ^ 1 " ) , s m f a c t n e g a t i v e , that i s , it is a t t rac t i ve w i t h respec t to in f in i te sepa ra t i on . C l e a r l y , s t rong so l ven t f o r c e s must be present to o v e r c o m e the s t rong c o u l o m b i c repu l s i on be tween t w o such s m a l l i ons at s m a l l s e p a r a t i o n s . W e f ind that th is w e l l g r o w s (i.e., b e c o m e s deeper and ex tends to longer range) as the ion s i ze is d e c r e a s e d (see Figure 61). In Figure 26 w e have a l so f ound that the s o l v e n t is drawn in much more t ight ly to s m a l l e r i o n s . Toge the r , t hese o b s e r v a t i o n s w o u l d sugges t that ions wh ich - 228 Figure 47. Po ten t i a l s o f mean f o r c e at in f in i te d i l u t i on fo r severa l pa i rs of l ike i o n s . The s o l i d , d a s h e d , do t ted and d a s h - d o t l ines are resu l ts for N a + / N a + , K + /K + , l / h and M + / M + p a i r s , r e s p e c t i v e l y , in the te t rahedra l s o l v e n t . - 230 -in teract v e r y s t rong l y w i t h the so l ven t w i l l s h o w a m i n i m u m in w ^ ( r ) a t s m a l l s e p a r a t i o n s {i.e., the s o l v e n t p re fe rs to s o l v a t e these ions as a d iva len t pair ) . W e w i l l d i s c u s s th is fea ture in w^( r) and its re la t i onsh ip w i th the i o n - s o l v e n t in te rac t ion in s e c t i o n 5 and aga in in more deta i l in s e c t i o n 6. A t th is po int w e shou ld say that a l though the present RHNC resu l ts f o r r) m a y not be exac t , w e do be l i eve that they are qua l i t a t i ve l y cor rec t fo r the present m o d e l s . Pet t i t t and R o s s k y [82], us ing more c o m p l i c a t e d m o d e l s and a re la ted integral equat ion theory (RISM) , ob ta ined s im i l a r resu l t s fo r ^( r ) fo r s m a l l an ions (i.e., ?- and C l ) . C lea r l y what is n o w needed are exact c o m p u t e r s imu la t i on resu l t s fo r these m o d e l s in order to test the present t h e o r i e s . C o m p a r i n g the present RHNC resu l ts fo r (^ r ) w i th t hose ob ta ined fo r the s a m e m o d e l s f r o m the RLHNC theory [80,81], we again f i nd poor ag reemen t . Gene ra l l y the R L H N C resu l ts fo r r) have an o s c i l l a t o r y behav iour s i m i l a r to that o f w ^ j ( r ) (only inver ted) , wh i le the f unc t i ons de te rm ined in the present s t udy s h o w v i r tua l l y no o s c i l l a t i o n s . Bo th t heo r i es do pred ic t an a t t rac t ive w e l l in w j j ( r ) a t s m a l l separa t i ons fo r s m a l l i o n s , a l though in the RHNC th is fea ture is much larger . For large i ons the RLHNC theory s h o w s larger h y d r o p h o b i c e f f e c t s (i.e., f o r an ion t w i c e the s ize of the s o l v e n t w^( r) is ac tua l l y nega t i ve at con tac t ) . A g a i n , the d i f f e r e n c e s w e see in w j j ( r ) f ° r the t w o theo r i es can not be exp la ined s i m p l y in te rms of the d i f f e r e n c e in e. In Figure 48 w e have exam ined f in i te concen t ra t i on resu l ts fo r 9 ^ ( r ) fo r the s a m e four ions c o n s i d e r e d in Figure 47 . Included in Figure 48 are the l i k e - i o n rad ia l d i s t r i bu t ion f u n c t i o n s of N a + , K + , I- and M + in 1.0M s o l u t i o n s of N a C l , KC I , C s l and M B r , r e s p e c t i v e l y . W e o b s e r v e that even at a concen t ra t i on of 1.0M much of the behav iou r demons t ra ted b y g ^ ( r) fo r these ions s t i l l c l o s e l y r e s e m b l e s that p red i c ted by the po ten t i a l s o f mean f o r c e at in f in i te d i l u t i on . For both K + and I- g ^ ^( r) s ta r ts at zero at con tac t and then r i s e s s l o w l y to a va lue of 1.0, s h o w i n g on ly a f e w re la t i ve ly s m a l l s t ructura l f ea tu res . For M + g ^ ( r) has a contac t va lue o f 0.5 and r i ses qu i ck l y to a m a x i m u m o f 2.5 at a reduced separa t ion o f 0 . 2 5 d s . Th is f i r s t peak in 9 ^ ( r ) f o r M + is f o u n d to g r o w w i t h inc reas ing c o n c e n t r a t i o n , p robab l y due s i m p l y to i nc reased ion ic sc reen ing . In Figure 48 w e see that a l though g ^ ( r) fo r N a + star ts out at zero at c o n t a c t , it a l so r i s e s qu i ck l y to a large peak w i th a - 231 -Figure 48. g ^ ( r ) f o r s e v e r a l ions in m o d e l e l e c t r o l y t e s o l u t i o n s at 1.0M. The s o l i d , d a s h e d , do t ted and d a s h - d o t l ines are g^ A r ) f o r N a + in a NaC l s o l u t i o n , K + in a KCI s o l u t i o n , l~ in a C s l s o l u t i o n and M + in a M B r s o l u t i o n , r e s p e c t i v e l y . - 233 -m a x i m u m at r - d ^ = 0 . 4 d s . The va lue o f g ^ ^( r ) then rema ins larger than 1.0 unti l r - d . = d (i.e., unt i l a s ing le s o l v e n t can just f i t in b e t w e e n the t w o N a + i ons ) , at w h i c h po in t g ^ ( r ) d rops sharp l y b e l o w 1.0. W e po in t out that the s i t ua t i on whe re a s ing le so l ven t is be tween t w o ions of the s a m e charge is ene rge t i ca l l y un favou rab le . A t ve r y l o w concen t ra t i on the f i rs t peak in 9 ^ ( r ) f o r N a * is f o u n d to inc rease s l i gh t l y w i t h i nc reas ing concen t ra t i on but reaches a m a x i m u m at about 0.1M (where it has a m a x i m u m value o f about 5.0). A s the concen t ra t i on is i nc reased a b o v e 0 . 1 M , the peak then d e c r e a s e s and sh i f t s s l i gh t l y o u t w a r d . It con t inues to dec rease up to a concen t ra t i on o f about 4 . 0 M , af ter w h i c h it s h o w s re l a t i ve l y l i t t le change . Peaks due to ion t r i p les can a l s o be i den t i f i ed in Figure 48 (e.g., fo r h and M + at a reduced sepa ra t i on o f 1.3d g ) . A l t h o u g h the i o n - t r i p l e s peak is qui te large fo r the C s l and M B r s o l u t i o n s , w e f i nd that it b e c o m e s much s m a l l e r fo r KCI and N a C l s o l u t i o n s . The c o n c e n t r a t i o n and c o u n t e r - i o n dependence o f the l i k e - i o n radia l d i s t r i bu t ion f unc t i on fo r C l - has been s h o w n in Figure 49 . S o l u t i o n s o f NaC l at 0 . 1 M , 4 .0M and 1 2 M , as w e l l as KCI at 4 . 0 M , have been inc luded in Figure 49 . A t 0.1M w e see that f r o m con tac t r ) r i ses s l o w l y w i t h inc reas ing ion s e p a r a t i o n , wh i l e at higher c o n c e n t r a t i o n 9 ^ ( r ) r i ses much more qu i ck l y . M u c h o f th is e f f e c t can be in terpre ted as resu l t i ng f r o m s i m p l e ion ic s c r e e n i n g . W e po in t out that f o r NaC l and KCI s o l u t i o n s at 0.1 M the cu rves fo r g^ ^( r ) f o r C l " are a lmos t i nd i s t i ngu i shab le . H o w e v e r , at 4 .0M the C h / C I -radial d i s t r i bu t i on func t i on s h o w s de f in i te c o u n t e r - i o n d e p e n d e n c e , as can be seen f r o m F igure 49 . Of c o u r s e , the peak due to ion t r ip les appears at a larger sepa ra t i on fo r KCI than it d o e s fo r N a C l . If w e sh i f t our a t ten t ion to sma l l e r s e p a r a t i o n s , w e obse rve that both NaC l and KCI have a peak in g . . ( r ) fo r C l " at a reduced sepa ra t i on o f about 0.4d Th is peak appears to g r o w w i th i nc reas ing c o n c e n t r a t i o n , a l though more qu ick l y f o r N a C l than fo r KCI . For the 12M NaC l so l u t i on this peak has b e c o m e quite la rge , ind ica t ing a re la t i ve high p r o b a b i l i t y o f f i nd ing t w o C l " ions c l o s e toge ther . It shou ld be no ted that th is peak must be due to d i f fe ren t e f f e c t s than the peak f o u n d in g ^ ( r ) f o r N a + , s i nce there the peak d e c r e a s e s w i th inc reas ing concen t ra t i on . The s t ructure r e s p o n s i b l e fo r the peak in the C l / C I " radia l d i s t r i bu t i on func t ion is not o b v i o u s ; h o w e v e r , one p o s s i b i l i t y is an ar rangement whe re a s ing le s o l v e n t ac ts as a br idge b e t w e e n t w o C l " i ons and the s a m e c a t i o n . Th is s t ructure is c o n s i s t e n t w i th the fac t that the peak is larger f o r NaC l than - 2 3 4 -Figure 4 9 . g ^ ( r ) fo r C h fo r seve ra l m o d e l e l e c t r o l y t e s o l u t i o n s . The s o l i d , dashed and d a s h - d o t l ines represent resu l t s f o r N a C l s o l u t i o n s at 1 2 M , 4 .0M and 0 . 1 M , r e s p e c t i v e l y . The do t ted l ine is g . . ( r ) fo r C h in a 4 .0M KCI s o l u t i o n . - 236 -Figure 5 0 . C h / C h par t ia l s t ruc ture f a c t o r s fo r m o d e l N a C l s o l u t i o n s . The s o l i d , d a s h e d , do t t ed and d a s h - d o t l ines are resu l ts f o r m o d e l NaC l s o l u t i o n s at 1 2 M , 4 . 0 M , 1.0M and 0 . 1 M , r e s p e c t i v e l y . W e po in t out that the part ia l s t ructure fac to r has been mu l t i p l i ed by the m o l e f r a c t i o n , x_, o f C h i ons . - 237 -- 2 3 8 -KCI , s i nce g + _ ( r ) w o u l d ind ica te that NaC l has a larger number of s o l v e n t - s e p a r a t e d unl ike ion pa i r s . Th is s t ructure a l so p red ic t s the cor rec t c o n c e n t r a t i o n behav iour fo r the peak. F i n a l l y , in Figure 50 w e have p lo t ted the C l " / C h par t ia l s t ructure f a c t o r , as g i ven by e q . (5.4), fo r a m o d e l NaC l so l u t i on at seve ra l c o n c e n t r a t i o n s . It is o b v i o u s f r o m Figure 50 that at l o w concen t ra t i on S _ _ ( k ) is dom ina ted by its s m a l l k d e p e n d e n c e , and hence by the l o n g - r a n g e behav iour o f g__ ( r ) . A s the concen t ra t i on is inc reased w e o b s e r v e s t ruc ture appear ing in S _ _ ( k ) at larger k v a l u e s . If w e c o m p a r e the C l / C h par t ia l s t ructure f a c t o r o f our m o d e l 12M N a C l s o l u t i o n w i th an exper imen ta l resul t f r o m 14.9 mo la l L iC l in ob ta ined b y neutron sca t te r i ng (see Figure 27 of Re f . 13), w e f i nd that they are qua l i t a t i ve l y s im i l a r . 5. E f f ec t s o f Including the RDMF H e r e t o f o r e , the resu l ts repor ted in this chapter have been fo r mode l aqueous e l e c t r o l y t e s o l u t i o n s w h i c h i nco rpo ra te a n o n - p o l a r i z a b l e s o l v e n t w i th a permanent d ipo le momen t equal to the e f f e c t i v e d ipo le m o m e n t , m of the pure te t rahedra l s o l v e n t . W e have s h o w n in s e c t i o n 1 that w i t h i n the S C M F theo ry the average loca l e l ec t r i c f i e l d in the bulk is cons tan t (to a ve ry g o o d a p p r o x i m a t i o n ) f o r the m o d e l s o l u t i o n s be ing examined in th is s tudy . C o n s e q u e n t l y , the m o d e l s o l u t i o n s c o n s i d e r e d in s e c t i o n s 2, 3 and 4 o f th is chapter are e f f e c t i v e s y s t e m s w i th in the S C M F theory c o r r e s p o n d i n g to m o d e l e l e c t r o l y t e s o l u t i o n s wh ich i nco rpo ra te a po la r i zab le w a t e r - l i k e s o l v e n t . In th is s e c t i o n w e w i l l examine the e f f e c t s o f t reat ing the po la r i za t i on of the s o l v e n t at the leve l o f the R D M F theo ry , as d e s c r i b e d in Chapter IV. W e e m p h a s i z e that up to th is po in t on l y the bulk average f i e l d s in our m o d e l aqueous e l e c t r o l y t e s o l u t i o n s have been c o n s i d e r e d . W e w o u l d expect the R D M F theory to be m o s t accura te at in f in i te d i l u t i on and at l o w concen t ra t i on . H e n c e , w e have repeated ca l cu l a t i ons fo r m o d e l s o l u t i o n s wh i ch inc lude the R D M F on l y f o r concen t ra t i ons at or b e l o w 0 . 1 M . The N a + , K + , C s + , M + , C l " , B r - , and I- i ons w e r e al l s tud ied at in f in i te d i l u t i o n , wh i l e s o l u t i o n s of N a C l and C s l w e r e i nves t i ga ted at f in i te c o n c e n t r a t i o n (i.e., f o r c ^ O . I M ) . It w a s found that at the l o w (but f i n i t e ) - 239 -c o n c e n t r a t i o n s e x a m i n e d , the i n c l u s i o n of the R D M F had a neg l i g ib le (~0.1% at 0 .1M) e f f e c t upon the average loca l f i e l ds o f the bulk. T h e r e f o r e , as w a s done in al l p r e v i o u s l y repor ted c a l c u l a t i o n s , the e f f e c t i v e d ipo le m o m e n t o f the pure te t rahedra l s o l v e n t w a s used in al l m o d e l ca l cu la t i ons repor ted in th is s e c t i o n . F i rs t w e sha l l cons i de r the add i t i ona l i o n - s o l v e n t in te rac t ion t e r m , u^g( r ) , as de te rm ined us ing e q . (4.29). In F igure 51 we have c o m p a r e d the va lues o f u ^ g ( r ) g i ven b y the R D M F theory fo r a N a + ion at in f in i te d i l u t i on w i t h t hose ob ta ined f r o m e q . (4.1c). W e reca l l that e q . (4.1c) c o n s i d e r s on l y the t w o - b o d y p r o b l e m of a s i ng le po la r i zab le so l ven t par t ic le at a d i s tance r f r o m an i on , and hence rep resen ts the l o w d e n s i t y l im i t . It can be seen f r o m Figure 51 that the R D M F theo ry p red ic t s v a l u e s fo r the add i t i ona l i o n - s o l v e n t in te rac t ion w h i c h are much sma l l e r than t hose de te rm ined f r o m e q . (4.1c). C l e a r l y , the la tera l s o l v e n t f i e l d s are hav ing a ve ry large e f f ec t upon r ) , X & e v e n at sma l l e r sepa ra t i ons . In Chapter IV w e have s h o w n that at long range the R D M F resul t fo r u ^ ( r ) is e s s e n t i a l l y 1/9 that of u ( r ) (when e is l s q p large). F r o m Figure 51 w e o b s e r v e that f o r N a + u^p( r ) is about s ix t i m e s larger than u^g( r ) at con tac t . W e note that a s i m i l a r resul t w a s f ound fo r a l l the i ons i n v e s t i g a t e d . The con tac t va lue o f u^g( r ) fo r N a + c o r r e s p o n d s to A p ( r = d ^ s ) =0.425D. W e f i nd that r ) d rops rap id l y f r o m con tac t , a l m o s t reach ing zero at a reduced sepa ra t i on of 0 . 5 d g , then inc reases s l i gh t l y to a s m a l l peak at r - d - = 0 . 9 d „ . None o f the other ions examined here s h o w e d i s S ^ such a peak , a l though fo r v i r tua l l y al l these ions u i g ( r ) d id f la t ten in the range 0.4d„ < r - d - < d „ . B e y o n d one so l ven t d iamete r r) w a s f ound to ° S 1S S I S rap id l y app roach its l o n g - r a n g e l im i t i ng behav iou r . The add i t i ona l a t t rac t ion be tween an ion and a so l ven t due to the R D M F w a s f o u n d to be re l a t i ve l y s m a l l at short range w h e n c o m p a r e d w i th the e l e c t r o s t a t i c t e rms o f the i o n - s o l v e n t po ten t i a l . Ve ry l i t t le change in 9 j g ( r ) due to u AP( r ) w a s o b s e r v e d , a l though c l o s e i nspec t i on revea l s a very s l ight 1 s i nc rease in g.:_(r) at a lmos t al l sepa ra t i ons . In Figure 52 and 53 w e have s h o w n the e f f e c t s o f u A P ( r ) upon unl ike ion po ten t i a l s o f mean f o r c e at in f in i te d i l u t i on fo r NaC l and M B r , r e s p e c t i v e l y . W e f i nd that , in genera l , the add i t i on o f the R D M F i m p r o v e s ion s o l v a t i o n . Th is w o u l d s e e m c o n s i s t e n t w i t h the fac t that the i o n - s o l v e n t in te rac t ion has been inc reased s l i gh t l y . For N a C l w e obse rve that w^j(r) has - 240 Figure 5 1 . A d d i t i o n a l i o n - s o l v e n t in te rac t ion te rm due to Ap ( r ) f o r a N a * ion at in f in i te d i l u t i on . The s o l i d l ine represents u (r ) as g i ven b y e q . (4.1c), wh i l e the A D dashed l ine is u i g ( r ) ob ta ined us ing the R D M F theo ry . - 242 Figure 52. E f f e c t of the R D M F upon w. .(r) fo r N a C l . The s o l i d l ine is /3w. .(r) fo r 1 A D 1 NaC l in the te t rahedra l s o l v e n t at 2 5 ° C w h e n u.„(r)=0, wh i l e the dashed l ine A i s represen ts the resul t w h e n u-Hr) is g iven by the R D M F theo ry . - 244 -Figure 53. Ef fec t o f the R D M F upon w ^ j ( r ) f o r M B r . A s in Figure 52, the s o l i d l ine is fo r u ^ ( r ) = 0 , w h i l e the dashed l ine is the R D M F resul t . - 246 -b e c o m e s l i gh t l y more p o s i t i v e near contac t w i th the add i t i on o f u ^ g ( r ) , a l though it then b e c o m e s mo re nega t i ve at the so l ven t sepa ra ted d i s tance and b e y o n d , as can be seen in Figure 52. For M B r the add i t i on o f u ^ ( r ) sh i f t s w^j(r) to more p o s i t i v e va lues at al l s e p a r a t i o n s . C o m p a r i n g the resu l ts in F igu res 52 and 53 w e d i s c o v e r that the R D M F appears to have a re la t i ve l y larger e f f ec t upon w^j(r) fo r larger i ons , in th is case M B r . In ve ry recent w o r k us ing M c M i l l a n - M a y e r leve l t heo ry , Pet t i t t and R o s s k y [82] have s h o w n that s o m e t h e r m o d y n a m i c p rope r t i es o f m o d e l aqueous e l e c t r o l y t e s o l u t i o n s (in par t icu lar that of <j>) are e x t r e m e l y sens i t i ve to s m a l l changes in w. . ( r ) . A D T h u s , w e w o u l d expect the sh i f t s in w. . (r ) due to u - „ ( r ) o b s e r v e d here fo r l} i s N a C l and M B r to resul t in s i gn i f i can t changes in the ac t i v i t y c o e f f i c i e n t s o f these t w o s o l u t i o n s at f in i te concen t ra t i on . The e f f e c t s o f the R D M F upon the l i k e - i o n po ten t ia l s o f mean f o r c e at in f in i te d i lu t ion fo r N a + and M + are s h o w n in F igures 54 and 55, r e s p e c t i v e l y . Here w e f ind that s m a l l and large ions s h o w o p p o s i t e e f f e c t s . For M + we o b s e r v e that the add i t i on o f u ^ g ( r ) sh i f t s w^ (^ r ) t o more p o s i t i v e va lues at a l l s e p a r a t i o n s , ind ica t ing i m p r o v e d s o l v a t i o n . Th i s behav iour is cons i s t en t w i t h the p r o p o s e d hyd rophob i c nature o f these large i o n s ; the i nc reased i o n - s o l v e n t i n te rac t ion w o u l d be expec ted to reduce the hyd rophob i c e f f e c t s , and c o n s e q u e n t l y i m p r o v e the s o l v a t i o n . For N a + w e see in Figure 54 that the add i t i on of u A P ( r ) sh i f t s w. . ( r ) to more nega t i ve va lues at al l sepa ra t i ons . 1 5 I X In s e c t i o n 3 w e have p r o p o s e d that the nega t i ve w e l l s in r ) f o r s m a l l i ons are a resul t o f their ve ry s t r o n g in te rac t ion w i t h the s o l v e n t . Thus , if the i o n - s o l v e n t in te rac t ion is i nc reased w e w o u l d expect the w e l l in w ^ ( r ) to g r o w deeper . Th is is in fac t what is o b s e r v e d in Figure 54. At In Chapter IV w e have s h o w n that u i g ( r ) w i " cont r ibu te to C j g (cf. e q . (4.98)) and have de r i ved an e x p r e s s i o n (cf. e q . (4.99)) f o r i ts con t r i bu t i on , S ^ * 3 , to the l im i t i ng s l o p e . The l o w concen t ra t i on behav iour o f C j g fo r mode l N a C l and C s l s o l u t i o n s w i t h and w i thou t the R D M F have been c o m p a r e d in F igure 56. A t very l o w concen t ra t i on we f i nd that our numer i ca l resu l ts do app roach their r e s p e c t i v e l im i t i ng l a w s . It is o b v i o u s f r o m Figure 56 that m a k e s an apprec iab le con t r i bu t i on to the l im i t i ng behav iour o f C j g , a l m o s t d o u b l i n g the to ta l l im i t ing s l o p e . C lea r l y then , u A p ( r ) can be expec ted to s i g n i f i c a n t l y a f fec t the l im i t i ng l aws of al l t h e r m o d y n a m i c p rope r t i es o f e l e c t r o l y t e s o l u t i o n s wh ich depend upon i o n - s o l v e n t c o r r e l a t i o n s . W e remark - 247 -Figure 54. E f f e c t o f the RDMF upon w ^ ( r ) f ° r N a + . The s o l i d l ine is 0w^( r ) f o r a N a + ion in the te t rahedra l s o l v e n t at 2 5 ° C w h e n u ^ ? ( r ) = 0 , w h i l e the dashed A i s l ine rep resen ts the resul t w h e n u i g ( r ) , s g i ven by the RDMF t heo ry . - 2 4 8 -- 249 -Figure 55. E f f e c t o f the R D M F upon A r ) fo r M + . A s in Figure 54 , the s o l i d l ine is fo r u ^ ( r ) = 0 , wh i l e the dashed l ine is the R D M F resu l t , l S - 251 -Figure 5 6 . E f f ec t o f the R D M F upon C j g . For ease o f c o m p a r i s o n w e have p lo t t ed the d i f f e rence b e t w e e n the in f in i te d i lu t ion v a l u e , C j g , and C j g i t se l f . The do ts and s o l i d t r i ang les are RHNC resu l ts fo r m o d e l C s l and NaC l s o l u t i o n s , r e s p e c t i v e l y , w h e n u ^ g ( r ) = 0 . The open c i r c l e s and t r iang les are resu l t s fo r m o d e l C s l and N a C l s o l u t i o n s , r e s p e c t i v e l y , w h e n u ^ ( r ) is g i ven b y the R D M F theory . The s o l i d l ine is the l im i t i ng s l o p e , S c , de te rm ined us ing e q . (3.48) and e = 8 8 . 3 . The dashed l ine rep resen ts the sum of S c and S ^ * 3 where the latter is g i ven by e q . (4.99). - 253 -that the l im i t i ng v a l u e s , C ^ g , are a l s o i nc reased d rama t i ca l l y by the R D M F . T h e r e f o r e , e v e n though u ^ ( r ) appears as a re l a t i ve l y s m a l l te rm in the 4 i o n - s o l v e n t pair p o t e n t i a l , i ts l o n g - r a n g e 1/r dependence and the fac t that it is a s p h e r i c a l l y s y m m e t r i c te rm a l l o w it to have a re l a t i ve l y large e f f e c t upon m a n y t h e r m o d y n a m i c p roper t i es o f e l ec t r o l y t e s o l u t i o n s . A s m e n t i o n e d a b o v e , mode l NaC l and C s l s o l u t i o n s wh i ch inc luded the R D M F were s tud ied at l o w c o n c e n t r a t i o n . The average energ ies of these s o l u t i o n s rema ined v i r tua l l y unchanged f r o m the ear l ier u ^ ( r ) = 0 r e s u l t s . Quan t i t i es such as the d ie lec t r i c cons tan t and K w e r e a l so on l y s l i gh t l y a f f e c t e d . A s w e w o u l d expect f r o m our ear l ier d i s c u s s i o n , the d e r i v a t i v e s o f l n y + s h o w e d great s e n s i t i v i t y to the add i t i on of the R D M F , even at the l o w concen t ra t i ons e x a m i n e d . For C s l the de r i va t i ve w a s found to inc rease (become sma l l e r in magn i tude) by about 20% at 0 . 1 M , wh i l e fo r NaC l it d e c r e a s e d by a p p r o x i m a t e l y 10%. F r o m our d i s c u s s i o n in Chapter IV it w o u l d s e e m o b v i o u s that the part ia l m o l e c u l a r v o l u m e of the so lu te shou ld be par t i cu la r l y s e n s i t i v e to the i nc lus ion o f the R D M F . Their in f in i te d i lu t ion va lues s h o w s t rong d e p e n d e n c e . For e x a m p l e , f o r C s l V ^ =75.6 c c / m o l e when u ^ ( r ) = 0 , w h i l e 7° =44.9 c c / m o l e w i t h the R D M F turned o n . W e f ind that the l im i t i ng s l o p e of V 2 (see Figure 36) is a l m o s t doub led due to a s im i l a r inc rease in S c _ Un fo r t una te l y , even if w e to ta l l y ignore the HNC con t r i bu t i on to S c and use on l y in de te rm in ing S v , w e d i s c o v e r that the theore t i ca l l im i t ing s l o p e s t i l l e x c e e d s the real resul t by mo re than a fac to r o f three, e v e n af ter co r rec t i ng f o r the d i f f e r e n c e in the c o m p r e s s i b i l i t i e s o f the te t rahedra l s o l v e n t and real wa te r . Thus , it w o u l d appear that eq . (4.99) d o e s not represent an exact resu l t and that the R D M F theo ry is s t i l l o n l y a p p r o x i m a t e , even in the l im i t s p 2 — ^ u a n c ' r — > ° ° . What is not c lear is w h y th is shou ld be the c a s e . In Chapter IV w e have s h o w n that at in f in i te d i lu t ion the R D M F resul t fo r the average loca l f i e l d at long range (cf. e q . (4.96b)) is equ iva len t to the e x p r e s s i o n of P o l l o c k et al. [167] fo r the s p e c i a l case w h e n the s o l v e n t is po la r i zab le but n o n - p o l a r . A ve ry s i m i l a r e x p r e s s i o n (cf. e q . (4.97)) w a s ob ta ined f o r the c a s e w h e n the s o l v e n t is po la r but n o n - p o l a r i z a b l e . H o w e v e r , s i nce there are no c o m p u t e r s i m u l a t i o n resu l ts ava i l ab le fo r the latter s y s t e m , the accu racy of e q . (4.97) is not k n o w n . One o f the mos t l i ke l y s o u r c e s of error in the RDMF theory w o u l d s e e m to be that aspec t o f our mean f i e l d approach in w h i c h w e - 254 -a s s u m e that on l y the average d ipo le m o m e n t s d i rec ted t o w a r d s the ion need be c o n s i d e r e d . One might a l s o expect the t h r e e - b o d y c o r r e l a t i o n s (e.g., d i p o l e - d i p o l e - i o n ) to be impor tan t in de te rm in ing the average e x c e s s loca l f i e l d . M o r e o v e r , s i nce u^P( r ) (and hence C j g ) depends upon < A E 1 ( R ) > 2 , s m a l l e r rors in the average e x c e s s loca l f i e l d w i l l have a re l a t i ve l y large in f luence upon C T C , and c o n s e q u e n t l y upon i ts l im i t ing s l o p e . 6. Results Obtained Employing Di f ferent So lven ts A l l resu l ts repor ted in the p rev ious four s e c t i o n s w e r e ob ta ined us ing the tetrahedral so l ven t m o d e l (descr ibed in Chapter V ) w i th an e f f e c t i v e permanent d ipo le momen t m e = 2 . 6 0 5 D and an e f f e c t i v e square quadrupo le momen t 0 g = 2 . 5 7 B . M o d e l aqueous e l e c t r o l y t e s o l u t i o n s e m p l o y i n g three d i f fe ren t but c l o s e l y re la ted s o l v e n t m o d e l s we re a l so s t u d i e d , w i th e s s e n t i a l l y al l ca l cu la t i ons be ing car r ied out at in f in i te d i l u t i on . In th is s e c t i o n we w i l l c o m p a r e these resu l ts w i t h t hose repor ted ear l ier in this chap te r . F i rs t we sha l l exam ine the e f f ec t o f t o ta l l y ignor ing the po la r i zab i l i t y o f the so l ven t and tak ing i ts permanent d ipo le m o m e n t as be ing 1.855D, w h i c h is the gas phase va lue f o r wa te r [118]. Th is nonpo la r i zab le te t rahedra l so l ven t has a d ie lec t r i c cons tan t o f 28.4 (as repor ted in Chapter V ) , w h i l e fo r the po la r i zab le m o d e l e =97.4. In F igure 57 w e have c o m p a r e d w j j ( r ) f ° r KCI in the po la r i zab le and nonpo la r i zab le te t rahedra l s o l v e n t s . A l s o inc luded in Figure 57 are the p r im i t i ve m o d e l po ten t i a l s o f mean f o r c e . C l e a r l y , the resu l ts f o r the two s o l u t i o n s d i f fe r m a r k e d l y . W e f i nd much bet ter s o l v a t i o n of the ions in the po la r i zab le s o l v e n t , w ^ j ( r ) be ing s h i f t e d to much m o r e negat ive va lues fo r the nonpo la r i zab le s o l v e n t . C l o s e r i n s p e c t i o n revea ls that the po ten t ia l s o f mean f o r c e have ve ry s i m i l a r o s c i l l a t o r y behav iour about thei r r espec t i ve l o n g - r a n g e a s y m p t o t i c l i m i t s . Thus , m o s t o f the d i s s i m i l a r i t y s e e n in w i j ( r ) f o r the t w o s o l v e n t m o d e l s c o m e s through the d i f f e rence b e t w e e n their d ie lec t r i c c o n s t a n t s . N e v e r t h e l e s s , w e w o u l d expect mode l s o l u t i o n s u t i l i z ing the nonpo la r i zab le tet rahedral s o l v e n t to have t h e r m o d y n a m i c p roper t ies w h i c h d i f f e r d rama t i ca l l y (for the m o s t part ) f r o m those repor ted in s e c t i o n 3 f o r s o l u t i o n s e m p l o y i n g the po la r i zab le te t rahedra l s o l v e n t . 2 5 5 -Figure 57 . Po ten t i a l s o f m e a n fo r ce at in f in i te d i lu t ion for KCI in po la r i zab le and nonpo la r i zab le te t rahedra l s o l v e n t s . The s o l i d and dashed l ines are RHNC resu l t s fo r w i j ( r ) de te rm ined fo r po la r i zab le and nonpo la r i zab le te t rahedra l s o l v e n t m o d e l s , r e s p e c t i v e l y . The do t ted and d a s h - d o t l ines represent their r espec t i ve p r im i t i ve m o d e l f unc t i ons ob ta ined f r o m eq . (2.99) us ing the d ie lec t r i c cons tan t s g i ven in the text . - 256 -- 257 -W e have a l so i nves t i ga ted the e f f e c t s o f us ing the fu l l quadrupo le tenso r o f wa te r in the s o l v e n t m o d e l . In the te t rahedra l so l ven t m o d e l it has been rep laced w i th a n e f f e c t i v e square quadrupo le momen t (as d i s c u s s e d in Chapter V) . C a l c u l a t i o n s we re car r ied out at in f in i te d i lu t ion fo r seve ra l d i f f e ren t ions in the C 2 v quadrupo le s o l v e n t . A s ing le c o m p u t a t i o n w a s a l so p e r f o r m e d at 0.5M fo r a m o d e l KCI s o l u t i o n . W e note that the e f f e c t i v e d ipo le m o m e n t and the d ie lec t r i c cons tan t o f the C 2 v quadrupo le so l ven t are the s a m e as fo r the te t rahedra l s o l v e n t . A s w e w o u l d expec t , the s o l v a t i o n o f the ions equal in s i ze but o p p o s i t e in charge (e.g., E q + / E q - and N a + / F " ) w a s no longer s y m m e t r i c , w i th p o s i t i v e ions r ece i v i ng a s l ight p re fe rence because o f the nega t i ve zz c o m p o n e n t of the quadrupo le t e n s o r . O n l y s l ight changes w e r e no t i ced in the i o n - s o l v e n t and i o n - i o n s t ructure due to the genera l i za t i on o f the quadrupo le momen t of the w a t e r - l i k e s o l v e n t , th is e f f ec t be ing a great deal s m a l l e r than that o b s e r v e d f r o m the add i t i on o f the o c t u p o l e m o m e n t to the so l ven t m o d e l , as d i s c u s s e d b e l o w . A t 0.5M the t h e r m o d y n a m i c p roper t ies o f the m o d e l KCI s o l u t i o n w e r e f o u n d to be ve ry s im i l a r (d i f fe r ing by l e s s than 1% in a l m o s t al l c a s e s ) to t hose o f the s a m e s o l u t i o n w i th the tetrahedral s o l v e n t . Thus , the s m a l l zz c o m p o n e n t of the quadrupo le tensor of wa te r appears to have re l a t i ve l y l i t t le in f luence upon ion s o l v a t i o n and upon the t h e r m o d y n a m i c p rope r t i es o f mode l aqueous e l e c t r o l y t e s o l u t i o n s . F i n a l l y , w e w i l l c o n s i d e r the e f f e c t s o f the add i t i on o f the oc tupo le momen t o f wa te r to the C 2 v quadrupo le s o l v e n t . In par t icu lar w e shal l examine i ts e f f e c t upon ion s o l v a t i o n and upon t h e r m o d y n a m i c p rope r t i es of mode l aqueous e l e c t r o l y t e s o l u t i o n s . The C 2 v o c t upo le so l ven t (as d i s c u s s e d in Chapter V ) has an e f f e c t i v e d ipo le m o m e n t of m e = 2 . 6 3 4 D and a d ie lec t r i c cons tan t o f 94.9, both va lues be ing ve ry c l o s e to those of the te t rahedra l s o l v e n t . The L i + , N a + , E q + , K + , C s + , F-, E q - , C h and I- ions w e r e a l l s tud ied at in f in i te d i lu t ion in the C 2 v o c t u p o l e s o l v e n t . F i r s t , let us examine the changes in the i o n - s o l v e n t s t ruc ture . In Figure 58 we have p lo t t ed 9 j , s ( r ) f ° r N a + a n c l F" ' o n s m t r , e C 2 v o c t upo le s o l v e n t a long w i th g- ( r ) f o r a N a + ion (or equ i va len t l y , f o r a F" ion) in the tet rahedral s o l v e n t . In the C 2 y oc tupo le s o l v e n t bo th ions s h o w a s l ight i nc rease in the occu r rence o f te t rahedral pack ing o f the s o l v e n t about the i o n s , as ind ica ted in Figure 58 by the i nc rease in 9 ^ s ( r ) a t a reduced separa t i on of about 0.65d . Th i s is not a su rp r i s i ng resul t s ince the add i t i on - 2 5 8 -Figure 5 8 . I o n - s o l v e n t radial d i s t r i bu t i on f unc t i ons at in f in i te d i lu t ion fo r the tet rahedral and C 2 v oc tupo le s o l v e n t s . The s o l i d l ine is 9 i s ( r ) f ° r a N a + (or F-) ion in the te t rahedra l s o l v e n t . The dashed and do t ted l ines represent resu l ts fo r a N a + and a F~ i on , r e s p e c t i v e l y , in the C 9 o c t upo le s o l v e n t . - 259 -- 260 -of the o c t u p o l e m o m e n t w a s f o u n d in Chapter V to s t ab i l i ze te t rahedra l pack ing w i t h i n the pure s o l v e n t . In Figure 58 w e see that the contac t peak in g^ s (r) f o r a F- ion in the C 2 v oc tupo le s o l v e n t is sharper than fo r a F- (or N a + ) ion in the te t rahedra l s o l v e n t , wh i ch in turn is sharper than fo r a N a + ion in the C 2 v o c t upo le s o l v e n t . A s im i l a r o b s e r v a t i o n can be made fo r the s e c o n d peaks in 9^J<r'- B e y o n d the s e c o n d peak we o b s e r v e that in the C 2 y oc tupo le so l ven t the ampl i tude of the o s c i l l a t o r y st ructure in 9 ^ s ( r ) rema ins larger f o r F~ than fo r N a + . Thus , the s o l v a t i o n she l l s t ructure appears to be more c l ea r l y de f i ned fo r a F - ion than fo r a N a + ion in the C 2 y oc tupo le s o l v e n t . M o r e o v e r , w e f i nd that in general f o r ions of equal s i z e the an ion is a l w a y s p re fe ren t i a l l y s o l v a t e d ove r the c a t i o n . The in f luence o f the oc tupo le m o m e n t , and hence the a s y m m e t r y in s o l v a t i o n , is found to dec rease w i t h i nc reas ing ion s i ze because of the s h o r t - r a n g e nature o f the i o n - o c t u p o l e in te rac t ion (wi th respec t to that o f the i o n - d i p o l e ) . In Figure 59 w e have c o m p a r e d <cost9 . _(r)> fo r N a + and F~ ions in the C 2 v o c t upo le and tet rahedral s o l v e n t s . It can be seen f r o m Figure 59 that the average o r ien ta t i on o f the d ipo le m o m e n t s o f the so l ven t pa r t i c l es around these s m a l l i ons is s t r ong l y i n f l uenced by the p resence o f the oc tupo le m o m e n t in the so l ven t m o d e l . In the C 2 y o c t upo le s o l v e n t , w e f ind a genera l dec rease in the o s c i l l a t o r y s t ructure o f < c o s t ^ s ( r ) > , pa r t i cu la r l y fo r N a + , i nd ica t ing an apparent dec rease in d ipo lar s t ruc ture . The p o s i t i o n s and shapes o f the peaks in < c o s # ^ s ( r ) > , par t i cu la r ly f o r F-, are o b v i o u s l y a f f e c t e d by the p resence of the o c t u p o l e m o m e n t in the s o l v e n t m o d e l . F r o m Figure 59 w e o b s e r v e that in the C 2 v o c t upo le so l ven t the con tac t va lues of < c o s t 9 ^ g ( r ) > have d ropped re la t i ve to the te t rahedra l s o l v e n t resu l ts f o r bo th F- and N a + i o n s . H o w e v e r , if w e examine the s tandard dev ia t i on of cost9. (r) as 1S V ; de te rm ined f r o m e q . (2.91)), w e f i nd that fo r F- the s tandard d e v i a t i o n , o, is s i g n i f i c a n t l y s m a l l e r (by about 20%) in the C 2 y oc tupo le s o l v e n t , ind ica t ing a na r rower d i s t r i bu t ion of ang les . In the c a s e of N a + , a ac tua l l y i nc reases s l i gh t l y f r o m i ts va lue in the tet rahedral s o l v e n t . E v i d e n t l y , the e f fec t of the oc tupo le momen t is such that the o r ien ta t ion of the s o l v e n t , at least near con tac t , has b e c o m e more d i r ec t i ona l l y s p e c i f i c fo r an ions than fo r ca t i ons o f the s a m e s i z e . A l l the e f f e c t s repor ted here fo r N a + and F- i ons we re a l so o b s e r v e d fo r larger ca t i ons and a n i o n s , a l though the in f luence o f the oc tupo le m o m e n t is again found to dec rease w i th i nc reas ing ion s i z e . - 261 -Figure 59 . < c o s 0 ^ s ( r ) > at in f in i te d i lu t ion for the te t rahedra l and C 2 y o c t u p o l e s o l v e n t s . The l ines are as de f i ned in Figure 58. - 263 -Figure 6 0 . Po ten t i a l s o f m e a n f o r c e at in f in i te d i lu t ion fo r L iF in the C 2 y o c t upo le and tet rahedra l s o l v e n t s . The s o l i d and dashed l ines are RHNC resu l ts f o r the tet rahedral and C 9 o c t upo le s o l v e n t s , r e s p e c t i v e l y . - 265 -Let us n o w cons i de r the s e n s i t i v i t y of the i o n - i o n s t ructure to the p resence o f the oc tupo le momen t in the so l ven t m o d e l . The po ten t ia l s of mean f o r c e at in f in i te d i lu t ion fo r L iF in both te t rahedra l and C 2 v o c t u p o l e s o l v e n t s have been c o m p a r e d in Figure 60 . W e o b s e r v e that fo r LiF w ^ j ( r ) is genera l l y more nega t i ve ( inc luding the con tac t peak and the f i rs t m a x i m u m ) in the C 2 v oc tupo le s o l v e n t . F rom Figure 60 w e see that the s e c o n d m i n i m u m of w j j ( r ) is sh i f t ed s t i l l further i nwa rd . Its m i n i m u m is n o w at a reduced sepa ra t i on o f about 0.75d ind ica t ing an inc reased p re fe rence fo r the s o l v e n t b r idg ing s t ructure fo r th is s m a l l pai r o f ions in the C 2 v o c t upo le s o l v e n t . W e again reca l l the w o r k of Pet t i t t and R o s s k y [82] in w h i c h it w a s s h o w n that s m a l l changes in w ^ j ( r ) can resul t in large changes in s o m e t h e r m o d y n a m i c p roper t i es (in par t icu lar <j>) o f m o d e l aqueous e l e c t r o l y t e s o l u t i o n s . C l e a r l y then, w e might expec t l n y + fo r a LiF (or even N a C l ) s o l u t i o n to s h o w d rama t i ca l l y d i f fe ren t concen t ra t i on dependence w h e n the te t rahedra l s o l v e n t is rep laced by the C 2 v oc tupo le s o l v e n t . It shou ld a l s o be po in ted out that f o r larger ion pa i rs the d i f f e rence b e t w e e n the C 2 y o c t upo le and tet rahedral resu l t s fo r w i j ( r ) is much s m a l l e r . The l i k e - i o n po ten t ia l s of mean f o r c e fo r L i + and F- ions in bo th the C 2 v o c t u p o l e and tet rahedra l s o l v e n t s have been s h o w n in Figure 61 . Here w e obse rve qui te d ramat i c e f f e c t s due to the add i t i on of the oc tupo le m o m e n t to the s o l v e n t m o d e l . For L i + w e f i nd that r ) b e c o m e s much more repu l s i ve at con tac t and the huge a t t rac t ive w e l l f o u n d w i t h the te t rahedra l s o l v e n t b e c o m e s much s h a l l o w e r in the C 2 v o c t u p o l e s o l v e n t . In Figure 61 w e see that the c o n v e r s e is true in the case of F \ A s d i s c u s s e d a b o v e , the behav iour o f ^ ( r ) at short range s e e m s to be c l o s e l y re la ted to the degree o f s o l v a t i o n o f that ion (i.e., how t igh t l y the s o l v e n t is he ld to the ion and c o n s e q u e n t l y h o w w e l l de f i ned the s o l v a t i o n she l l s t ructure is) . Thus , f o r F - in the C 2 v o c t upo le so l ven t the con tac t peak in 9 ^ s ( r ) b e c o m e s sharper (see Figure 58) and the a t t rac t i ve w e l l in w ^ j ( r ) g r o w s deepe r , wh i l e fo r L i + the o p p o s i t e re la t i onsh ip is t rue. H o w e v e r , w e e m p h a s i z e here that it may w e l l be the l o n g - r a n g e changes in the i o n - s o l v e n t s t ruc tu re , a l s o ev iden t in F igure 58 , wh i ch have the largest in f luence upon the s h o r t - r a n g e behav iour of r ) . Once a g a i n , w e no te that the e f f e c t s o f the oc tupo le m o m e n t upon are m o s t p ronounced fo r s m a l l i o n s . - 266 -Figure 61. L i k e - i o n po ten t i a l s of mean f o r c e at in f in i te d i l u t i on fo r L i + and F- in the C 2 oc tupo le and tet rahedra l s o l v e n t s . The s o l i d and do t ted l ines are RHNC resu l ts f o r L i V L i + and F / F " , r e s p e c t i v e l y , in the te t rahedra l s o l v e n t . The dashed and d a s h - d o t l ines represent /3w^( r ) fo r L i + and F", r e s p e c t i v e l y , in the C 2 v oc tupo le s o l v e n t . 268 -TABLE X. A v e r a g e i o n - s o l v e n t energ ies per ion at in f in i te d i l u t i on . Resu l t s fo r the C 2 v oc tupo le and tet rahedra l s o l v e n t s are c o m p a r e d . The va lues g i ven are in kT uni ts and are those o f the e f f e c t i v e s y s t e m s . ION Tet rahedra l C 2 y Oc tupo le S o l v e n t S o l v e n t L i * -424.1 -393.5 N a + -363.8 -342.1 E q * -316.2 -301.7 K + -296.4 -284.7 C s + -255.8 -249.6 F- -363.8 -394.7 E q - -316.2 -336.5 c i - -278.7 -291.7 1- -230.5 -235.1 W e w i l l n o w examine the in f luence that the oc tupo le m o m e n t has upon s o m e of the t h e r m o d y n a m i c p roper t i es o f e l e c t r o l y t e s o l u t i o n s at in f in i te d i l u t i on . The average i o n - s o l v e n t energ ies per ion at in f in i te d i l u t i on are g iven in Tab le X , where resu l ts f o r seve ra l ions in bo th the te t rahedra l and C „ 2v o c t u p o l e s o l v e n t s have been i nc luded . W e f i nd that the add i t i on of the oc tupo le momen t to the s o l v e n t mode l has a fa i r l y large e f f e c t upon the average i o n - s o l v e n t ene rg i es . A s w e w o u l d expec t f r o m our p rev ious d i s c u s s i o n s , U- _ / N . is more p o s i t i v e {i.e., s m a l l e r in magn i tude) fo r a ca t i on X O X in the C 2 v o c t upo le s o l v e n t than fo r the s a m e ca t i on in the te t rahedra l s o l v e n t , w h i l e the c o n v e r s e is true fo r a n i o n s . Hence , if w e c o n s i d e r t w o ions of the s a m e s i z e w h i c h are equal and o p p o s i t e in charge (e.g., E q + / E q - or Na + / F~ ) , w e f ind that the an ion in teracts more s t r ong l y w i t h the C 2 v oc tupo le s o l v e n t than d o e s the c a t i o n . W e a l so o b s e r v e f r o m Tab le X that th is a s y m m e t r y in s o l v a t i o n is largest fo r the s m a l l e s t i o n s . If we examine the ind iv idua l i o n - d i p o l e , i o n - q u a d r u p o l e and i o n - o c t u p o l e con t r i bu t i ons to the i o n - s o l v e n t energy fo r any o f the ions l i s ted in Tab le X , w e again f i nd s o m e in te res t ing resu l t s . For an ions in the C 2 y o c t upo le s o l v e n t the i o n - o c t u p o l e energy is nega t i ve and the i o n - d i p o l e and i o n - q u a d r u p o l e energ ies have d e c r e a s e d (i.e., i nc reased in magn i tude ) w i th respec t t o their va lues in the tet rahedra l s o l v e n t . For ca t i ons in the C 2 y o c t u p o l e s o l v e n t , h o w e v e r , the i o n - o c t u p o l e energy is p o s i t i v e ( ind icat ing a net - 269 -average r epu l s i on be tween the ion and the oc tupo le m o m e n t ) and the i o n - d i p o l e and i o n - q u a d r u p o l e ene rg ies have b e c o m e less nega t i ve . C l e a r l y , w h e n a C 2 V o c t u p o l e so l ven t par t i c le in teracts w i t h a p o s i t i v e i o n , none o f the m o s t f a v o u r a b l e d i po le and quadrupo le o r i en ta t i ons c o r r e s p o n d to f a v o u r a b l e o r i en ta t i ons o f the oc tupo le m o m e n t w i t h respec t to the i on . W e note that the magn i tude o f the i o n - o c t u p o l e energy is a l w a y s less than 5% of the to ta l i o n - s o l v e n t energy fo r al l the ions l i s ted in Tab le X . F i n a l l y , let us f o c u s upon the par t ia l mo la r v o l u m e . In Chapter III w e have s h o w n that at in f in i te d i l u t i on v"2 can be sp l i t into t w o independent t e rms (of c o u r s e th is is not the c a s e at f in i te concen t ra t i on ) . C o m p a r i n g e q s . (3.12) and (3.64a) w e i m m e d i a t e l y have V ? = k T x J d - PsC°is) , (6.4) where V.° is an ind iv idua l i on i c par t ia l mo lecu la r v o l u m e . Then us ing e q . (6.4) w e have ca l cu la ted va lues fo r V.° fo r seve ra l ions at in f in i te d i lu t ion in both i the C 2 v o c t u p o l e and tet rahedra l s o l v e n t s . These va lues fo r V.° have been reco rded in Tab le XI a long w i t h exper imen ta l resu l ts f o r ions in real wa te r . It is i m m e d i a t e l y o b v i o u s f r o m Tab le XI that the t rends in the present resu l ts are much larger than those o f exper imen t . If w e r e - e x a m i n e e q . (6.4), w e see that V.° depends upon the i so the rma l c o m p r e s s i b i l i t y , x£, o f the pure s o l v e n t . W e have repor ted ear l ier (in s e c t i o n 3) that x£ fo r the tet rahedra l so l ven t is f i v e t i m e s larger than that o f wa te r at 2 5 ° C . For the C 2 v oc tupo le so l ven t X,^ is 50% larger s t i l l . Thus , in order to a l l o w a more reasonab le c o m p a r i s o n {i.e., one w h i c h d o e s not depend upon the d i f f e r e n c e s in x^) be tween both se ts o f theore t i ca l resu l ts and those o f expe r imen t , w e c o m p u t e corrected va lues f o r V.° us ing o n l y the i so the rma l c o m p r e s s i b i l i t y o f real water in e q . (6.4). T h e s e c o r r e c t e d va lues f o r V.° are a l so g i ven in Tab le X I . It can be c lea r l y seen f r o m Tab le XI that the oc tupo le m o m e n t has a very large e f f e c t upon ind iv idua l i on ic part ia l mo la r v o l u m e s . W e f ind that V . 0 is much s m a l l e r {i.e., more nega t i ve ) f o r a ca t i on in the C 2 v o c t upo le so l ven t than f o r the s a m e ca t i on in the tet rahedra l s o l v e n t , wh i l e the c o n v e r s e re la t i onsh ip ho lds fo r a n i o n s . M o r e o v e r , f o r t w o ions equal in s i ze and equal and o p p o s i t e in charge (i.e., E q * / E q - or N a + / F - ) there is a dramat ic d i f f e rence b e t w e e n their r e s p e c t i v e va lues o f V? in the C 2 v o c t u p o l e s o l v e n t , w i th the an ion a l w a y s hav ing the larger va lue (i.e., appear ing as though it is larger in - 270 -TABLE XI. Ind iv idual i on ic par t ia l mo la r v o l u m e s at in f in i te d i l u t i on . Resu l t s f o r the tet rahedra l and C ? o c t u p o l e s o l v e n t s are c o m p a r e d w i th those f o r real wa te r at 2 5 ° C . The exper imen ta l par t ia l mo la r v o l u m e s are f r o m Tab le 6 on page 376 o f Re f . 5. The va lue g i ven in pa ren theses are c o r r e c t e d resu l ts as d i s c u s s e d in the text . A l l va lues f o r V.° are g i ven in c c / m o l e . ION Tet rahedra l S o l v e n t C 2 v O c t u p o l e S o l v e n t Expt . L i * -103.1 (-20.0) -226.5 (-29.3) -11.2 N a * -62.7 (-12.2) -177.5 (-22.9) -7.4 E q * -29.8 (-5.8) -136.1 (-17.6) — K* -14.5 (-2.8) -116.6 (-15.1) 3.4 C s * 22.6 (4.4) -68.3 (-8.8) 15.5 F- -62.7 (-12.2) -2.0 (-0.3) 3.3 E q - -29.8 (-5.8) 46.0 (5.9) — ci- 0.3 (0.1) 87.9 (11.4) 23.7 | - 53.0 (1°- 3 ) 158.5 (20.5) 41.4 so lu t i on ) . C l e a r l y , fo r ions of the same s i ze any d i s s i m i l a r i t y in NA0 must be to ta l l y due to d i f f e r e n c e s in their apparent electrostriction [5] o f the s o l v e n t (that is to the to ta l v o l u m e change or c o m p r e s s i o n o f the s o l v e n t due to the p resence of the ion) . N o w one might expec t that the e l e c t r o s t r i c t i o n o f the s o l v e n t shou ld be p ropo r t i ona l to the average in te rac t ion be tween the ion and the s o l v e n t . C o n s e q u e n t l y , w e w o u l d then expec t that the larger the magn i tude of the average i o n - s o l v e n t ene rgy , the greater the degree of e l e c t r o s t r i c t i o n , and hence the s m a l l e r V.° shou ld be . Un fo r t una te l y , th is deduc t i on is con t rad i c t ed by the va lues in T a b l e s X and X I . Let us t ry another a p p r o a c h . One v i e w cur ren t ly he ld fo r e l ec t r os t r i c t i on [5] sugges t s that m o s t o f the e f f e c t o c c u r s ve ry near an i o n , wh i ch f o r s m a l l un iva lent ions usua l l y i m p l i e s just the f i r s t ' s o l v a t i o n s h e l l . If w e r e - e x a m i n e Figure 58 w e o b s e r v e that the f i rs t s o l v a t i o n she l l o f a F- ion in the C 2 y o c t upo le s o l v e n t is he ld in more t i gh t l y (i.e., expe r i ences greater e l e c t r o s t r i c t i o n ) than is the f i rs t s o l v a t i o n she l l o f a N a * i on . Fur the rmore , if w e compu te c o o r d i n a t i o n numbers f o r the N a * and F- ions (using e q . (5.2)), w e f i nd that in the C 2 y o c t upo le s o l v e n t it is the F- ion w h i c h has a s l i gh t l y larger number o f s o l v e n t s in i ts f i r s t s o l v a t i o n s h e l l . A g a i n , w e w o u l d conc lude that the a n i o n , in th is c a s e F-, shou ld have the s m a l l e r va lue of V . ° , but aga in th is w o u l d be con t ra ry to the resu l ts in Tab le X I . In our d i s c u s s i o n o f Figure 58 w e had men t i oned apparent - 271 d i f f e r e n c e s in Q ^ g f r ) a t longer range fo r F- and N a + i ons in the C , y oc tupo le s o l v e n t . T o g e t h e r , the a b o v e o b s e r v a t i o n s w o u l d s t rong l y sugges t that it is these changes in the l o n g - r a n g e pack ing st ructure of the s o l v e n t around the ions w h i c h accoun ts fo r m o s t of the g r o s s d i s s i m i l a r i t i e s in V.° seen in Tab le XI f o r ions o f equal s i z e . C l e a r l y then , the l o n g - r a n g e i o n - s o l v e n t s t ructure (apart f r o m the l o n g - r a n g e i o n - d i p o l e co r re l a t i ons w h i c h g i ve r ise to S y ) appears to be an impor tant f a c t o r in de te rm in ing the va lues of ion ic par t ia l mo la r v o l u m e s , at least f o r the m o d e l aqueous e l e c t r o l y t e s o l u t i o n s be ing c o n s i d e r e d in th is s tudy . W e conc lude th is d i s c u s s i o n by no t ing that the i on i c charge and s i ze dependence of V.°, as s h o w n in Tab le X I , fo r ions in the C 2 y oc tupo le so l ven t is cons i s ten t w i th the exper imen ta l resu l ts g i v e n . - 272 -CHAPTER VII CONCLUSIONS In th is thes i s w e have examined the s t ruc tu ra l , t h e r m o d y n a m i c and d ie lec t r i c p roper t i es o f m o d e l aqueous e l e c t r o l y t e s o l u t i o n s w h i c h e x p l i c i t l y inc lude the s o l v e n t as a m o l e c u l a r s p e c i e s . The d ie lec t r i c and st ructura l p roper t i es o f the w a t e r - l i k e s o l v e n t s e m p l o y e d in this s tudy w e r e a l so i nves t i ga ted . The ion and so l ven t m o d e l s u s e d we re s i m p l e ones i nco rpo ra t i ng k n o w n m i c r o s c o p i c (mo lecu la r ) p roper t i es w i t h no f r ee l y ad jus tab le pa rame te rs . W e have c o n s i d e r e d o n l y un iva lent i o n s , p r i m a r i l y f o c u s i n g upon the a lka l i ha l i des , and hence have m o d e l l e d these ions s i m p l y as charged hard s p h e r e s . Ionic c r ys ta l radi i we re used to de te rm ine the h a r d - s p h e r e d i a m e t e r s . The w a t e r - l i k e s o l v e n t m o d e l s w e r e a l s o t rea ted as hard spheres into wh i ch measured va lues of the l o w - o r d e r mu l t i po le m o m e n t s and po la r i zab i l i t y t enso r of wa te r w e r e i nc luded . Our m o d e l s y s t e m s were s tud ied us ing integral equa t ion m e t h o d s , the RHNC theory [68] be ing e m p l o y e d a l m o s t e x c l u s i v e l y . In o rder t o app ly the RHNC theo ry , it had to be f i rs t gene ra l i zed fo r a m u l t i - c o m p o n e n t s y s t e m . In our gene ra l i za t i on of the O Z equat ion w e have examined the s i m p l i f i c a t i o n s wh ich resul t w h e n al l the s p e c i e s present in the s y s t e m have at least C 2 y s y m m e t r y . A compu te r p rog ram wh i ch uses genera l f o r m s o f the mu l t i po le po ten t ia l and the O Z and RHNC equa t ions w a s wr i t ten and then used to generate al l the resu l ts p resen ted in th is t h e s i s , both f o r the pure s o l v e n t and fo r the s o l u t i o n s y s t e m s . In th is s tudy w e have exp lo i t ed the f o r m a l i s m o f K i r k w o o d and Bu f f [104] in order to der ive genera l r e l a t i onsh ips be tween in tegra ls ove r h f l ^ ( r ) and cer ta in t h e r m o d y n a m i c p roper t i es o f e l e c t r o l y t e s o l u t i o n s . The usual K i r k w o o d - B u f f e x p r e s s i o n s can not be app l i ed d i rec t l y because app l i ca t i on o f the charge neut ra l i ty c o n d i t i o n s leads to inde te rmina te resu l t s f o r the t h e r m o d y n a m i c quan t i t i es . B y de f i n ing k - d e p e n d e n t ana logs o f the K i r k w o o d - B u f f equa t ions and tak ing the appropr ia te k—>0 l im i t s a n a l y t i c a l l y , w e w e r e ab le to ob ta in exact de te rmina te e x p r e s s i o n s . T h e s e re la t i onsh ips are d i rec t l y app l i cab le to real s y s t e m s s ince the s y s t e m w e have c o n s i d e r e d - 273 -i n c o r p o r a t e s the s o l v e n t as a true mo lecu la r s p e c i e s . A l t h o u g h w e have repo r t ed resu l ts f o r a t w o - c o m p o n e n t s a l t / s o l v e n t s y s t e m o n l y , the me thod w e have used is genera l and can e a s i l y be app l i ed to more c o m p l i c a t e d s y s t e m s . The l o w concen t ra t i on l im i t i ng behav iour of our t h e r m o d y n a m i c e x p r e s s i o n s w a s a l s o exam ined and c o m p a r e d w i t h the m a c r o s c o p i c resu l ts o b t a i n e d through D e b y e - H i i c k e l theory [6]. Not su rp r i s i ng l y , the exact D e b y e - H i i c k e l l im i t i ng law fo r l n y + w a s ex t rac ted f r o m the mo lecu la r t heo ry . M o r e o v e r , the m i c r o s c o p i c l im i t i ng law fo r w a s f ound to be f unc t i ona l l y equ iva len t to the m a c r o s c o p i c e x p r e s s i o n . H o w e v e r , w h e n the HNC a p p r o x i m a t i o n fo r the l im i t i ng s l o p e o f is c o m p a r e d w i th the exact m a c r o s c o p i c resu l t , w e d i s c o v e r that the HNC theory is rather inaccurate f o r th is quant i ty . For the m o d e l aqueous e l e c t r o l y t e s o l u t i o n s w e have c o n s i d e r e d , the HNC theory appears to o v e r e s t i m a t e the l im i t i ng s l o p e fo r by about an o rder o f magn i tude . In the present s tudy w e have d e s c r i b e d t w o l eve l s o f t heory w i t h wh i ch e l e c t r o l y t e s o l u t i o n s con ta in ing a po la r i zab le s o l v e n t may be e x a m i n e d . The f i r s t o f these is the S C M F app rox ima t i on [67] in wh i ch the m a n y - b o d y p r o b l e m of po la r i za t i on is reduced (by ignor ing f l uc tua t i ons ) to a p r o b l e m i n v o l v i n g an e f f e c t i v e pa i rw i se add i t i ve p o t e n t i a l . In th is t hes i s w e have s h o w n h o w the S C M F a p p r o x i m a t i o n can be app l i ed to the present e l ec t r o l y t e s o l u t i o n m o d e l in o rder to de te rmine the average to ta l d i po le momen t of the s o l v e n t at f in i te sal t c o n c e n t r a t i o n . W e have a l s o d e v e l o p e d a s e c o n d and m o r e de ta i l ed leve l o f f o r m a l i s m , the R D M F theo ry , w h i c h a l l o w s us to e x a m i n e the average loca l e lec t r i c f i e l d expe r i enced by a s o l v e n t at a d i s tance R f r o m an i on . The R D M F theory is a m e a n f i e l d approach ( l ike the S C M F a p p r o x i m a t i o n ) and g i v e s r ise to an e f f e c t i v e spher i ca l po ten t ia l be tween an ion and the s o l v e n t pa r t i c l es around it. Fu r the rmore , th is spher i ca l po ten t ia l w a s s h o w n to have an e f fec t upon the l im i t i ng laws o f t hose t h e r m o d y n a m i c p rope r t i es w h i c h depend upon i o n - s o l v e n t co r re l a t i ons at l o w concen t ra t i on . It shou ld aga in be e m p h a s i z e d that m o s t of the f o r m a l i s m s d e v e l o p e d in th is thes i s are genera l and cou ld be used w i t h re la t i ve ease in the i n v e s t i g a t i o n of other s y s t e m s b e s i d e s the m o d e l aqueous e l e c t r o l y t e s o l u t i o n s c o n s i d e r e d in the present s tudy . M o r e genera l m o d e l s fo r bo th the ions and the s o l v e n t cou ld be e x a m i n e d , e.g., where bo th the so l ven t and the ions p o s s e s s more c o m p l i c a t e d s h o r t - r a n g e po ten t i a l s and have higher mu l t i po le - 274 m o m e n t s . S o l u t i o n s of more than one sa l t w i th ions of seve ra l d i f fe ren t charges c o u l d be e a s i l y s t u d i e d . W e po in t out that the genera l f o r m o f the RHNC theo ry p resen ted in th is thes i s c o u l d , in p r i nc ip le , be app l i ed to any m u l t i - c o m p o n e n t s y s t e m charac te r i zed by ang le -dependen t pair p o t e n t i a l s . S e v e r a l c l o s e l y re la ted w a t e r - l i k e s o l v e n t s we re e x a m i n e d in th is s tudy us ing the RHNC theo ry . The HNC b a s i s set dependence fo r t hese m o d e l s w a s o b s e r v e d to be qui te s im i l a r t o that p r e v i o u s l y repor ted [71,110] fo r h a r d - s p h e r e m o d e l s w i t h d i p o l e s and l inear quadrupo les . M o d e l s y s t e m s fo r l iqu id wa te r w e r e inves t iga ted ove r a large range o f t empera tu res and p r e s s u r e s . These s y s t e m s w e r e f ound to have d ie lec t r i c c o n s t a n t s w h i c h agree w e l l w i th expe r imen ta l v a l u e s , par t i cu la r l y at higher tempera tu re . The RHNC resu l ts fo r the d is t r ibu t ion f u n c t i o n s of our w a t e r - l i k e f l u ids are in rather poo r agreement w i t h exper imen t , pa r t i cu la r l y at l o w tempera tu re . H o w e v e r , w e we re able to ob ta in the correct s t ruc ture f o r l iqu id wa te r at 25 °C through the s i m p l e add i t i on of a spher i ca l po ten t ia l to the h a r d - s p h e r e m o d e l . M o r e o v e r , the add i t i on o f th is so f t potent ia l a l so i m p r o v e d the l o w tempera tu re d ie lec t r i c p roper t i es o f our w a t e r - l i k e m o d e l . The oc tupo le m o m e n t w a s s h o w n to have re l a t i ve l y l i t t le e f f e c t upon the d ie lec t r i c cons tan t , a l though it w a s f ound to have a s o m e w h a t larger e f f ec t upon the s t ructure w i th in the m o d e l f l u id s y s t e m s . V i r t ua l l y al l o f the m o d e l aqueous e l e c t r o l y t e s o l u t i o n s w e have i nves t i ga ted at f in i te concen t ra t i on e m p l o y e d a po la r i zab le s o l v e n t m o d e l w i t h on l y d ipo le and square quadrupo le m o m e n t s . For these s o l u t i o n s and w i th in the S C M F a p p r o x i m a t i o n , the ave rage l oca l e lec t r i c f i e l d in the bulk w a s s h o w n to be e s s e n t i a l l y independent o f sal t c o n c e n t r a t i o n , and hence the average to ta l d i po le momen t o f the so l ven t w a s taken to be a cons tan t . The equ i l i b r ium d ie lec t r i c cons tan ts ob ta ined fo r these m o d e l s o l u t i o n s are in qua l i ta t i ve agreement w i th the exper imen ta l va lues for the d ie l ec t r i c cons tan t s o f aqueous e l e c t r o l y t e s o l u t i o n s , par t i cu la r l y at higher c o n c e n t r a t i o n s . W e a l s o po in t out that al l the m i c r o s c o p i c l im i t i ng l aw e x p r e s s i o n s d e r i v e d in th is s tudy w e r e c o n f i r m e d by our numer i ca l resu l ts at l o w c o n c e n t r a t i o n . Our r e l a t i ve l y s imp le m o d e l f o r aqueous e l e c t r o l y t e s o l u t i o n s d e m o n s t r a t e d a remarkab le d i v e r s i t y o f behav iour through s i m p l y va ry ing the h a r d - s p h e r e d i ame te r s of the i o n s . Pa i r s of sma l l e r i o n s , such as N a C l , we re f ound to be e x t r e m e l y s o l u b l e , whe reas pa i rs of larger i o n s , such as M ' l , we re - 275 re la t i ve l y i n s o l u b l e . M o r e o v e r , ion s o l v a t i o n and many t h e r m o d y n a m i c p roper t i es w e r e s h o w n to be qui te s e n s i t i v e to the ion s i ze a s y m m e t r y of a sa l t . The ion s i ze dependence of seve ra l t h e r m o d y n a m i c quant i t ies w a s e x a m i n e d , i nc lud ing the i so therma l c o m p r e s s i b i l i t y , the par t ia l mo la r v o l u m e s and the mean a c t i v i t y c o e f f i c i e n t . S o m e of the behav iour o b s e r v e d is c o n s i s t e n t w i t h that demons t ra ted by real aqueous e l e c t r o l y t e s o l u t i o n s , e.g., the s l o p e of v^ w a s found to change s ign at ve r y l o w concen t ra t i on fo r s o l u t i o n s of large i o n s . S o m e of the behav iou r o b s e r v e d d i sag reed w i t h exper imenta l r e s u l t s , e.g., the va lues fo r l n y + f o r our m o d e l N a C l s o l u t i o n s w e r e less than t hose o f C s l and KCI . H o w e v e r , even in the c a s e of the NaC l s y s t e m , the resu l t s repor ted in th is s tudy m a y p rov ide s o m e ins ight into unusual behav iour exh ib i ted by other aqueous e l e c t r o l y t e s o l u t i o n s . W e f ind that fo r t h e r m o d y n a m i c p roper t ies such as the mean ac t i v i t y c o e f f i c i e n t , both i o n - s o l v e n t and s h o r t - r a n g e i o n - i o n s t ructure can have a large in f l uence , even at re la t i ve l y l o w concen t ra t i on . In th is s tudy w e have made a de ta i l ed i nves t i ga t i on o f the i o n - i o n , i o n - s o l v e n t and s o l v e n t - s o l v e n t s t ructure w i t h i n our m o d e l s o l u t i o n s . S m a l l e r ions were f o u n d to d isrupt the s o l v e n t - s o l v e n t s t ructure to a much greater degree than larger i o n s , a l though th is e f f ec t o n l y b e c a m e o b v i o u s at higher c o n c e n t r a t i o n s . A s we might expec t , the s o l v a t i o n st ructure around an ion b e c o m e s more c l ea r l y de f i ned as the s i ze o f the ion d e c r e a s e s , a l though both concen t ra t i on and c o u n t e r - i o n e f f e c t s we re o b s e r v e d . Near con tac t the i o n - s o l v e n t s t ruc ture s h o w e d l i t t le c o u n t e r - i o n d e p e n d e n c e , yet s t rong dependence w a s found in the s e c o n d s o l v a t i o n s h e l l , pa r t i cu la r l y at higher c o n c e n t r a t i o n s . The mo lecu la r nature of the s o l v e n t w a s seen to have a ve ry s t rong in f luence on the s h o r t - r a n g e i o n - i o n s t ructure for both l ike and unl ike ion pa i rs . Th is w a s e s p e c i a l l y true in the c a s e o f ei ther ve ry s m a l l or ve r y large ions . For the m o s t par t , the c o n c e n t r a t i o n dependence o f the i o n - i o n co r re la t i ons w a s d o m i n a t e d by s i m p l e ion ic sc reen ing e f f e c t s , a l though s o m e c o u n t e r - i o n dependence w a s d e m o n s t r a t e d b y l i k e - i o n co r re l a t i ons at higher c o n c e n t r a t i o n s . F i n a l l y , we remark that p r e v i o u s RLHNC resu l ts us ing the same m o d e l s at in f in i te d i lu t ion w e r e f ound to be in rather poo r agreement w i th the present resu l t s . The e f f e c t s o f the R D M F theory we re e x a m i n e d fo r seve ra l mode l aqueous e l e c t r o l y t e so l u t i ons at in f in i te d i l u t i on and at l o w concen t ra t i on . The - 276 -e lec t r i c f i e l d due to the ion ic charge w a s s h o w n to be subs tan t i a l l y reduced by the lateral s o l v e n t f i e l d s , even at s m a l l s e p a r a t i o n s . C o n s e q u e n t l y , the R D M F had o n l y a s m a l l e f f e c t upon the s h o r t - r a n g e i o n - s o l v e n t s t ruc ture , a l though it appeared to have a larger impac t on i o n - i o n c o r r e l a t i o n s . A s e x p e c t e d , the R D M F w a s o b s e r v e d to have a large in f luence upon s o m e t h e r m o d y n a m i c quan t i t i es , m o s t no tab l y V 2 . H o w e v e r , at least f o r s o m e quant i t i es the R D M F theory turns out to be re la t i ve l y i naccura te , even in the l o n g - r a n g e and l o w c o n c e n t r a t i o n l i m i t s . For e x a m p l e , the R D M F con t r ibu t ion to the l im i t i ng s l o p e fo r e x c e e d s the k n o w n m a c r o s c o p i c resul t f o r real aqueous e l e c t r o l y t e s o l u t i o n s . N e v e r t h e l e s s , w e have been ab le to c lea r l y demons t ra te the impo r tance o f po la r i za t i on in de te rm in ing the st ructura l and t h e r m o d y n a m i c p roper t i es o f e l e c t r o l y t e s o l u t i o n s . O b v i o u s l y , fur ther s tudy o f these po la r i za t i on e f f e c t s is war ran ted . W h e r e a s ion s o l v a t i o n w a s f o u n d to be re la t i ve l y i nsens i t i ve to the s m a l l zz c o m p o n e n t o f the quadrupo le tensor o f wa te r , the add i t i on o f the oc tupo le m o m e n t to our w a t e r - l i k e so l ven t m o d e l w a s s h o w n to have a large impac t . The o c t u p o l e m o m e n t w a s found to have a par t i cu la r l y large e f f ec t upon g^ ^( r ) . The ind iv idua l i on ic par t ia l mo la r v o l u m e s at in f in i te d i lu t ion d e m o n s t r a t e d ex t reme s e n s i t i v i t y to the de ta i l s o f h o w an ion is s o l v a t e d . Fu the rmore , the present resu l t s s t r o n g l y sugges t that the l o n g - r a n g e i o n - s o l v e n t pack ing st ructure is ve ry impor tant in de te rm in ing the va lues o f V.° . S e v e r a l s u g g e s t i o n s fo r further s tudy have been made i m m e d i a t e l y above or in ear l ier d i s c u s s i o n s in th is t h e s i s . The ve ry in te res t ing behav iour exh ib i ted by our s o l u t i o n s o f larger ions c l ea r l y requ i res more de ta i l ed i nves t i ga t i on . Our e x a m i n a t i o n o f the in f luence o f the oc tupo le m o m e n t of wa te r upon ion s o l v a t i o n and t h e r m o d y n a m i c p roper t i es needs to be ex tended both to higher m o m e n t s (e.g., hexadecapo le ) and to f in i te c o n c e n t r a t i o n . Ions w i th higher charges (e.g., d i va len t ) and w i th l o w order mu l t i po le m o m e n t s (e.g., C N ) shou ld a l so be i n v e s t i g a t e d . In this s tudy w e have c o n s i d e r e d m o d e l aqueous e l e c t r o l y t e s o l u t i o n s o n l y at 2 5 ° C . C l e a r l y , the tempera ture dependence o f these s y s t e m s shou ld be e x a m i n e d , par t i cu la r ly at re l a t i ve l y high tempera tu res whe re the present m o d e l s w o u l d be expec ted to work much bet ter . The resu l t s p resen ted in th is thes i s can ve ry e a s i l y be used to tes t the a s s u m p t i o n that on l y a p a i r w i s e add i t i ve po ten t ia l need be e m p l o y e d in M c M i l l a n - M a y e r leve l theory [25,28] in order to s tudy p r i m i t i v e mode l - 277 -e l e c t r o l y t e s o l u t i o n s . Th is i nves t i ga t i on is cur rent ly be ing car r ied out [187]. A n o t h e r e x t e n s i o n to the present s tudy w h i c h is a l so in p rog ress [133] is the exam ina t i on o f large c o l l o i d a l pa r t i c l es and e lec t r i c doub le l aye rs in the s a m e mode l e l e c t r o l y t e so l u t i ons c o n s i d e r e d here. F i n a l l y , more c o m p l i c a t e d but h o p e f u l l y m o r e rea l i s t i c m o d e l s might a l s o be i nves t i ga ted us ing the me thods ou t l i ned in th is t h e s i s . 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Phys., 7 2 , 5763 (1980). - 287 -APPENDIX A TREATMENT OF POTENTIAL TERMS IN c(12) In genera l , care must be taken in handl ing the l o n g - r a n g e ta i l s in m n l ( r ) a n c j Jnn^- ( r ) w h e n numer i ca l l y p e r f o r m i n g both f o r w a r d and uv,a(i 'uvjaP ' ^ a b a c k w a r d Hankel t r a n s f o r m s (cf. e q s . (2.32), (2.34) and (2.46), (2.47)). For the h a r d - s p h e r e mu l t i po la r f l u i ds be ing c o n s i d e r e d in th is s t u d y , it is the l o n g - r a n g e con t r i bu t i ons due to / 3 u a ^ ( l 2 ) that are of p r imary c o n c e r n , s i n c e c ^ 1 2 ) — > - B u a( 1 2 ) as r—>°>. W e f i nd it conven ien t to de f i ne the ap ap short-range f unc t i ons m n l ; S / _ \ mnl / \ . > mnl / \ /» « \ C M v ; a 0 ( r ) = c M , ; a / 3 ( r ) + \ x , ; a / 3 ( r ) > ( A ' 1 a ) where fo r m n l ^ O X m n l „ ( r ) = / 3 u m n l „ ( r ) , f o r r>d . H uv, a/3 ' a/3 0 , f o r r<d „ ' a/3 ( A . 1 b ) 000 and d f l £ is the h a r d - s p h e r e contac t d i s t a n c e . W e note that \ ) Q . a ^ r ^ ' s de f i ned b e l o w . In e q . (A.1b) w e a s s u m e that al l po ten t ia l t e rms that are not mu l t i po la r are s h o r t - r a n g e d , and there fo re are not inc luded in X m n ^ " „ ( r ) . 3 • uv,ap Firs t let us c o n s i d e r a s y s t e m w h i c h con ta ins no charged s p e c i e s . For th is s y s t e m w e need to cons i de r on l y t e rms f o r wh i ch 1 ^2 . It f o l l o w s f r o m e q s . (2.10b) and (3.38a) that none o f the Hankel t r a n s f o r m s , c u u . af£k), w i l l ~ 1 1 2 have d ivergent behav iour at s m a l l k. In f ac t , al l but ^ Q Q . ^ k ) w i l l go to - 288 -zero as k—>0. Fu r the rmore , the integral t r a n s f o r m s o f the func t i ons X^"^ a^(r) can be p e r f o r m e d a n a l y t i c a l l y , and it can be s h o w n [103] .that fo r r^d „ X „(r ) = 0 . Hence , the po ten t ia l te rm makes no con t r i bu t i on to ap uv;ap ' ^ - m n l / j f F o r ^ m n l , j e v a | u a t e s t o a cons tan t t e rm uv;aP a/3 a/3' uv;a,p w h i c h can then be added to the hat t r a n s f o r m o f the s h o r t - r a n g e c to ob ta in the fu l l a 1 ™ * „ ( r ) . For a s y s t e m con ta in ing both ion ic and d ipo la r s p e c i e s , t w o add i t i ona l t e rms require s p e c i a l t rea tment , n a m e l y c 00-<x/3^ r "^ a n c ' C00^ap^ r^" ^ e P ° ' n t out that there are n o r m a l l y no integral t r a n s f o r m s a s s o c i a t e d w i t h these t e r m s . It can be s h o w n [61,135] that as k—>0y C Q Q ] k) w i l l d i ve rge as 1/k (of c o u r s e , th is is o n l y the case w h e n par t i c le a has a charge and par t i c le j3 has a d i p o l e m o m e n t ) . W e take advantage of the l inear i ty o f Four ier t r a n s f o r m s and t r a n s f o r m o n l y the s h o r t - r a n g e c n u m e r i c a l l y ; O^OIQ.^ 1") C A N ' 3 E t r a n s f o r m e d a n a l y t i c a l l y . U s i n g e q s . (2.10b) and (A.1b) w e wr i t e that $0;affr) = A / r 2 ' ( A ' 2 ) where A w i l l depend upon the charge and d ipo le m o m e n t o f pa r t i c les a and /3, r e s p e c t i v e l y . Then inser t ing e q . (A.2) into e q . (2.34b) and eva lua t ing the integral y i e l d s * o J ! w k ) = 7 % s i n ( k v * { a - 3 ) a/3 k where i=/-T. W e note that ^QO'a/3^^ ^ a s t' i e c o r r e c t [1^5] s m a l l k behav iou r . F r o m the de f i n i t i on o f ^ ^ ( 1 2 ) (cf. e q . (2.38)) it c lea r l y f o l l o w s that *?00 • a/3^  ^ ^ w ' " a ' s o c'' v e r9 e a s ^ a s ^— > ®- F ° r t u n a t e l y , th is d ivergent behav iour w i l l p o s e no numer ica l p r o b l e m s in the back Hankel t r a n s f o r m (cf. e q . (2.46b)) because the d ivergent t e rms in the integral w i l l cance l exac t l y at - 289 -s m a l l k. W h e n bo th pa r t i c l es a and /3 are c h a r g e d , it can be s h o w n [61,135] that C Q Q ^ a ^ ( k ) d i v e r g e s as 1/k 2 as k—>0. T h u s , w e again sp l i t c o o ^ a / / r ^ ' n t ° a s h o r t - r a n g e f u n c t i o n , wh i ch can be e a s i l y Four ie r t r a n s f o r m e d n u m e r i c a l l y , and a l o n g - r a n g e f u n c t i o n , ^ o o ^ a / / r ^ ' w n o s e t r a n s f o r m can be d e r i v e d a n a l y t i c a l l y . F o l l o w i n g p r e v i o u s w o r k e r s [ 188 -190 ] , w e de f i ne >000 . v _ a q a q($ , . _ - a r , X 0 0 ; a / 5 ( r ) " P " T ~ ( 1 6 ] ' ( A . 4 a ) wh i ch can be Four ie r t r a n s f o r m e d ana l y t i ca l l y to g ive 2 miaif" - * " v % [ 7 7 ? 7 7 7 ] • < A - 4 B ) where q ^ and q ^ are the charges on the i ons . The cons tan t a must be c h o s e n w i th care s o as not to cause ^ Q o ^ a p ^ r ^ t C > b e c o m e , a r 9 e at s m a l l r, but it must a l so a l l o w A g Q / ^ C r ) —> / J q ^ q ^ / r at s o m e reasonab le va lue of r. It is again the c a s e that * ? Q t V a / / k ^ w ' " n a v e the s a m e d ivergent behav iour as c22^ 0( k): w e note that at f i n i te concen t ra t i ons h29^  J r ) U U ; a p U U ; a p must be s c r e e n e d , and hence its Four ier t r a n s f o r m w i l l not con t r i bu te . Thus , w e de f i ne the short-range f unc t i on ~ 0 0 0 ; S / - x ~ 0 0 0 * r O O O K * wh i ch can be Four ie r t r a n s f o r m e d numer i ca l l y w i thou t d i f f i c u l t y . For an i on i c s y s t e m at in f in i te d i l u t i on , the fac t that the f u n c t i o n s mn "1 ^uv a^T' a r e n ° l o n 9 e r s c reened requ i res that spec ia l a t ten t ion be pa id to cer ta in f u n c t i o n s . F i r s t , the l o n g - r a n g e ta i l o f h ( ) ( V a / / r ^ must be co r rec ted fo r the k = 0 po in t w h i c h w a s ignored in the numer i ca l in tegra t ion {i.e., by the FFT [126] ) o f * ? § 0 ' a j ? k * - A ' S ° ' t h e l o n 9 - r a n 9 e t a i l o f h 0 0 ' a / / r ^ ' W h i c h - 290 -has a 1/r dependence at large r [61], w i l l a f f ec t both CQQ . A ^ R ^ ANC' 0 2 2 C Q Q , a ^ ( r ) . W i t h i n the HNC t h e o r y , both f unc t i ons have a dependence upon [ h.QQ ] r ) ] 2 at large r. The resu l t i ng 1/r* ta i l s are t runcated dur ing a numer i ca l ca l cu la t i on (both in the c losu re ca l cu la t i on and in the Hankel t r a n s f o r m ) , but ana ly t i ca l e x p r e s s i o n s w h i c h cor rec t f o r th is t runca t ion can be d e r i v e d . W e remark that these c o r r e c t i o n te rms are s m a l l and re l a t i ve l y un impor tan t (except when de te rm in ing C ^ s ) f o r un iva lent i o n s , but can b e c o m e qui te s ign i f i can t fo r larger cha rges . - 291 -APPENDIX B REPRESENTATIVE EXAMPLES OF EXPONENTIAL INTEGRALS Fi rs t w e w i l l cons i de r an integral o f the genera l f o r m F = a H L (1+zcr)^ 'dr , (B.1) where a is s o m e cons tan t . It is the behav iour o f F as K—> 0 that is o f in terest here. It is conven ien t to wr i te e q . (B.1) in the f o r m where F = a [ l n + I, + I 2] , 0 ~ J 2 u d rz (B.2a) (B.2b) and -2nr •dr d r 2 0 0 -7KT I, = K j e dr z d (B.2c) (B.2d) Then us ing s tandard tab les o f in tegra ls [165], it is p o s s i b l e to s h o w that -2/cd -2/cr and I* = - 2K S d dr K -2/cd 2 e (B.3a) (B.3b) W e can then insert e q s . (B.2c) and (B.3) into e q . (B.2a) to ob ta in F = a -2KA (B.4) and expand ing the exponen t ia l y i e l d s - 292 -F = a | g - | / c + / c 2 d - - - - J . ( B . 5 ) T h e r e f o r e , in the l imi t K—>0. w e have that F = a [a " I"] ' ( B . 6 ) W e w i l l n o w examine the K—>0 dependence of an integral o f the general f o r m R+d F = a f ( 1+/cr)-R-d -/cr (r 2+R 2-d 2) 2 - ( 2 r R ) 2 dr , ( B . 7 ) where again a is s o m e cons tan t . Equa t i on (B.8) can then be wr i t ten as F = a where R + d 3 -/cr I, = K J r J e K t dr , 6 R-d - J • ( B . 8 a ) ( B . 8 b ) R+d 2 -KT I o = / r" e dr , ^ R-d ( B . 8 c ) 9 , R+d _ r I. = -2 K(R^+d^) / r e K r d r 1 R-d ( B . 8 d ) 0 O R + d - r-I Q = -2(R^+d Z) J e K r dr , R-d ( B . 8 e ) and 7 9 0 R + d P~Kr I_1 = *(R 2-d 2) 2 / £ dr R-d r .2 j 2 , 2 R + d e - K r I_2 = (R*-d*r J ^ - 2 — dr . R-d r ( B . 8 f ) ( B . 8 g ) Us ing s tandard f o r m s for the in tegra ls in e q s . (B .8b-e ) and (B.8g) as g i ven in tab les [165], w e can rearrange e q . (B.8a) to ob ta in - 2 9 3 -F = a -Kt\( 3 4 2 8 r 8 x L K K + 2 ( R 2 + d 2 ) (v+-) - l ( R 2 - d 2 ) 2 K L R + d jR-d ( B . 9 ) If w e then eva luate e q . (B.9) at i ts l i m i t s , w e f ind that F = a e - K R - Kd r~ H J _ 8 R _ 8 d 8 R d ] I 3 2 2 „ J L/C K K K J - e ./cd r _ [^8 _ 8 R 8 d I 3 2 2 L /C K K 8Rd' K J ( B . 1 0 ) w h i c h w e can wr i te as F = a e -KR 8 + 8 R 3 2 K K • /cd - /cd e - e ] - [ 8 d + 8 R d j j - e K d + e - K d j ( B . 1 1 ) N o w , it is the K—> 0 l im i t i ng behav iour o f e q . (B.11) w e w i s h to de te rm ine . By expand ing the e x p o n e n t i a l s , w e can s h o w that at s m a l l K and /cd - / c d e - e /cd , - K d e + e 2/cd. + -j/c d 2 + « 2 d 2 ( B . 1 2 a ) ( B . 1 2 b ) Inser t ing e q s . (B.12) in to e q . (B.11) and s i m p l i f y i n g y i e l ds F = ( l + / c R ) e k R 3 d J ( B . 1 3 ) - 294 -APPENDIX C TRANSFORMATION OF THE ROTATIONAL INVARIANT <1>123(12) A s in the text , we w i l l make use of the no ta t i on o f S te inhauser and Ber tagno l l i [166] in our d i s c u s s i o n . U s i n g the ro ta t ion matr ix g i ven by e q . (39) of Ref . 166, it is e a s y to s h o w that in the (x, y , z ) f r ame of re fe rence (see Figure 6(b)) x 2 = *2 " z 2 = c o s a 2 c o s ^COS72 - s ina2s in72 sina2 c o s ^ COS72 + cosa2Sin72 - s i n / 3 2 c o s 7 2 - c o s c ^ c o s ^ sin72 _ s i n a . 2 C O S 7 2 - s i n a 2 c o s / ? 2 s i n 7 2 + c o s a 2 c o s 7 2 s i n ^2 s i n 72 s i n p ^ c o s c ^ sin/L, s i n a ^ cos/3 0 (C.1a) (C.1b) (C .1c ) In the ion re fe rence f r ame w e take advantage of the res t r i c t i on on as g iven by e q . (4.66a) and wr i te - I X 2 = - s i n a 2 I sin72 I cosa^ s i n 7 2 I -cos *2 7 2 - s i n a ^ cos72^ c o s c ^ C 0 S 7 2 ^ (C.2a) s i n ?2 (C.2b) - 295 -c o s < z 1  z 2 s i n i ( C . 2 c ) In order to take the unit v e c t o r s , a r | d 4 f r o m the ion f rame of re fe rence to the (x, y, z ) re fe rence f r a m e , w e mu l t i p l y the unit v e c t o r s by the ro ta t i on mat r ix R, as g i ven by e q . (4.69), w h i c h y i e l d s x~ = - s i n c u s in7^ cosco + c o s *2 sincj ° 2 S i n 7 2 c o s a ^ s i n 7 2 - s i n a 2 I s i n 7 2 I sinco - c o s 7 2 c o s 6 j ( C . 3 a ) * 2 - s i n a 2 c o s 7 2 c o s u c o s a 2 c o s 7 2 -s incuf c o s 7 2 sincj s i n 7 2 sinco s i n 7 2 COSCJ ( C . 3 b ) c o s °2 sm< c o s "2 c o s w sina) ( C . 3 c ) W e can n o w ob ta in e x p r e s s i o n s re la t ing the t w o se t s o f Euler ang les a s s o c i a t e d w i t h the t w o d i f fe ren t f r a m e s of re fe rence by equat ing c o m p o n e n t s of the t w o f o r m s fo r x 2 > y 2 and z 2 . F r o m e q s . (C.1c) and (C.3c) w e have that s i n p\, = and c o s t COSCJ sin< coscu, I s i n ° 2 cos/3 2 = cosc^ sincj ( C . 4 a ) ( C . 4 b ) S i m i l a r l y , we use the z - c o m p o n e n t s of x 2 and y 2 > as g i ven by e q s . (C.1a), (C.3a) and (C.1b), (C.3b), to ob ta in - 296 -( C . 5 a ) and r j COS7? COSC0 - i cos7 2 = sint^ s i n 7 2 s ino + = L s i n a 2 r j s i n 7 2 c o s a ) -i s i n 7 2 = sint^ -cos7 2 sinco + = . ( C . 5 b ) L sinc^ -I Then us ing the t r i gonome t r i c iden t i t ies [165] 2 2 c o s 2 7 = c o s 7 - s in 7 , ( C . 6 a ) c o s 2 7 = 2sin7COS7 , ( C . 6 b ) and the requ i rement that s i n 2 7 2 = 0 , wh i ch f o l l o w s f r o m e q . (4.66b), it can be s h o w n that I . 2 c o s 2 7 2 = c o s 2 7 2 s i n 2 0 -1 C O S O) . 2 _ N 5—f- - s i n w ( C . 7 a ) s i n -I and s i n 2 7 , = - 2 c o s 2 7 9 I s i n 2 o 9 [ s ino ;cosJ | # ( c > ? b ) ^ L sine- -I If one subs t i t u tes e q s . (C.4) and (C.7) into e q . (4.64b) and takes =0 , one can ob ta in the resul t $ 1 2 3 (12) = V% 2 2 1 2 I 3 c o s / 3 1 ( c o s CJ - s i n s i n t o ) c o s 2 7 2 I T I . p 3 1 1 1 ^ 2 + 2 s i n / 3 1 c o s 2 7 2 c o s c ^ c o s c ^ s i n w j — c o s co L Ls inc^ o "I o 1 ( C . 8 ) I . 2 1 . 2 - s inc^sinc^sinco + 2 s i n cu, sincocosco | From e q . (C.4a) we have the iden t i t i es s i n c u c o s c u C O S C J c o s c ^ = : j ( C . 9 a ) s i n a ^ and - 297 -coseu sincip s i n c u = ^ — . ( C . 9 b ) C O S C I 2 c o s c o U s i n g these r e l a t i o n s h i p s , together w i th s i m p l e t r i g o n o m e t r i c iden t i t ies [165], w e s i m p l i f y e q . (C.8) wh ich can even tua l l y be wr i t t en as the e x p r e s s i o n fo r <^ ( 1 2 ) g i v e n in e q . (4.70). 

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