THE STRUCTURAL, THERMODYNAMIC AND DIELECTRIC PROPERTIES OF ELECTROLYTE SOLUTIONS: A THEORETICAL STUDY By PETER GERARD KUSALIK B.Sc, The University M.Sc, A The University THESIS SUBMITTED of of IN Lethbridge, 1981 British Columbia, 1984 PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E (Department of We accept to required THE UNIVERSITY conforming standard OF BRITISH COLUMBIA February, ® Peter Chemistry) this thesis as the STUDIES 1987 Gerard Kusalik, 1987 In presenting degree this at the thesis in University of freely available for reference copying of department publication this or of partial fulfilment of British I agree and study. by this his or her The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 that the representatives. may be It thesis for financial gain shall not Department of requirements I further agree thesis for scholarly purposes permission. DE-6(3/81) Columbia, the is an advanced Library shall make it that permission for extensive granted head by the understood be for that allowed without of my copying or my written - ii - ABSTRACT In traditional theories for electrolyte solutions the solvent is treated only as a dielectric continuum. A more complete theoretical picture of electrolyte solutions can be obtained by including the solvent as a true molecular s p e c i e s . In this thesis we report results for the structural, thermodynamic, and dielectric properties of model electrolyte solutions which explicitly include a water-like molecular solvent. The ions are modelled simply as charged hard spheres and only univalent ions are considered. The water-like multipole solvent is also treated as a hard sphere into which the low-order m o m e n t s . a n d polarizability tensor of water are included. The reference hypernetted-chain theory formalism of Kirkwood and Buff is used to study the model s y s t e m s . is employed to obtain general expressions relating the m i c r o s c o p i c correlation functions and the thermodynamic of electrolyte The properties solutions without restricting the nature of the solvent. The low concentration limiting behaviour of these expressions is examined and compared with the macroscopic results determined through Debye-HiJckel The influence of solvent polarizability is examined at two theoretical theory. levels. The more detailed approach, the R-dependent mean field theory, allows us to consider the average local electric field experienced by a solvent particle as a function of its separation from an ion and is shown to have an effect the limiting laws of some thermodynamic properties. upon 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 well with experiment. dilution M o d e l aqueous electrolyte solutions are studied both at and at finite concentration, but only at 25°C. dielectric constants of these solutions are qualitatively of experiment. A remarkable diversity of infinite The equilibrium consistent with those behaviour is obtained for our solutions by s i m p l y varying the hard-sphere diameters of the ions. model In many cases the behaviour observed for thermodynamic quantities is in accord with experiment. The i o n - i o n , ion-solvent and s o l v e n t - s o l v e n t correlation of the solutions are examined in detail, revealing a wealth of information. functions structural Ionic solvation is generally found to be very sensitive to details of the interactions within the s y s t e m . the - iii - TABLE OF CONTENTS ABSTRACT ii T A B L E OF C O N T E N T S iii LIST OF T A B L E S v LIST OF FIGURES vi ACKNOWLEDGEMENTS ix CHAPTER I. INTRODUCTION CHAPTER 1 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 38 6. Averages and Potentials of CHAPTER Numerical Solution Mean Force 41 III. THERMODYNAMIC THEORY FOR ELECTROLYTE SOLUTIONS 51 1. Introduction 51 2. General Expressions 52 3. Limiting Behaviour 63 CHAPTER IV. M E A N FIELD THEORIES FOR POLARIZABLE PARTICLES 74 1. Introduction 74 2. The S e l f - C o n s i s t e n t Mean Field Theory 76 3. The R-Dependent 83 Mean Field Theory - CHAPTER iv - V. RESULTS FOR W A T E R - L I K E MODELS 108 1. Introduction 108 2. Choice of 110 3. Results for Hard-Sphere 4. Results for Soft CHAPTER Basis Set Models 114 Models 139 VI. RESULTS FOR MODEL A Q U E O U S ELECTROLYTE SOLUTIONS 150 1. Introduction 150 2. Dielectric 156 3. Thermodynamic 4. Structural 5. Effects 6. Results Obtained CHAPTER of Properties Properties 162 Properties Including 202 the RDMF 238 Employing Different Solvents 254 VII. CONCLUSIONS 272 LIST 278 OF REFERENCES APPENDIX A. T R E A T M E N T OF POTENTIAL TERMS IN APPENDIX c(12) B. REPRESENTATIVE E X A M P L E S OF EXPONENTIAL APPENDIX 287 INTEGRALS 291 C. T R A N S F O R M A T I O N OF THE R O T A T I O N A L INVARIANT $ 1 2 3 (12) .. 294 - V - LIST OF TABLES I. II. Reduced ion Experimental pressures III. Numbers IV. Projection diameters, densities examined of terms of in unique Maximum in VI. VII. VIII. IX. X. XI. the HNC binary Basis dependence Model for aqueous Average Individual per the study 23 temperatures and study in 109 terms required in HNC basis sets . n projection 111 iteration of on e , the an FPS average 164 array energies and processor g(r=d) . . . . 112 113 139 R electrolyte solutions energies partial 110 111 Ug ( r ) ion-solvent ionic for this product required Parameters in =2 basis sets max n o n - z e r o terms for any given of C P U time set this included numbers water projection 1 V. d . * , used molar per studied 152 ion at infinite dilution volumes at infinite dilution . 268 270 - 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 3. The angle 4. A n illustration of the method used in determining 5. A n illustration 6. A n illustration 7. The mean dipole moment of water-like particles as a function of temperature and pressure 115 The dielectric constants of water and of water-like functions of temperature and pressure 118 8. 9. 6- for (a) a positive 21 ion and (b) a negative ion . . . . 46 <AE^j(R)> . . . 87 of the method used in determining <AE^p(R)> ... 92 of the method used in determining <AE^Q(R)> ... 98 models as Radial distribution functions for water-like fluids at 2 5 ° C 121 10. The projection h Q Q ( r ) 124 11. The projection h g Q ( r ) 126 12. The projection h ^ d r ) 128 0 2 1 23 13. The projection (r) 130 14. The projection h Q ( r ) 132 15. The projection (r) 134 16. The projection (r) 136 17. Soft 18. Radial distribution functions for soft water-like models at 2 5 ° C and m *=2.75 e Radial distribution functions of water and of water-like fluids at 2 5 ° C and m *=2.75 e Structure factors of water and of water-like fluids at 2 5 ° C and m *=2.75 e The concentration dependence of Y 19. 20. 21. 22. 2 potentials 4 at 2 5 ° C Comparing theoretical and experimental values for the dielectric constant of aqueous KCI solutions 140 142 145 147 ... 154 157 23. 24. The dielectric constants electrolyte solutions as Average root 25. 26. 27. total ion-ion of real and functions of energies per of m o d e l aqueous concentration ion as functions 160 of square concentration 163 Average ion-dipoie root concentration energies energies average 29. The square of concentration as functions of square energies the per ion as functions of square 168 Average solvent-solvent concentration Total ion 165 Average ion-solvent root concentration 28. per energies per solvent as functions of 170 as functions Debye of screening concentration parameter as a 173 function of 176 30. The product 31. Cjg as a function of square root concentration 182 32. G. as a function of square root concentration 184 33. G as a function of salt 34. Isothermal 35. Partial molar concentration 36. +s ss P 2 ^ _ as volume of as l n y 38. Solvent-solvent solvent and of <cos0 a of square root concentration concentration as the of a 187 function solvent of as a concentration function 190 of the solute as a function of of square root concentration 199 radial d i s t r i b u t i o n f u n c t i o n s o f the several model electrolyte solutions for square the pure solvent and for model pure 203 electrolyte solutions 206 40. Ion-solvent 41. <cos#. 42. Ion-solvent 43. <cos0. 44. Potentials of radial (r)> s 179 195 function (r)> s s function 193 Partial molar volume root concentration + a compressibility 37. 39. + at distribution infinite radial (r)> of oppositely of functions infinite dilution dilution distribution for Cl" charged at ions infinite 208 214 216 . . force . . . . 211 function Ch mean at dilution for several pairs 219 - viii - 45. Concentration dependence of g _(r) f o r KCI 223 46. Concentration dependence of g _(r) for M'l 225 47. Potentials mean at infinite of like of force dilution for several pairs 228 g^ ^( r ) for several ions 49. ^( r ) for C h for several 50. CI-/CI" 51. Additional partial structure ion-solvent i o n at + + ions 48. Na + in m o d e l model factors for interaction infinite dilution electrolyte solutions electrolyte solutions model term NaCl due to at solutions Ap(r) 236 240 of the RDMF upon 53. Effect of the RDMF upon 54. Effect of the RDMF upon r) 55. Effect of the RDMF upon w. . ( r ) 56. Effect of the R D M F upon C 57. 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 polarizable and unpolarizable tetrahedral solvents I o n - s o l v e n t radial 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 the tetrahedral a n d C 2 o c t u p o l e s o l v e n t s ^( r ) w^(r) I f o r NaCl 242 for 244 MBr for N a for M 247 + 249 4 251 g for V 59. 60. <cos0. ( )> infinite octupofe solvents t s Potentials of C2 octupole V 61. a dilution m e a n f o r c e at and tetrahedral Like-ion potentials a n d F- in t h e C 9 231 for a Effect r . 234 52. 58. 1.0M f o r the tetrahedral andC 2 258 V ^ infinite dilution solvents 255 261 f o r L i F in t h e 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 octupole and tetrahedral s o l v e n t s 263 f o r Li + 266 ACKNOWLEDGEMENTS I wish to thank my academic advisor, Dr. G.N. Patey, for his guidance and support throughout the the University of British Columbia and the Research Council of grateful to past six years, and the Chemistry Department Canada (NSERC) for their my family, particularly encouragement. Most Natural of all I would like to Sciences and Engineering financial support. my parents, for thank tremendous sacrifices she has made during the of all their I am also help and my wife, Sheila, for preparation of the this thesis. - 1 - CHAPTER I INTRODUCTION The study of electrolyte solutions, has been one of solutions, particularly the most active areas of aqueous electrolyte physical chemistry. The considerable attention received by aqueous electrolyte solutions appears to have two an innate in principal motivations. water, or in this The first arises from case, in water as a solvent. Of course, the water as a chemical substance cannot be overstated. only naturally properties [2,5] occurring inorganic liquid on earth [1], but [2-4] and the fact all life on this planet. role charged species play electrochemical processes [5]. It in many of is water the only its unusual physical as a solvent The second arises chemical reactions is little wonder solutions have received so much A n electrolyte importance that all biological processes use it make water essential to the central Not interest from and then that aqueous electrolyte attention. 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. is water. A n electrolyte conductance [3]. solution is often In this study we which can be assumed to solutions have been the great deal of experimental dielectric remain ionize) in solution. subjects of macroscopic properties to interpret and the rather Unfortunately, details of poorly understood [5-12]. the of measurements are somewhat are difficult has proven know that the solute exists as a [13-16] have direct in solution (i.e., the i o n - s o l v e n t , and solvent-solvent distribution these this data underlying microscopic properties We the microscopic structure apparent dependence upon salt much of free ionic species in solution, but only recently measurements and a has been such as density, vapour pressure, and [6-9]. those Aqueous numerous experiments constant have been measured and their concentration determined difficult consider only strong electrolytes, i.e., data on their Properties solvent defined as one having a high fully dissociate (or electrolyte accumulated [6-8]. For aqueous solutions the to perform. ion-ion, functions) been possible. Moreover, the results ambiguous [13,16] and provide only limited However, obtained information. Thus, - 2 - the details of how they the microscopic structure relate to needed then solutions macroscopic properties is a more in order to is to this point of complete aqueous electrolyte remain solutions and poorly understood. microscopic understanding of more fully understand the What is electrolyte macroscopic observations. It that this thesis is addressed. th The study of field of electrolyte [17] were intrigued become known by the conductors but not, of fact experimental colligative results which clearly results were the distinct interpreted solute. In At those of data Many by Planck [19] present it gained general attempts develop equilibrium electrolyte interactions properties [21] of to be and Hiickel theory of made dilute in very [22] electrolyte during the derive Debye and Huckel for of who electrolyte in must published first [21] with experiment. [23] solutions became, and probably remains, the century of the many ion-ion of the results were of as landmarks. structure of to describe The theories stand out today the equilibrium century. this laws for These theoretical in support of a this decades of limiting ionization published his theory idea, but with the first These dissociation in The long-range nature solutions. Onsager [18] dynamical theories, to behaviour. good agreement and of Hoff as a possible indication of day theory theories, and later it possible to van't acceptance by the turn of were made solutions and their transported non-conducting solutions. the time this seemed like a radical wealth of electrochemistry. and concluded that there 1887 solution is usually credited to Arrhenius [20] 1887. in that indicate that conducting solutions possess from However, the century course, in electronic conductors. Clausius charged particles present. properties as 19 that matter was noted that ionic solutions obey Ohm's law be electrically of solutions began during the chemistry which has now Investigators [17] electrolyte found Debye The electrolyte standard approach [3,5,6] used in discussing or describing them. Much of solutions the work done on the equilibrium [6,21,24,25] in the 50 years after the was concerned with justifying theory advent and improving the theory basic approach to electrolyte which the is treated simply as a dielectric essentially solvent unchanged. showed that the It solvent was of solutions used by Debye actually need not McMillan be explicitly of electrolyte Debye-Hiickel itself. theory However, the and Huckel [22], in continuum, remains and Mayer [26] who formally considered as a molecular - 3 - species if an effective solvent averaged ion-ion interaction potential Of course, implicit in the above statement is used. 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) J 1 = u?.(r) 1 J + er , (1.1) 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) ion-ion interaction. is the short-range 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 , + not currently accessible through computer simulation. V^, etc.) are 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 molecular properties of the water molecule with no freely parameters. (measured) adjustable 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]. integral equation theories can be written as two coupled equations. Most 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. closures exist. At present only approximate 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] anisotropic potentials. possible in the investigation of systems possessing 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 as well as hard ellipsoids and spherocylinders [99,100]. [98] 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 C 2 V symmetry [4], we have restricted ourselves to model systems in which all species have at least C 2 V symmetry. consider the simplifications that result from this restriction. We shall 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 multicomponent system employing the rotational outlined in section 2. invariant language [68,101-103] 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 theory. [104] - 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 ,X ,...,X ) 1 2 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 u(X ,..-,X ) —i —w = 1 N .S.u >(X. ,X.) i<j —i —j (2 as an expansion + . 5 u i<j<k ( 3 ) (X. , X . , X . ) + ~ ~J ~ 1 , (2.2) 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. agreed that for most liquids the pair interaction potential [27,33]. It is generally is the dominant term In most statistical mechanical studies higher order terms are usually neglected or the pair potential higher order terms [27,33]. is modified in an attempt to take into account 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 (l2) e , (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. A point charge model uses a small set of discrete charges in place of the continuous >ntir charge distribution of the molecule of interest. Hence u (12) is given by •q (^) i ir(i2) = qA2) (2.4) z. i<3 th where q. is the i charge on particle 1, cjj is the j and r^ is the separation between points i and j. th charge on particle 2, 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 (12) e = Z mnl 1 U av Vn,l|_2m! f mnl 2nlJ (2.5) 1+1 v-v 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 invariants, ^ " ^ ( f l , , 0 2 , f ) , are defined below. rotational 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 a a a a (rotating) frame of reference. The generalized spherical harmonic [72,108] v "So'*-*' 7 L2m+1J m L(m+ hu ) !J where m ^ ( 0 , 0 ) is a spherical harmonic [106] and P ^ c o s f l ) (2.6b) 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. ^1 i&2 f ? ) , which fulfill the requirement (2.5), are known as rotational ^>,, a , r ) . t™\zF 2 where m , n , l invariants. J a each particle, f and of rotational invariance in eq. They are defined by [68,101] ^ « , ) H g j O , ) ^ » are positive integers, ^J>&) spherical harmonic [108], 0= (8, c6,v//) The functions > s (2.7) again a generalized Wigner is the set of Euler angles [87,108] for is the orientation of the vector from particle is the usual 3-j , symbol [109]. The orientations R 1 to particle 2 1 f £2 rf a r e t n e 2 sets of angles of rotation from the lab fixed frame to the molecular fixed . frame [87]. The sum in eq. (2.7) which the 3-j [109] is only over those values of a,P,y for symbol evaluates to a non-zero value. The triangle condition 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\ In eq. (2.7) < and m | i>| < n . f -'- can.be any non-zero constant. mn use of two different (2.8b) In this thesis we will make definitions: f™ 1 = I/ 0 K f m n l 0 , (2.9a) 0' = [(2m+1)(2n+1)J* . (2.9b) We will alternate between eqs. (2.9a) and (2.9b) when it is convenient to do so. - The rotational 13 - 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. rotational invariant basis set. Equation (2.5) is an expansion in this 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 > r ). Using f ^ as given by eq. (2.9b), we mn 2 rewrite eq. (2.5) as u (12) e = L nv u " (r) m mnl rt,,fl ,f) X M»> (2.10a) , 2 MP where v2i+iH r MP v r y v " i i Q£ Qri 1+1 °m+n,lL(2m+1)!(2n+1)jj We can expand eq. (2.3) in a similar manner. (2.10b) * r 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 u (12) unchanged. e It has been shown [101,110,111] that this condition requires that the invariant expansion coefficients satisfy u ^(r) m = (-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,M e = 0 F , (2.12a) ii) for linear symmetry in both particles H,v = 0 , (2.12b) - iii) for C 2v 14 - symmetry in both particles (2.12c) = even a,v and mnl u uv ( r ) ( Blum and Torruella [101] mnl = u' ±u±v ( (2.12d) also showed that since u (12) must be real then (2.13) Together eqs. (2.12c), (2.l2d) and (2.13) imply that at least for particles of C symmetry m+n+1 = even and u (r) must be real. (2.14) 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) expansion. is really a spherical tensor form [113] of the same In Cartesian notation the first four moments (n = 0,1,2,3) of a discrete charge distribution are [107,112,114] a a (2.15a) - 15 - u- = Ee (2.15b) ia = A s e_ (3r. 0:, i] 2 'a a - r 5 - .) r. (2.15c) 2 i a ]a a i ] and fl i jk " 2 2 e a[ 5 r - ia ja ka r r r (r, 5... + r . •a ia"jk where q is the net charge, V i 8... ' "ja^k + r. 5, ' "ka^ij'] (2.15d) ' 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 and r., x. Q 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. symmetry symmetry. However, this number can be greatly reduced by molecular [107,114]; of most importance here is the reduction under C 2 From Kielich [107] we have that for symmetry, the dipole, quadrupole and octupole moments have 1, 2 and 2 mutually components, respectively. Thus, the dipole moment V independent is given by the scalar u and the quadrupole tensor has the form 0 0 0 0 0 0 XX 0 = 0 (2.16a) 0 yy zz where we require [107] that 0 xx +0 yy +0 (2.16b) zz To specify the octupole tensor it is sufficient to give & x x z , ^y a y 2 n d ^ z z z 16 - where 0 xxz +0 yyz +0 zzz - =0. The Cartesian representation is the form most often used in the literature to specify the multipole moments of specific molecules. study it is convenient to work with multipole In this 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] 1 J and the relationships z = r COS0 a K and x + i y =. r s i n © e ^. a a a 1 a J It is then easy to show that in general Q~0 = Q , Q° = M Q° = 0 QJ and , 2 , 2 2 - f ( 0 x ± iu ) x z + i 0 Q (2.18a) , y ( 0 [107], we have immediately yz yy ± ) ( 2 i V ' symmetry where u= u =0 ( 2 v = ©„_= 0 ' 2 - l 8 b ) 1 8 c ) =0 from eqs. (2.18) that QJ = Q ^ 2 (2.17c) ' ^f xx- 2 If we restrict ourselves to C„ Q (2.17b) = j ^ U 1 *V Qi (2.17a) 1 = = 0 , (2.19a) = 0 (2.19b) and Q2 " Q 2 = ^ 2 ( 0 xx" V In a similar fashion we can show that for C 2 v ' ( 2 ' 1 9 C ) symmetry the components of the octupole moment are given by Q° = Q = Q3 3 Q3 = fi zzz 1 , (2.20a) = 0 , - ^ ( n (2.20b) xxz- ° y y 8 > ( 2 - ° 2 C ) - and =1 - 3 i 3 17 - =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 C 2 symmetry we have the required 1, 2 and 2 independent V 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 u^.2) where $QQ ( 0 , 2 , f ) = 1 r 1 ions. 2 (2.21 ) = u? r) j ( is understood and q., q^ are the charges of the 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 B = 10~ esu. c m . and the molecular axis system is defined as it appears in X X 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 ^ fl = -1.34F, where F = 10 esu. c m . Their calculated values of the dipole zzz ' and quadrupole moments are in reasonable agreement (within 8%) of the x x z z r - 18 - Figure 1. Molecular axis system for the water molecule. The stars indicate the atomic centres. - measured values. 19 - 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 reports multipole moments up to and including the n=6 [117] 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 0 s 0 0 0 0 by setting © z z (2.22) 0 0 to zero in eq. (2.16). Such a quadrupole tensor is totally specified by the single parameter 0 , which we shall refer to as a square g quadrupole moment (it is also known as a tetrahedral quadrupole moment - [67,72]). 20 - 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. addition to eqs. (2.12c) and (2.12d), we require that (M + v + 21) MOD In this case, in [72] 4=0 (2.23) 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, U ^( r ) , FL between two particles a and /3 is given by [27,33] °°; r<d 0; r>d a/3 a/5 (2.24a) - Figure 2. 21 - A charge distribution possessing (a) a square quadrupole and (b) a dipole and a square quadrupole. - where , d d with d a-3 = J L 22 , , + d T~ - f l > l being the diameter of particle a. f l ( u a ^( r ) ' 2 4 b ) It is clear from eq. (2.24a) that the hard-sphere interaction is a purely repulsive potential. HS with 2 In approximating u*(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. validity of these assumptions and their influence upon properties of The 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 s structure as measured by diffraction experiments 0-0 [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 Morris [124] [80,81], we have chosen to use the radii of as determined in this manner. the ion diameters used in this study. Table I summarizes the values of They are expressed as reduced ion diameters (i.e., in terms of solvent diameters), d.*= dj/d where d =2.8A. s The values have also been rounded to the nearest 0.04 to accommodate the grid width used in the numerical calculations (i.e., 0.02d ), as discussed below. s 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. ions, M' + and M , are almost twice the size of the solvent. + in size to tetraalkylammonium ion effects. The two other They are similar ions and will be useful in investigating large - TABLE I. 23 - Reduced ion diameters, d.*, used in this study. ION 0.68 0.84 1.08 1.16 1.28 1.44 1.80 1.96 1.00 Na\FRfcr.CICs ,Br+ Eq ,Eq+ 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 , ,J2 r£) = 9(12), 2 is of fundamental importance [27,33]. It is a measure of the probability density of finding particle 1 with orientation fl a t 2 t n e and particle 2 with orientation 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 over all orientations of particles 1 and 2. g(12) 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 (2.27) 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] (2.28) 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 - 25 have spherical symmetry, then the integration over 0 is trivial and eq. (2.28) 3 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 a and P fl (12) = Z b ir) mnl MP;a/3 m n l ^juf (a, nl 1 ,Q ,r) ' 2 (2.29a) 2 V' -Si^"W>C "- • 2) k 1(G n - J) <2 m nv 29b) where b „(1 2) can be c „( 1 2) or h „ ( 1 2 ) , b „ ( 1 2 ) can be c ( 1 2) or ap ap ap ap ap 0 h ^ ( l 2 ) , and tP^Hsi ^ ,Q , r) a The and ^ " ( J 2 1 2 coefficients # r) 1 , f l , k) are defined by eq. (2.7). 2 are given by ; b ( 12) [^(o, ,n ,f)j* dfi,dn a/3 C;a/5 and (r) - - ^ o . ^ ^ ) the k-space projections, where j-^(kr) 2 [^n n ?)]*ao.do l f * ^l ^' ^. 5 c a : a r e 2 f t n e ' ( 2 ' 3 0 ) 2 Hankel transforms 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 • #W'> - £ -^-#r/.> CU c (r> m n (s) l as for 1 even, (2.32a) and c -mnl /„\ _ mnl , x r mnl o P s r -1 - 2 t + 3(x) r $ = 2 for 1 odd, (2.32b) O and P^(x) fc 1 x 2 1 i=0L i ! ( t - i ) ! , P°(r/s)]ds S Q in which the polynomials P^(x) + 1 x £ p £ ^ r e , , t p2 ;t .+ 1 ( x ) = 2 .1 / - 1 x I i = 0 L i ! (t-i)! are given by t + i + 3/2 — M t +i n (i + 1 / 2 ) ! J + 5/2) —— (i + 3 / 2 ) ! J , 2.33a , 2.33b for t>0 and P°(x) = PQ(X) = 0 , (2.33c) where we use the general definition of z! [128]. ~mnl (k) fo uvjaji ' Hankel transforms of c c a C}a^ n k) t n e n = wr e 4 7 r ( The k-space projections, j t t e n [102] as zeroth order (Fourier) and first order ' ( r ) . Explicitly we have L J v a r 2 ^0 ( k r ) C;a/J ( R ) D R F O R 1 E V E N ( 2 ' 3 4 A ) and CW k) 4 d • ( r 2 V k r ) w/ r ) d r ' - for 1odd (2 34b) where j (x) 0 and = (2.35a) - 27 - j,<x> - s i n x We note that transform expansions After obtains h transforms (2.34) c a n be (2.35b) computed using fast Fourier techniques. The (2.28). the cos x for performing c ^( 1 2 ) and fl the h ^( 1 2) can fl necessary angular then be integrations inserted and into eq. simplifying, one [102] m n l uv;ap ~mnl c ik) /, \ k) . A = z [z ^ 1 p Z v lal.iL i .(-u w m h c mil n 1 i ( k ) c i n l Jk)l ? , (2.36a) where 1 1,1 _ mni ' 2 z and is introduce , 1 the time to usual 6-j symbol the m sum define the n ; (k) x over = 1 is been unlike hard h core shown i l f 1 i n l z l . l , l f m 1 [109]. n Following 1 H 0 Blum l . l i l ) 0 (2.36b) 0 [102] we now „(12) models. [102] Z ( ) c |m-n| to m from n 1 m n (k) l m+n. (2 It is convenient at 37) this function -(12) 'ap for m L2i + 1 j f ™ 7) which f [ 2 1 + 1] ; X'transform the C in w h i c h .xin+n+i 1 that or c = h (l2) ap - fl , , ( 1 2 ) , will For the choice we can rewrite c be of f -(12) ap a smooth mnl given e q . (2.36) as , (2.38) continuous by function e q . (2.9b), it has of r - N m n '* z (k) X p 7 7 28 - Nm i ; x (k) 1 z z 1 x Cin;x -uv; where N " j (k) nvjap m is the X x-transform of By comparing eqs. (2.36) and (2.39), we general OZ equation be easier to greatly invert into smaller m i ? X MCJ; ay (k)l J ,(k) (2.39) 7/3 T? „{k) livjap m n i as defined by eq. (2.37). / - i v / see that the x-transform independent sets of numerically. has split the equations which should A l s o , the numerical constant Z^ } ^" 2 has been 1 mm simplified. Again following whose + C ' nu>; ay CJ=-I (i,j) v , J / elements Blum [103], we define the matrices N are N . o( k) uvjap and C ' % ( k ) , respectively, where nv;ap ' '' X F I and C ,, X K i = m(m+1) + u + 1 (2.40a) j = n(n+1) + (2.40b) and Equations (2.40) follow of M allowed from for each m. v + 1 the fact that there are, in general, (2m + 1) values The general OZ equation (2.39) can then be written as N Z(-1) [N X X a/3 I ay y where p^ = p^I + C X 1 M U |p PC ayj y H is a diagonal matrix. (-1) X , X (2.41) 7/3 The elements of matrix P are given by f o r ri=m(m 1 + f o [j=m(m+1 r (2.42) otherwise Now, for an n-component —• Y whose (a,/3) elements £ and are N * ^ system, we introduce the matrices N and C*^, respectively. = diag(p A and C ~~ Y 1 ,p , • • • ,p ) 2 n We also define (2.43a) - 29 - P = diag(P,P,•••,P) . (2.43b) We then rewrite eq. (2.41) as N = X (-l) |g + C j j p P C x X X , X (2.44a) which we can rearrange to obtain N = C x X £ P C |^(-1 ) X 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 Once the values of N A m n matrix. ' ,,(k) x are known the projections Tj m n ik) ^ are easily obtained using the inverse x-transform 95 (k) m n l = (21+1) I ( m n 1 )N m n ; x (k) in which the sum over x is from -min(m,n) to min(m,n). inn 1 projections a ^ r ^ a r e (2 45) Finally, the found by inverting the transforms defined by eqs. (2.34) and (2.32). The inverse Hankel transforms are given by CU (r) C;a/3 (r) "y( = k 2 V CU f7-C ^1 k kr) (kr) C;a/3 (k)dk (k)dk '' for 1even f ° r 1 (2 ° d d 46a) ' ( 2 ' 4 6 b ) and the inverse hat transforms by CU - C U " 3 ^ C^»> I<^» = < '« M r ) P d 1 (2.47a) where P^(x) and P°(x) are defined by eqs. (2.33). (2.47b) - 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 which satisfy the general OZ ff ) r a equation (2.27) for a mixture of non-spherical particles. We should point out ~ V that in general C ~ and N Y A will be complex matrices and complex arithmetic must be used to solve eq. (2.44b). In principle, the expansions in rotational infinite sums. invariants (cf. eq. (2.29)) are 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. set of rotational However, the finite 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 ' 48b) 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 point we recall that we have also truncated the electrostatic pair potential s (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 + 1} {(X-D = X 2 2 . (2.49b) We eliminate these zero rows and columns from the matrices by defining new indices i and = m(m+1) + M- X = n(n+l) + v - x A j 2 + 1 (2.50a) + 1 (2.50b) A to replace those defined in eqs. (2.40). The new dimensionality is D X = { n max + 1 ) 2 " * 2 For an n-component system, the matrices N • ( and C x x 2 ' 5 1 ) will then have dimensions (nD )x(nD ). We define the indices A A g and h such that the (g h A x = D(a-1) + i = D (0-1) + j A x (2.52a) A (2.52b) x element of N * will be N m n ' x f l ( k ) , and similarly for C . Also, eq. (2.42) is now replaced by i„=m(m+1)-M"X +1 m(m+1 )+*z-x + 1 2 x X J X . (2.52c) otherwise Let us now consider the simplifications that arise if we restrict ourselves to systems in which all the particles have at least First we will examine the results of imposing eq. (2.14). symmetry. Immediately, the properties of the 3-j symbol allow us to rewrite the x-transform, eq. (2.37), as - 32 - pmn;x and ( k ) = z n m 1 ~mnl ( } (2 53) ( k ) 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 ~mn;-x/ k) and eq. similarly for N ^ ' ^ O O . = symbols that £mn;x ( k ) (2 5 4 b ) Clearly then, the OZ equation as expressed by (2.39) will be invariant to the sign of x for systems with C symmetry. 2 y 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 CW > - < "^x<x-; o>CS^ • k 21+ k> <2 - 55a) where the sum over x is from 0 to min(m,n) and a v X = r1 L for r |_2 for x=0 « x>0 , (2.55b) . Tun * y * mn * Y We have already pointed out that in general C uv,' ap„ ( k ) and Nuv,. ap„ ( k ) are complex. However, for m+n + l=even we find that there can be no mixing of w 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 '" ,(k). uv,ap m n X Now let us examine simplifications that result from imposing eqs. (2.12c) and (2.12d). We immediately obtain pn;x r n;x m ( k ) = ( k ) (2 56) Then using eq. (2.56) and eq. (2.12c), we can rewrite the general OZ equation, - 33 - as expressed by e q . (2.39), as N m n ; uu; (k) = 2 p ? i (-D a 7 i co=0 7 x [ > x a/3 i ; + C (k) x M^;a7 x C i n m i ; x (k)l /uco;a7 J 7/3(k) ; (2.57a) x op; where r 1 ,, a L = In matrix forms for CJ=0 , for CJ>0 . « L2 w notation, the general OZ equation of the matrices will dimensionality change. upon applying only D X = [ ( n max + l ) 2 (2.57b) rt (2.44b) w i l l be unchanged but the If w e examine the changes in e q . (2.12c), it is easily shown that + 1 ] / 2 ' £X 2 + 1]/2 • (2.58a) c To determine the dimensionality, D under both C 2 v symmetry conditions, w e observe that in e q . (2.57a) w e have essentially decreased the number of terms in the sum over co by a factor of 2 , except f o r the (n +1) terms f o r ' ' ^ max which co =0. For x=0 w e have v D 0 " = [ D 0 + [(n n max m a x + + 1 / ] 2) ]/4 2 2 . (2.58b) We then generalize to obtain D X = [ ( n max + 2 ) 2 ] In a similar fashion w e determine i / " Hx+D ]/4 4 2 (2.59a) that = [(m+D ]/4 + [ / 2 ] - [(x+D ]/4 + 1 (2.59b) = [(n+1) ]/4 (2.59c) 2 x . 2 M and £ We remark integer 2 + [v/2] - [(x+D ]/4 2 + 1 . that the d i v i s i o n s in e q s . (2.58) and (2.59) are to be taken as divisions. For s y s t e m s of C replace e q s . (2.50) and (2.51). 2 y s y m m e t r y , e q s . (2.59) can be used to However, the indices g and h will still be - 34 - C -C given by eqs. (2.52a) and (2.52b) with D _ j l A respectively. and j A replaced by D A l A and A 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). now be diagonal, given by P.c .c = for M=0 for M>0 r1 12 Matrix P will , (2.60) , •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) where /3=l/kT. = h(12) - lng(12) - /3u(12) + B(12) , (2.61) 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(l2) - lng(l2) - /3U(12) , (2.62a) includes contributions to c(12) from classes of diagrams known [33] chains, netted chains and bundles. 2 short-ranged, having a h(12) as simple It is believed [27,87] that B(12) is 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) - where rj(12) is defined in eq. (2.38). 0u(12)] ~ n<12) - 1 , (2.62b) - 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. [62] The LHNC 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 = -17 (1 2) W(12) + j3u(12) , (2.63) we obtain 9 c ( 1 2 = -h(l2) ) 9r Later we shall see that W(12) 9 W ( l 2 ) 9r - ^ U ( 1 2 ) 9r . (2.64) 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) = fh(l2) r 9 W ( 1 2 9r ) dr - 0u(12) . (2.65) 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] ^1,11,1, ^ t i n l 2 Mi i'l Uzv in which the numerical coefficient 2 2 2 = z have shown that pmnl^nnl mnl " v * v (2.66a) - P™ 1 f m ' n i l i r 2 l 2 m n and { • • • } (2^l)(2n l)(2hl)(-ir l t + 1 is the usual 9-j M l M 1 _ " M c,V 1 symbol [108]. [ M^i 36 - ( 0 0 O' U.bbbJ Thus we have 3W * ^" (r) _ ^ 3r m pninl » uv ^^^^ h m 1 n 1 l l { r M1P1 J R ) J n2 2 i d r A2P2 - ^ ^ ( 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) for 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 To r<d. 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 reference and perturbation parts. and correlation functions into Explicitly, we write X(12) = AX(12) + X (r) , R (2.69) where X(12) can be c(12), h(12), r?( 2), or u(12) and X (r) is the same function 1 R 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) + lng^r) - lng(12) - /3AU(12) + c ( r ) This method assumes that exact results for g ( r ) R obtained. R and c ( r ) R . (2.70) can be readily One can then proceed as before and derive the result - Ac m n l av (r) = j" J r 37 - m r l L 111,0,1, 2 9AW ( r ) -, 3r 2 . Bin - Ah^^r) UV dr 9r (2.71) - UuP^r) . The RHNC closure is easily generalized to mixtures. we see that there can be no coupling between different in the HNC equation. m 2 n 2 l 2 2 2 2 mnl , n Z m n l l/MP M f 9AW (r) h^r) ^ 3r m Clearly, in eq. (2.70) pairs of components 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 <3 p( r ) , are determined using the Lee-Levesque [130] generalization of the a Verlet-Weis [131] fit to Monte Carlo data. potential, we only apply eq. (2.71) for rxi^^ Again, because of the hard-sphere and use eq. (2.68) when Expanding eq. (2.68) in rotational invariants, we have that for c OO;a0 ( r ) " 1 T J 00;a/3 r< r < ^ ^. a &p a (2.72a) ( r ) 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 ) = 9af ( r ) e x p[ V A r ) " **W ] ' W r) 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<d ^ (cf. eqs. (2.72)). When a r>d 0, we can show [81] that the RLHNC is given by fl - 38 c 000 / x HS/ x [A 000 / 00;a/3 = %(f P[^00;a(S ( r ) r ) e x { r x O " ) 000 A / ^ 00;a/3 J u rP^Jr) (r) 1 - (2.74a) and A mnl _ ,000 / xf.mnl , v _ flAi] mnl " ^C;a^ r , »"] ) ( 2 ' 7 4 b ) 000 HS We point out that in the RLHNC theory, h^n j r ) = h I r ) for UU;ap ap for mnl*000. 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 oo^a// ' r ' n t ' i e ^ H N C closure approximation. Therefore, the angle-dependent terms of the pair interaction potential components a/3 can have no effect on 9 ( ) 0 ^ a / / ^ ' "^' r consequences will be discussed later. s between deficiency and its 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 coefficients. mnl / j 'uv,ap } ,,(r) uv;ap u^jap The equations must be solved numerically and so the projections, j mn (r-), must be represented with a set of discrete points anc T ? m n ^ fl (r) and c m n uv;ap on a numerical grid of width A r . In this study we will use a value of Ar=0.02d , which is consistent with previous work [68,70,71,79-81]. s 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. chosen . r). In this study we have This guess is usually a converged result from a previous calculation at slightly different conditions (e.g., concentration, total density, temperature, etc.). coefficients We then solve the OZ equation to determine a set of n ^ J. r ) . This set along with our initial guess for c ^ " J. r ) 'uv;a(l uv;ap is used as input into the RHNC closure which returns a new estimate for c m n 1 uvjaf „( r ) . m n mn 3 3 Direct substitution of the new estimate for the initial guess will 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. mnl /^\(i + 1) _ The (i + 1) approximation _ mnl , + , mnl mnl is given by /,-\(i) mnl V;a/3 p ; a 0 c , ( r ) inew) M where the mixing parameter, a and new estimates parameters m n ^ „, determines ' , _ ( 2 K * x 7 5 ) how much of the previous are taken and it satisfies 0<a o < 1 . Separate uv; ap ^ mixing 3 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;a0 ( r ) I !81 o r b o t h V;a0 ( r ) a n d c Z)aP {r) [6 *J°' 71] 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 ^ ( r) ' uv;a(S m n K Q with the potential K terms subtracted, as described in Appendix A . This method has several obvious advantages. c's contain no long-range tails due to the pair potential. can be readily truncated to reduce storage requirements. These short-range As a result, they The long-range tail - 40 - of C ^( 12) will always be given exactly, and hence it is relatively q obtain solutions after making adjustments to the potential. solutions, particularly at low concentration, 0 ^ ( 1 2 ) easy to For electrolyte will change very little with a small shift in the concentration, whereas r? J, 12) 'ap will show much 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 ( r ) is being mixed. The one obvious disadvantage is that c ( 12) m n uv,afS 3 0 aj3 3 a is 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., —& n) f ° r order rule [134] was used in this region for these systems. The r a (n=6) some of the systems studied here. Hence, a higher numerical derivatives needed in the HNC closure were computed using a standard 4** order central difference formula [134]. Comparison with results nd obtained using only a 2 order formula showed almost no change. 1 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 - to calculate a complete set of 41 - 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 ' ' " mn „( 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 electrostatic potential. a (r) due to the This is particularly true here, where the ion-ion and some ion-solvent c ^ " „( k)'s will have divergent behaviour at small k mn [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. Averages and Potentials of Mean Force 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 ^ ( l 2 ) and a 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, <3 p( 1 2 ) , is a sufficient to completely describe the thermodynamic properties of that system. It can be shown [27,33] from the definition of g ^ ( l 2 ) fl value, that the average of any mechanical quantity, m ^ ( l 2 ) , associated with the pair a/3 a is given by the general expression M a/3 = ( 8 7 r 2 ) 2 J g ^ J m ^ J d O ^ a r v (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 u a ty N \ tot = i u ij ( l 2 > ) <' > 2 77 where N and N^ are the numbers of particles of species a and /3 in the f l 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 'V i * & W j ^ r f r 1 2 * V^d^dP^dr , (2.78) where N is the total number of particles in the system, p^,= N/V is the total number density and X = N /N is the mole fraction of species a. Expanding fl <3 p( 1 2) a fl and u ^( 1 2) a in terms of rotational invariants, and using the orthogonality condition [110,111] /^ (fi ,n ,f)[#^; Mn ,n ,f)]*dn dn 2 l l 1 2 1 2 1 mnl 2 , „ 2,2 (_1jm+n+l 5 6 5,, 6 5 [ 1 (2.79a) mm, n n , 11, M M i f e , |_(2m+1 ) (2n+1 ) (21 + 1 )J { } x ( 8 7 } r - 43 - and 0 we have UpGT (2.79b) ^ jinn 1 ) 2 — E Ap a/3 N 47rr d r 0 a 0 m n l (2m+1)(2n+1)(21+1) x f r g*uv;apIv) uuv;ap ' ( r ) d rJl 2 m n l m n 1 J (2.80) 0 Here we note that g m n l Jr) = h „( r ) m n l 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 ^ , of f l single component pairs within the system. these energies per N U & N q It is usually convenient to express rather than N . After eq. (2.80) we .write ^^mn1)2 = 2*a *P P a P (2m+1)(2n+1)(21+1) mnl x f 5 f r 2 mnl g where 1 if 2 if / \ mnl Ar ) u , \ , Ar) dr a=0 (2.81a) (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 - 9u „ ( 1 2 ) 1 6kT h ^ Xfl /rg (8TT ) 2 2 a / 3 (l2) ap 9r dn,dfi clr 2 (2.82) - 44 - Equation (2.82) is often referred to as the pressure or virial equation. We and U ^ ( 12) determine Z by again expanding 9 ^( 12) q a 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) — ^ = -kT5(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 M a/3 = (r) \ ^ m as the average total value of m°^( 12) ensemble average. m°^( 12) aV ) 12) ( 2 ' 8 4 ) at r, where <•••> denotes the is some property of the particles of species /3 to be evaluated at a distance r from a particle of species a, number of particles of species /3 at the separation r. and N ^( r ) is the a Using eqs. (2.76) and (2.79b), and recalling that [13,27] « S 4 9 o o ? a ^ r ) d r = <V r ) > ' ( 2 - 8 5 a ) we have the expression M a0 ( r ) " T T T ^ W - T T 9 0;a/3 ( 8 7 r ) S 9 a / / 1 2 ) n £ (12) d ^ d O , . (2.85b) (r) 0 If we assume there is very little correlation between M i r ) and <N „(r) > ap then the average a/3 < *(r)> ~ m r) ap ap per particle is given by <N i r ) > — l 2,2 000 1 a , , J9 ^2)m° a (12)dn Q l d 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 ^ ) r a > a s being the average orientation of a particle of species /3 at a distance r from a particle of species a. Expanding g ^( 12) in eq. (2.86) and taking advantage a of the orthogonality of rotational invariants as given in eq. (2.79a), we obtain ,mnl «ijnnl M^-a/J ( (f ) (2m+1)(2n+1)(21+1) m n 1 r ) > = / \ 2 where we have assumed that m+n + l=even. 0 0 6 , 900;a/3 * (2.87) ( r ) 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 <P ( c o s 0 £ )> 1 (2.88a) = <cos0^ > s and <P (cos0 )> 2 where & s i s choosing 0. g - 2 i s 1 , (2.88b) is the angle between the dipole vector and the vector joining the ion, i, and the solvent, s. cos0. = !<cos 0 > g In Figure 3 we have illustrated our convention in for positive and negative ions. This convention guarantees that will be positive for favourable dipole orientations for both positive and negative ions. For the choice of f forms for the rotational m n l given by eq. (2.9a), and using explicit invariants [61,81], it is easy to show that eqs. (2.88) can be written as h <P (cos0 .)>(r) 1 + £ 00;+s o„000 3 (cos0_ )>(r) 1 T T -00;+s hh ° <P ( r ) 1 1 (2.89a) ( r ) (r-( r \) " 00;-s s (2.89b) and 2 <P (cos0 J>(r) 2 where + and - I £ ^ i s ( r > (2.89c) denote the positively and negatively charged ionic species. It - 47 - is also easy to demonstrate that for a random distribution of dipole orientations <P ( c o s 0 ) > ( r ) 1 = <P (cos0 )>(r) i s 2 = 0 i s . (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 >-<y> ) 2 where 2 2 , (2.91) a is a measure of the width of the distribution. Therefore we can easily compute the standard deviation of (cos0. ) using eqs. (2.88) and (2.89). g The equilibrium, or static, dielectric constant is another quantity we will calculate since it is an important solutions [8,10,81,137]. property of polar solvents and of electrolyte 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 , is theoretically well defined [61,81], but g 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) co—> 0 * £ ] L , (2.92) 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. understood and e experimentally g These dynamical contributions are not well can not, at present, be unambiguously determined for electrolyte solutions. Thus, for electrolyte would not expect exact agreement solutions we 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 how large the dynamical contributions to e indication of may be. cl In this study we will make use of three different determining the dielectric constant of a pure solvent. Kirkwood [138,142] relationship expressions for The first is the - 48 - = yg , (e-l)(2e+l) (2.93a) 9e where . 2 4irp M y = * 9kT (2.93b) , in which u is the dipole moment of the solvent. The Kirkwood g-factor can be expressed [61,62] as 4tfp 9 = 1 It has been shown [143] + oo . . 7 W h n OO;ss ( r ) d r * ( ' 2 9 3 c ) that the dielectric constant can be obtained through the limit h OO;ss ( r ' 3 47rp yer -* ) A { € ] ) 2 a s r -*°° • ( ' 2 9 4 ) s We will also determine 1 ' • 1 e using the relationship [74] " %KJ?ss= JJ;ss<=°>] k( 0) + £ 2 jry 7: - T[ ii? s s ? k 7T T . (2.95a) < - ° > " Soo?s. -°>] k (k where 1 19 ? o o 2 - s s ( k = 0 ) 47TM 2 us" - ( 2 ' 9 5 b ) and it follows from eq. (2.31) that 5 For 00?ss electrolyte dielectric constant. ( k = 0 ) " 4 f l r -C r 2 c 00?ss ' ( solutions, eq. (2.95a) is still a valid route [74] 2 ' 9 5 c ) to the However, eqs. (2.93a) and (2.94) are no longer valid because of the Debye screening of ^ Q Q ^ S S ^ ^ et al. [74] ( r ) d r a n c ' b OO^ss^ ^' r '- e v e s c 1 u e have shown that for a screened ionic system ^S-l = yg ' where y and g are still given by eqs. (2.93b) and (2.93c). (2.96) Chan etal. [135] - 49 - have also found that e 4TT Fr s Z p- p. Q. a. Iv • "3 i i j "i y *3 (2.97a) 1 ] where the sums are over ionic species and / in which n ^j( ) h. .(r) r d r 4 0 1 J , (2.97b) is understood to be the spherically symmetric ion-ion pair r correlation function. Equations (2.97) are known as the Stillinger-Lovett second moment condition. [144] 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] (2.98) where 9 ^ j ( ) r ' s t n e infinite dilution, ion-ion radial distribution function and |3=l/kT. j( r ) At 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. Rossky [82] Pettitt and 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 er Thus this effective as r —> 00 (2.99) ion-ion potential could be used to perform McMillan-Mayer level theory for model electrolyte solutions at finite concentration, as was - done by Pettitt and Rossky - 50 [82]. In the HNC theory, we rewrite eq. (2.98) using the HNC equation (2.62a) to obtain /5w.j(r) = Pu.j(r) - T?.j(r) (2.100) However, since all our calculations are done using the RHNC approximation, we will report the ion-ion potentials of mean force as given by (2.101) 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. - 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] the thermodynamic properties. [87-89]. often provides a convenient route to The Kirkwood-Buff approach is well known It uses grand canonical concentration fluctuation relationships in order to relate certain thermodynamic functions to integrals of the type a/3 G = 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 solutions. electrolyte 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 (3.2) 'a'p a/3 ' where is given by eq. (3.1a) and p^. N / V is again the number density of species a. If we consider a mixture of m species and denote the Q chemical potential of species a by M , the partial molecular volume by V q and the isothermal compressibility of the system by Xip, then the relevant relationships given by Kirkwood and Buff [104] al _V_ 3N kT xr. kT L = 1 r 9 M can be expressed as follows: a] |B| a /3 9 N «JT,P,N kTL PfljT, 3 kTXrp (3.3a) (3.3b) - 53 - ~ = PsS-l = V y L 9 N kTx JT,P N = ^ |B| P«I I B a (3.3C) ' 0=1 ^pi-lye S f T " i (3.3d) , where S |B| is the determinant B p. a . J . i '.'/llSLd ' = - < 3 3 e ) of B, and \0L\ p indicates the cofactor of the element a 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 p a is given by pn 1 1 m kTL9P JT, A where the elements of B' B dp 0=2 a "P P i r l ± 2 * L (3.4a) , |B are defined by = ?a ae 8 8 , P _ M M L | + a p aP p p G ' 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. parameter and p + v = v + =(v /v_)p_. + have that q + Also, it is convenient to introduce the v_, as well as the relationships p + + = f P 2 , P_ = + ~P2 V Also, since the salt molecule is electrically neutral, we = -(v_/u )q_. + For electrolyte solutions, we have charge neutrality conditions which can be expressed as [74,135,148,149] - 54 - f p i q i i j G ~ j = (3. 5a) q and ? Pi G = 0 . i s (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 = G_ + s . s As mentioned earlier, the charge neutrality (3.5d) 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 thermodynamic properties. expressions are obtained for the One way of doing this in a general systematic manner is described below. We begin by realizing that G afi = V k =0 ) ' E ( 3 * 6 a ) where ~ h follows immediately Air a0 ( k ) 0 if S = 0 r h 0 a/3 (r) s i n ( k r ) d (3.6b) r from eqs. (2.34a) and (2.35a). At finite ion concentration h ^( r ) is screened and decays exponentially at large r (as will be discussed below). Hence at small k, h ^ k ) fl V k ) = can be expanded in the form [73,74,135] *a¥ + * *aV 2 ' + ( 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)=p8 / r t + pp„h / J (k). (3.7) 55 - - T h i s then be ailows obtained determinate by taking 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 t h e k—>0 l i m i t of the appropriate properties to k-dependent quantities. For the /solvent h p p_h _(k) p p P P_ h _( k) p_ h!_( k) p_ p h_ ( k ) P P h p_p h_(k) p h+(k) p B(k) = s y s t e m w e c o n s i d e r , B ( k ) h a s the e x p l i c i t 2 h* (k) + + + + s + S + 2 + + + (k) s form (k) (3.8a) s 2 s where h*aa ( k ) = haa ( k ) + 1 and w e have m a d e u s e o f the r e q u i r e m e n t 1 (3.8b) that h „{ r ) = ( r ) , in w h i c h ap a,/3 = +,-,s. know In o r d e r the s m a l l to take the required k behaviour where again | B ( k ) | ^ (3.7) t o g e t h e r that = p_ p 2 ++ |B(k)| _ + »r(0)c;t(0) 2 h _ a a n d o f the s u m (3.9) p o f the e l e m e n t conditions h s r ^ ° h - [h s (2) given B ^(k). Using eq. f l b y e q s . (3.5c) a n d + 2 ] s 0) (2) ) (2) (3.1Oa) + + = p p_p + c [h<°>] h ! [ h i h ] - h< h +s +s -s 0) |B(k)| o ^ \ B ^ ) \ ["ht<°>h h<+-h ss- 2h<° ss +s h-sl J = |B(k)|_ L a at s m a l l k + + k^ P w i t h the c h a r g e n e u t r a l i t y (3.5d), o n e f i n d s |B(k)| Z = d e n o t e s the c o f a c t o r a k—> 0 l i m i t s , it is n e c e s s a r y t o of the determinant SU) pa 2) (2) + 0) (2) 2 - h^htJO) - ht(0)K| l 2) + + k 2 ( 3 . 1 Ob) 56 - |B(k)|_. = p 2 + p 2 K (0) t(0) . K [ K - (0),2 (3.10c) |B(k>| = |B(k)| = P p/p + • |B(k)l - |B(k)L. K^'thl^-hl ') 2 (3.10d) = p . V p , k [h<°>[h< >-h< > 2 2 2 • e't^'-hl ')] • 2 (3. I0e) and l< >l s k S - > <>2 k [Ki? [hi2) h!?>-2hi? ]] 2 2 ) We remark that in the k—»0 limit |B(k)| = |B(k)| ~\~ S explicit (3.1 Of) - ) + + = |B(k)| s - 0. The s s forms for the cofactors can then be used to show that as k—> 0 |S(k)| - S(k) P P 2 + 2 + pJV[ht<°>h<0> P- 2 P c 2 ht(0) ss + K - [h<°»] ]Dk 2 (0) _ +- 2 K (0)l 2 +s J D k : , (3.11a) (3.11b) where D It = h<J> + h!?> - 2h<? ) (3.11c) is also obvious from eqs. (3.11a) and (3.11b) that as k—>0, both S(k) and |B(k)|-»0. - First w e s h a l l d e r i v e For by the p r e s e n t explicit s y s t e m the p a r t i a l 57 - 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 molecular volume properties. o f t h e s a l t , V ^ , is d e f i n e d [6] % where V + = = ^ V . + v V and V _ are given k-dependent quantity b y e q . (3.3c). It then , follows (3.12) that t h e a p p r o p r i a t e is V (k) = v V (k) + v_V_(k) V 1. ( k ) = L 2 + + (3.13a) , where Now \p expression contains denominator. W e then the f a c t o r simply Dk cancel this in b o t h common Jjt(O) limit V (k) = which p 2 in the c o n v e n i e n t V Clearly 2 = s 1 . (3.13b) 2 P [1 • 2 , S ( G S S inserting fashion w e define G + volume is e a s y t o s e e that , (3.14) + S )] • < - ' 3 5 > expression for V^. quantity V ( k ) . Then t h e k—>0 l i m i t , w e o b t a i n t h e o f the s o l v e n t , +- " - — — (G + G^ G + sc _ _ .^0)3 the k-dependent G V to obtain _£(0) _ -2G + e q s . (3.10) a n d (3.11b) a n d t a k i n g molecular and its gives e q . (3.15) is the d e s i r e d d e t e r m i n a t e In a s i m i l a r factor -5(0) [ h t ( ° > notation that the its numerator ——L§ s ? o It n u s i n g e q s . (3.10) a n d (3.11b) a n d c o l l e c t i n g t e r m s , w e f i n d 2 resulting partial |B(k)|. S (( kk )) o.= a=+,-,sL S +,-,s ' i a j = 1 + p s ss s +- r 2G^) +s e q s . (3.15) a n d (3.16) s a t i s f y "s s V + 22 P V = 1 ' . the r e q u i r e d (3.16) relationship ( 3 - 1 ? a ) - 58 - From eq. (3.15) we immediately have that limit V 0 = (3.17b) Also, since in eq. (3.16) only G _ is divergent in the limit + P 2 — > 0 (as shown below), it follows that (3. 17c) Obviously eqs. (3.17b) and (3.17c) represent the correct single component results for the partial molecular volumes. particular case when f + Finally, we note that for the = 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 (3.18) 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. k-dependent By analogy with eq. (3.3d), we define the isothermal compressibility kTx (k) T T = ^ — S(k) |B(k) | " . (3.19) The k—> 0 limit follows immediately from eqs. (3.11a) and (3.11b), and yields the relationship G _+ P (G .G + kT X t r = — 1 + <> S + S + ( G s ss + ss G + - " G 2 ) ±2-r 2C , (3.2C W 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. - The chemical potential 59 - 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 where a f l and = u° + kTlna = M° ; c a = a a and y a J + m kTln( a f l (3.21) are concentrations expressed are activity coefficients a m ) 7 a , a and c a and molarity, 7 ° . M + kTln(y c ) is the activity of species a , m as molality 0 u a ' and u° are the chemical potentials of the standard states. be independent of the concentration scale in which a chemical potential f l u° a' arm Since u must is expressed, the 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;c 'a a ^ 'arm M 2 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. that the molarity activity coefficient, y We point out 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), JL^, is given by M2 = v ++ u + v - u - (3.22a) > where the single ion quantities are defined by eq. (3.21). mean activity If we introduce the coefficient of the salt defined [6] such that y = y+ y!" , + ± (3.22b) then it follows from eqs. (3.21) and (3.22a) that M2 = + kTln(j^ /-) We take the partial derivative p s + + ^kTln(y p ) ± of eq. (3.22c) with respect to p ox 1 and P, fixed to obtain . 2 2 (3.22c) holding T and 60 - r91ny ± ^ p s or TTTf P 2 bp. p or s (3.23) P 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 [9p J 2 one immediately & j = + _ f I ]L9PjJp relationship = + or - finds that - *+ feL* & 2 feL * * - & + 2 + where in addition to the variables specifically indicated T and p also held constant. The partial derivatives, (da^/b constant volume case are given by eq. (3.3a). determinate (3.24a) k 5 4 J • (3 + g 24b) or P are required in the p^), Again, in order to obtain results it is necessary to define 1 «| ( k ) [a£. ^ = kT (3.25) |B(k)| 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 1 2i k |_9 POJT, T + p G SS p/[G _ + + P (G s s s (3.26) G _+ G + J)] From eqs. (3.3b), (3.12) and (3.24b) we can show that the constant pressure derivative can be expressed as _V_ kT ^2] 2 J T , P,N 9 N k T L P 2 J T , p, 9 kTx^ Inserting results from eqs. (3.15), (3.20) and (3.26) we obtain (3.27a) - 61 - kTL 2.lT,P,N (3.27b) 9 N c 2 p [ 1 + 's ss ( G + G + -" 2 s G + ) ] 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 k [9p J 2 T where N / vl"-^ ! 2 kTL 2JT,P,N L 2-rT,P,N 9N T f P 3N ' (3 28a) is being implicitly held fixed on the left-hand side of the equation. However, p can JLfi^l = g can not be held fixed because the volume is allowed to vary. It easily be shown that [ w H = i < L 2 J T , P, N 9N 1 " P? ?> v 2 2 v = ^Pe • v V s s c (3.28b) Then combining eqs. (3.27b) and (3.28), and using eq. (3.16) we obtain 1 1 r 2] k L9p jT,: 9 M T (3.29) P ^(G _ 2 2 + - G + s ) Expressions for the mean activity coefficients now follow from eqs. (3.23), (3.26) and (3.29). r-ainy+n _ Explicitly, we have 1 ] vp [G _ 2 + immediately + + P G ^s ss P (G s s s - G _+ G p] 1 (3.30a) + and r L 9 1 n 9 y±i ^2 J T , I vp (G _ 2 + - G + s ) 1 (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 +- JT, k T LL9P °P J T,p S s * 2 p p [G, s +- + p (G s G ss +A - G. *) ] (3.31a) +s and _Lf_^2 kT L P J T , p kTL P2-l ' s T 9 9 0 - G +s P [G _ 2 + + p_<G,s s + - " G (3.31b) G + s ) ] 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 \— - P2-JT,M i=+," " i L Pi. 9 g (3.32) C where (3II/3p^) is defined by eq. (3.4a). (3.32) leads to an indeterminate Again direct substitution into eq. 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 |_3p J T For 2 P +- • G T/ (3.33) 2 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^— °f >u 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) + where w _ ( r ) + = exp[-/3w _(r) ] (3.34) , + 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 " K r e (3.35a) , 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 l + -l = [jZT K q q lY vp ' (3.35c) 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 _ + = — up + - — = • + ••• 2 STpZ , (3.36a) where 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 - equation, both o f which will require must effective From the long-range behaviour ourselves to systems which pairwise For are d e s c r i b e d in t h e p r e v i o u s that w e k n o w restrict 64 - additive the present chapter. Our of 0 ^ ( 1 2 ) . analysis Therefore, w e can be described b y pairwise or potentials. p u r p o s e s , the the OZ equation, as given only relevant b y (2.36a), w e projections have are hgo^a// "^" 1 that y jmmO 5 00m z m L (-D h° ( k ) c -uOjyP„ ( k )Jl O C J ; ay .u=-m where small Z^ ^ is a n o n z e r o m coefficient k dependence o f this reduce . „( k) given take b y e q . (2.36b). advantage (cf. e q . (2.31)). ix v ap m 0 m m m expression which e q . (3.37) f u r t h e r , w e tranforms, b w is required O b v i o u s l y , it i s t h e here. o f a property It c a n b e s h o w n (3.37) In o r d e r t o o f the Hankel (by using explicit r forms that f o r the if b m n l spherical Bessel „ ( r ) decays faster £ Of that hg^a/^ r ) than M";a/3 course, f o r all electrolyte ensures functions ( k = 0 " ) f o r notational o r at f i n i t e „(k = 0) = 0 1 > 0 (3.38a) ' concentrations, screening hence for m>0. o f e q . (3.37), w e - C as = I p G 7 (3.39a) i s a n e x a c t C a7 7s c o n v e n i e n c e w e have can then (3.38b) apply eq. (3.39a) ' introduced 2 000 -000 potentials f the expression as Equation 0 ap T h u s , if w e c o n s i d e r t h e k—> 0 l i m i t terms) then d e c a y s e x p o n e n t i a l l y , and m m where 1/r , 3 solutions h USP ; (3.38b) t o o b t a i n and expanding the s i n and c o s relationship subject describing our s y s t e m be pairwise (3.39b) only additive. t o the restriction that t h e - 65 - For terms t h e s y s t e m s b e i n g c o n s i d e r e d here w e c a n e x p l i c i t l y i n t h e s u m o f e q . (3.39a) a n d then G +s " ( 1 s p s s C ) = ( 1 out t h e rearrange t o obtain P+ ++ + write G ) c + P- +- - + G s (3.40a) c s and G_ (1 s which, when - P C S ) S S = (1 + p _ G _ _ ) C _ + P G _C s + c o m b i n e d w i t h the charge neutrality + condition + , S (3.40b) (3.5c), y i e l d t h e result r" C + °+S Equation = G "S (3.41) is in f a c t + v_C_ -, + 1 ' = the origin 2 P C S SS G + - ' ( o f the charge neutrality condition A l s o , s i n c e e q . (3.41) i s o b t a i n e d w i t h t h e a i d o f the c h a r g e condition At small (3.5c), it h o l d s o n l y this point for 4 1 ) (3.5d). neutrality p >0. 2 it i s i n t e r e s t i n g k, w e h a v e a r e l a t i o n s h i p ' 3 t o note that if w e e x a m i n e e q . (3.37) at a n a l o g o u s t o e q . (3.39a), n a m e l y h0 0 ; a s (k) = [Tc (k) + o+ h (k)c (k) 00;as 0 0 ; a+ 0 0 ; +s ' 000 000 000 c { K ) n + l ; p trOOO - p K b O O ; a - KK x-000 ( k ) g 000 n ; c l K / , xl T O O ; - s ( k 1 ] [ ) 1 _ y p The small k behaviour dominate the this - -I s • 0 00 000 s c 0 0 ; s s ( k ( 3 ' 4 2 a ) ) of h . .(k) (an a p p a r e n t 1/k d i v e r g e n c e ) w i l l ' J ~000 e x p r e s s i o n , a n d s o it f o l l o w s that o o « a s ^ ^ ' nn k n same small k dependence. m u s t T h e r e f o r e , at large r w e c a n w r i t e 000 0 0 ; i s h! ( r ) " * a i s 2 p h + - ( r a s o n a v e that > (3.42b) and C ? s s where a as W e remark r ) a p 2 2 h + - ( ^ . (3.42c) c?5 u n n n UU;TS a a s = °- n UU, S T n e long-range o f e q s . (3.42b) a n d (3.42c) w i l l chapters. s s is a c o n s t a n t d e p e n d e n t o n t h e v a l u e s o f „ ( k ) at s m a l l k. „„„ .„ Uujas 000 000 ' that if c .. (r) = c . _ ( r ) (i.e., t h e i o n s a r e s o l v a t e d e q u i v a l e n t l y ) , then result ( tails in ^ 0 0 * a s ^ ^ b e d i s c u s s e d in m o r e r t h a t detail a r e a in later 66 Inspection of eqs. (3.41) and (3.36a) shows that at low concentration G + s = G_ s will have a term due to P G _ which varies like \fp^. 2 also expect C ^ s (i = + We might + or -) to have a \/p~^ dependence at low concentration, so we write ' I S = " + C + + S ^_C_ = S (v cl + + p_cf ) s + s + where the superscript o indicates the infinite dilution result. at present, to obtain an exact expression for the slope S ... , (3.43) It is not possible, 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 ^[h ap a / 3 (12)] - 2 0u a/3 (12) We then expand c(12), h(12), and u(12) in rotational It as r — ( 3 . 4 4 ) invariant as in Chapter II. is possible to deduce, with that aid of eq. (3.42b), that as p —> 0 and 2 ,000 1 6 -oo r > [ h 0l1 , 00;is . .2 fll " ( r ) ] ,000 ., ^ 00;is ( r u ( 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 ^ . It is known from the work of Hcfye and Stell [61,152] that in g the limits p — > 0 and 2 r—>t» t + /cr h 00;is ( r ) where y has been defined in eq. (2.93b). - /cr (3.46) 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 p —>0, 2 is is C? „ [e-\' 2 ? i 2 «2 9/ 2 M K + (3.47) - - 67 C o m b i n i n g e q s . (3.43) a n d (3.47) a l o n g w i t h e q . (3.35c), a n d t h e n rearranging gives - Kv* ( e - 1 ) c S A g a i n w e note fluids will 2 • oye = (3.48) that e q . (3.48) is a p p r o x i m a t e , a n d f u r t h e r m o r e of nonpolarizable particles. holds only for T h e a c c u r a c y o f the H N C e s t i m a t e for S c be discussed below. From e q s . (3.36a), (3.41) a n d (3.43) w e h a v e the l i m i t i n g l a w s - -s • < [^ ^<" °s "- -s'K ' -«> G G + + C + + c ^ . (3 a + where G° We point only for = = G° out that limit G = X_±s + p —>0+ i s t h e a p p r o p r i a t e 2 limit (3.49b) here s i n c e e q . (3.41) holds P >0. 2 It is i n t e r e s t i n g e q s . (3.40) t o t o a p p l y the i n f i n i t e dilution limit (i.e., p = p_ = 0) t o + obtain 1 — P L S and G° e = 1 SS 2— P L ^S SS . (3.50b) It is o b v i o u s that t h e s e e x p r e s s i o n s d o n o t a g r e e w i t h e q . (3.49b), a n d h e n c e Gj_ and G must be discontinuous +S s (3.49b) a n d (3.50) that _ < C l e a r l y , G?_ terms a n d G°_ = G° are just s at p = 0 . 2 = (, +G : s + the w e i g h t e d ~000 o f the F o u r i e r t r a n s f o r m s , ^ o O ' * i s ^ ^ behaviour M o r e o v e r , it f o l l o w s ' 0 k c a n be e x p r e s s e d in the f o r m ,_Gf )A (3.5! ) . S a v e r a g e s o f G°_ ^ = + o r ~)> ' ' t from eqs. ^ 1 s a n d G°_ In discontinuous - limit p ^ 0 2 limit h°S?.(k) k-^0 ' U U 1 * 68 - limit k-^0 S limit P -^0 2 h°°°. ' U U 1 (k) . (3.52) 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). (p.C, G S s s = + + It can be shown that + p C - c S S " 1 p )G. + + C SS- , e s (3.53) s ss C from which it follows that at low concentration C° ss G = , 1 o p C S S - + 0 ( r ^S "2> ' ( ' 3 5 4 ) ss We also point out that 1 1 1 — 'o = 1 + » s G • s s p ' s 4 k T ' ( 3 5 5 ) P ^S ss 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 p —>0 limit of eq. (3.20). 2 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 p —>0, 2 « ] _ « ] _ . ( 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. - 69 - Now let us consider the limiting behaviour of the derivative of the osmotic pressure. Combining eqs. (3.33) and (3.36a) we immediately T-\¥l-\ ~* ~> -F= V v(\-h}/VpZ) obtain . (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. frequently coefficient used measure of the solvent chemical potential A more is the osmotic [6,7], <t>, defined by where M is the molecular weight of the solvent s 3 molality and a g is the solvent activity. m is the concentration in 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 n = j>kTM I r - 4>m iooov . (3.59) s The limiting law for <p (cf. eq. (3-5-12) of Ref. [6]) can be written as <p —» 1 - | A i / p p 2 as p — » 0 . (3.60) 2 At very low concentration eq. (3.59) can be rearranged to give n = i>kT0p 2 . (3.61 ) Now substituting eq. (3.60) into (3.61) and taking the derivative with respect to P , we again obtain eq. (3.57). Clearly, the limiting behaviours of the osmotic 2 pressure and of the osmotic coefficient are simply related. The limiting behaviour of the partial molecular volume, V , is of 2 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 - 1 V„ p or 70 - + ^s pG ss s 2 +- •]" p G +s (3.62a) PG_ 2 + using eq. (3.41) f 1 + P S S5l G v, C. + v C 2 (3.62b) L 11 -- ps s C s LP P G _J S n - -s P V j H +S Bs _ + H Now applying eqs. (3.36a), (3.43) and (3.55) we obtain the limiting law expression V 2 = 2 V + (3.63) \^~2 where V° = * k T J ( l - p _ C J ) + r . k T x J O - P.C° ) s +s s-s + X pkTx^ " pkTx° {p C + + (3.64b) v_Q°_ ) 0 ? + (3.64a) and S„ = AkTp • We remark that as one would expect, (3.65) 2 kV 2 splits into two independent terms which depend upon the interaction of the positive and negative ions with the solvent (cf. eq. (3.64a)). as It is also interesting to note that V?, can be written 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 S v which can be expressed in the form = AkTV 2 •x£ + 3 L 9 P [6] (3.66) JT 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 - fainel L 9 P If we introduce the (3.67) JT identity [oVlne] L 9P J = 1 [3e 1 = e[dp ] s T P T S X T (3.68) ' which follows from eq. (3.18), we can then rewrite eq. (3.67) as (3.69) 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 then we obtain the differential (i.e., eq. (3.48)) is substituted into eq. (3.69), c equation (3.70) 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] solvent. e. of the pure Of course, eq. (3.71) is not a very accurate theory and overestimates 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 obtained. In the LHNC theory, it clearly follows from eq. (2.74a) and eqs. (3.39b) and (3.43) that S = 0 . c 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. of Rasaiah era/. [153] This is consistent with the observation that the LHNC approximation does not predict - 72 electrostriction in polar fluids. We now examine the low concentration behaviour of the partial molecular volume of the solvent, V V s First we rewrite eq. (3.17a) as "oV • = (3 2 - 72) Then inserting eq. (3.63) into eq. (3.72) one immediately has the limiting law V Clearly, V g s = 7T " ^ ( 1 s has a linear dependence on p * 2 { 2 ' 7 3 ) at low concentration and its limiting slope will be almost totally determined by P 3 unless p g has a strong 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 p 2 dependence in the limit 2 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 e - Nj where UJJ/N^ N. € i ] C T l q + q . l * ^ . is the total average ion-ion energy per ion and average ion-dipole energy per ion. (3.76) U ^ / N ^ is the In a similar manner, we can obtain an expression for the dipole-dipole energy by inserting the long-range low 11 2 concentration form for hfjO'SS^ ^ 1 (cf ec i- (2.35a) and Ref. 61) into eq. (2.81a). Integrating and then simplifying yields the result - X - X + - 73 - '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 In the current theoretical framework Chapter II) is difficult to treat. (i.e., the RHNC theory as described in 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. equivalent to the For purely dipolar fluids, the SCMF approximation is 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 being studied here. its application to the systems - 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 below. as described 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. the SCMF appproximation, the systems characterized by this effective Within 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 m At finite concentrations we must simply be that of the pure solvent. moment, 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 between the charge and the induced moment is follows that the interaction - 76 - u (r) = - - j . q p H o w e v e r , in s o l u t i o n These give an i o n molecules will rise to local concentration order is s u r r o u n d e d ion will its c h a r g e and a g a i n alter expect e q s . (4.1a) and (4.1c) t o In t h i s which chapter develop we we average As in the an local SCMF may for field local be p o o r one by other electric solvent molecule. i o n , and this a r o u n d the ion. At must finite ions which will field. approximations The in s e c t i o n 2 . detailed theory, we will first in tend Thus, we would a dense system This R-dependent levels of i s the In s e c t i o n 3 of at mean field the is f o u n d ion to this which theory in to chapter w e we can and the solvent h a v e an e f f e c t depend will estimate an a p p r o a c h {i.e., w e (RDMF) theory q u a n t i t i e s , s u c h as V ^ , w h i c h which approximation a distance R from a mean field between SCMF through a solvent follow spherical potential different formalism experienced by thermodynamic ion-solvent field describe two spherical potential M o r e o v e r , this the be studied. outline fluctuations). effective laws will a s e c o n d and m o r e the ignore will than solution. a polarizable solvent [67] electric a l s o be s u r r o u n d e d screen s u c h as an e l e c t r o l y t e more in s o m e f a s h i o n a r o u n d the c h a n g e s in the the by (4.1c) ion. will gives rise at R. on the upon to limiting the correlations. 2 . The S e l f - C o n s i s t e n t M e a n Field Theory The problem SCMF of theory polarization potential. It following is s i m p l y octupole of does so and interest In + of into by field in t h i s s p e c i e s , one charges ion C a r n i e and P a t e y a problem ignoring an e x t e n s i o n involving fluctuations of contributions the thus will r e d u c e s the an e f f e c t i v e in the SCMF local making it many-body pairwise electric approximation consider a system which these being [67] a p p l i c a b l e to additive field. to the higher order c o n t a i n s three a polar-polarizable solvent. and - , are a s s u m e d to and n o [67] The include systems study. general, we designated of multipole be s i m p l e moments. spherical The t w o ions molecular ionic species, possessing only The p o l a r i z a b i l i t y of these - 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 th moment of the j solvent molecule as mj where p. = Mj + 2j > (4.2a) is the permanent dipole moment of the solvent, Pj = a-fB^j (4.2b) is the instantaneous induced dipole moment and (E^)j instantaneous electric field felt by the solvent j. is the total 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 For molecules of C 2y <E-^> . (or higher) symmetry in an isotropic fluid, we know that will be non-zero only in the direction of that fx, <E^> and m' are all in the same direction. <E > 1 where the scalar C(m') inserts eq. (4.4) (4.3) This immediately implies As a result, we can write = C(m')m' , (4.4) will depend upon the properties of the system. If one 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 ' as being a renormalized polarizability. (4.5b) 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 + (<p > - <2> ) <m > = m' 2 2 2 2 + 3a'kT , = m' 2 (4.6a) where a' (4.6b) = jTra' 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 U N "HS = " + U I I IQ + U 5mj-(E )j 1 Q + "lO + -Ln..(E 1 0 iffflj- £l >j MgO MDO + + ). - ^ y ( E ( D n ) . + - ^ ' ( E ^ (4.7) , in which the sums over j are over the number of solvent molecules N and s' where { *l>j = (=ID>J —ID' — 1Q' —10 a n C ' —II (E + a r e t ' i e field contributions, respectively. hard-sphere energy and U j j , U -lQ>j + ( 5l0>j + % i > j ' <'> 4 8 ^'P ' "- puadrupolar, octupolar and total ionic 0 31 In eq. (4.7) u^g is the instantaneous JQ, ^ Q ' ^0 a n C ' HDO a r e t ' ie 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 - ^TOT U t = "K " < ^ 1 D > " lN <2-E S l Q N 79 s<^^lQ - " > " K < 2 - E > 1 N s<i^lO> 0 > + U " N s<if5ll> " i s<e-En> , N (4.9a) where °t = + U II + U IQ + U I0 + U QQ QO + U 00 • ( 4 ' 9 B ) In the SCMF theory one ignores fluctuations in the electric field and assumes that <P_*E^>= < £ > • <E^>. molecules of C 2 v Using this assumption and the fact that for symmetry the average electric field must always be directed along the permanent dipole moment, we can rewrite eq. (4.9a) in the form ^OT = U t " i N s" < E lD " ^ S ^ ' ^ ^ ^ I Q ^ ^ I O ^ ^ I I ^ > • ( 4 ' 1 0 ) If we now let C(m') = ( y m ' ) + C (m') + C (m') + Cjdn') Q Q , (4.11) then it clearly follows from eqs. (4.4) and (4.8) that <E_ > = CpdnOm' , (4.12a) <E 1 Q > = C (m')m' , (4.12b) <E 1 Q > = C (m')ni' ( 4 . 1 2c) <E 1 : > = CjdnOm' . ( 4 . 12d) 1D Q 0 and Inserting eqs. (4.12) into eq. (4.10) yields - ^N m'(m'+M) [c (m')+C (m') +Cj(m')J s Q 0 . (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 Pi 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 u 00;ss (r) = -/nJ73[-ij^] (4.15a) r "oiU - -^KH ' r) U -I5B) and n1 where m. and 1 q -i m k 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 , J where m g is an effective K (4.16) 6 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 and g (for the systems we will consider this difference is only 2-4%) 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 7 m. e We have yet to determine an explicit form for C(m'). For the effective system we know that -i s e = N m "DQ = - s e "DO = - = - s e N N m s m < E < E e < lD lQ ' > e 1 0 E > ' > e { 4 ( E ' ' 4 ( 1 7 a ) 1 ' 1 ' 1 4 7 b 7 ) C ) and °ID N m < lI E > e ' ( 4 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 and similarity for 1 Q < E ^ Q > and > E = < E 1 Q (4.18a) > < 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- s e , (4.19a) - C(m') Q 82 - = _^Q_ , N m m' s e C (m') = D (4.19b) (4.19c) O n s e and -u C (m') = T 1 e N m m' s e . (4.l9d) We note that although the effective and polarizable systems have the same structure, their total average energies must be different system we have ignored the polarization energy. since in the effective Substituting eqs. (4.19) into eq. (4.13) we find that the energy of the polarizable fluid is given by «TOT - " t * • ^ . [ ^ e • l £ • 0« ] . D (4.20, 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 parameters are held fixed. g while all other 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 = N / N , eqs. (4.5), (4.6) and (4.19) are solved iteratively, to give values s s of m', a ' and m that are consistent with the given molecular and state g parameters. - 3 . The R - D e p e n d e n t The RDMF solutions of systems may spherically a tetrahedral symmetric possible. solvent solutions symmetric. A l s o , all hard-sphere contact The into only is in C h a p t e r a single particles salt will are will be II. RDMF theory field which are a result of the presence of symmetric ion allows a spherically the fluid surrounding Thus, we the need only distance in the at produced will or ion and an be all point We now from the to electric particles the field solutions given out take ion being or may not in the advantage the the the they in the system. local experienced by both electric The of fixed at average case that the ion. upon R, generated directed are the being ion. We average a distance interaction R from can then fields. a solvent ion centre c o n s i d e r i n g the particle of average fields the the local field as though a solvent local fact viewpoint i o n , and h e n c e a p p e a r of contain average of average additional R by the an be at again Considering only from presence of fields and a molecule and s o l v e n t c o n s i d e r e d h e r e , the average at a distance R components. local electric For the field at by = <E (R)> + <E (R)> + < E ( R ) > 1D that t h i s rewrite of changes isotropic to spherically ion, must average being will the local dipole ourselves a dipole are may ( s c r e e n i n g ) c h a r g e s p l a c e d at the other which dependence of the general e q . (2.24b). an i o n . us t o be only ions be that the F u r t h e r m o r e , all average ^ ( R ^ We the i o n w i l l , in g e n e r a l , h a v e electrolyte R will the related The ion. must f l u i d , due t o between be e a s i l y from the additional determine ion examine away by interaction the from surrounding radially the examine the by solvent The more restrict with to as hard s p h e r e s w h e r e given ions. we to only For s i m p l i c i t y , we in w h i c h other of will extensions model addressing will a multipolar-polarizable applicable derivation be t r e a t e d a we directly a solvent d i s t a n c e s , & p, problem immersed following m o d e l , that of b e l o w , is i o n s , although In the s q u a r e q u a d r u p o l e , as d e s c r i b e d consider - M e a n Field T h e o r y t h e o r y , as o u t l i n e d be 83 expression e q . (4.21) in the 1Q is j u s t n an R - d e p e n d e n t (4.21) . analogue of e q . (4.8). form <E^(R)> = < E > + <AE (R)> , X 1 (4.22a) - where the <E^> SCMF is j u s t theory. solvent particle change in the the present the average <AE-^(R)> is at we in the SCMF molecular dipole moment. clearly In the we then write Ap_(R) is the present intermolecular vector. theory we reference Hence, instead (4.1b). the We symmetric <AE + determine R-dependent gives the by a ion. the For as < A E the N . ( R ) > average (4.22b) total have , 1 + = m' + A2(R) = will of (4.23) quantity. a-<AE (R)> 1 U s i n g e q s . (4.3) and , (4.24a) = out that [164], and t h e r e f o r e present for theory , moment a.<AE (R)> it in w h i c h the water express z - a x i s is along <AE^(R)> the in an ion-solvent . (4.26) a solvent molecule molecule a e q . (4.1b) w i l l will to use a<AE (R)> the by (4.25) convenient 1 given . 1 find (4.24b) induced e q . (4.25) w e point be s h o u l d be r e p l a c e d e v e r y w h e r e permanent moment. is is n e a r l y a very with defined with m , the e by eq. spherically good be u s e d in c o n j u n c t i o n approximation, m' dipole term when s a y , it presence of (R)> a-<E (R)> average polarizability, a , of A l s o , s i n c e the 1 Q field is to context, we = m' frame AE(R) where the correction present average excess A£(R) In the bulk as g i v e n as m'(R) where the solution that m'(R) which bulk an i o n , that w i s h to = JX + appropriate follows + the to R due t o (R)> 1 D of correction at theory, we m'(R) (4.22a) it field <AE As is the field can express this = 1 m'(R) is the local <AE (R)> where local - a distance R from average system 84 approximation. the SCMF effective - 85 - We now define the additional ion-solvent interaction term due to Ap_(R) as U i s ( l 2 ) = -^e(R)-< I (R)> - Ap_(R) • |^<AE (R)> + A 1I 1D + ^Ap_(R) • <AE (R)> 1 <AE (R)>J 1Q (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 <AE (R)> ) 1 (4.28) . Finally, combining eqs. (4.26) and (4.28) yields the expression uff(R) = - [ A p ( R ) ] / 2 a 2 It is interesting to note that ^g(R) u is a . (4.29) 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 < A E ^ 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, < A E ^ g ( R ) > , due to the charge on the ion. The other two terms, <AE^ (R)> + and < A E ^ _ ( R ) > , are essentially screening terms and are due to all other positive and negative ions in the system. Therefore we write <AE (R)> 1 I = <AE (R)> l g + <AE 1 + (R)> + <AE _(R)> 1 . (4.30a) It is obvious that <AE (R)> l g = q i /r 2 , (4.30b) where q. is the charge on the ion, which we have labelled i, and <AE^ (R)> g 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 other ions (i.e., excluding q.). molecule has particle (see Figure 4), and all reference Now we know that because the solvent symmetry, the average orientation of the total dipole moment of the reference solvent will be in the direction of < A E ^ 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 (R) f = m'(R) < ^ ! u i s (R)> = m'(R) < c o s 0 . ( R ) > (4.31) i £ 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 illustrated is IS in Figure 3. Again, we stress that in the intermolecular frame this will be the only non-zero projection of m ' ( R ) . an expression for the interaction between m'(R) reference If we can obtain 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. - 87 - Figure 4. A n illustration of the method used in determining case where q^ is a negative particle. T ion is shown. The dipole moment, nr , located at a distance R from the solvent < A E , ( R ) > . The t ion, q., is that of the reference - From 88 - e q s . (2.5) a n d (2.17) it i s e a s y t o s h o w [79,81] that t h e i n t e r a c t i o n b e t w e e n m ' ( R ) a n d q. is given b y J U m j ( l 2 ) = r 2^*00 mj ( 1 2 ' ) ( 4 ' 3 2 ) mn 1 when w e take f a s g i v e n b y e q . (2.9a). in F i g u r e s 3 a n d 4 , a n d u s i n g a n e x p l i c i t F o l l o w i n g the c o n v e n t i o n s form f o r the rotational A p p e n d i x A o f R e f . 81 o r A p p e n d i x B o f R e f . 6 1 ) , w e r e w r i t e given invariant (see e q . (4.32) in t h e form m ( R ) q . -, T r u m j ( l 2 ) -sign(q )| = ^Icosfl i r where From the s i g n f u n c t i o n equals , (4.33) mj 1 if q . i s p o s i t i v e a n d - 1 if q . i s n e g a t i v e . t h e l a w o f c o s i n e s [165] w e h a v e t h e r e l a t i o n s h i p s r = r 2 m 2 + R 2 - 2rRcostf» * , (4.34a) and cose = R " r c o s (4.34b) mj which, when u substituted .(R,r,tf,) into = -sign( e q . (4.33), y i e l d )m (R)q. T q i R [r T h e n the a v e r a g e spherical shell interaction o f ions r c o s ^ between . i +R - 2rRcos«] energy, U ^ ^ ( R , r ) , (positive ~ (4.35) 2 rn^(R) and the or n e g a t i v e ) at a d i s t a n c e r f r o m q. is given by UJj!?(R,r) in w h i c h t h e e l e m e n t = P j Jg s j (R,r 0,ifOi^(R,r,*) dAdr R , (4.36a) o f area dA = r sin0d0di// 2 . T h e l i m i t s o f i n t e g r a t i o n f o r d^j a r e 0 t o 27T (a f u l l r e v o l u t i o n ) , w h i l e t h e y a r e 0 „ t o ir, w h e r e m (4.36b) f o r d0 - 89 - ^ r W - d , cos - , -L?- , 2rR J 1 L g u a r a n t e e s that Now, i o n q. and the in p r i n c i p l e , w e know that the (valid only be the distribution when independent q. of do of reference not know ions the r approximation ' s J u s t t h ion-ion e radial should b e c o m e exact i n t o e q . (4.36a) and ^(R,r) solvent integrating over (r) large which spherical we shell a s s u m e it we can to write , (4.38) function. Inserting This e q s . (4.35) and (4.38) obtains j i (R-rcos»)sin^ [r + R - integrals in the we 2 2 integrating However, since = - 2 7 r r p g j ( r ) s i g n f q ^ q^m^R) d r x J* Then s interpenetrate. s o l v e n t , then r. one d\p, not i o n ) and if distribution at do -(R, r , 0, \//). g i : j J 1 reference s 9 j j( ) i f |r-R|<d. be u n i f o r m g j(R,r,4>,<//) = g where (4.37) sJ must position j s 2 is a s p h e r i c a l l y s y m m e t r i c R, the |r-R|>d if 0, e q . (4.39) o v e r may be f o u n d d# 2rRcos0] 2 (using in t a b l e s standard forms d ^ 0 ( 4 > 3 9 ) 2 for the [165]) and s i m p l i f y i n g trigonometric yields the expression Tjjj!?(R,r) = -27rr 2 P : j g (r) sign(q )q m (R) t i : J i L r 2 " j R c o s * rf ^ • 4r[r^+R - 2 r R c o s ^ ] ^ (4.40) 2 j If one carefully e v a l u a t e s e q . (4.40) at uS(R,r) = 0 mj its l i m i t s , one f o r r^R+d. j s finds , m that (4.41a) - 3-i ^ (r)sign(q )-l_ 9 -V, U^(R,r) m = -27rr^p g j R 2 - r 1 + uP^(R,r) 90 - i j 2 2rd. = -47rr^p g j i j R ) i - d , 2 - ^ I d r for R-d j s remark which that (r)sign(q ) ^ - L _ from an internal electrically nonexistent electrically equivalent to a point contained all , (4.41b) point of view (cf. e q . (4.41a)); f r o m r<R-d DS (4.41c) e l e c t r o s t a t i c t h e o r y [140] a spherical shell o f charge is an e x t e r n a l p o i n t of view it i s c h a r g e w h o s e c h a r g e i s e q u a l t o that 2 i n t h e s h e l l (cf. e q . (4.41c) w h e r e N j = 47rr p ^ g ^ _j( r ) d r ) . The t o t a l other j s J d r i 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 s t a t e s that + js for We ^r<R d average interaction i o n s , q . , in t h e s y s t e m v a l u e s o f r. energy, U j(R), between m'(R) and all m is found by simply integrating T h e i o n s are n o t a l l o w e d t o i n t e r p e n e t r a t e e q s . (4.41) o v e r s o w e write (4.42) and i n s e r t i n g e q s . (4.41) o n e o b t a i n s t h e n e c e s s a r y result q_; m (R) T U mj ( R ) = - 4 1 r p j s i g n ( q i R R R Now distributed taking + d j s is dr ij R [r 2 9 i j (r)[ 1 + ' js advantage of the fact be non-zero only U j(R) m that r -d 2 2rd d in s p h e r i c a l s h e l l s a b o u t (j = + o r - ) w i l l d ) ^ L _ d 1 _ all other 2 (4.43) DS ions will q . , it c l e a r l y f o l l o w s ' that t be u n i f o r m l y <AE, .(R)> a l o n g m ' ( R ) , a n d c o n s e q u e n t l y w e have that = -m "(R)<AE (R)> 1 U s i n g e q s . (4.43) a n d (4.44) o n e i m m e d i a t e l y l j obtains . (4.44) - 91 dr R d. • R+d. ^ p 9 i j 1 ( r ) [ R' + where as j = + or-. To <AE-> (R)> has the s a m e d i r e c t i o n a l e n s u r e that <AE^j(R)>, w e m u s t m u l t i p l y (4.45) are c o m b i n e d to give the (4.45) dr R-d. e q . (4.30b) b y desired sign(q.). sense E q u a t i o n s (4.30) and relationship (4.46) <AE^j(R)> w i l l a l w a y s be d i r e c t e d a l o n g the i o n - s o l v e n t where We Let us p i c t u r e average the now all the same mutually the we first spherical the the shells of have the solvent point (we It excess of the which dipole local 2 directly to will them to interaction dipole to between that derive our m o m e n t s , rn'(r), at illustrated the situation or We ion. the find directly that on away C l e a r l y , all as b e i n g to <AE^^(R)>. lateral) from dipoles will local dipole field due examine. a v e r a g e p r o j e c t i o n s , m^( r ) , In o r d e r at f i e l d , and h e n c e similar field, symmetry. v contribution we dipole c h a r g e o n the shall refer quite C either is t h i s lateral be c o n s i d e r e d . e x a m i n e the to sign of moments in a m a n n e r is o n l y 5 we necessary another. determine need to Figure u p o n the dipole theory, A g a i n , it moments spherical shell lateral present attention in a p o l a r dipole repel one We that an i o n ion, depending in the to turn our vector. <AE-^(R)> in employed of the for total <AE^j(R)>. dipole an e x p r e s s i o n f o r reference moments <AE^^(R)> d i p o l e , rn^(R), and a distance r from the being examined along with ion. all variables. F r o m e q . (2.5) a n d (2.17b) w e c a n s h o w t f b e t w e e n rn'(r) and m'(R) is g i v e n b y [67] that the the interaction all In the Figure 5. A n i l l u s t r a t i o n o f the m e t h o d u s e d in d e t e r m i n i n g c a s e w h e r e q . is a n e g a t i v e i o n is s h o w n . <AE^_(R)>. T h e - 93 - u m m ( ) = -m (r) t ^ i i T m 1 2 ( R ) 2 ( l 2 ) f ( 4 > 4 7 ) mm r where be w e again take written [61,81] f 2 0^ a n d 0 azimuthal always ^ a in the 4>JJ( 1 2 ) where m n s given b y e q . (2.9a). angles. = 2005^00502 - sin0 sin0 cos(</> -0 ) 2 1 1 , 2 in F i g u r e 5 a n d <j>^ a n d 0 It i s e a s y t o s e e f r o m be equal. invariant c a n form are the a n g l e s i n d i c a t e d 2 The rotational F i g u r e 5 that <j>^ a n d # 2 (4.48) are the 2 must A n a l o g o u s t o e q s . (4.34) w e h a v e the r e l a t i o n s h i p s mm = r COS0, r 2 = + R r " 2 ~ r C r O 2 S mm r R c o s ^ ' (4.49a) ^ (4.49b) and c o s « = RCOS0- r (4.49c) m mm r From the l a w o f s i n e s [165] it is e a s y t o s h o w s i n0 that = 2L|iM (4.49d) mm and sin« = R | i H i . (4.49e) mm Substituting e q s . (4.49) into e q . (4.48) a n d c o m b i n i n g this result w i t h e q . (4.47) yields Unn/R'T'*) = m ( r ™ ) ( R ) |^3rR - 2(r +R )cos0 + 2 2 rRcosV] mm . (4.50) r 4. W e n o t e that t h i s e x p r e s s i o n i s i n v a r i a n t t o s i g n ( q . ) b e c a u s e b o t h m ( r ) a n d t ' m'(R) w i l l r e v e r s e d i r e c t i o n if the c h a r g e o n the i o n i is r e v e r s e d (i.e., 0 —»18O°+0 , 1 1 The . m'(R) will 0 — > 180° + 0 ) . 2 average 2 lateral dipole-dipole interaction energy, U^^R,!:), mm a n d the s p h e r i c a l s h e l l o f d i p o l e s at a d i s t a n c e r f r o m be given b y between the s a m e i o n - 94 - U where mm t ( R ' r = ) p s^9 ' R s r ' ^ d A is e x p r e s s e d in e q . (4.36b). f o r d\p a n d <p t o it f o r 27T ( s m d0, ) u m m ( R The limits cos spherically We will symmetric also one A d ' r l p r 2 2_ 2 : s_ 2rR + R '| (4.51) are still f o r <AE^j(R)>, but n o w s . d i f | -R|<d f 0 to r (4.52) s that g s s ( R , r , 0 , ^ ) i s distributed in the s h e l l ) . a s s u m e it i s i n d e p e n d e n t o f R a n d t a k e substituting s s ( R , r , ^ ) = g. (r) s e q s . (4.50) a n d (4.53) i n t o = ggjj° <r> is . (4.53) e q . (4.51) a n d i n t e g r a t i n g o v e r d0, obtains U^ a t (R,r) = 27rr p g (r)m (r)m (R) dr 2 t s x f t i s [3rR - 2 ( r + R ) c o s 0 + rRcos fl] s i n ^ ^ 2 2 2 integration manipulation (4 2 [ r + R - 2rRcos0] The d (i.e., the s o l v e n t s are u n i f o r m i l y 9 Now ) of integration f o r <AE^j(R)>, w e k n o w A g a i n , as w a s the case 0 i f |r-R|>d _ = ' r a s w a s the c a s e 0, <t> ' 2 o f e q . (4.54) i s n o n - t r i v i a l 54) 2 t o p e r f o r m , but a f t e r c o n s i d e r a b l e (again m a k i n g u s e o f s t a n d a r d t a b l e s o f t r i g o n o m e t r i c integrals [165]) it c a n b e s h o w n that T J ^ V r ) = 2 ^ m it IJ^R^) and t t L then f o l l o w s from = 0 °° g. ( r ) m ( r ) m ( R ) [ — S * m Jdr . [ r + R - 2rRcos4>P 2 2 (4.55) J e q s . (4.52) a n d (4.55) that f o r | r-R| >d s (4.56a) - 95 - S 1 S 2rR 3 for Finally, the total m (R) and all other T all lateral dipole v a l u e s o f r, a l t h o u g h We U average obtain (R) ,p.«t(H) 2R d 2 Now ion moments only interaction is found those energy R t m ( 1 other to between e q s . (4.56) |r-R|<d r ) [ ( 2 r R ) 2 _ ( r will g + R 2 2 ) over contribute. 2 , 1 0 advantage of the fact that t h e s o l v e n t field will c o m p o n e n t s t o the a v e r a g e f i e l d e q . (4.44) w e w r i t e _ (4.57) about t h e projected v e c t o r , it m u s t b e in t h e d i r e c t i o n must r distribution o f the a n g l e , a n d b e c a u s e a l l t h e a v e r a g e dipole d J s s a v e r a g e lateral follow that of m ( R ) . A l l average to zero. Thus, analogous that U which when 2 3 m o m e n t s , m'(r), a r e d i r e c t e d a l o n g t h e i o n - s o l v e n t total (R) b y integrating shells f o r which ( r ) R-d L R-d s taking sr + d ; 3 is independent the U (4.56b) the e x p r e s s i o n = mm dipole |r-R|<d = (R) M M -m (R)<AE T L D , (R)> (4.58) c o m b i n e d w i t h e q . (4.57) y i e l d s R+d <AE (R)> = 1 D Y 3 * S 2R d S U (r)g R-d L (r)[(r +R -d 2 f i s 2 2 s ) 2 - (2rR) ]|dr . 2 (4.59) s s t We point the bulk Ap_(R). o u t that in e q . (4.59) m ( R ) w i l l average dipole F r o m e q s . (4.24b) a n d (4.31) w e f i n d m (R) w e have taken contributions m o m e n t , m ' , and the average e x c e s s T where contain = m'<cos0. due to both induced that ( R ) > + Ap(R) , advantage o f the fact moment, (4.60) that A r j ( R ) , a n d h e n c e <AE-,(R)>, t will case be n o n - z e r o only for <AE-^J(R)> along m ' ( R ) . This has already been shown to be the and < A E ^ ^ ( R ) > find this t o b e a l s o true and u s i n g e q s . (2.89) w e o b t a i n in t h e p r e s e n t for <AE-^Q(R)>. Inserting the relationship theory and below w e e q . (4.60) into e q . (4.59) - < A E 1 h s s - ' 00 -is m r ( R ) > D 96 1 , °°', s ( r ) , 1S [(r +R -d ) 2 2 Ap(r)g._(r) + 2 - (2rR) 2 s which is the desired result. Inspection contributions to c a n be other Ap(R). due to The similar D excess to consequence of that used for <AE-^Q(R)> the e q . (4.61) r e v e a l s that identified, one This separation will local quadrupole fashion to contribution the <AE^ (R)> of prove due o n l y useful (4.61) two to We will D to lateral fields. a v e r a g e p r o j e c t i o n s , ©£(r), o f distinct m ' , and in a again consider These lateral the the in d i s c u s s i o n s b e l o w . f i e l d , < A E ^ Q ( R ) > , is d e t e r m i n e d <AE^ (R)>. due dr 2 fields quadrupole very the are a moments of solvent p a r t i c l e s a r o u n d the i o n i. T h e p r o j e c t i o n s © ' ( r ) are a n a l o g o u s t p r o j e c t i o n s m'(r) and w i l l be d e f i n e d b e l o w . W e e m p h a s i z e that t h i s to S the derivation applies strictly (2.22) and illustrated First let to square quadrupole in F i g u r e as defined by eq. 2(a). us d e f i n e [67,72] $ 0 2 2 = * 1 2 3 ( 12) moments the functions $° (n ,n ,f) + * ° ? | ( 0 22 1 2 1f O f) (4.62a) 2f and *J (n! 23 (12) = ,0 2 r + #J^(fl B r) f) 1 f 2 l , (4.62b) inn 1 w h i c h , if f is g i v e n b y e q . (2.9a), c a n be w r i t t e n [81,166] in the explicit forms $ 0 2 2 $ 1 2 3 (12) - ^ [ ( x . f ) - (y .f 2 2 1 2 2 ) ] (4.63a) 2 1 2 and (12) ^6[5[(x .f 2 1 2 ) 2 - (y .f 2 - 2[ ( x - z ) ( x . f 2 Then matrix u s i n g the definition (see e q . (39)) of of 1 the 2 1 2 ) ] ( z -z ) 2 1 2 1 2 ) - ( y . z )(y -r 2 1 2 1 2 Euler a n g l e s a, p\ 7 as w e l l R e f . 166, it can be shown that ) ]j . as the (4.63b) rotation - 97 - $ (12) 0 2 2 = »/6 s i n p \ C O S 2 T = Ve (4.64a) 2 7 and $ 1 2 3 0 2) x As [cos/3 in the interaction 3cos/3 cos27 2 case of . 2 s i n /3 c o s 2 7 1 2 + sin27 cosia^-a^) 2 <AE^ (R)>, we D . 2sin0 + 2 will all a v e r a g e p r o j e c t i o n s , 0^( r ) , in a s p h e r i c a l s h e l l at a distance r from m^R). In indicating can Figure 6 w e the dipole-quadrupole e q s . (2.5), (2.19c) and interaction [79,81] is g i v e n u i Q most energetically ion i which is p i c t u r e d the moment i s the symmetry and projection of it the positive, which corresponds only is i s the case that quadrupole must the moments dipole geometrical derivation. follows % favourable z 2 / 3 will not for cos27 This moment problem, However, before first define ion-square we @ ( r). g quadrupole It that is . 2 (4.65) orientation orientation average to orientation moment. 2 quadrupole in F i g u r e 6(a). this quadrupole the (4.64b) average q. and the this interaction, we i = ^ y ^ s i n o n e that onto following of the . by (l2) is the represent (4.64a) it q It to v a r i a b l e s u s e d in the e x a m i n e the From have tried ion 2 s i n f c ^ - c ^ ) ]J 2 determine between the . sin/3 1 we zero for determine clear f r o m a water-like solvent with of the respect quadrupole m o l e c u l e s of the C 2 v average e q . (4.65) that [72], t h i s to for 0 g projection to p*2 = 90° (4.66a) and cos27 where the frame ( s e e F i g u r e 6(b)). therefore write that superscript ©g(r) I 2 i n d i c a t e s the We is a l l o w e d point to out = -sign(q^) , a n g l e s as d e f i n e d that spin freely U^Q(12) about the is (4.66b) for the ion independent axis. We reference of can and then Figure 6. An illustration case where reference of the method q. is a p o s i t i v e frames a n d their u s e d in d e t e r m i n i n g ion <AE^g(R)>. is s h o w n . F o r c l a r i t y rotations the are s e p a r a t e l y g i v e n The various in (b). 99 ©I(r) = - <* sign(q ) (r)> 0 2 2 , (4.67a) i which, when combined w i t h e q s . (4.62a) and (2.87) Now using dipole-square 022 - 0 ej(r) becomes 52Tis WO 5g™? (r> 8 h sign( ) g i ( r ) (4.67b) u e q s . (2.5), (2.17b), (2.19c) and (4.62b) w e quadrupole interaction, ^ 0 ( 1 2 ) , can show c a n be e x p r e s s e d that [67,72] the in the form u H o w e v e r , the terms m© = ( 1 2 ) ,123, $ (12), function ^ as g i v e n by ( 1 2 (4.68) ) e q . (4.64b), is not expressed in 2 b y e x p r e s s i n g the I I I cu, , $2 and 7 Now in o r d e r must rotate apply the unit vectors x\~, y associated with 2 to go about rotation from the the y^ the ion axis b y matrix and 9 ion reference terms of vectors the = 0 (given associated with expressions choice of of into the 2 and two (see ^ in A p p e n d i x two to the Euler angles (see A p p e n d i x the frame C). (x,y,z), Therefore we we Appendix 0 e c a (4.69) 0 COSU. n a for xV^, y C) between different f r a m e ) , and - s i x\co' ' s o e x P and 7 2 , as d o n e 2 forms e q . (4.64b), f o r reference simplification the frame frame 1 0 E u l e r a n g l e s a ^ , /3 components relationships xV^, y of an a n g l e co (see F i g u r e 6(b)). .sincj unit in t e r m s [166] R the z~ reference COSCJ the * o f a n g l e s s h o w n in F i g u r e 6. T h u s , b e f o r e p r o c e e d i n g w e m u s t f i r s t 123 I I $ (12) in t e r m s o f t h e s e a n g l e s , i.e., p^ , 7 , P^ and co. W e b e g i n write to 4 m© i/6 r, reference which after C) we we 2 two frames. take considerable obtain e s s * > 2 Y in A p p e n d i x and ^ the r yields sets of We =0 a r , 2 d C. 2 in Equating several Euler substitute (this z follows manipulation and angles these from our - $ 123 100 - T 2 2 = v/6 3cos/? ( c o s u> - s i n (12) 1 1 2 s i n u) 2 + 2sin/3 sinco C O S C J ( 1 + s i n 1 Finally, w e replace (4.70) b y n o t i n g b y 0^ a n d e l i m i n a t e that I "I 02 ) the c o s 2 7 I cos27 . 2 dependence 2 (4.70) in e q . the products cos/3 1 cos27 2 = cos0 1 (4.71a) sin/? 1 cos27 2 = sin0 1 (4.71b) and are independent yields of the sign the necessary result et(r) m (R) T u J Tn0 v o f q . . C o m b i n i n g e q s . (4.68), (4.70) a n d (4.71) 1 2 ) = 3cos0 s ' r 4 m0 + We have quadrupole cannot from already 2sin0 pointed 1 sina> cosco( 1 probable (integrating) over advantage (at t h e m e a n them. After ^ ^ 4 = r field of the fact . (4.72) o f the that a l l a n g l e s , l e v e l ) and s i m p l y performing this r cos0 1 in t h e c a s e , 2 sin0 1 w e employ e q s . (4.49) i n o r d e r to express functions 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 o n e h a s t h e result s i n w) m0 of <AE^j-j(R)>, , are angle-averaging integration 2 ( c o s (J + As 2 T T 12 I 2 o u t that t h e a v e r a g e p r o j e c t i o n 3m (R)©Jr) m0 +sin a )j is a l l o w e d t o spin freely about the z a x i s , and h e n c e w e I I the angle. However, w e can remove the dependence e q . (4.72) b y t a k i n g u 1 2 s i n w) moment specify equally 2 2 ( c o s GD - s i n 1 1 sincucoscj relationships . (4.73) analogous to o f 0^ a n d co a s f u n c t i o n s o f r, R a n d - 101 - <*> u m© 31^(11) rf(r) r ( R ' r ' 0 ) =T = 2 R |_ m0 3 " 5 " 1 2 3 <f r R + r ) c o s 0 Z 3 r (r R+|R )cos 0 2 + The average lateral found by over d^ j dA. U s i n g the 2r R - s is ^0^ ^ K R energy, r (4.74) ' s again (4.53) a n d i n t e g r a t i n g approximation 2 £ s - (f r R + r ) c o s 0 1 is g i v e n b y m requires a great trigonometric in the eq. deal of integrals 2 3 £—_ 2rRcos0] [ r +R </> (r R+f R )cos20 + 3 *m written . 2 2 where 3 6*r p q- {r)d (r)J{R)ar = 2 x over interaction - ^|-cos 0 2 o n e has ^ ( R r r ) v quadrupole-dipole integrating 2 3 2 - ^5-cosV ± sin0d0 , (4.75) 2 (4.52). E v a l u a t i n g the i n t e g r a l o v e r d<p in e q . (4.75) e f f o r t , but with and after [165] the aid of tables of much simplification standard the result can be form U ^ ^ r ) - 3*r p g (r)0t(r)mt(R) 2 s is 2 r x | ( r-Rcostf>) (1 -cos 0 )- Applying eq. ! ^ | 55 [r L 2 2rRcos0] R" 2 + . r (4.76) (4.52) i m m e d i a t e l y y i e l d s LT^I^R, r) = 0 f o r |r-R|£d and . . 7 2 3 , r p g s m (R)0jr) ( r ) - - ^ - [ l r i s (4.77a) s . T I&W) d 2 s rr +R -d 1 l ] 2 - 2 2 2 d dr We remark lateral that t h e s e e x p r e s s i o n s bear s t r i k i n g dipole result. f o r |r-R|<d similarity to . (4.77b) e q s . (4.55), the - The found total average b y integrating shells). m0 R interaction energy (i.e., all values o f r is again summing over Explicitly, w e obtain R + d r ©t( ) T ( quadrupole-dipole e q s . (4.77) o v e r 3rrp_m (R) U lateral 102 = ) r V-5-J" H H ^ i s ^ s R - d ^s 8R'd^ x [(2rR) One can use arguments that < A E - ^ Q ( R ) > will very also (r +R -d ) ](r -R +d )]dr 2 - 2 2 2 2 2 2 s similar only 2 . s to those used f o r be n o n - z e r o to show < A E ^ Q ( R ) > in the direction of (4.78) m ^ ( R ) , so we write U Finally, the combination <AE, (R)> = i Q Q 3TT R + 0 S_s 5sign(q. ) R ^ d x is interesting = ( R ) - m T ( R ) < A E 1 . ( R ) > Q (4.79) o f (4.67b), (4.78) a n d (4.79) g i v e s t h e d e s i r e d P • It M that 2 s 2 i s 02, is R-d L b c o [(2rR) t o note s ; L d rh22 (r) r o - (r +R -d ) ](r -R +d )]dr 2 2 2 2 2 2 2 s both and < A E ^ Q ( R ) > 1/R dependence. appearing < A E 1 + t o be the result ( R ) > We Given < A E n o w have 1 of effective However, since as given ( R ) > . O n c e the total of In Chapter potential, the present terms, u f ^ ( R ) , study II, t o d e t e r m i n e to <AE^(R)>. < A E ^ j ( R ) > and u s i n g e q s . (4.46) a n d (4.80), r e s p e c t i v e l y . A p ( R ) , excess contributing functions, b y e q . (4.61), m u s t which local field, d e t e r m i n e d , e q s . (4.26) a n d (4.29) w i l l ion-solvent terms o f the correlation directly it d e p e n d s o n t h e v a l u e D screening charges, analogous to the case f o r e x p r e s s i o n s f o r all three can be evaluated < A E ^ ^ ( R ) > , as additional an a p p a r e n t _ ( R ) > . the n e c e s s a r y projections < A E ^ Q ( R ) > < A E ^ and c a n be v i e w e d (4.80) as given by < A E ^ Q ( R ) > , (4.61) a n d (4.80), r e s p e c t i v e l y , c a n b e p e r c e i v e d a s h a v i n g Thus, they . s eqs. 2 result in turn d e p e n d s < A E ^ ( R ) > , immediately f o r the R D M F be f o r s o l v e d give iteratively upon has been us the e f f e c t i v e theory. w e employ the RHNC t h e o r y , a s d e s c r i b e d in the correlation functions for a solution. These - correlation which functions in turn functions. of with system using principle, this and m g infinite the value symmetric from potential behaviour contributing First to In o r d e r w e will examine at p 2 Clearly, w e then 0 2 ; i s constant. have N n o w turn h a v e that investigate for A p ( R ) . In values W e point ' s of m ' , a' o u t that at solvent. spherically a to the l o w c o n c e n t r a t i o n this possibility w e need to d e p e n d e n c e s o f the t e r m s R behaviour ( r ) 1 Q / that at i n f i n i t e ( R ) > = 0 r 3 a insert of s r - <AE^Q(R)>. It this ™ ' > form the i n t e g r a l c a n be (4.81) for evaluates as R — our attention behaviour to a n d K—>0, into t o an i d e n t i t y . 0. tail, becoming (4.82) zero r behaviour, and consequently limiting 022 • (r) U2; l s dilution has no long-range i t s large a s r—>°> a If w e t h e n that the l o w c o n c e n t r a t i o n We we <AE, (R)> attains t h e large one finds < A E to ^g(R) a contribution to II, f o r several values o f the o f t h e pure u 5 of Chapter =0 e q . (4.80) a n d i n t e g r a t e , # that because it m a y m a k e estimate approximation. R and l o w concentration a is s o m e hQ2 ^g(r) is j u s t is s o l v e d in s e c t i o n the c u r r e n t system we <AE^(R)>. [61] that Therefore, 022 g correlation For a fixed set theory the s e l f - c o n s i s t e n t the S C M F m evaluate be s o l v e d in manner. outlined be r e p e a t e d e q . (3.45) that term h where of of C- e x a m i n e the l a r g e method to allow to Therefore, for a given a n d a , the R H N C g then using must in the f o l l o w i n g m dipole, m to be evaluated It f o l l o w s shown must theory p o t e n t i a l , and hence the theory c y c l e w e update calculation dilution limiting theory the i t e r a t i v e iterative permanent pair approximation. parameters, including still in t h e R D M F a b o v e , the R D M F the S C M F n o w at e a c h effective be used the i o n - s o l v e n t the S C M F / R D M F / R H N C numerically but modifies A s mentioned conjunction solve c a n then 103 - of as s o o n as it c a n n o t contribute U^(R). <AE^j(R)>. From e q s . (3.34) a n d (3.35a) - K is g i v e n where expand b y e q . (3.35b). and integrate small K limit <AE (R)> 104 - W e insert (see A p p e n d i x B). Then collecting e q . (4.46) a n d terms and a p p l y i n g the yields sign^ ) 1 X e q . (4.83) i n t o 4*p = D= ," ^(R -d, ) 3 j q j 3 + 2e k T + 2i-5l[e k R KekT If w e n o w u s e e q . (3.35b) P , w e obtain (R+1//C) and charge neutrality, I*' <AE (R)> - - 2 1 1 1I note r e p r e s e n t s the R — a n d that 1/ ]1 K (4.84) J dropping any terms linear in the i n f i n i t e limit , KR limiting K—>0 dilution e {^+KR) behaviour of (4.85) <AE^j(R)>. (4.86) 1I is c o n s i s t e n t with e q . (4.30). In c o n s i d e r i n g the l o w c o n c e n t r a t i o n we find parts; upon that it c o n v e n i e n t <AE^ (R)> M Ap(R). to split due o n l y First w e w i l l as p — ^ - 0 2 We {i.e., / c = 0 ) e q . (4.85) b e c o m e s <AE (R)> which in the r e s u l t 2 which - limiting behaviour of <AE^Q(R)> < A E ^ ^ ( R ) > , a n d h e n c e e q . (4.61), into t w o t o m ' , and <AE^p(R)> examine <AE-^ (R)>. M directly From dependent e q . (3.47) it only follows a n d R—>°>, iQfc I r _ n R + cL r (4.87) If one performs B), then the integration it c a n b e s h o w n that <AE (R)> = lm in e q . (4.87) a n d c o l l e c t s t e r m s in the l i m i t -2|qil 3R" K—>0 e k R (see A p p e n d i x and R — (1+/CR) . (4.88) - 105 We remark that e q s . (4.85) a n d (4.88) a r e v e r y that at l o w c o n c e n t r a t i o n and long will be c a n c e l l e d b y the lateral dielectric c o n s t a n t , e , i s large indication of h o w poor similar. range n e a r l y dipolar field an a p p r o x i m a t i o n t w o thirds due t o m (as is the c a s e It i s c l e a r l y the c a s e of the i o n f i e l d w h e n t h e pure g for water). This solvent is an o b v i o u s e q . (4.1a) i s f o r a q u e o u s electrolyte solutions. In o r d e r to determine the l o n g - r a n g e l o w concentration behaviour of <AE^p(R)> w e m u s t k n o w Ap(R), but Ap(R) d e p e n d s u p o n <AE^(R)>, which in turn d e p e n d s u p o n (R)>. T o o b t a i n <AE, a first approximation for ™ (1 ) Ap(R), w h i c h w e w i l l d e s i g n a t e a s Ap (R), w e u s e e q s . (4.22b) a n d (4.26), c o m b i n e d w i t h the results A p T h i s result c a n then which performing , = _lL^irii2] -«R 3R^ L - I e be s u b s t i t u t e d P p follows ( R e q s . (4.82), (4.85) a n d (4.88), w h i c h into from s 2 e q . (4.61) w h e n the n e c e s s a r y . ( 1 + K R ) 2 ( ) - (2rR) ]ldr J 2 2 S 2 g . (r) = 1 ( v a l i d integration yield 4 . 8 9 ) the e x p r e s s i o n R+d * S_ J [Ap(r)[(r +R -d 2R^dJ* R-d L s s <AE, (R)> = i (i) from and simplifying at large r). , ( 4 . 90) After (see A p p e n d i x B ) , o n e has that <AE (R)> l p =^[il2](-j) - ( l ) e where n o w c o m b i n e this obtain a second estimate Ap ( 2 ) (R) t 3 result w i t h namely = ^ I f l l ^ l e ' ^ d + z c R ) d-£) L e (4.91b) e q s . (4.82), (4.85), (4.88), (4.22b) a n d (4.26) t o for Ap(R), 3R^ (4.91a) <1+icR) , 8 7T p a. = — . i We K R J By repeating the above procedure t o obtain of Ap(R), it i s p o s s i b l e t o s h o w that higher . a n d higher (4.92) order estimates - Ap (o5) 106 - (R) = -I^ir.£l2]e3R L J KR e Z ( R)( ^^ -^ ...) . (4.93) 2 1+K 3 1 + For | £ | < 1 (i.e., f o r ap <3/87r), w e r e c o g n i z e the last f a c t o r the Taylor series expansion for 1/(1+£). W e note that i n e q . (4.93) a s for water at 2 5 ° C , 1=0.403. T h e r e f o r e , w e h a v e the s e l f - c o n s i s t e n t result Ap(R) = [lllle-^d+KR) (3+87rp a)R^ L J . , i!f!L! (4.94) e s which will be valid only at l o n g range a n d l o w c o n c e n t r a t i o n . Let u s n o w c o n s i d e r the a v e r a g e e x c e s s infinite dilution l o c a l f i e l d , <AE^(R)>, at (i.e., p^ =0) f o r t h e s p e c i a l c a s e w h e n the s o l v e n t i s polarizable but non-polar. In s u c h a c a s e m'=0, a n d f u r t h e r m o r e <E_ > = <AE^ (R)>=0. S t a r t i n g w i t h an i n i t i a l g u e s s f o r <AE^(R)> a s g i v e n 1 m by e q . (4.86), a n d u s i n g e q . (4.90) in t h e s a m e we c a n s h o w that f o r this iterative scheme outlined above, s y s t e m at large R |q. | <AE,(R)> = —4R 1 Substituting 2 s . (4.95) s the C l a u s i u s - M o s o t t i r e l a t i o n s h i p °° „ = ^irp_a e +2 J 3 + 87rp a [139], , (4.96a) OO into e q . (4.95) i m m e d i a t e l y y i e l d s the r e l a t i o n s h i p <AE,(R)> = l i - i ^ l 2 . R 3e oo (4.96b) f 1 where i s t h e high f r e q u e n c y polarizability. system. dielectric exactly theory predicted compared with reported obtained from essentially t o molecular computer in this M o r e o v e r , P o l l o c k etal. [167] f o u n d a different result. W h e n the t w o e x p r e s s i o n s values f o r the average simulation exact results, while effects the s a m e e x p r e s s i o n f o r t h e large R d e p e n d e n c e o f the a v e r a g e l o c a l f i e l d . were due o n l y P o l l o c k etal. [167] h a v e a l s o s t u d i e d p o l a r i z a t i o n They obtained that c o n t i n u u m constant local field [167], e q . (4.96b) w a s f o u n d the c o n t i n u u m at large R t o give expression w a s only accurate f o r - small 107 - values of p a . g It dilution is a l s o when (4.88) it interesting the s o l v e n t to examine is polar is e a s i l y s h o w n that the b e h a v i o u r but not polarizable. R 1 low it point to affect it limiting the l i m i t i n g B ) , that (4.97) in f o r m to the s a m e extent indicating in e i t h e r that the f i e l d system provided constant. behaviour behaviour as Q] i ^ ^' r s of C- anc * T , 1 U S W E W O U L C L E X P E C T U s i n g e q s . (3.39b), (3.45), (4.29) a n d is a g a i n p o s s i b l e t o s h o w , after Appendix F r o m e q s . (4.86) a n d Ap(R), as g i v e n b y e q . (4.94), h a s that s a m e l a r g e - R out that concentration (4.94) b e screened h a v e the s a m e d i e l e c t r i c We infinite . 3e C u r i o u s l y , e q s . (4.96b) a n d (4.97) are e q u i v a l e n t that t h e y <AE^(R)> at at large R <AE,(R)> = iSi due t o a c h a r g e w i l l of c o n s i d e r a b l e m a n i p u l a t i o n (see p^—>0, as 2 ( 3 ++8 7 r p _ an) ) 1 S where is the c o n t r i b u t i o n IS define Cjg (as g i v e n S c " as being J J t o C- 2 is due o n l y to e q . (3.35c) F u r t h e r m o r e , it f o l l o w s P - = from If w e t o the t o t a l t o u^P(R), then limiting slope, S combining of e q s . (3.43) yields 1 2 ^ ^ e q . (3.41) that s l o p e o f C j g , then u^?(R) a n d fl = 1 A T . IS the c o n t r i b u t i o n in e q . (3.43)) d u e o n l y A limiting € IS and (4.98) a l o n g w i t h of L - now the l 2 it m u s t ^ if also t ( 4 ; g g ) u^P(R) m a k e a c o n t r i b u t i o n t o influence the l i m i t i n g behaviour G.„=G „. T h e r e f o r e , u^?(R) w i l l a f f e c t the l i m i t i n g l a w s o f a l l +S - s thermodynamic ion-solvent t h e o r y , i.e., properties correlations that t h e v a l u e long-range i s of S A p of electrolyte solutions at l o w c o n c e n t r a t i o n which have a dependence on (e.g., N^). W e a g a i n e m p h a s i z e s h o u l d b e a c c u r a t e l y g i v e n b y e q . (4.99) s i n c e the l o w concentration behaviour o f Ap(R) e q . (4.94), s h o u l d b e an e x a c t result predicted b y the R D M F at the m e a n f i e l d level. - 108 - CHAPTER V RESULTS FOR WATER-LIKE MODELS 1. Introduction In C h a p t e r We will report described the full find II w e h a v e d e f i n e d t h e w a t e r - l i k e results obtained in C h a p t e r s C 2 v water quadrupole II a n d IV. F o r t h e m o s t t o refer moments octupole m o d e l using the R H N C / S C M F m o d e l , w i t h and w i t h o u t it c o n v e n i e n t t o the C a s the will be used when quadrupole) w a s [67,168]. in o u r s t u d y discussed discuss find the octupole (i.e., a m o d e l simplified o f model model o f the RHNC basis sets theory basis set was a dipole in e x p l o r i n g of non-linear used t o obtain A modified n m a x present virtually results version all results (as w i l l b e the b a s i s s e t obtained o f this a reasonable incorporates potential, will 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 the and dielectric structural properties o f the model Hence, in s e c t i o n s f o r the multipolar model, which W e will chapter. W e requirements. reported work as the symmetry. = 4 represents and a previous extensively solutions convergence and computational In s e c t i o n 3 w e w i l l models. employed C^^ included. comparison with useful for models only upon dipole and a n d b a s i s s e t d e p e n d e n c e in s e c t i o n 2 o f t h i s between focus W e will only has also been aqueous electrolyte that t h e b a s i s s e t c o r r e s p o n d i n g t o compromise will moment. includes containing was in C h a p t e r V I ) . It a l s o p r o v e d dependence which which w a s study the octupole model y part, this also examined to allow M o r e o v e r , this solvent 2 theory investigate. quadrupole m o d e l , w h e r e a s t h e t e r m H o w e v e r , t h e tetrahedral m o d e l square models w e will systems this 3 and 4. hard-sphere a soft focus spherical mainly on being investigated. In t h i s temperatures calculations w e have examined w a t e r - l i k e and pressures. were experimental phase study done densities. diagram o f liquid are given water in p r e v i o u s w o r k dielectric constant refer in T a b l e The points examined shall The temperature different at w h i c h II, a l o n g w i t h t h e c o r r e s p o n d i n g large portion chosen to correspond with Of importance h a s b e e n m e a s u r e d [48,169,170] to points at s e v e r a l and pressure points sample a relatively and were [67,168]. models at 1 a t m . o r at v a p o u r here is the fact o f the those that t h e at a l l t h e s e p o i n t s . W e p r e s s u r e a s b e i n g at normal - TABLE II. 109 - E x p e r i m e n t a l d e n s i t i e s of w a t e r f o r p r e s s u r e s e x a m i n e d in this s t u d y . Temperature the Pressure Density ( 9 / m I) 1 atm. 1 atm. Vapour Press. Vapour Press. Vapour Press. 5000 bars 5000 bars 5000 b a r s 5000 bars 0.99707 0.9653 0.865 0.710 0.452 1.106 1.051 0.993 0.931 (°C) 25 90 200 300 370 100 200 300 400 pressure, while those at temperatures 5000 bars s h a l l be r e f e r r e d and [169] [48] [48] [48] [170] [170] [170] [170] [170] to as high p r e s s u r e points. In the Chapter present models experimental models. average [171] were The have outlined through value of the individual the polarizability experimental The f o l l o w i n g tensor the 3 tensor of the Murphy were by [164] components may be A g a i n , it which we cm , reported results values of polarizability approximation. c o m p o n e n t s of -24 to of have included is in an u s e d in determined the using the E i s e n b e r g and K a u z m a n n find the their relative polarizability tensor obtained: = 1.501X10~ cm , a 2 4 X A. a a This form note that energies for the given calculation. the The In our convenient to y zz = 1.390X10 -24 = 1.442X10" tensor w a s SCMF by y results dielectric 3 3 cm , 2 4 cm . 3 employed we will S i m p s o n ' s rule (2.93), a g a i n e m p l o y i n g we how SCMF p o l a r i z a b i l i t y , a = 1.444X10 and the values. IV w e in all report S i m p s o n ' s rule e x p r e s s all p a r a m e t e r s will to c a l c u l a t i o n s , a n d in s o m e s h a l l c o n s i d e r c a n be t o t a l l y are t h o s e d e t e r m i n e d integration constants we of SCMF calculations. of e q . (2.80) are give perform the were the in r e d u c e d u n i t s . characterized by the also when the u s e d in the determined required following We through integration. d i s c u s s i o n , it The w a t e r - l i k e following eqs. is fluids reduced - 110 - parameters: p * = d* d model. It c l e a r l y (5.1b) d <0M /d 0* = (/3© /d n * = ( ^ / d g = a / d =2.8A s (5.1a) = * and d ' 3 M a /3 = 1 / k T s " / s ' * where pd follows 2 5 2 3 s 3 s s 7 )* )2 , (5.1c) , (5.1d) ) ^ (5.1e) , (5.1f ) , is the h a r d - s p h e r e from diameter * e q . (5.1b) that d = 1 . o f the w a t e r - l i k e 2. C h o i c e o f B a s i s S e t In t h e p r e s e n t which is computationally properties unique study it is i m p o r t a n t terms determine an H N C b a s i s s e t p r a c t i c a l , y e t g i v e s reasonable c o n v e r g e n c e f o r the w e shall consider. projection to first which In T a b l e must III w e h a v e p r e s e n t e d t h e n u m b e r o f be i n c l u d e d in the H N C b a s i s s e t s f o r both C» a n d t e t r a h e d r a l m o d e l s f o r n =2,3,4,5,6. In T a b l e IV w e h a v e 2v max ' ' ' ' e x p l i c i t l y g i v e n t h e unique p r o j e c t i o n s f o r n = 2 for both m o d e l s . A set of TABLE III. N u m b e r s o f unique p r o j e c t i o n t e r m s r e q u i r e d in H N C b a s i s s e t s . Both tetrahedral and C „ m o d e l s are c o n s i d e r e d . max 2 3 4 5 6 Tetrahedral Model C 2 v Model 19 49 130 262 532 TABLE IV. P r o j e c t i o n t e r m s ' Model i n c l u d e d in n # of Unique Projections Tetrahedral '2v max =2 basis sets Projections Included (mn\;uv) 12 (000;00),(022 00),(110 00),(112;00), (121;02),(123 02),(220 00),(220;22), (222;00),(222 22),(224 00),(224;22) 19 all t h o s e a b o v e p l u s (011 ;00),(022 ;02),(121 ;00),(123 ;00), (220;02),(222;02),(224,02) TABLE V . M a x i m u m numbers of n o n - z e r o terms for any given projection in the H N C b i n a r y p r o d u c t . V a l u e s are g i v e n f o r e a c h b a s i s s e t i n c l u d e d in T a b l e III. unique projections symmetry is as n tetrahedral general for is Tetrahedral Model 2 3 4 5 6 28 150 1000 3200 12000 c o n s i s t s of requirement obvious from rapidly max max Table is model model C 2 v linear In T a b l e V w e HNC binary projection number terms III that the symmetry m a x = 6 in the very double rapidly of number than unique of half be related by might consist of far projections terms the A s one some number fewer (2.23). grows required by ' It very the needed by e x p e c t , the maximum double sum as n sum will cannot a HNC basis sets projections when n m a x [71,110]). in a s p e c i f i c b a s i s set grows the h a v e r e c o r d e d the p r o d u c t , or which number more symmetry. n Model v m o d e l , i.e., e q s . (2.11), (2.12), (2.14), or increased, with large (e.g., 8 4 f o r in the the 2 48 225 1800 6200 all t e r m s being slightly of systems of of C (cf. number of e q . (2.67)), that can ever have. A s we non-zero any would terms given expect, this is i n c r e a s e d s i n c e the t o t a l n u m b e r m 3x i n c r e a s e as the s q u a r e o f the n u m b e r of of - 112 - T A B L E V I . C P U t i m e r e q u i r e d per i t e r a t i o n o n T i m e s are g i v e n f o r e a c h b a s i s s e t are in C P U s e c o n d s . T h e v a l u e s in c a l c u l a t i o n s w h i c h l i m i t the n u m b e r b i n a r y p r o d u c t (as d i s c u s s e d in the Tetrahedral Model max 2 3 4 5 6 projections Tables III terms are general in the and V model 2.5 7.0 30 90(80) 470(330) basis set. of 2 RHNC set (i.e., the theory the time for savings smaller report study. small t e r m s b a s i s set of the values given c o n c l u d e that m o d e l , and t h i s Chapter Also in the figure basis sets. times at least rises to the required complete II). product n =6 for one V a l u e s are i n c l u d e d in the binary e x a m i n e d , the ' (~30%), a l t h o u g h the required) to (see s e c t i o n 5 o f when largest we C P U time c o n s i d e r e d in t h i s result 3.0 9.5 50 (230) attempted in 85% of these 90% for a symmetry. y F i n a l l y , in T a b l e VI processor Model v not l e a d s us t o a tetrahedral C 2 However, inspection immediately zero for C an F P S 164 a r r a y p r o c e s s o r . i n c l u d e d in T a b l e III and p a r e n t h e s e s are f o r of t e r m s c o n s i d e r e d in the text). an F P S 164 full given table iteration for are the are i g n o r e d . tetrahedral there of the each basis times We array which s e e that for is a c o n s i d e r a b l e max amount of C o n s e q u e n t l y , this time saved b e c o m e s much truncation procedure was smaller extensively u s e d f o r c a l c u l a t i o n s i n v o l v i n g o n l y the t w o l a r g e s t b a s i s s e t s (i.e., the n = 6 t e t r a h e d r a l and the n = 5 C» ). It s h o u l d a l s o be n o t e d that there max max 2v y was of no this detectable additional The constant, to The those there n m a x e , the of of the the w a t e r - l i k e basis set Fortunately, we the RHNC sets models obtained results contact for value the of fluid being examined were at 25°C. dependence when that there as a result product. a s p e c i f i c tetrahedral model find solution HNC binary a v e r a g e e n e r g i e s and the parameters of the dependence of function, g(r=d), for is s t r o n g = 4. truncation b a s i s set distribution VII. c h a n g e o b s e r v e d in the It is o n l y from slight the radial is g i v e n c h o s e n to is o b v i o u s going dielectric n from in be Table similar T a b l e VII that = 2 to n = 3 to m sx m sx basis set dependence - 113 - T A B L E VII. B a s i s s e t d e p e n d e n c e o f e , the tetrahedral fluid f o r which average energies and g(r=d). A p*=0.7317, M * = 2 . 5 0 a n d 0 * = O.94 i s g considered. nm a x I 2 3 4 5 6 | 89.1 83.7 66.5 66.4 66.2 -IL^/NkT | 9.19 9.42 9.17 9.15 9.14 -U D Q /NkT| 5.93 6.49 7.01 7.06 7.07 -U Q Q /NkT| 1.49 1.50 1.76 1.76 1.79 10.14 11.24 11.95 12.07 12.16 1 6 9(r=d) | with the n = 5 and n = 6 systems. max max For 1 Table V I I . there results. similar The required 2% difference r between basis set dependence w e observe t o that [71,110]. i s l e s s than all the properties reported for models C o n s i d e r i n g the t o converge fact that a solution with 100, a n d and tetrahedral linear sometimes set f o r a given c o n s i d e r e d in . the n = 4 and n =6 max max f o r this dipoles r model model is quadrupoles m o r e , iterations are s y s t e m , the n m a x = 4 basis s e t w o u l d s e e m to be a reasonable c o m p r o m i s e between computational r e q u i r e m e n t s a n d a c c u r a c y . T h e r e f o r e , the n = 4 basis set w a s used ' max e x c l u s i v e l y t o o b t a i n a l l t h e r e s u l t s p r e s e n t e d in s e c t i o n s 3 a n d 4 . M 7 One observation when given mn 1 (r) b e c o m e s very less than 10~ . 3 be t r u n c a t e d sets Thus one done = 6 . and t h e o t h e r ' obtained solution with were not represent Upon re-examination the values (i.e., w h e n f o r large might expect was different (i.e., w h e n values d o n o t d e c r e a s e f o r large (2.9a)) reasonably these basis sets the required j of h s given could condition. T w o m o d e l ; one w a s cases (in a n u n s y s t e m a t i c f -'- b y eq. that n „ =4, max the solution manner) f r o m the of the basis a p p r o a c h f o r t h e m o d e l s w e are mn i s that v a l u e s are o n 1. C l e a r l y , t r u n c a t i o n o f the contact representation basis sets ^ js g i v e n n = 4 ,1 = 4 . In b o t h max ' max markedly a viable f m n 1, e.g., f o r 1>7 a l l c o n t a c t using the same tetrahedral n o truncation does [101-103] small f o r t h e larger o n 1. E q u a t i o n (2.48b) r e p r e s e n t s calculations were max can be made in Cartesian representation h 1 which ( r ) , this set o n 1 considering. time in B l u m ' s b y e q . (2.9b)), w e find that 1. O b v i o u s l y , it i s t h e 1/1! f a c t o r in - 114 - the Cartesian representation these systems m a point = 2,3,4) w a s x density The the same reduced quadrupole one would by the a C moment expect, we tetrahedral for a C dipole 2 was quadrupole v large 1 terms f o r fluid at the same a s the above tetrahedral taken as being behaviour m o d e l , as given the b a s i s s e t d e p e n d e n c e (i.e., f o r moment found causes small. o u t that p a r t i a l determined and w i t h As ( r ) which to appear t o b e c o m e Finally, we n of h very that similar in Table VII. o f water t o that reduced model. at 2 5 ° C . demonstrated The b a s i s s e t dependence o f 2v o c t u p o l e f l u i d f o r nm„a x— A andmna x = 5 w a s a l s o e x a m i n e d . W e r 0 a slightly stronger in t h e m a g n i t u d e tetrahedral octupole dependence o f the total system. moment and h e n c e (e.g., about This average energy) is not a surprising t o i n c r e a s e the i n c r e a s e their a 4% drop magnitudes in e and a 2 % i n c r e a s e than w a s result found the case f o r the since w e would o f t h e higher order expect the projections, importance. 3. Results f o r H a r d - S p h e r e M o d e l s Calculations temperatures u s i n g the and p r e s s u r e s studied at a l l p o i n t s , w h i l e normal p r e s s u r e at 2 5 ° C with a tetrahedral previous work model a p p r o x i m a t i o n , are Yet, found which dipole obtained ranges. moment here fluid 2 v where II. T h e C octupole and 3 0 0 ° C . fluid carried 2 v was The c a l c u l a t i o n 0 *=0.94, this moments, m', value o u t at a l l t h e quadrupole examined at 2 5 ° C being model was o n l y at was repeated consistent was theory. at 2 5 ° C increases with o f m ' i s still The a d d i t i o n with gave examined In the over studies present find SCMF that m ' increasing well above o f the octupole average agreement earlier w i t h the surprisingly, we i n c r e a s e i n the in s t r i k i n g model but t h e value o f water. a slight are as d e t e r m i n e d in F i g u r e 7 . N o t H o w e v e r , these o f the RHNC tetrahedral the C temperature the tetrahedral pressure instead shown t o cause only results in T a b l e increasing temperature, e v e n at high permanent listed were [67,72,168]. The a v e r a g e d i p o l e decreases with n = 4 basis set max dipole with previous the that study, we moment. The work employed found identical of the moment is [67,168] in same temperature [67,168] values f o r m ' a l m o s t pressure. the that t h e to those and RLHNC - The mean dipole and p r e s s u r e . T h e lines are the moment values SCMF of of results water. The normal the for the soft o p e n s q u a r e s are pressure, while C„ model the - Figure 7. water-like stars particles moments the p r e s s u r e , r e s p e c t i v e l y . The d o t t e d of 115 C 2 y are in D e b y e s . The quadrupole line represents values obtained represent d i s c u s s e d in the as a f u n c t i o n next SCMF model the for C 2 y normal the dashed and dipole octupole at temperature s o l i d and permanent results section. at of fluids high moment at same points for - 116 - 117 obtained for determine for ice the at or are At consistently RLHNC studies e = 105 f o r the even though u s u a l l y the 25°C of real w a t e r structural dielectric the different model [67], w h e r e U s i n g the theory that o f fluids become with experiment low t e m p e r a t u r e , i.e., 25°C, affects its will be e x a m i n e d in d e t a i l easily demonstrated. moment C 2 v [49] to If be s i m p l y in s e c t i o n polarizability we the ignore f l u i d at 2 5 ° C . and T I P 4 P [41] [45,46] t o Both models It models, two give dielectric appear m o r e remark model system. present At it radial unique RHNC give Later in that w e the obtain good at lower packing structure of water properties. This at both not dielectric this models temperatures, but the distribution s t r u c t u r e d , and h e n c e The fact The to hard-sphere higher this fluid), same. this (i.e., the earlier with quadrupole v models in that expect at hypothesis 4. in the present water-like and take the obtain fluids c a n be effective dipole e =28.4 f o r is a l s o interesting to popular water-like m o d e l s , have constants of obtained respectable. point out the that the M C Y recently 34 and 5 3 , r e s p e c t i v e l y , at are n o n - p o l a r i z a b l e and h a v e d i p o l e fluids 2 high obtain would polarizability c o m p a r i s o n w i t h these results, those hard-sphere case theory, we C gas phase value, then w e quadrupole shown the results and good agreement e at higher t e m p e r a t u r e s the of the we the less temperatures, normal e s s e n t i a l l y the significant. s u g g e s t s that importance for being hard-sphere not RHNC real w a t e r . strongly The was for was for become less for at f i n d that the packing structure RHNC from at higher data, both model constant [37]. are c o n s i d e r i n g g i v e that t h i s moment we calculated model c o n s i d e r e d in R e f . 67 (we dipole b y the differences temperatures f i n d that remark v a l u e , e =97.4, f o u n d s h o w that a n d our agreement We models we 25°C. fluid moments polarizable s y s t e m s accurate theory, therefore the predicted We e v e n at the more is quite the with experimental tetrahedral effective section we will function) the than estimate in F i g u r e 8. e. tetrahedral the for temperatures, however, we found is larger polarizable water-like obtained agreement of The a v e r a g e d i p o l e w i t h v a l u e s that have b e e n hard-sphere overestimate was better simple well another constants lower experiment value for are s h o w n in g o o d pressure. is reported here system. compare very > 1 0 0 ° C , the which quadrupole v dielectric investigated i.e., 2 25°C [172] The C moments here at 25°C of for about our 2.2D. been 20°C. In polarizable - The dielectric temperature quadrupole and s t a r s constants of water and p r e s s u r e . The fluids refer lines represent respectively. at to normal the same experimental 118 - Figure 8. and o f dots and t r i a n g l e s and high models values water-like are models SCMF as f u n c t i o n s results for pressure, r e s p e c t i v e l y . The open as in F i g u r e [48,169,170] at 7. T h e normal solid and and high of C2 V squares dashed pressure, - 119 - - In g e n e r a l , w e f i n d quadrupole We to have model causes also found the m o d e l and C 2 octupole v The t o t a l these This terms which repulsive fluids a n d the C 2 W e point the e x p e r i m e n t a l reference The e f f e c t s For the t w o m u l t i p o l a r minimum evident from 2d s > 1.65d If w e c o m p u t e v -16.7. some o f the and s h o r t - r a n g e W e remark at 2 5 ° C is a g a i n 2 v Both also and C 2 y octupole distribution function be the R L H N C result are c l e a r l y evident. o f g(r) h a s i n c r e a s e d the position is a very sharp o f the f i r s t first peak in g(r), a s has also sharpened and shifted However, its maximum f o r real w a t e r corresponding the c o o r d i n a t i o n in Figure 9 . v c l o s e t o the S C M F interactions value 2 o f the o c t u p o l e quadrupole would multipolar f o r the C e 9 is t h e r a d i a l (which The result whereas C value, while systems. 2 m * = 2 . 7 5 , and hence any g in F i g u r e 9 . T h e s e c o n d peak of about fluid are s h o w n of m * is very f l u i d s , the c o n t a c t inward. a p p e a r s at a b o u t [121,122]. o f the s t r o n g multipolar expect t o the energy. to the influence in F i g u r e system the h a r d - s p h e r e has m o v e d for both separation shown computed for [41] o f (e.g., d i s p e r s i o n moment, o f 2.74 a n d 2.77 f o r t h e p o l a r i z a b l e Also value s i n c e w e might at 2 5 ° C dipole value quadrupole v theory. fluids are d u e s o l e l y 2 to of -16.4, - 1 6 . 8 and -18.1, f u n c t i o n , g(r)=gQQ^( r ) , obtained o u t . that t h i s tend the t e t r a h e d r a l , t h e C values contributions give constants. A t 25°C in the p o t e n t i a l v moment should the C 2 moment. the t w o e f f e c t s dielectric fortuitous octupole y in s t r u c t u r e dramatically moment, which that give b y the R H N C distribution dipole e n e r g i e s , U-j-Q-p/NkT, w e r e with large t o the C of the octupole [67] o f - 1 6 . 9 f o r the t e t r a h e d r a l g(r)). peak quite fairly given the h a r d - s p h e r e inward similar fluids well ignored for is octupole v result fluids, respectively. of quite have the s a m e e f f e c t i v e moment. results 2 to make The r a d i a l differences W e find u s i n g e q . (4.20). is p e r h a p s t o the v a l u e systems give systems terms) quadrupole the a d d i t i o n C o n s e q u e n t l y , in Figure 8 w e s e e that w e have that t h e R L H N C close average dipole constant. moment in e f o r a f i x e d in a larger These compare agreement drop 7 ) that a n d the C respectively. of the o c t u p o l e average configurational polarizable quadrupole a noticable dielectric one another. addition (see Figure results r i s e t o a larger cancel that 120 - still the m a x i m u m t o t h e tetrahedral numbers, occurs at a in the s e c o n d distance - - 1 2 1 Figure Radial dashed distribution functions l i n e s are R H N C results models, respectively, when hard-sphere radial for 9. water-like for the C 2 m * = 2 . 7 5 . The distribution e function for fluids v at 2 5 ° C . The quadrupole dotted that line and the represents density. solid C 2 y the and octupole 1.45-. 16.-i 1.30H g(r) 1.15H 1—I—r—I 0.0 0.08 1.00-^ to 0.85H 0.700.0 0.16 i—r n—T 0.4 0.8 1—r 1.2 n—i n — r ~i—r 1.6 2.0 2.4 (r-d )/d s s - CN = R f r 47r p R r e p r e s e n t s the integrand, w e find A g a i n , these do CN^4.5 [122]. that not repulsive these two fluids The effects contact both cores of of quite the g(r) distance. inward. structure quadrupole result by g(r) similar to projections which for C 2 v to try from the about for at the water effects of octupole real might due t o s e c o n d peak moment of for g(r) for the be e x p e c t e d , the C2 V extra terms shows virtually in its the effects the no has d e v e l o p e d at influence of 25°C. the peaks appear to does magnitude 5.7 at to the s e c o n d peak the water. moment As of 2 5 ° C , where RHNC results that steeper minimum values of t h i r d and f o u r t h octupole first structural m o d e l s , the v o n the and improve both only basis set We except to that of generate RHNC moments of the of the be shifted packing are still our of results for the i n c r e a s i n g the found that C2 subset u s e d in the water i n c r e a s i n g the little to the quadrupole change steeper contact a p p e a r unable distribution values of theory, hard-sphere models have s h o w n V m o d e l s , as w e l l a small an e v e n radial in the peak. containing give a only tetrahedral water. of some quadrupole have been plotted moderately quadrupole effects moments. 10-16 we functions least the in the M o r e o v e r , the multipole correlation at result s e e n in F i g u r e 9. shoulder l e a s t w i t h i n the In F i g u r e s HNC give the 15% and 50%, r e s p e c t i v e l y , p r o d u c e s v e r y for order structure represent 2 of The m a x i m u m and octupole T h e r e f o r e , at terms for different a l s o e x a m i n e d the and o c t u p o l e low (5.2) small. function, we the the becomes somewhat C l e a r l y , the In an a t t e m p t RHNC C in a d e s i r a b l e w a y , but relatively with addition change; however, a small slightly models the can a l s o be potential. tetrahedral the compare well of , s e p a r a t i o n c o r r e s p o n d i n g to are s t i l l model peak multipole g(r) dr Thus, even accounting unrealistic quadrupole 2 s d where 123 - of the and C 2 V projections octupole are all t h o s e w h i c h as h g Q ^ ( r ) . the t o t a l calculations. W e remark number Most of of the unique the fluids. contain that of pair The potential these projections projections correlation functions a f f e c t e d b y the a d d i t i o n o f the o c t u p o l e m o m e n t 123 224 m o d e l ; b.QQ ( r ) and YIQQ ( r ) c h a n g e m a r k e d l y , w h i l e in the are t o the 123 (r) - 124 Figure The p r o j e c t i o n the C 2 v h Q Q ^ ( r ) . The quadrupole m *=2.75. e and the solid C 2 v - 10. and d a s h e d octupole lines represent RHNC m o d e l s , r e s p e c t i v e l y , at results 25°C and for - 126 Figure The projection h 11 2 (r). n n The - 11. curves are defined as in Figure - 128 Figure - 12. 1 23 The projection h'00 ( r ) . n n The curves are defined as in F i g u r e 10. 0.08-1 1.0- 0.06H 0.5H 0.04H 0.0- 1 1 1 1 0.08 0. 0.0 .123 '00 1 0.02H 0.00- -0.02 0.0 I 1 0.4 1 1 0.8 1 1 1 1.2 1 1.6 (r-d )/d s 5 1 1 2.0 1 2.4 - 130 Figure - 13. 1 23 The projection h 02 J no (r). The curves are defined as in Figure 0.05-1 o.ooH - Q . Q 5 - J 123 - 0 . 1 0 - - 0 . 1 5 H - 0 . 2 0 0 . 0 - 132 Figure The p r o j e c t i o n 224 h.QQ ( r ) . The 14. curves are defined as in Figure 0.06-. 0.04H 0.02^ u224 "oo o.oo- -0.02H -0.04- 0. - 134 Figure - 15. 224 The projection h'02 ( r ) . no The curves are defined as in Figure 0.08-n 0.30-1 —\ \ 0 . 0 6 H 0.15- \ 0.04-^ 0 . 0 0 \ \ \ u224 02 0 . 0 n 0 . 0 8 0 . 1 6 0 . 0 2 H CO A 0 . 0 0 - - 0 . 0 2 0 . 0 ~ i — i — i — r n — i 1.6 2.0 2.4 - 136 Figure The projection 224 ( ). r The - 16. curves are defined as in Figure 10. 0.05-1 0.02H 0.0 0.4 0.8 1.2 1.6 (r-d )/d s s 2.0 2.4 - a n d Y122 ( ) r no appear to be the variation in the systematic different angle h.QQ^(r) and the dependence. h^^it), correlation moment. constant and the octupole forces of Before fluids, we true point out d i s c u s s e d in solvent model theory. This the arbitrarily have the found less of the study of It in the observed drop within find of in the energy. the a by we addition that the dielectric C l e a r l y , the fluid. This is, its results hard-sphere s i m p l i c i t y , the aqueous is o b v i o u s from concentration where In electrolyte Tables III in the model solutions and VI to case a much previous water-like tetrahedral resources required consideration grid. for of larger studies (as that solve was will this the RHNC electrolyte number of [67,72,79-81] in tetrahedral model set the value of e m p l o y e d , the p a r a m e t e r 0 was somewhat —26 2 ^ 2.50X10 e s u c m . In the p r e s e n t s t u d y , w e this value of the to that is [67]. model numerical cases w i t h the structure computational low that there each represents In b o t h dipole-dipole d i s c u s s i o n of b e c o m e s an i m p o r t a n t at the dipolar that b e c a u s e o f r e d u c e s the interest. with since find c o r r e l a t i o n s , as g i v e n structured quadrupole this However, we functions average the C h a p t e r VI). is r e q u i r e d which disrupt in our solutions, particularly points particular is c o n s i s t e n t concluding used extensively be to affected. dipole-dipole d e c r e a s e in the act course, also The become This - correlation are o f functions octupole least 138 was - square quadrupole moment underestimates effect o f the f u l l q u a d r u p o l e t e n s o r o f w a t e r . H o w e v e r , w e d o f i n d that —26 2 © =2.57X10 esu cm ( w h i c h is j u s t s l i g h t l y larger than h a l f the s u m o f the magnitudes of 0 and © ) w o r k s v e r y w e l l as an effective s q u a r e q u a d r u p o l e xx yy s moment. In t h i s identical average results for energies. Thus, this solvent c a s e the value model ail tetrahedral average properties, The radial of the employed and C distribution square in our quadrupole study of 2 y quadrupole including functions moment aqueous the dielectric appear was models give almost constant and indistinguishable. used electrolyte in the tetrahedral solutions. - 4. Results f o r Soft In temperatures, will systems show s e c t i o n w e have results for found h o w the spherical potential dielectric properties different correct (i.e., b y Ug (r) R with the r a d i a l from that structure experiment this o f real w a t e r . the m o d e l water-like at lower discrepancy might distribution c a n be o b t a i n e d making o f this F i r s t , let us d e f i n e for hard-sphere W e have s p e c u l a t e d that that the s t r u c t u r e , i.e., i s quite that e d o not a g r e e w e l l < 100°C. i.e., due t o t h e f a c t model Models the p r e v i o u s f l u i d s , the RHNC 139 - function, of In t h i s W e will these section w e by modifying soft). be o n l y the then examine the new model. an e m p i r i c a l = r[he short-range + Be ar b + C e r c x potential + De 2 d y "] , (5.3a) where x = r - X y = r - y (5.3b) q and This potential w a s a d d e d t o the C symmetric term parameters a , b, c a n d d w e r e -35d - 2 - 4 the useful potential. T A B L E VIII. model empirically w a s r e d u c e d t o 0.92d . T h e -1 -1 a s s i g n e d the v a l u e s - l O d , - 4 0 d T w a s taken sand s x =1.16d„ o three in e x a m i n i n g In Figure Parameters different i I a s 1.34X10 _ 13 S forms of of turning Ugp^ ") 1 a r e given. o n , or o f f , c e r t a i n the three forms o f the Uc (r). 0 | c 1 5 | 0 1X10 1 5 | 0.18 I D I o I o 1X10 1 5 | 0.18 | A I 40000 | 1X10 40000 | 40000 | S ergs per In T a b l e VIII the v a l u e s o f A 17 w e h a v e s h o w n Potential | 4>> y =1.65d„. o the e f f e c t s for as a s p h e r i c a l l y diameter , respectively, and D u s e d t o g e n e r a t e prove (5.3c) octupole v a n d the h a r d - s p h e r e and - 7 0 0 d molecule, while 2 . Q B -0.08 These parts ' B C will of potential - 140 - Figure Soft potentials potentials of these at 2 5 ° C . The s o l i d , dotted /3u^j^(r), potentials 17. j3u^ \r) are R defined and by and d a s h e d l i n e s /3u^ '\r), R the represent respectively, where parameters given in T a b l e the VIII. the forms 9 . 0 7 . 0 5 . 0 - (r) - 3 . 0 - 1 . 0 - - 1 . 0 0 - 142 Figure Radial The ^ U distribution for s o l i d , d a s h e d and d o t t e d S R ^ ^ ' g(r) functions r for the ^ ^ R ^ 1 model ^ a n d system soft - 18. water-like l i n e s are R H N C ^ ^ R ^ using 1 ^ ' r e s P e |3u^ \r) 3 c t ' models results v e l but v - T n e with for at 25°C the models dash-dot no and m line octupole *=2.75. e employing represents moment. - which we will for 18 w e have s h o w n RHNC 4 different various terms soft in the We e m p l o y e d , the first than is c o n s i s t e n t 2.5. This water at 25°C. where the find peak D within the integration in e q . (5.2). r e d u c e the the we H o w e v e r , the with results first for 25°C. In o r d e r R position correct repulsive the term The first (r). distribution function stop of the still for minimum spherical in g(r) n o w that has a m i n i m u m is the first the is c l o s e t o less [121] number we is a d d e d t o minimum and C N = 4 . 7 w h i c h being approximately coordination position is R maximum appears moment |3u^ \r) have b e e n r e p o r t e d in g(r) upon octupole like b r o a d , its which distribution The d e p e n d e n c e potential M o r e o v e r , the to radial as u p o n the soft minimum the range 6 - 8 , d e p e n d i n g u p o n w h e r e C N , another (2) radial models a simple should be. use 0Ug correct results in g(r) b e c o m e s quite H o w e v e r , the somewhere about out p o t e n t i a l , as w e l l that w h e n s e c o n d peak p o t e n t i a l , i.e., water-like soft c a n be s e e n . and t o point functions. In F i g u r e the We g smooth function 25°C. S R simple at (2) that j 3 u u ( r ) m a y be hard (3) b e c a u s e it is i n d i s t i n g u i s h a b l e f r o m / J u u - ( r ) for r<1 35d and for r > 1 . 4 5 d . It is c l e a r f r o m Figure 17 that t h e s e p o t e n t i a l s are to detect ( 1) /3Ug (r) investigate - 144 appears of at real in water. about r = 1.7d w i t h s m a l l p e a k s o n e i t h e r s i d e . T h e s e t w o s m a l l p e a k s are d r a w n into a s i n g l e peak c e n t r e d at r = 1.65d w i t h the a d d i t i o n a l a t t r a c t i v e t e r m c o n t a i n e d (3) in / 3 U g included the the in the effects would s e e m to tetrahedral The Levy agreement two 2 v and L e v y in the 18 that is c l e a r l y that the structure. In the octupole if the affected. s e c o n d peak indicate radial R octupole The m o s t at the t e t r a h e d r a l moment is significant distance. in shall refer and i n c l u d e s the (r) moment is i m p o r t a n t discussion below we (3) u t i l i z e s Ug distribution at 25°C in F i g u r e w i t h the in v e r y model g(r) Figure not of This stabilizing to the octupole moment as model. fluids [121] clearly drop model which soft C octupole also see from m o d e l , then is the water-like the We (r). R s poor 19. functions of the soft are c o m p a r e d w i t h the The R H N C experimental agreement. result for curve, while In g e n e r a l , the the v soft that o f In F i g u r e 20 the partial 2 and o f experimental s y s t e m s are a g a i n c o m p a r e d w i t h the [121]. C structure C the structure 2 model 2 y 2 y Narten is in octupole result is d e f i n e d [13] of and good fluid f a c t o r s , S(k), of experimental factor C g(r) o f y C the is the Narten by - 145 Figure Radial distribution m *=2.75. e the C 2 v The solid octupole experimental contact functions peak for water 19. and of water-like and d a s h e d l i n e s are R H N C model result of - of the for s y s t e m s , r e s p e c t i v e l y . The d o t t e d Narten C- results fluids and L e v y octupole [121]. W e n o t e model d o e s not that the line at 25°C soft is most appear on the and C2v the of the plot. and r/d - 0.92 s 2.4 - 147 Figure Structure curves factors of water are a s in F i g u r e 19. and o f - 20. water-like fluids at 25°C and m * = 2 75 e The - 149 S a0 ( k ) = 1 soft C S(k) of the C octupole evident in the We C v models 0 entirely in the hard-sphere in the RHNC Full the soft dipole the 2 experiment. low experimental model by of its the amplitude between (r) R C of at is a l s o [173] u^p most correcting s u g g e s t s that then structure the importance influence for the C its there 2 y moments at here large. inadequacies a of deficiency the problem spherical 300°C and dielectric 25°C the average result, while good using at that the agreement packing structure 300°C dielectric with of water properties. is n o t h i n g unique merely c o u l d be fit potential. and is t h e . f a c t unique dielectric 25°C At octupole Our p u r p o s e w a s water soft peak is t o o for part d o n e at 2 5 ° C are in v e r y a p p e a r that the (r). oscillation compensating for average dipole both temperatures does split The theory. larger than Of experiment. 0 is s i m p l y it work the with characteristic in F i g u r e s 7 and 8 , r e s p e c t i v e l y . e m p h a s i z e that (3) adjusting Ug The r e s u l t i n g T h u s , it w o u l d p o t e n t i a l , i.e., show calculations were is f o u n d . temperature agreement o c t u p o l e and the s o f t (3) the s p h e r i c a l p o t e n t i a l , u u ^ , ( r ) . However, what (3) RHNC is s l i g h t l y Again, we soft difference the model. obtained and Very recent are s h o w n opposite curve m o d e l , or w h e t h e r lie w i t h v d o e s not is whether theory. moment constants at clear is a g a i n in g o o d fluid addition RHNC/SCMF C constants only is the actually model v the is not may 2 experimental s t r e s s that (5.4) a C l e a r l y , the 2 J" ^ ( g ^ ( r ) - l ) s i n k r dr , + about to with show our choice of that a simple a the multipolar - 150 - CHAPTER VI RESULTS FOR MODEL AQUEOUS 1. a single Chapter finite study salt V. we have dissolved The vast concentration) shown that effective solvent when in o n e utilize the of of the full model the the tetrahedral II quadrupole and V . tensor unchanged. properties The ions hard s p h e r e s , a s d i s c u s s e d in C h a p t e r II. their in T a b l e hard-sphere investigated were Model studied this at reasonable not electrolyte dilution work the [79-81], that for well ions for is not alkali due between to ions solutions the of 2M with that terms the degree around the ion) structure. of no of the an water-like simply ions as were model charged considered; solutions soluble in the pair solvation is d e t e r m i n e d Obviously then, we of almost would at of exclusively expect by the dilution, KCI w e were able were to find solutions RLHNC theory radial distribution that of is coupling in the the theory RLHNC solvent hard-sphere RLHNC it 0.125M), direct packing the finite give aqueous the no to C l e a r l y , this real and the (i.e., the of Csl we difficulties). allows at of finite (e.g., f o r Consequently, we not infinite solutions behaviour now appears to at (e.g., f o r potential attempting R L H N C , but theory previously an e x t e n s i o n first a concentration closure an i o n As by is o b s e r v e d f o r RLHNC II. study insoluble apparent the had b e e n mean force up t o This unrealistic f u n c t i o n , a s d i s c u s s e d in C h a p t e r theory pure [79-81]. RLHNC quite only with what [169]. fact anisotropic the type electrolyte be be quite this theory the aqueous appear to consistent halides All u s i n g the potentials model appear to reach a concentration although ion-ion numerical obtain I. have is r e p l a c e d b y of at its Chapter V w e univalent present found able to largely b e g a n the We Small the we of RLHNC s o l u t i o n s , again work behaviour w i t h the model results larger solutions same concentration. while are g i v e n those because of are t r e a t e d Only all in 25°C. aqueous concentration. does at infinite earlier examine diameters water consisting described (virtually model A l s o , in of solutions solvents examined solvent m o m e n t , the essentially electrolyte water-like solutions in C h a p t e r s square quadrupole remain investigated majority s i m p l i c i t y , as d e s c r i b e d of SOLUTIONS Introduction In t h i s of ELECTROLYTE to packing give - 151 - good results where there solvent, as is the case The model ions, was increased QHNC strong found sharply). influence theory upon to greatly structure o f these s y s t e m s be r e q u i r e d in o r d e r model electrolyte described study in t h e d e g r e e o f s o l v a t i o n and indicated properties These results that a v e r y accurate results clearly obtained r s had a s h o w e d the accurate theory would models. Chapter e s s e n t i a l l y a l l the results were 9i ( ) o f the s o l u t i o n , f o r the present using the RHNC The HNC b a s i s s e t dependence o f the properties was examined. The particular A s was dependence w a s the case case of a model f o r t h e pure reported theory, as o f interest when at 0.5M w a s basis set going from only slight = 4 . Fortunately, w e again found in t h i s KCI s o l u t i o n solvent, strong observed f o r all properties = 3 to n IT13X n m a x = 2 to basis set IT13X dependence volume for almost derivative properties We change values o f in C h a p t e r II. considered. n (i.e., t h e c o n t a c t (i.e., g^jCr)). solutions e x a m i n e d f o r the s a m e The ion s o l v a t i o n , particularly f o r improve o f this the i o n and the solutions. the t h e r m o d y n a m i c to obtain T h e r e f o r e , in t h e r e m a i n d e r for solutions. not only between [63] w a s a l s o b r i e f l y Furthermore, this also the i o n - i o n sensitivity interactions f o r aqueous electrolyte f o r aqueous electrolyte smaller but reference are s t r o n g all results when o f the activity examined, remark resources that the t o their attempted. coefficient a s e n s i t i v i t y , i n c r e a s i n g 2.6 c c / m o l e . x l i m i t , and h e n c e t h e n = 6 calculation was max n m a 1%). O f t h e = 5 calculation w a s pushing our computational n m = 5 (e.g., t h e c o n s t a n t increased b y about s h o w e d the greatest Thus, the projections) was going to n x = 4 HNC basis set (now used exclusively to obtain containing a l l other results not even 9 5 unique reported in this chapter. In t h i s solutions study (employing w e have examined several m o d e l the tetrahedral concentrations, including that t h e m o l a r i t y chapter.) infinite concentration Aqueous solutions although three other constant pressure conditions studied, w e have whenever salts water-like solvent) over dilution, as given scale aqueous in T a b l e electrolyte a range o f I X . (We remark has been used universally throughout o f alkali halides have r e c e i v e d the m o s t have also been c o n s i d e r e d . under w h i c h used the experimental p o s s i b l e in o u r m o d e l most real densities calculations. In o r d e r electrolyte attention, t o m i m i c the s o l u t i o n s are [174,175] o f t h e real For both this systems M B r and M'l w e have - 152 TABLE IX. M o d e l aqueous electrolyte solutions studied. Those c o n c e n t r a t i o n s g i v e n in p a r e n t h e s e s are f o r s o l u t i o n s w h i c h are b e y o n d the s o l u b i l i t y l i m i t o f their real c o u n t e r p a r t s . The c o n c e n t r a t i o n s i n d i c a t e d w i t h a star r e p r e s e n t t h o s e b e y o n d w h i c h n u m e r i c a l s o l u t i o n s c o u l d not be o b t a i n e d . Salt | Concentration LiF I 0 LiCI I o0, 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) NaCl KCI 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 Csl 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)* MBr 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 M' 0, 0.025, 0.05, 0.075, 0 . 1 , 0.25, 0.5, 0.6, 0.65. 0.7. 0.725. 0.74* EqEq employed MBr the and M'l dicussions the EqEq both have data we of same will we examine + have In the present model limits of were still their + as f r o m we the value effects of solution of found d _. + which corresponds to these systems were point out for the our mole obtained the model salts by M'l by the of We e q . (2.24b)). ion size o f aqueous Thus, results M'l point out In that the a salt. NaCl from since model s o l u t i o n s , c a n be For EqEq compared asymmetry. that the at RHNC theory concentrations counterparts, as indicated study of densities M B r and ion [175]. (as g i v e n as m e a s u r e and C s l s o l u t i o n s real d _ employed same study NaCl able to that the of use d _ a g a i n h a v e the to a ^ H ^ N B r value s o l u t i o n s , as w e l l in o r d e r our 0, 0.025, 0.05, 0.075, 0 . 1 , 0.25, 0.5 the solutions and NaCl for | density below salts (moles/litre) in T a b l e NaCl solution at fractions X;=0.25 and extrapolating and C s l there the could above IX. the For x_ = 0 . 5 0 ! solved solubility example, we a concentration experimental were be of 16M, Densities results. concentrations We beyond for - which we were Table IX). The discussed alkali numerically behaviour below. halides It very this rule to with LiCl may (but low) concentration salt at quite theory. IX) this be by Li solvent Hence, a full SCMF in p r i n c i p l e , at Of course, for including its We every found even than 1% f r o m the in the pure bulk it clearly will be shown follows related Y for solvent clearly to value. The indicated. we would close to the SCMF electrolyte from the s for initial Li all high given the value It expect solvent a p p r o x i m a t i o n , it solution Y is local N the from would out the in Table behaviour may polarizable. of on SCMF required, investigate. the solvent, those of the a KCI s o l u t i o n the theory) at average changed by effective tetrahedral less dipole solvent result kT bulk. examined In the least e v e n at in the a s the the has a l s o been concentrations bulk current to remain models concentration solvent have pure e for d e c r e a s e in the theory u s i n g m * = 2 . 7 4 , the that field SCMF F i g u r e 21 w e a 2 . 0 M KCI s o l u t i o n a p p e a r that be to concentration in the local T h u s , at i n c r e a s e d , the s F i g u r e 21 average result. for wish (4.19) that w i t h i n field were obtained is o b v i o u s still pure of we C o n s e q u e n t l y , the kT same method are unchanged f r o m carried by that extreme IV) w o u l d properties relatively which the given study in C h a p t e r e q s . (4.12) a n d average for finite salts. + remain s reach a set those {i.e., because of dilution N to encountered g u e s s ( i n p u t ) into each solution value. kT several solutions 4.0M, the (as solvent 2N Y this In solution o b t a i n e d , m * = 2 . 7 3 4 , d i f f e r s f r o m the pure ' e than 0.25%. If w e d e f i n e the q u a n t i t y less then origin. the apparent order calculation was at An part in outlined moment, most in be difficulties c o n s i d e r e d in t h i s (as will (as d e s c r i b e d b e l o w ) , t h i s of infinite dipole that field by at SCMF electric have concentration full local moment we a solution A solutions calculation effective solvent. 2.0M. models solvent. investigated dilution the indicated H o w e v e r , the Unfortunately, infinite (also points for salt, a numerical salts theory these that tetrahedral numerical other model IX s e r v e d as the well. at applicable to The pure always scheme worked + be RHNC near Table be L i C l . a given F o r all the demonstrated not dilution the solutions in the possibly - solve from appears to of to these soluble exception RHNC of is o b v i o u s are infinite unable 153 of of very within an contribution to - 154 - Figure 21. The concentration dependence for which of Y . R e s u l t s are tetrahedral solvent m * = 2 . 7 4 . The represents a 2.0M KCI s o l u t i o n . Its e given point significance for indicated four with salts a is d i s c u s s e d in the in circle text. - 156 the average contribution effective local electric from the dipole independent of and h e n c e all determined a very good of the results m *=2.74. also concentration least a factor be n o t e d for every three) in the In C h a p t e r IV w e have separation solvent at t h i s results obtained All sections the 2 through m =2.605D. In e infinite results dilution nonpolarizable quadrupole attention which tetrahedral 2 v ion-solvent number of the RDMF of and at low u s i n g the consider SCMF which at (by at us to allows particle as a function polarization we will solutions of the present tetrahedral model reported solvent electrolyte solvent different in model solutions with at m o d e l s , including as the d e s c r i b e d in C h a p t e r V . the calculation required. g effects equal reduction (i.e., m =u = 1.855D), as w e l l solvents and i o n - i o n moment concentration. aqueous electrolyte several different the dipole in s e c t i o n 5, w h e r e determined octupole becomes the study, were then theory being present chapter computations treating will in the a full The e f f e c t s solvent be p a i d t o a permanent a solvent dilution employ model experienced by model 5 were as done performing introduced section 6 we and C will for can be taken represents a substantial be e x a m i n e d infinite the in t h i s solvent not total an i o n . level will at that local field from presented with solution of average model was an i n c r e a s e d approximation, solvent is w h a t M o r e o v e r , the g a non-polarizable e x a m i n e the pure t e t r a h e d r a l concentration each 2. Therefore, to finite should upon c o m p e n s a t e d by This to its exactly concentration. to of is a l m o s t salt using It ions. moment equivalent m field water-like C the 2 y Particular solvents have structure. Dielectric Properties The solutions equilibrium were t h e s e three determined formulas F i g u r e 22 w e dielectric s e e that n u m e r i c a l l y extrapolate calculated for the pure (2.97) are c o n s i s t e n t l y to of our model u s i n g e q s . (2.95), (2.96) and s h o u l d all y i e l d the e q s . (2.95) and (2.96) g i v e and w h i c h constants, e the agreement results which an infinite solvent. smaller s a m e value about for is not v a l u e that The dielectric (by (2.97). are e s s e n t i a l l y dilution constants 5%) than electrolyte those e exact. in In principle, H o w e v e r , in We mutual that agreement is c o n s i s t e n t determined obtained find with from from the those eqs. - 157 - Figure 22. Comparing aqueous results g the for model from trapezoidal f o r the required are values very from the results etal. o f Behret etal. integrations. close t o the etal. represent A t finite solvent and from concentration star a n d (m *=2.74) g e q s . (2.95) solvent using dielectric The experimental + [176] f o r a q u e o u s K C I , w h i l e o f Harris dashed line the open and O'Konski [179], r e s p e c t i v e l y . The RHNC respectively, ^ ^ c u r v e . The s l o p e o f P2^ _. are measurements [178] a n d H a g g i s constant of e q s . (2.96) and (2.97), t h e d o t i s t h e pure the limiting and triangle ^ and f o r the pure t e t r a h e d r a l rule, respectively, while square, circle Giese labelled with e q s . (2.93) u s i n g S i m p s o n ' s rule obtained represents values f o r the dielectric calculated from b y e q s . (2.95) f a l l s o l i d triangle obtained lines solutions S i m p s o n ' s rule as given constant and experimental K C I s o l u t i o n s . The employing e theoretical [177], line is e s for a model C s l solution studied with the RLHNC theory. - other two routes pure solvent. pure solvent This due t o limiting factors the for experimental is c o n s i s t e n t a c c u r a c y in the between found numerical the to the M o r e o v e r , the the with al_[ the limiting d i s c r e p a n c y appears variation employed, values of routes qualitative well with grid w i d t h smaller various for Hankel t r a n s f o r m s , w i t h discrepancy between is c o m p a r a b l e results. than t h o s e Unfortunately, this systems with s e e that the constant e=88.3, which F F T and the solution lower value d o e s , h o w e v e r , agree very numerical agreement Figure 22 w e dielectric of - a value in s e c t i o n 3 . b e i n g the that parameters In lack to constant, reported the (We r e m a r k extrapolated dielectric law behaviour be and e x t r a p o l a t e 159 to the e the Ar=0.02. better.) various routes found between behaviour of e the electrostatic is m u c h s to to the typical for our model s solutions is independent constants we Also will of report included KCI. note are q u i t e similar s o l v e n t , the constants at the in s u r p r i s i n g l y results the curve all dielectric values model good (when less with to of 1M for the The e for g a model experimental solutions the being the for solutions the pure dielectric c o n s i d e r e d here U n f o r t u n a t e l y , the for Csl curve T h u s , as w i t h obtained) for those dielectric e q . (2.96). [176,180]. experiment. a c c u r a t e that it. a q u e o u s KCI and C s l c a n be aqueous electrolyte agreement are p r o b a b l y similar below from results properties they determine obtained is v e r y concentrations RLHNC of were in F i g u r e 22 are R L H N C This theoretical that route used to below solution. We the RHNC for are RLHNC the present models. In Figure 23 w e constants with electrolyte e g are c o n s i s t e n t l y dielectric our found for fact than real depends upon e electrolyte experiment. solution does Even has v e r y the [79]. At very similar present low of pure for higher experiment. We This g be RHNC has a value results {i.e., > 1 M ) a larger e for the those the limiting pure slope results agreement a real L i C l for in F i g u r e 23 that of (i.e., 8 . 0 M ) our of aqueous are s t e e p e r than in q u a l i t a t i v e that dielectric is s i m p l y d i s c r e p a n c y s i n c e the high c o n c e n t r a t i o n to the observe larger concentrations behaviour e the model solvent also A g a i n , the appear to for concentration slopes for this values of study tetrahedral pure w a t e r . limiting account solutions at the counterparts. can partially at than t h o s e that solutions their in the W e f i n d that larger the constant model solvent model of compared experimental determined solutions. consequence for those have for our with model NaCl solution. For our - 160 - Figure 2 3 . The dielectric constants of real and functions of c o n c e n t r a t i o n . The solutions of N a C l , KCI a n d C s l , w h i l e experimental model the NaCl density values curve were [176] for indicates used. solid of model lines the LiCI, NaCl the are aqueous electrolyte RHNC r e s u l t s dashed lines represent a n d K C I . The concentrations for for dotted which solutions model the portion extrapolated of the values - 162 - model also solutions e consistent decreases more g w i t h what properties of our model qualitative agreement slowly with c f o r larger is o b s e r v e d experimentally. aqueous electrolyte with i o n s , w h i c h is Thus, the dielectric solutions are in reasonable experiment. 3. Thermodynamic P r o p e r t i e s First w e will examine the average energies o f the m o d e l solutions being effective systems were order investigated to determine within the S C M F in t h i s study. The average energies o f the c a l c u l a t e d u s i n g e q s . (2.80) and the t o t a l average electrolyte In p r i n c i p l e , in (2.81). e n e r g i e s o f the p o l a r i z a b l e approximation, w e must employ systems e q . (4.20) w h i c h w e c a n rewrite as UpO-r = ^roT where U p Q ^ is the total that to a very salt concentration present (when study good polarization approximation related, never <E^> the energies w e will differing U POL energy. and w e h a v e t a k e n applicable), since those + e ' (6.2) H o w e v e r , w e have already (and h e n c e U p Q ^ ) i s i n d e p e n d e n t o f m to be constant. e report are those than an a d d i t i v e Therefore in t h e ' of the e f f e c t i v e of the polarizable s y s t e m by more found are very constant system closely or multiplicative factor. In F i g u r e 24 w e h a v e s h o w n for model e q . (3.75). with N a C l , KCI a n d M B r s o l u t i o n s The ion-ion negative energies law are always smaller A t higher s e e n that limiting l a w . larger the average slope given by i n c r e a s e in m a g n i t u d e limiting concentration in m a g n i t u d e (i.e., e q . (3.75)). o f both energies per ion behaviour w e find at l o w that KCI has t h e f o r a_H t h e s a l t s e x a m i n e d t h e a v e r a g e and smaller ion-ion than W e remark i n c l u d e d in Figure 24, l i e s b e t w e e n solutions solutions a n d a p p r o a c h their v a l u e s , although relationship been be {i.e., < 0 . 1 M ) . ion-ion along with the limiting e n e r g i e s o f a l l three increasing concentration concentration most the average total those predicted i o n s than b y the limiting that t h e C s l l i n e , w h i c h t h e K C L and N a C l more has n o t curves. For KCI (e.g., N a C l energies deviate ion-ion quickly and M B r ) , it c a n f r o m the - 163 Figure Average total ion-ion c o n c e n t r a t i o n . The electrolyte slope negative solid solutions determined energies energies per l i n e s are R H N C studied, while from which ion the e q . (3.75) u s i n g have been - 24. as f u n c t i o n s results for d a s h e d line e =88.3. W e plotted. of three square of represents point out root the model the limiting that it is the - 164 - 165 Figure Average The ion-dipole solid and C h o f limiting lines comparison the are NaCl, C s slope subtracted energies from effective as f u n c t i o n s results for four of C s l and M of + infinite all ion 25. RHNC determined the per the systems from dilution + of of the values M B r . The have (which also been are point plotted. square ions concentration. investigated, Na* line represents e =88.3. For ease negative) out root being dashed e q . (3.76) e m p l o y i n g energies. We which - that it have is the the of been energies of - 167 The in the average present concentration study take the at o n the has b e e n very low limiting concentration is by range concentrations e q . (3.76). A g a i n , it 25 w h i c h deviate out that u n l i k e ion show linear the four of would ion-dipole the the model size concentration become concentration indicate in ]/c Na is m o r e allowing concentrations. true. In t h e s e the At solutions preferential the also be e x a m i n e d b y two ions will larger The at the are s h o w n + interact on small by Figure pointed energies different per size) Na Cs + + (i.e., increase the quite ion-solvent ion-dipole F o r C h at term. moderate ion-solvent energies at the disrupting strongly (i.e., C l " ion. and B r by curves - the with the seem a C l * ion converse Counter-ion in F i g u r e 2 6 . Br- i o n , w h i c h is more effects solvent. at and b e d i s c u s s e d in tetrahedral are solvent > 1 . 0 M ) the (as w i l l in rapidly This would more per have e x a m i n e d . counter-ion. ion the negative can These In this is p a i r e d with energy. e n e r g i e s p_er s o l v e n t energy energies, again a v e r a g e be c l o s e r t o g e t h e r , in Figure 2 7 . solvent-solvent of we negative that the K concentrations average s o l v e n t - s o l v e n t concentration to symmetrically more those a s h o u l d be ions dominant find expense of moderate IVh i o n , has the Csl solutions the than the c o m p a r i n g the be s o l v a t e d f i n d that the It as values given ion-quadrupole concentration K , is the ions w i l l of than for d e c r e a s e in m a g n i t u d e concentrations solvation the energies f i n d that c o n s i d e r e d in solutions + effective solvent We ion-solvent more low than higher in s e c t i o n 4 ) is at case we At 0.75M) w e N a , rather from ion-dipole negative ions slope for average because of + seen the than considered concentration. energies become 0.25M to + these h e n c e the average aqueous electrolyte increased. low slope. c a n a l s o be s e e n in F i g u r e 2 6 . when that s t r u c t u r e , thus is (i.e., negative salt is d e c r e a s e d , but linear effects low O n l y the o b s e r v e that largest a different ions more we + limiting e n e r g i e s , the (i.e., e x p e c t , the as i o n Counter-ion and from h a v e p l o t t e d the energies the rapidly the e q . (3.76). positive and the In F i g u r e 26 w e as the detail smallest a non-universal magnitude) to most by Cs v a l u e s are m o r e c at we more is the for of <0.025M) energies become law, although the four c a n be c l e a r l y dependence given dependence upon ion, for As limiting It {i.e., concentration law for in F i g u r e 2 5 . illustrated. i n c r e a s e d the predicted of the e n e r g i e s p_er j o n have been plotted behaviour F i g u r e 2 5 that do ion-dipole It for model N a C l , KCI c a n be s e e n that w i t h d e c r e a s e s in m a g n i t u d e . The increasing most 168 Figure Average ion-solvent c o n c e n t r a t i o n . The for been model which per ion of its have 26. as f u n c t i o n s solid, dashed, dash-dot solutions labelled with systems energies - of and dotted square lines root are R H N C N a C l , K C I , C s l and M B r , r e s p e c t i v e l y . E a c h appropriate been i o n . It illustrated. is the energies of the line results has effective - 170 Figure Average RHNC in t h i s solvent-solvent results study effective for three have been systems. energies of the per model - 27. solvent as f u n c t i o n s aqueous electrolyte i l l u s t r a t e d . The e n e r g i e s shown of concentration. solutions are t h o s e considered of the - rapid d e c r e a s e is observed for indicates that small We that the note included only study in total is magnitude surprising average energy In C h a p t e r of this to III type energy the for in the low is w h y the over s u c h a large of The accuracy u s e d the real From h ^(r) our our is more higher must for all of rapid. total in the ions We densities in h to the at the total non-linear degree. cancellation dipole-dipole What energy least that a large large). have is not should 4M). persist However, for the this heats of [174]. were We than calculated to f i n d that even 10% f r o m the appear to our indicate to of is a and a v e r a g e is rate observe some experimentally solutions. We model have RHNC at 1.0M pure been the solvent approximately again point calculations level out that in o r d e r we to conditions. d i s c u s s i o n in C h a p t e r all to electrolytes electrolyte e magnitude) energies This observed average (evident less the also average another energy total (2.82). solutions C l e a r l y t h e n , the have solutions investigated i n c r e a s e s in contributions when pressure conditions pressure been C s l , being the smaller cancel one (only changed by model structure. h a v e not solutions concentrations. ion-dipole pressure equation (a,P* = + , - , s ) are the is o b s e r v e d model clearly solvent those {i.e., negative energy range strong experimental constant our f i n d that more limit linearity some pressures have for average with what Hence, constant mimic f l total pressures of maintained have least u s i n g the calculated value. at We in c o n c e n t r a t i o n . concentration dilutions the model Furthermore, with energies apparent is c o n s i s t e n t from the approximation concentration linearity disrupting and s o l v e n t - s o l v e n t linear total the clear of e q s . (3.76) and (3.77)) w e (cf. This s i n c e F i g u r e s 2 4 , 26 and 27 w o u l d ion-ion, ion-solvent are not four c , e v e n at result Csl. concentrations. for total a good average contributions lower becomes the at for indistinguishable increased. of slowest a MBr solution, which in Figure 2 8 . dependence upon somewhat at energy F i g u r e 28 that t o linear almost - effective energies for energies shown concentration increase a average N a C l , the more negative are e x a m i n e d , the from average more The total as the are in Figure 2 7 , are slightly in t h i s ions 172 III s c r e e n e d at Jr) ap we large = ^S.e~ r Kt know that at r and s o w e > finite can concentration write (6.3) - 173 Figure Total average represent that it RHNC is the energies results energies as f u n c t i o n s for of model the of - 28. c o n c e n t r a t i o n . The four N a C l , K C I , C s l and effective curves shown MBr solutions. We systems which have been note plotted. - 175 - where is a constant screening fitting parameter, K , is given the long-range electrolyte well solutions long-range w e have e x a m i n e d . of h m (r) g s s i x v a l u e s that w e r e concentrations l e s s than numerical usually became fits shorter w a s o v s e t y r (r), °* t n form small h _(r), h + solutions e given were s 0.5M. were always (i.e., t h e y were ( r ) , as investigated the in v e r y A t concentrations determined ( r ) and h d e p e n d e n c e (cf. n o t u s e d t o c a l c u l a t e an a d d i t i o n a l obtained by e q . aqueous b e c a u s e o f i t s p£ became t o o difficult ranged l o w concentration. B y V a l u e s f o r K * = K6 (r), h + + 1 e q . (3.42c)), and h e n c e w a s The h o f the s y s t e m . T h e values f o r K f o r the m o d e l numerical hfjO'SS^ ^" tail b y e q . (3.35c) at v e r y Q each o f the functions 110 as from upon the parameters of h ^( r ) t o the functional tails (6.3), w e h a v e o b t a i n e d from dependent good higher to perform than value o f K*. a g r e e m e n t at 0.5M the as the long-range decaying faster tails because K was increasing). In Figure 2 9 w e h a v e s h o w n study. A t very parameters toward from l o w concentration approach the limiting larger most solutions the values o f K * rapidly the limiting values that law. tails to that determine highest smaller for NaCl. Both values) positive f o r these solutions the long-range of K * were values those given s l o w l y ) than those r ) were predicted sufficiently obtained are longer values f o r the screening parameter and K C I by e q . o f h ^ C r ) are predicted by f o r M'l, these Thus the ranged b y the limiting long s t u d i e d , K* given tails b y e q . (3.35c). a concentration those a r e s c r e e n e d m o r e q u i c k l y ) than t h o s e *2 s m a l l e r than f o r M'l than (tend deviations For both NaCl larger of h ^ ( r ) f o r the M'l solution are s c r e e n e d m o r e remark b y e q . (3.35c). are consistently The smallest always being long-range in t h e p r e s e n t the calculated screening (tend t o w a r d with concentration *2 r a n g e d (i.e., t h e y shorter l a w given determined l a w c a n b e s e e n in F i g u r e 2 9 . O f the s o l u t i o n s increases (3.35c), i n d i c a t i n g f o r K* w e f i n d that v a l u e s ) and n e g a t i v e the limiting results (i.e., t h e y law. W e ranged s o a s to enable us up t o 0 . 7 4 M ( w h i c h is the studied f o r this solution). Over the concentration range • 2 0 . 6 M t o 0 . 7 M w e d i s c o v e r that K * s t o p s i n c r e a s i n g a n d r e m a i n s a p p r o x i m a t e l y • 2 constant. For concentrations increasing c . W e a l s o f i n d that h + + (r) apparent to and h _ _ ( r ) actually long-range attraction contradict above 0.7M at t h e s e K actually begins to decrease with concentrations the long-range tails of c h a n g e s i g n and b e c o m e p o s i t i v e , i n d i c a t i n g a n the usual n o t i o n s between of ionic like ions. This behaviour s c r e e n i n g and o f i o n i c would seem interaction - 176 Figure The square The solid of lines s t u d i e d . The when the e =88.3 line used. 29. screening parameter are r e s u l t s dashed is Debye - for four represents of the the as a function model limiting aqueous law of concentration. electrolyte as g i v e n by solutions e q . (3.35c) - 178 within solution. dielectric ion-ion constant) are making correlations in g r e a t e r solution eq. Clearly, solvent detail in the M'l s o l u t i o n . below. EqEq solutions, which to those contributions W e will are j u s t discuss these solvent slightly that s m a l l e r than those given b y of A t a given concentration In III, w e m u s t We also examine C j g as defined integrations were transforms and first determine performed the contributions to G at l o w c o n c e n t r a t i o n . analytically whenever by that that G + product value _ will P 2 ^ _ which + of 1/f=0.5 concentration with very tail of h ^( r ) the values o f P G _ 2 rapidly of ^( ), n r a determined conditions very + at 4 . 0 M t o w i t h i n for G _ . + given 0.1%). It i s c l e a r f r o m always rapidly limit w e have e q . (3.36a) and has a limiting investigated. approach the limiting from solutions b e h a v i o u r , that limiting is P G _ 2 + Atlow what w e shall increases more l a w . F o r NaCl the P G _ appears to s h o w Atlow l a w , although behaviour. all demonstrated 2 representing the limiting + were tails the charge neutrality solutions b y the limiting 2 ^ w a s successfully fit to e q . f l in t h e P 2 — > 0 the M ' l , M B r and NaCl 2 . 0 M . For M'l P G _ f l T h u s , in F i g u r e 3 0 w e h a v e p l o t t e d t h e f o r all the model c r o s s t h e line rule a n d G always s a t i s f i e d t o a reasonable level of remains finite than p r e d i c t e d eventually about These contributions 2 t o a s super limiting-law The required C a r e w a s t a k e n in ^ due t o the l o n g - r a n g e a s p —>0. results f o r M'l deviate refer f l examine our results diverge b y e v a l u a t i n g e q . (3.1a). b y e q s . (3.43) a n d (3.39b). (e.g., at 0 . 1 M t o w i t h i n 0.01%, w h i l e concentration the values f o r numerically e q s . (3.5c) a n d (3.5d) w e r e First w e w i l l e x p r e s s i o n s given in of these transforms. the long-range Finally, w e remark accuracy asymmetry. evaluated with trapezoidal t h e k—>0 l i m i t s particularly (6.3). increasing u s i n g t h e t r a p e z o i d a l rule b e c a u s e a l l t h e F o u r i e r in our calculations were represent computing the value o f K decreases within t o make use o f the t h e r m o d y n a m i c Chapter will asymmetry + i o n s i z e , but increases w i t h order close to the behaviour o f K depends not only {i.e., t h e v a l u e o f d _ ) , b u t a l s o u p o n t h e s i z e increasing for M B r and in Figure 2 9 , l i e v e r y upon the i o n size the t w o ions. effects f o r the C s l o u t that t h e v a l u e s o f K * have not been included Therefore, w e find described b y the t o the l o n g - r a n g e F r o m Figure 2 9 it c a n b e s e e n that It s h o u l d a l s o b e p o i n t e d KCI curve. (in a d d i t i o n significant the screening parameters (3.35c). the effects + curve rapidly does s l o p e at a c o n c e n t r a t i o n o f divergent behaviour. w i t h j / c , r e a c h i n g a v a l u e o f 2.55 (not s h o w n It i n c r e a s e s in F i g u r e 3 0 ) at - 179 Figure The product represent considered P G _ 2 results in t h i s e q s . (3.36) u s i n g as a function + for five of of 30. square the'model root concentration. The s o l i d aqueous electrolyte s t u d y . The d a s h e d line e =88.3. - is the limiting lines solutions slope determined from 181 0.74M. The values of for the that at to the highest than t h o s e initially constant. given by follows solution (not ion behaviour It for both is P G _ of h _ ( r ) , at least 2 + although both P G _, MBr the P G _ for + increases M'l short-range 2 upon At w i t h the appear the details be the different the most low the the curve quite the larger h a v e no simple (but still b y the sharply EqEq than t h o s e for dependence limiting-law the the fact that low an i m p o r t a n t behaviour We value concentration for of for K increasing the in d e t e r m i n i n g structure i o n s o l v a t i o n , and hence u p o n for peculiar decrease with ion-ion that (although behaviour its role find behaviour same divergent begins to short-range here. be true s o m e of apparent actually to concentration long-range limiting-law opposite with finite) investigated super show The can play h a v e c o n s i d e r e d the concentration we find slope, S , determined c a slight obtain discrepancy limiting of for smaller will the the depend nature of itself. limiting value solution). M o r e o v e r , this + the very to show do and then low solutions predict f l structure down slightly determined the h ^(r) at C o n s e q u e n t l y , e v e n at r e l a t i v e l y In F i g u r e 31 w e (3.43). of an M'l P G _. solvent that entirely K would Csl o b s e r v e super find increase appears always the results We ions. out some of tails just Furthermore, we point slowly ion-ion of strongly for the appears to + is, however, consistent concentration. the for relatively behaviour to values of found 2 and MBr solutions long-range behaviour 2 NaCl their + P G _ and s m a l l are not + that of are + c l o s e l y , then turns in Figure 3 0 ) are o n l y interesting rate P G _ 2 concentration be d i v e r g i n g . W e s e e in F i g u r e 30 that W e note large increasing M B r the values of 1M. of with a p p e a r to for about asymmetry. values of studied quite behaviour or not law included size do limiting of the 2 increase rapidly F o r KCI the e q . (3.36). the T h u s , the upon also + concentrations a concentration NaCl. 2 M B r s o l u t i o n , h o w e v e r , they have b e c o m e at P G _ - from the calculations behaviour. that from our C ^ g as d e f i n e d numerical e q . (3.48). d i s c r e p a n c y b e t w e e n C ^ g , the extrapolation is e a s i l y a c c o u n t e d for p a r t , the quantity larger involved. the by We i o n s the C g the difference T to There infinite of infinite rapidly Cjg eq. do agree does, however, dilution dilution. in n u m e r i c a l also observe from more results by v a l u e , and This accuracy F i g u r e 31 that, deviates from its of for - 182 - Figure Cjg as a f u n c t i o n have plotted the itself. The s o l i d the present using of square root difference in e q . (3.48). the results s t u d y . The d a s h e d line e =88.3 concentration. For ease between l i n e s are R H N C 31. infinite for five represents the of comparison dilution v a l u e , C j g , and of solutions the limiting we Cjg examined slope, S in determined - 183 - - 184 Figure G + s as a f u n c t i o n of square limiting v a l u e G° ease c o m p a r i s o n . RHNC of solutions have s been (as defined 32. root c o n c e n t r a t i o n . The by e q . (3.49b)) and G results included. - for model difference + g between has b e e n p l o t t e d N a C l , KCI, C s l , MBr and M'l the for - The solutions d e p e n d e n c e of has b e e n appears to be solution is c l e a r l y e q . (3.49a). no simple while C s l it actually of Both of these we relationship + At ion begins to + are size or from F o r the have already found with decreasing function of being for NaCl NaCl, curve at a and M B r s o l u t i o n s G . decreases + s r a p i d l y , and in f a c t a p p e a r s t o be d i v e r g i n g f o r M ' l . T h e a p p a r e n t d i v e r g e n t behaviour of G . f o r the M'l s o l u t i o n i s an e x p e c t e d r e s u l t . T h i s is b e c a u s e +s in F i g u r e 30 w e 1.5M. g a variety asymmetry the + each consistent demonstrates g G limiting slope for i n c r e a s e and c r o s s e s about electrolyte concentration observations is a m o n o t o n i c g model low the find G to five very y/c, a l t h o u g h of concentrations For e x a m p l e , G concentration u p o n v/c f o r s in F i g u r e 3 2 . function indicated. for =G_ g different. higher behaviours, with + illustrated a linear At G - 186 e q . (3.41) w e know that G + M'l that P G _ 2 PP a + e d e p e n d s u p o n the g a r s t diverge o product for M'l and P G _. 2 + KCI In F i g u r e 33 w e h a v e s h o w n r e s u l t s f o r G „ obtained for model N a C l , ° SS C s l M B r and M'l s o l u t i o n s . A t v e r y l o w c o n c e n t r a t i o n w e f i n d that G „ „ SS has a linear from dependence on c, which F i g u r e 33 that G KCI s o l u t i o n s , w h i l e concentration. s is a m o n o t o n i c g for is c o n s i s t e n t the e q . (3.54). decreasing function C s l , M B r and M ' l The s l o p e of with it strictly curves for G the increases with appears to 3 for We see NaCl and increasing increase with O i n c r e a s i n g i o n s i z e , b e c o m i n g v e r y large that G a p p e a r s t o d i v e r g e f o r the M'l approaches behaviour G + 0.74M is c o n s i s t e n t observed for s Before p_G^ 2 r G^ +s +-' and G s s M'l and G of were indicated ss about found e q . (3.53) and the in T a b l e values of s h o u l d be p o i n t e d also demonstrated 9M. to Although the o with was we the OS For this IX, w e c o u l d not 2 + do e x a m i n e d ) like we to It is interesting and G g not of the reach a concentration to divergent nature note that for g s of the of two Csl solution at of 2 for the (at solution. 1.0M w i t h no +Q above 8.5M. RHNC of with M B r and M ' l , d _ + c the Consequently, for apparent 9.1M, theory. rapidly least for G + concentrations in a P G _, increase very diverge salts behaviour reach a concentration also M'l divergent magnitudes solutions appear to those our able to numerical + by that with c for only obtain P G _, G M B r s o l u t i o n , they able were out s o l u t i o n the increase sharply concentrations were apparent in F i g u r e 3 2 . above which for G p r o c e e d i n g , it concentration As (where f o r M B r and M ' l . A g a i n , w e o b s e r v e s o l u t i o n a s the c o n c e n t r a t i o n 3 has a v a l u e o f 4 . 0 8 d ). W e r e m a r k that this MBr difficulties. (i.e., the i o n - 187 Figure G of as a f u n c t i o n the model of salt - 33. concentration. We aqueous electrolyte solutions have s h o w n examined RHNC in t h i s results study. for five - 189 - s i z e ) is the respective same. ions which The divergent appears to sources remain entirely clear. that the a/3 = come behaviour through unclear. tails long-range aj3 solutions similar tails but h^^r). of must r e p u l s i v e or = h ^( r ) of the functions in s i g n . the not at the show were of long the the in F i g u r e 3 4 . solvent solvent more the G ^ f o r our Q at We First w e not we fl Bar find for the [169]. At low x.p d e c r e a s e s m o r e [181]. those rapidly the However, contrary Xm d e c r e a s e s m o r e rapidly (i.e., of 5 to of be p l a y i n g model aqueous III real NaCl were a to for what for C s l than for Clearly, calculate from result the e q . (3.74). for but are of real As x T is solutions d e c r e a s e are F r o m F i g u r e 34 w e is c o n s i s t e n t experimentally KCI. the water values of model of eq. remaining determined our some isothermal three KCI w h i c h is f o u n d major compressibility rates solutions. than rapid electrolyte to the w i t h the the 10d ) s study. compressibilities of for order intervals detailed c in a c c o r d a n c e w i t h of + + M o r e o v e r , the concentration larger the and M B r s o l u t i o n s . isothermal compare well diverge. h ( r ) is of negative (the Results for several than a phase begin to s or must concentration f i n d that [6,181], the g s h a l l c o n s i d e r the increasing concentration, although f i n d that h ^ ( r) magnitude in C h a p t e r decrease with experiment of is a l s o h a v e p l o t t e d kTXrp a s o b t a i n e d finite d o e s not dependence upon observed experimentally that actual c a n be s e e n in regions C s l , M'l require properties. T times the undergo and G g behaviour expressions given studied + to m o n o t o n i c , separated by behaviour kTx = 45.7X10 a linear + N a C l , K C I , and C s l s o l u t i o n s . tetrahedral 25°C, tails exclusively positive show behaviour c a n u s e the solutions which pure their solutions behaviour h__(r) at l o n g r a n g e . c o m p r e s s i b i l i t y , Xip. In F i g u r e 34 w e included this f i n d that the s t r e s s that determined thermodynamic (3.20) f o r of H — , s s and n e g a t i v e —, exotic of rather Again, we having solutions we C s l and M ' l be p r e p a r i n g There w e appear s y s t e m s a n d their Now more are fl in d e t e r m i n i n g not the long-range ++, may larger than that a t t r a c t i v e ) , but which their by e x a m i n e the for asymmetry (see e q . (3.37)) but 2 consistently of O Z equation we size effect. (e.g., s a l t p r e c i p i t a t i o n ) a s P G _, G long-range these in the The p h y s i c a l s i g n i f i c a n c e o f these MBr somewhat role only {i.e., at l o n g range l i k e s p e c i e s a t t r a c t , u n l i k e r e p e l ) . T h u s , it as t h o u g h changes differ demonstrated the H o w e v e r , if +s, - s separation over salts c l e a r l y has a large v a l u e s are p o s i t i v e appears For These two [6], Finally, we see with we point - 190 Figure Isothermal model were compressibility alkali as a function halide s o l u t i o n s obtained from - 34. of concentration. Results for c o n s i d e r e d in t h i s e q . (3.20). study are s h o w n . T h e the three values - 191 - - out that the i s o t h e r m a l 192 - compressibility o f the M'l solution Figure 34) s h o w s none o f the divergent rather forms curve which a smooth C l e a r l y , the apparent would The partial and r e s u l t s e q . (3.73). molar volume w e find that V W e o b s e r v e that o f the s o l v e n t it i n c r e a s e s o r d e c r e a s e s w i t h determined (values for increasing concentration the b e h a v i o u r c ) , at l e a s t from In of V (i.e., g at l o w c o n c e n t r a t i o n , e x a m i n e d here w e f i n d c a n be easily computed f o r NaCl for C s l and MBr. e q . (3.73) that b y V ^ . F o r a l l the s o l u t i o n s u s i n g e q . (3.16) the r e s u l t s will this t o be f o r V.° g i v e n in T a b l e X I ) . study The partial molar are s h o w n in F i g u r e 3 6 . A t v e r y partial molar volumes e q . (3.63). more rapidly slopes volumes from limiting these t w o solutions been behaviour those molar universal is the case in o r d e r w e find that t h e the HNC limiting l a w given have turned results experimentally salts [182]. [183] a s t o w h e t h e r solutions T h u s , unlike d o in f a c t o f the l o w concentration, hold w e are able t o perform experiment, w e know o b e y the [6]. F o r m a n y at v e r y still there h a s o r not the difficult a c c u r a c y at l o w e n o u g h law does solutions, In t h e p a s t salts linear an apparent for some C o n s e q u e n t l y , it b e c o m e s v e r y that t h e l i m i t i n g Fortunately, f o r our m o d e l with by Debye-Hiickel theory [184] o f s u f f i c i e n t to demonstrate dilution. slope together V2 a p p e a r s t o turn o v e r for MBr. begins to decrease again appear t o b e c o m e of these tetraalkylammonium l a w predicted here sign and o v e r , they in the literature salts deviates W e a l s o o b s e r v e that o n c e the c u r v e s f o r (i.e., t h e n e g a t i v e volumes limiting experimental in t h e p r e s e n t F o r the C s l a n d M B r s o l u t i o n s the change of tetraalkylammonium tetraalkylammonium infinite studied obey l a w behaviour. actually considerable debate partial examined l o w concentration in \/c) f o r V2 h a s b e e n f o u n d particularly o f the s a l t s In F i g u r e 36 w e s e e that w i t h ' i n c r e a s i n g i o n s i z e increasing concentration. linearity of four o f a l l the s a l t s of the curves f o r in v/c. T h i s as c a n c e l in the in Figure 3 5 . A t l o w increasing concentration whether for the KCI r e s u l t . in c o n c e n t r a t i o n , a s p r e d i c t e d b y increases with g III w e h a v e c o n c l u d e d f r o m with b y G ^ , but w a s determined are s h o w n b e c o m e s linear g Chapter by above f o r M'l must solutions V and K C I , but d e c r e a s e s w i t h below lie just d i v e r g e n c e s in t h e for several model concentration true demonstrated f o r Xrp (cf. e q . (3.20)). expression be behaviour (not i n c l u d e d in the value to obtain concentration f o r these salts. c a l c u l a t i o n s at o f \i° for a - 193 - Figure 3 5 . Partial molar obtained included. from volume of the e q . (3.16) f o r solvent model as a f u n c t i o n of N a C l , K C I , C s l and concentration. Results MBr solutions have been - 195 - Figure 36. Partial molar The solid Csl and M B r between volume the lines represent its (3.48). The dotted electrolyte solutions Xm(tetrahedral results infinite determined slope solute s o l u t i o n s . For e a s e V2 a n d slope, S limiting of from line which solvent) / of from from Xm(real square root e q . (3.15) f o r value, V ^ . t h e limiting [6], of comparison we e q . (3.65) u s i n g 25°C results obtained dilution is the at as a f u n c t i o n the and water). S real dash-dot multiplication model the of line c is the as g i v e n 1:1 limiting by eq. aqueous is the by NaCl, KCI, differences d a s h e d line slope, s j , for while the e =88.3 report concentration. the corrected ratio 197 - given the s a l t , and h e n c e f o r low concentration It is o b v i o u s concentration for counterparts. The approximately factor of isothermal the real our model the results for solutions. We an a p p r o x i m a t e pure isothermal limiting that of for have v the molarity (3.30b). here the since we We corrected it S S one or c a n then tetrahedral constant two slightly carried limiting discuss is the the effects were of of determined pressure derivatives have c h o s e n to mimic difficult, however, because deal w i t h and F r i e d m a n this [151]. problem sufficient constant of the in a m a n n e r The singularity is us w i t h to by repeating The solvent to a similar obtain conveniently to the position This reduction IV we to of present in the by move the mean study lny using eq. (we + values to note derivative that removed prove at used by by that integrate pressure conditions) would very obtain S which will c in d e t a i l in s e c t i o n 5. appropriate singularity an obtained to 25°C. logarithms in the can s i n c e in C h a p t e r RDMF in o r d e r are the solvent contribution of accurate results than at that corresponding tetrahedral water the the the for included densities. the tetrahedral an a d d i t i o n a l Integrating these values directly constant more the solvent give solvent a n d the for c slope for pressure derivatives coefficients S in A limiting slope provides determine is a d e s i r a b l e r e s u l t makes difference a given higher out 25°C. s m a l l e r than for [6] at has b e e n tetrahedral used to This error solvent RDMF We will much real is Numerically we the in F i g u r e 36 f o r corrected the into a c c o u n t for be the with their solutions corrected s h o w n that overestimates magnitude! the A e q . (3.66). T for and h e n c e d o e s not c > cf. than by is s t i l l was This procedure w a s of activity solvents. slope (31ne/3P) at about variation aqueous solutions r e c a l l that t h e r m o d y n a m i c s HNC theory that S . v The very of shown increase difference for slope reported below S this determining ambiguity larger model accounted for takes calculation that the our real two C h a p t e r III also of much solutions for the compressibility an o r d e r shows 2 of overestimates value changes. indicate In for solvent pressure which than that immediately H o w e v e r , this route V aqueous electrolyte larger c a n be no N^. limiting slope determined 5 c a n be S . of F i g u r e 36 that model HNC theory alternative about our solutions model the from compressibilities in F i g u r e 3 6 . s u c h a s M B r there behaviour 20 t i m e s about salts - p 2 =0. Rasaiah subtracting the - limiting l a w f o r the d e r i v a t i v e ( 9 1 n y / 9 p 2 )p. The resulting + of lny given + lny (as g i v e n b y e q . (3.56)) f r o m + difference P 2 = 0 , a n d h e n c e c a n be e a s i l y of 198 - function integrated. W e then b y the D e b y e - H u c k e l l i m i t i n g w h i c h w e shall report were is s m o o t h simply out aqueous electrolyte c a l c u l a t e d in t h i s that the l i m i t i n g slightly solvents. much more solutions slowly results up more much curve. NaCl solution the current than Et^NBr with values of lny Debye-Huckel alkali limiting large as being ions. solution such large evidence our earlier. salts up m o r e rapidly than alkali our r e s u l t s will for l n y b e an a t t r a c t i v e hydrophobic ions. than t h o s e w e shall refer that a n d Et^NI M'l solution Clearly, many force forces (and t o a l e s s e r o f the p r o p e r t i e s + for f o r an counterpart is a l w a y s + smaller c a n be o b s e r v e d f o r g i v e n b y the t o this f o r M'l (results as super f o r it h a v e n o t b e e n rapidly than for M B r . Super for some [186]) a n d h a s b e e n natures [5,186] of these that interpreted relatively in a q u e o u s (due t o the s o l v e n t ) b e t w e e n t w o In the p r e s e n t attractive + U n l i k e E t ^ N B r , h o w e v e r , the decreases even more (e.g., P r ^ N I turns d o e s the m o d e l s e e m to be a reasonable halides. less + + of both C s l and KCI. a consequence o f the hydrophobic for long-range model + If w e curves, F u r t h e r m o r e , it h a s b e e n h y p o t h e s i z e d there electrolyte Like the experimental Similar behaviour W e remark deviates + real counterparts. behaviour has been o b s e r v e d experimentally tetraalkylammonium [5,186] f o r the m o d e l lny solutions. would law. For l n y in F i g u r e 3 7 ) l n y Debye-Huckel their in g e n e r a l W e s e e that f o r M B r l n y t o the real Debye-Huckel b e h a v i o u r . included of differ lny f o r M B r are a l w a y s + solutions h a l i d e s , w e f i n d that turns halides. f o r our W e point in F i g u r e 37 are the m e a s u r e d v a l u e s o f l n y which + alkali than t h o s e MBr system. respect solutions. constants e x p e r i m e n t , w e f i n d that of Et^NBr f o r the m o d e l obtained + a n d o f the real it d o e s f o r t h e i r and m o d e l f o r the real less included of real law behaviour KCI s o l u t i o n are c o n s i s t e n t l y aqueous limiting f o r the r e a l quickly lny F i g u r e 37 w e o b s e r v e that H o w e v e r , unlike Also for from f o r our m o d e l + those in the d i e l e c t r i c b e i n g c o n s i d e r e d here than compare Csl From lny fashion. l a w s l o p e s o f the m o d e l because o f the difference respective lny solutions with a d d o n the v a l u e l a w . A l l the v a l u e s f o r In F i g u r e 3 7 w e h a v e c o m p a r e d the v a l u e s o f model a n d e q u a l s z e r o at study between w e have seen both a n d unlike like strong i o n s in degree f o r M B r ) , as w a s d i s c u s s e d exhibited b y the M ' l a n d M B r s o l u t i o n s - 199 Figure lny as a f u n c t i o n + RHNC results dotted lines for four square concentration. The model solutions are e x p e r i m e n t a l results [185,186] f o r solutions also included at for of root 37. the electrolyte been of - 2 5 ° C . The b o t h the limiting model solid investigated law a n d real several lines in t h i s 1:1 represent s t u d y . The aqueous s l o p e s (labelled with solutions. L.L.) have - are c o n s i s t e n t w i t h w h a t tetraalkylammonium It lny a n d M'l follows all that o n l y a n d M'l for e q . (3.30b) that the + Debye-Huckel G + that behaviour show the for tend + to d e c r e a s e the lny . We sufficiently + large t o for lny (which s strong Let is w h e t h e r that or not matter) s h o w s mean activity the of solutions the [6,7] solvent to lny + d o e s not of examine that in e q . (3.61) h o l d s low model value will solution G tend (or P2G _. G^ limiting move NaCl values of determine immediately u p o n the s e e that reciprocal aqueous attention (as and limit at p — > 0 , and 2 that o f finite defined deal [6,7,185]. activity of electrolyte little coefficients the for of simple relationship II f r o m E q u a t i o n (3.58) c a n be e m p l o y e d the like t o means v a l u e s are a v a i l a b l e in the than is h a v e r e c e i v e d a great between is g O b v i o u s l y , the + osmotic solutions + to any a l k a l i h a l i d e a convenient the MBr concentration. would e q . (3.33), w e behaviour of see an ion) c a n h a v e a a n d have r e c e i v e d r e l a t i v e l y only super s much s m a l l e r of pressure measurements for of e it NaCl, MBr Debye-Huckel very NaCl U n f o r t u n a t e l y , the d e d u c e the the pressure depends only aqueous electrolyte M negative NaCl effect behaviour for represent we perform does exist. model What w e coefficients concentration. P m in Figure 37 w e that this around e v e n at hand m e a s u r e m e n t s o f be u s e d t o ' C l e a r l y , h o w e v e r , the relationship cannot P2^+- large e n o u g h to that a g e n e r a l t h e r m o d y n a m i c a n d <f> g i v e n for have found 1.0M), the Figure 3 0 . and n u m e r o u s t a b l e s [6,7] in e q . (3.30b), then g is g e n e r a l l y g structure limiting-law osmotic + C s l and K C I . aqueous solution super + + are not a real H o w e v e r , if other in e q . (3.58)) f o r attention G. again to are d i f f i c u l t O n the of ( 9 1 n y / 3 p 2 )p, and h e n c e turn up f r o m return Unfortunately, osmotic + [7]. this. derivative P2G _. values of coefficient investigating pressure derivatives 3.0M), and c o n s e q u e n t l y w i l l to + upon results for us n o w G note F o r our s m a l l e r at cause l n y d e p e n d u p o n the influence We that e v e n though G above those of + ignore behaviour value of of 40 t i m e s U n f o r t u n a t e l y , the curve G find (about we is a l w a y s n e g a t i v e . g increase NaCl of behaviour, although super D e b y e - H u c k e l behaviour. + aqueous solutions c u r v e s are s u p e r D e b y e - H u c k e l . For our a concentration law. If constant In Figure 30 w e + (up t o for . g limiting-law lny . positive GL. + limiting-law M B r and M'l will emphasize super super solutions G s for salts. is o b v i o u s f r o m immediately - is o b s e r v e d e x p e r i m e n t a l l y d e p e n d u p o n b o t h G _ and G + 201 We of note osmotic between II consequently <f> at finite concentration if the - behaviour of one. point We not only V at potential. no In our temperature, model for answer 2 + counterpart. would suggest that t h i s However, we point explanation solutions fluoride would for the of small salts [6]. provide ion-ion First aqueous a very we shall of the structure evident from which at solvent. of was of allowed to be h e l d II w i l l reflect whether solution are consistent is not the case large present for unless behaviour for osmotic route for of to NaCl reversal not pure Therefore, the with some present those of of its NaCl at the (bU/bp2^- may demonstrated order the solutions contributions results cation aqueous with fact. or is chemical at this approximate in e q . (3.33) vary fixed as t o and d i r e c t provide by for y other aqueous and <t> f o r + pressure obtaining some measurements information about solutions. ions distribution 25°C. at We this generally of model is more structure. If we value minimum centred at effective pure examine of g s s ( ) has s h i f t e d r=2d for a s the pure packing an o b v i o u s is t o s a y , the KCI at g _(r) c It the outward. The the solvent, now The peaks also disrupting for from is NaCl that decreased substantially pure structure influence. as deep. than solutions s o l v e n t , although has changed dramatically n for c c are not now r the g (r) the is h a v i n g wells NaCl structure of model KCI a n d N a C l be d a m p e n e d , that a n d the our solvent-solvent r e s e m b l e s that concentration of have plotted that the appears to 38 that The contact orginally find smaller solvent first functions structure 12M, respectively, along with packing the solvent-solvent In F i g u r e 38 w e at 4 . 0 M s t i l l Figure s e e that the position been 0 c o n s i d e r the have b e c o m e solvent-solvent pure has s given make solutions. solvent (r) solvent for the simple radial solutions solvent constant a g a i n e m p h a s i z e that at 4 . 0 M and 4 . 0 M and presence at an Properties solvent-solvent in b o t h pressure values in e l e c t r o l y t e electrolyte tetrahedral NaCl the i o n s , e.g., osmotic normally be unconventional We structure 4. Structural that M only the also would g above out e x p r e s s i o n is s t i l l derivative Experimental mentioned we M a model discrepancies in g calculations - of but can currently for P G _ this derivative v a l u e ) , and h e n c e our results 25°C that the (experimentally, definitive real out constant concentration solvent is k n o w n , a l t h o u g h g 202 the at of 12M the and s e c o n d neighbor appears the peak, as - 203 - Figure Solvent-solvent model electrolyte the pure for a model lines are radial distribution solvent KCI s o l u t i o n results functions s o l u t i o n s . The s o l i d tetrahedral for at model at 38. line of i s the 25°C, while the a concentration NaCl solutions the at pure radial solvent distribution d a s h e d line of 4 . 0 M . The 4 . 0 M and and o f function represents dotted several and 9 of ( ) r s s dash-dot 12M, respectively. 1.45-1 16-n 1.30H l A 5 - \ 1.00—i 0.85H 0.70 i 0.0 — i — 0.4 i — r ~ 0.8 "1 1.2 I—I—I—I—I—I—I 1.6 2.0 2.4 ( r - d J / d - 205 - three distinct p e a k s at r=1.8d 2d 5 and 2 . 1 5 d f o r 12M N a C l . The these peaks (the o n e at r = 2 d ) i s w h a t remains of solvent. The t w o remaining p e a k s appear at s e p a r a t i o n s t h e pure correspond peak to solvent is t h e sharper surrounding a Na for clearly 9 ( ) r s s i o n are more + indicate in the model comprise the first r = 1.85d and 2 . 1 5 d solution the angle tetrahedral of 9 disrupting W e find ( )- that opposing <cos0 that correlations Hence, these s = l8O°) s dipole w e shall examine solutions. i o n size shows model a p e a k s at NaCl between r ) , where 6 UUfSS (r)> case 5 5 at 4 . 0 M a r e s h o w n behaviour in s o l u t i o n solvent similar to is found t o dipole-dipole more particles, can f o r t h e pure oscillatory is much of <cos0 moments correlations. effective particles. than C s l at If w e e x a m i n e in t h e v a l u e s of the t o the diameters ( r ) > appear o f t w o solvent the i o n - s o l v e n t Theion-solvent o f the ions is much (i.e., t h e c o n t a c t has a very minimum large structure o f the t o b e due particles to the separated minima in t h i s in Figure 4 0 . expect. contact shell grow of solvent study The functions (spanning ions A s shown then r s very larger around range 9ig( ) r ' s more and t h e in Figure 40, g;_(r) quickly (i.e., r - d - ) o f 0.2d particles a large 9i ( ) peaks b e c o m e aqueous at i n f i n i t e dependence of For s m a l l deeper). value w h i c h separation o f our model distribution and a l l s u b s e q u e n t at a r e d u c e d that t h e f i r s t radial considered a s w e might and a l l s u b s e q u e n t indicate r small ion. f o r three structured deep (r)> NaCl features i o n s i z e ) have been c o m p a r e d Na o Csl solutions corresponding (i.e., 0 electrolyte for g s and c a n be s e e n at s e p a r a t i o n s Next + NaCl c l o s e l y , drops a single first f g s and C s l m o r e present. upon results o functions of + of the solvent of t w o solvent Results f o r < c o s 0 c o r r e l a t i o n s , in t h i s (r)> f o r NaCl dilution vectors A g a i n , the presence of the ions the dipolar <cos0 by majority o u t that ( ) s Na in i o n s o l v a t i o n {i.e., r s The The present <cos0__(r)> = 5S the t w o dipole From Figure 39 w e o b s e r v e ions a vast b e s e e n in 9 and f o r m o d e l solvent-solvent ID or C l " ion. + in p o s i t i o n . involved which that t h e s o l v e n t s Finally, w e point the quantity u s i n g e q . (2.87). r s s study between solvent in F i g u r e 3 9 . disrupt held are directly can already g suggest f o r 12M N a C l shell). a Na peak at 4 . 0 M . be d e t e r m i n e d that solution o f the s e c o n d neighbor b y either would rigidly that solvation In t h e p r e s e n t is just separated o f the t w o which particles g particles smallest of o *D a Na drops This + to a would i o n i s h e l d in - 206 Figure <cos0 s s (r)> for the <cos0 pure solvent (r)> determined for solid line is while the dashed and dotted s o l u t i o n s , r e s p e c t i v e l y , at - 39. and f o r model the lines represent a concentration electrolyte s o l u t i o n s . The pure t e t r a h e d r a l results for of 4 . 0 M . model solvent NaCl at 25 and C s l - 208 Figure Ion-solvent radial and d o t t e d l i n e s are r e s u l t s respectively. distribution functions obtained at 40. infinite in the dilution. The s o l i d , dashed present study for N a , Ch + and M , + - 210 - very tightly. smaller. r-d^ s the first F o r large The first =0.5d are found solvent around reduced separation that at l e a s t second solvation first for small first separations solvation the first solvation Ch w e find 15. For M is slightly numbers less obvious shell the value simulation dilution M* is better, study (We again note s e c o n d peak function that unfavourable <cos0. (r)> is similar Neutron using e q . (5.2). ones.) to stop diffraction number o f about examined ions such as M the Na* and o f 10 t o size. reported b y although studies f o r C h the [13,16] a high 5.8 f o r C h . the values of <cos#. in Figure 3. _(r)> Infinite in F i g u r e 41 f o r t h e N a * , C h a n d values respect it in e q . (5.2) (i.e., in the range usually + o f < c o s 0 . _(r)> t o the ion, w h i l e behaviour o f g. ( r ) , a l t h o u g h represent negative values demonstrated b y the p o s i t i o n s o f the first 15 and s e c o n d m a x i m a in t h e c a s e do not coincide o f g-_(r), w e f i n d exactly that t h e s t r u c t u r a l with those features are g e n e r a l l y more t h e C h and o f g. ( r ) . A s of <cos0. (r)> 1 b 1 3 favourable that because the F o r both is d e f i n e d The oscillatory t o that W e remark integrating CN =5.4-7.3, are given with o f the o f s o l v e n t s in defined defined). the values the angle 1S both is distorted r e a s o n a b l e b e c a u s e o f i t s large than positive at a diameter. (i.e., t h e n u m b e r s poorly w e have a l s o orientations indicate minima this one solvent [50,51,53,54,58], a coordination f o r this packing o f the has its maximum the C N is s o m e w h e r e larger e q . (2.89a), w h e r e dipole point Thus, o f t h e p e a k s in H o w e v e r , f o r larger of its C N appears results favourable + C N =5.6-8.4. report from ions. than more for M studies In t h e p r e s e n t obtained C l ions o f the ions at w h a t out to H o w e v e r , w e s e e in F i g u r e 4 0 - less becomes CN^7.5, while concentration and much S loosely. A s s o that t h e a v e r a g e r e d u c e d s e p a r a t i o n The C N f o r N a * is s o m e w h a t agreement diameter. + g;_(r) i s s o d e e p . much computer always I minimum). b y the hard-sphere shell) can be determined of becomes + dictated such as Na* the C N is r e a s o n a b l y w e l l minimum r and h a s m o v e d sovent, the positions solvent Na of 9 i g ( ) of the hard-sphere The s e c o n d peak o f one shell shallower value i o n appears t o be held r e l a t i v e l y + to be strongly coordination ions much tetrahedral f o r the smaller smaller the is also of a M the ions. towards The shell o f t h e pure g. ( r ) + is also the p o s i t i o n solvation in t h e c a s e such as M , the contact minimum (which s ions distinct f o r smaller M * ions, the average at a l l s e p a r a t i o n s from ions. From dipole the ion. Figure 41 w e s e e that f o r orientation o f the solvent is H o w e v e r , f o r N a * the a v e r a g e < c o s 0 . _(r)> at infinite - 211 - Figure 41. d i l u t i o n . The lines are a s d e f i n e d in Figure - 213 - dipole orientation separations around r - d - < c o s # . _(r)> were to larger result that if > at <c9. we angle. does M Na contact approximately and f r o m more of very near values of particularly at electrolyte concentration shallower). is <cost9^ (r)> value of gives + in the a we much first interesting in o r d e r s is at finite is a f f e c t e d the by structural e s s e n t i a l l y half with results to to point compute 9£ ( ) r f° s r at in the features of the obtain tetrahedral computer C h ion finite the presence of As from a concentration the of simulations case 9^ ( ) s infinite dilution concentration. packing of other ions 9 ( ). of We the solvent in the r w e system, s e from e ss become r at d a m p e n e d at finite (e.g., the c o n t a c t peak d r o p s and the f i r s t m i n i m u m b e c o m e s Comparing g. X has c h a n g e d m o s t (r) for NaCl a n d KCI at 4.0M w e find that g. 1 5 markedly again suggest from its that s m a l l e r infinite i o n s , in t h i s dilution result for case N a , are + NaCl. more (r) 3 This effective than i o n s o f m o d e r a t e s i z e , s u c h as K , at d i s r u p t i n g the s o l v e n t s t r u c t u r e . N a C l at 1 2 M , g - ( r ) f o r C h b e a r s s u r p r i s i n g l y l i t t l e r e s e m b l a n c e t o the X s + infinite dilution result. The first the second neighbor the two is w h a t Ch ion. C l e a r l y , at t h i s single shell. corresponds to the C h ion and the for both ion in our agreement we find of NaCl with of drop into peak The larger solvent. where from a 12M NaCl + ion infinite solution distinct the second solvation counter-ion a value [13]. we compute of 5.5 w h i c h Moreover, for is a p p r o x i m a t e l y dilution a N a * ion value. in solvents in a Ch good same solution represents a in i t s g smaller, C N of T h e r e f o r e , in the has f e w e r 0.8d between much is a a about the this 4, which of have o n l y is a l s o e v i d e n t , t h o u g h If of shell is s i t u a t e d For while The s m a l l e r a reduced separation of at 4 . 0 M . obtain peaks. C h ion appears to the estimates a Na its at This feature solution, we C N of two the peak and KCI s o l u t i o n s has b e e n d i s p l a c e d o u t w a r d , due t o concentration experimental that the the arrangement 12M NaCl considerable model the minimum peak has s p l i t remains solvation the an [13]. solutions high c o n c e n t r a t i o n . Figure 42 that solvents It of N e v e r t h e l e s s , if shell, N a contact. experiments have plotted o b s e r v e that around a C h ion the range contact dependence. solvation 5 4 ° , which diffraction several model immediately a small (i.e., a s s u m i n g the d i s t r i b u t i o n s are v e r y n a r r o w ) , w e In F i g u r e 42 w e and f o r size first This value compares favourably [50,53,58] would the or for o b s e r v e that the ion s i n c e far + are at + u s e the contact an a n g l e o f for s of We show strong <cos0^ > than shell becomes unfavourable =0.35d d o e s not compute solvation out actually present first - 214 Figure Ion-solvent dilution solution for radial distribution result, while at model the solutions for 42. C h . The s o l i d d a s h e d line r e p r e s e n t s a concentration NaCl function - of at 4.0M. T h e d o t t e d 4.0M and 9jg( ) r line f° r infinite C h in a m o d e l and d a s h - d o t 12M, respectively. is the lines are KCI results - 216 - Figure 43. <cosc9. (r)> of C l " . The lines are as defined in Figure - 218 - solvation s h e l l than strongly with each o f the solvent We have a l s o calculated surrounding dilution a C h ion. along with solution at 1 2 M . demonstrates dependence At does a C h i o n , even though slight (even 4.0M correlations <cosc9^ (r)> ranges of favourable value o f of dipole alignment varies with of <cos#^ (r)>. s negative highly screened. o s c i l l a t e s about such f o r NaCl a s N a * are around more a C h ion. changed markedly alignment zero (except effective This again at d i s r u p t i n g over a large smaller 0.9d where 9^J- ^ • r drops A sharp s the c o u n t e r - i o n aqueous force The NaCl in < c o s 0 ^ ( r ) > We will now electrolyte at i n f i n i t e solutions. dilution results (e.g., L i F ) w e f i n d that depending For pairs o f ions much result. drop smaller 0.85d First very w counterat r - d - at 4 . 0 M s h o w — 0.2d can be similar but the C h i o n and the solvent. structure o f our m o d e l w e shall consider the potentials charged ions. f o r LiF, N a C l , EqEq and M B r . ^j( ) r (e.g., M B r ) i s v e r Y structured upon the molecular w ^j( ) oscillations. asymptotic ( r ) > has c o r r e s p o n d i n g t o the Na* peak of oppositely strongly smaller ions o f the solvent i n t h e v a l u e o f < c o s t 9 . _(r)> in b e t w e e n for /3w^j(r) obviously its long-range that is generally at s e p a r a t i o n s c o r r e s p o n d i n g t o t h e a r r a n g e m e n t f o r pairs pairs with s implies indicating orientation. <cos0^ (r)> turn o u r a t t e n t i o n t o t h e i o n - i o n plotted function dipole the ordering and KCI s o l u t i o n s is situated have larger contact) range o f s e p a r a t i o n s , s t a r t i n g at a r e d u c e d s e p a r a t i o n o f a b o u t in near Strong dipole dilution o f 4.0M the in F i g u r e 4 3 that at from the infinite indicating separation as shown f o r <cost9. is evident identified on f o r KCI. [13,16]. respect to the A t 12M the shape o f the curve a n d p e r s i s t i n g until seen experiment with W e see as ranges o f unfavourable than no counter-ion A t a concentration C o m p a r i n g t h e t w o c u r v e s at 4 . 0 M w e o b s e r v e that more s with favourable as well <cos0^ (r)> d e p e n d e n c e and a l m o s t is a l w a y s become now s at 4 . 0 M and a N a C l orientation behaviour at i n f i n i t e f o r a C h i o n is always positive, s the degree particles < c o s 0 . _(r)> is again consistent <cos#^ (r)> the oscillatory shown the contact more does the C h ion. and KCI s o l u t i o n s concentration at 4 . 0 M ) , w h i c h infinite dilution ion-dipole f o r NaCl i o n interacts + f o r the s o l v e n t s W e o b s e r v e that that t h e a v e r a g e d i p o l e by <cos0^ (r)> In F i g u r e 4 3 w e h a v e results only i o n , although p a r t i c l e s than the N a r at s h o r t nature becomes a much T h u s , f o r large behaviour, as given of mean In Figure 4 4 w e For small i o n range, o f the solvent. less i o n pairs structured w ^j(r) b y e q . (2.99), v e r y takes quickly - 219 - Figure 44. Potentials of charged mean force at infinite dilution ions. The s o l i d , dashed, dotted LiF, NaCl, EqEq and and M B r , r e s p e c t i v e l y , in the for several pairs dash-dot lines tetrahedral of oppositely represent solvent. results - 221 - {i.e., w i t h i n j( r) takes asymptotic the three form. quickly numbers of observe but except (i.e., for this w ij( ) r behaviour ij( ) f° r r before (~0.26) a n d N a C l the s a m e is repulsive is c o n s i s t e n t small the l e a s t with with For deep NaCl and b r o a d maximum respect that and is centred not as deep, while minimum value separates two oppositely larger nature of charged the i o n - q u a d r u p o l e ions hard-sphere p a c k i n g o f the s o l v e n t . corners of geometry quadrupole structure. The and EqEq structure the s o l v e n t for smaller moment of (where pairs Similar results effects always of were ion size w that d _ is the s a m e + more positive jj( ) in ' r model separation). the i o n s ions W e note and R o s s k y ij( ) s r Y [82] of favourable at r - d - • < d stable of attractive) role a solvent dipolar more the important short-range ion-dipole b y the near solvent. in that a the tetrahedral C l e a r l y , the in s t a b i l i z i n g and EqEq. salts. this and R o s s k y ion s o l v a t i o n for NaCl particle the s o l v e n t - s e p a r a t e d b y Pettitt these of and has its indicates be l o c a t e d than upon NaCl t o that that the s e c o n d m i n i m a an c r u c i a l r o n l y 0.8d shallow where LiF is r similar (because o f might asymmetry for both e very in t h e t e t r a h e d r a l f° v f° r is also f a v o u r e d again reported ij( ) w r e s p e c t t o the The fact is m o r e r (i.e., l e s s with i o n pair be p l a y i n g in F i g u r e 4 4 b y c o m p a r i n g noted M B r has b y becomes be e x p e c t e d t o b e c o m e all appear particle) must w is a v e r y interaction The solvent-separated bridging that actually The situation solvent interaction). solvent in diameter. w i t h the p r e s e n t f o r L i F , NaCl (excluding f o r all ion pairs by Pettitt f o r M B r it h a s b e c o m e at o n e s o l v e n t w. . ( r ) compute decreasing ion size to infinite observed we salts at a r e d u c e d s e p a r a t i o n [79,81], a n d h e n c e w o u l d ions If much ion pairs. LiF o n l y for rises of In F i g u r e 4 4 w e increases with first a n d E q E q the s e c o n d m i n i m u m configuration ions. (~0.04). In F i g u r e 4 4 w e s e e that the s e c o n d m i n i m u m very ij( ) reduced separation this ions its r (e.g., 0.1M), w e f i n d j( r) LiF and NaCl small of the m a g n i t u d e w ion pairs f o r these concentration of of pairs approaching at c o n t a c t f o r pairs contact maximum for smaller ion pairs, although value o r near at a p p r o x i m a t e l y M B r . For both positive that contact number w diameters for smaller its c o n t a c t that the f i r s t appears solvent l o w but f i n i t e the largest diameters), while F r o m F i g u r e 4 4 w e s e e that from L i F ) at s o m e far or f o u r increases slightly w^j(r) more o n e or t w o s o l v e n t than [82]. c a n a l s o be s e e n W e have W e find bridging previously that for EqEq. w ij( ) r i s Obviously - 222 - increased ion observation model from asymmetry differs from c o m p a r e the we RLHNC theory r peaks. w^j(r) much improves only better b e t w e e n the that explained simply of RHNC from concentration The particular that g _ ( r ) for case of is With and of essentially retained. solution at concentration three we of were 9+_( ) again when drops when going to Before jj( ) w r s u c h as L i F , the {i.e., has larger much more ions. result We the M'l. the for in the w poor both result contact of for values indicating rather ij( ) r pure being a n less qualitative n solvent RHNC agreement poor c of for that the i o n s the o be t dielectric g _ ( r ) has b e e n s h o w n simple has b e e n c o n s i d e r e d . concentration up t o other observe d e p e n d e n c e ; as the only + g _ ( r ) is s h i f t e d solutions the shape investigated dependence. long-range The one major MBr case). tail of study exception In the g _ ( r ) grows + is previous with have plotted g _ ( r ) for the + reach, specifically 0.74M. for is in t h i s s t u d i e d , i n c l u d i n g the 0.1M to downward + In F i g u r e 46 w e from We 4.0M) g _ ( r ) essentially shows l e s s e r d e g r e e the the in Figure 4 5 . + dependence found going obtained and p o s i t i o n i n g w^j(r) now results concentrations able to those n theories. concentration for t RHNC For larger that t h i s KCI s o l u t i o n a much asymmetry. f i n d rather negative for note difference least of ' solution o s c i l l a t i o n s ) than that - 1 2 k T ) clearly small This same behaviour o s c i l l a t i o n s d e c r e a s e s , although same concentration r w model. earlier d i s c u s s i o n s . For increasing concentration Most concentration solution behaviour two have d i s c u s s e d h o w increasing the the the the (and t o for present same models, we our dependence of same simple section we of i n c r e a s e d (at magnitude results and R L H N C a model screening effects. o f size RHNC KCI has a f a i r l y + concentration of the R L H N C . in t e r m s obtained The see ion pairs structured s l i g h t l y , w i t h the agreement M'l dependence upon RHNC ion solvation than M'l the RLHNC g i v e s about structured the for same basic oscillatory small more L i F the provides the [81] The RLHNC a l s o predicts (for the results be e x p e c t e d f r o m is g e n e r a l l y constants the [80,81] u s i n g the However, for RLHNC. w^j(r) present s h o w s the ij( ) w for RLHNC If the show previous s h o w e d e s s e n t i a l l y no theories the solvation which a g r e e m e n t , as m i g h t the improves KCI. 0 . 5 M , but At At then highest short long range we range the clearly tail increases 0.74M. p r o c e e d i n g , let demonstrated by us return t o NaCl for Figure 30 a n d the P G _. 2 + If we examine super limiting-law g _ ( r ) for + NaCl 223 - Figure Concentration lines dependence of are r e s u l t s for model g _(r) + KCI 45. for KCI. The s o l u t i o n s at s o l i d , d a s h e d and 0 . 1 M , 1.0M dotted and 4 . 0 M , r e s p e c t i v e l y . - 225 Figure Concentration lines are dependence results for of model g _(r) + M'l - 46. for M ' l . The s o l u t i o n s at s o l i d , d a s h e d and 0 . 1 M , 0.5M dotted and 0 . 7 4 M , r e s p e c t i v e l y . - we f i n d that that it s h o w s v e r y observed for model appears in g _ ( r ) some behaviour. + in fact Na /Na + Now solutions. dilution solvent m o d e l s , ions equivalently decreasing (i.e., w + + slightly surprisingly equal almost ( r ) =w__( r ) ) . by e q . (2.99)). large behaviour striking. from This its contact respect to Figure to in w ^ 1 t o longer ions. 2 + o f its w " ) s m smaller separation. f of Ar ) w ji( ) r ions 1 from clearly considered " ) i becomes coulombic repulsion W e f i n d that t h i s well range) a s the i o n size forcing t w o appears the most its minimum rapidly at r - d ^ — 0 . 3 d . g i s , it i s a t t r a c t i v e between grows (at a r e d u c e d H o w e v e r , it i s the which forces repulsive, repulsion. decreases very solvent value (as g i v e n less of the solvent Ar ) For a its contact near c o n t a c t than t h e s o l v e n t with s behaviour ions closer together. n e g a t i v e , that s increases with and h ) w ^ + be a result broad well i f o r L i , s e e Figure 61). very + ion and + asymptotic ££( ) Clearly, strong must with be present t w o such small i o n s at (i.e., b e c o m e s d e e p e r a n d i s d e c r e a s e d ( s e e F i g u r e 6 1 ) . In that t h e s o l v e n t Together, these F o r the four i o n s at + be solvated dependence of r minimum must a c t f o r the present t o the decreasing c o u l o m b i c hydrophobic , in o u r m o d e l + decreases rapidly is increased 26 w e have also found smaller for P G _ behaviour in c h a r g e w i l l (e.g., K o n its long-range f o r ions the strong separations. extends which value to a very to infinite overcome small the s o l v e n t 0.256",) w h i c h ^( r ) that value W e s e e in Figure 4 7 that f o r N a minimum + is n o t the c a s e has a local and somewhat of show conditions f o r N a , K , |- a n d M The ion size near c o n t a c t , d u e i n part separation o f about such take actually does b y NaCl correlations |3w^(r) the contact this larger than A s ion size ^( r ) (r) + + o f the unconventional a s that o f w ^ j ( r ) . (although immediately g its super linked t o o n e another. in s i z e but o p p o s i t e featureless function to that explain the charge neutrality in F i g u r e 4 7 . W e a g a i n r e m a r k ion size ioris o n l y M , from might dependence demonstrated W e have s h o w n in F i g u r e 4 7 w e o b s e r v e that For which anomaly function. to be not as simple + solutions are inextricably a manifestation distribution infinite particularly NaCl let us turn o u r a t t e n t i o n t o l i k e - i o n electrolyte found + limiting-law be more radial + + dependence to (see Figure 45). N o o b v i o u s and w e know (cf. e q . (3.5c)) that G _ a n d G may and concentration H o w e v e r , w e shall s e e b e l o w unexpected behaviour T h e r e f o r e , the super behaviour KCI s o l u t i o n s f o r our m o d e l + limiting-law similar 227 - is drawn observations would in m u c h m o r e s u g g e s t that ions tightly which - 228 Figure Potentials of mean force solid, dashed, dotted M /M + + at infinite and d a s h - d o t p a i r s , r e s p e c t i v e l y , in the 47. dilution lines are tetrahedral for several results solvent. for pairs of like i o n s . The N a / N a , K /K , l / h + + + + and - 230 - interact very strongly small separations pair). W e will ion-solvent At the s o l v e n t {i.e., the s o l v e n t d i s c u s s this interaction this point w e should the present models models. and a related for ^( r ) exact computer present integral for small anions simulation results models agreement. behaviour from G e n e r a l l y the R L H N C similar t o that of in the p r e s e n t do an a t t r a c t i v e predict although theory in the R H N C shows solvent j j( ) f° difference In for larger r w well this radial 1.0M m u c h ^j( ) in hydrophobic more for correct complicated similar results i s n o w n e e d e d are in o r d e r to test show w four ions For both K to a value jj( ) the effects the separations larger. functions F o r large Both theories for small ions, i o n s the R L H N C (i.e., f o r an i o n t w i c e at c o n t a c t ) . for poor no o s c i l l a t i o n s . small t obtained an o s c i l l a t o r y inverted), while is m u c h examined the s i z e A g a i n , the d i f f e r e n c e s + of o f the we see in t e r m s o f N a , K , I- a n d M + 1.0, s h o w i n g maximum o f 2.5 at a r e d u c e d is f o u n d ionic to grow screening. only has a contact with + We observe starts o f the b y g ^ ( r) separation increasing 9 ^ ( ) r in 1.0M s o l u t i o n s at a for these at c o n t a c t ions at of 0.25d . s This first infinite it a l s o r i s e s q u i c k l y t o a large rises structural peak concentration, probably although still a n d then small of concentration o f 0.5 and r i s e s q u i c k l y In F i g u r e 48 w e s e e that o u t at z e r o at c o n t a c t , + of mean force a f e w relatively for in F i g u r e 4 8 are the that e v e n at z e r o value results Included b y the p o t e n t i a l s a n d I- g ^ ^( r) g ^ ( r) concentration demonstrated predicted For M + finite in F i g u r e 4 7 . functions features. increased r) have virtually a with those c a n not be e x p l a i n e d s i m p l y o f the b e h a v i o u r dilution. + for r considered distribution r e s e m b l e s that slowly ^( r ) [80,81], w e a g a i n f i n d (only negative the t w o t h e o r i e s closely starts results in e. the s a m e M t w i t h the RHNC (RISM), obtained for theory N a C l , K C I , C s l and M B r , r e s p e c t i v e l y . for a as a divalent are q u a l i t a t i v e l y models ) r in s e c t i o n 6. the p r e s e n t they theory results feature Figure 48 w e have like-ion of ions detail [82], using results r study is actually w^( r) r RHNC the R L H N C determined in that for these ^ ( w theories. same w these (i.e., ?- a n d C l ) . C l e a r l y w h a t C o m p a r i n g the p r e s e n t the in m o r e although and R o s s k y in a n d its r e l a t i o n s h i p 5 and again equation a minimum to solvate in w^( r) s a y that Pettitt show prefers feature in s e c t i o n will m a y n o t be e x a c t , w e d o b e l i e v e r) for with to a in 9 ^ ( ) r due s i m p l y g ^ ( r) peak for N a with a to + - 231 Figure g^(r) for several dashed, dotted and ions in dash-dot model lines - 48. electrolyte are g^ A r ) 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 solutions for + Na + at in in a M B r 1.0M. The a NaCl solution, solid, solution, K + respectively. - maximum until at r-d^ =0.4d . The value s (i.e., until r-d. =d a single i o n s ) , at w h i c h point situation a single solvent where energetically for a maximum at outward. 4 . 0 M , after which c a n a l s o be of 1.3d ). At of this that screening. for We g ^ ^( r ) radial from larger for c a n be out function for smaller separations, we g. .(r) for C l " at KCI. F o r the relative noted The that for c o u r s e , the structure it peak due t o does for NaCl. both NaCl solution probability peak of must acts finding + responsible for is c o n s i s t e n t peak two be due t o N a , s i n c e there as a bridge this the different peak the in the possibility w i t h the two fact that is quickly. curves 4 . 0 M the Ch/CI- our a attention NaCl to in appears than to for large, indicating increasing It should peak where be in function a single same cation. for found a concentration. radial distribution larger be a p p e a r s at than the and the peak Figure ionic 0.1 M the for is an a r r a n g e m e n t the in more close together. Cl/CI" NaCl increasing This peak effects C l " ions with shift quickly decreases with peak between we has b e c o m e quite C l " ions separation d e p e n d e n c e , as can 0.4d more shifts radial included at As triples and KCI h a v e a peak about reaches Solutions of ion triples If r solutions. like-ion H o w e v e r , at counter-ion a reduced separation of o b v i o u s ; h o w e v e r , one solvent definite 9^( ) about ion simple and KCI s o l u t i o n s indistinguishable. o b s e r v e that 12M NaCl this structure is not NaCl from is 5.0). of + C s l and M B r rises much as resulting increasing concentration, although high g^( r) for KCI than the rises slowly 9^( ) the but KCI and N a C l r that a reduced the in F i g u r e 4 9 . r) Na in about due t o for dependence of interpreted peak of at + smaller contact shows Of Peaks and M two s a m e charge a concentration h 1.0 d e c r e a s e s and large f o r concentration that value as KCI at 4 . 0 M , h a v e b e e n from higher first is q u i t e has b e e n s h o w n - C l " are a l m o s t separation with Cl at Figure 49. grow and c o u n t e r - i o n the peak then little change. peak the out increasing concentration 0 . 1 M , the relatively 1 2 M , as w e l l point with of than the We point ions concentration b e c o m e s much s e e that distribution seen it for effect two larger in b e t w e e n 1.0. d e c r e a s e up t o ion-triples fit below in F i g u r e 48 (e.g., f o r the function to remains has a m a x i m u m shows ion s e p a r a t i o n , while Much it it find 0.1M w e (where continues 0 . 1 M , 4 . 0 M and 49. low It concentration distribution very then c a n just is b e t w e e n increased above identified solutions, we The 0.1M g ^ ^( r ) sharply is Although g At - solvent increase slightly about concentration slightly at to of drops unfavourable. N a * is f o u n d the g^(r) 233 NaCl This than - 234 Figure g^( r) dash-dot for C h for lines several represent r e s p e c t i v e l y . The dotted model results line for - 49. electrolyte NaCl is g . . ( r ) s o l u t i o n s . The s o l i d , dashed solutions for Ch in at 1 2 M , 4.0M and a 4 . 0 M KCI and 0.1M, solution. - 236 Figure Ch/Ch partial structure dotted and d a s h - d o t factors l i n e s are for results 1.0M and 0 . 1 M , r e s p e c t i v e l y . W e been multiplied by the mole model point for out - 50. NaCl solutions. The model that f r a c t i o n , x_, o f NaCl the Ch solutions partial ions. solid, at structure dashed, 12M, 4.0M, factor has - 237 - KCI, since g _ ( r ) would + solvent-separated concentration unlike indicate - 238 - that NaCl ion pairs. behaviour This h a s a larger structure given is obvious its small the concentration larger b y e q . (5.4), f o r a m o d e l from 12M NaCl obtained they Figure 50 that NaCl the C l " / C h solution partial at l o w c o n c e n t r a t i o n S__(k) is i n c r e a s e d w e o b s e r v e s t r u c t u r e If w e c o m p a r e the C l / C h solution with b y neutron are q u a l i t a t i v e l y the correct scattering factor, behaviour structure result of g__(r). in S _ _ ( k ) factor from It is d o m i n a t e d b y appearing partial an e x p e r i m e n t a l structure at s e v e r a l c o n c e n t r a t i o n s . k d e p e n d e n c e , a n d h e n c e b y the l o n g - r a n g e k values. model predicts of f o r the p e a k . F i n a l l y , in Figure 50 w e h a v e p l o t t e d as also number As at o f our 14.9 m o l a l LiCl ( s e e F i g u r e 2 7 o f R e f . 13), w e f i n d in that similar. 5. E f f e c t s o f Including the R D M F H e r e t o f o r e , the r e s u l t s aqueous electrolyte solutions a permanent moment pure dipole tetrahedral theory solvent. the a v e r a g e approximation) f o r the m o d e l solutions section we will at the l e v e l that electrolyte We dilution shown field which in s e c t i o n being examined of treating the bulk (to a v e r y in t h i s good study. corresponding to solvent. the p o l a r i z a t i o n average fields the R D M F which theory this model In t h i s o f the s o l v e n t IV. W e e m p h a s i z e in our m o d e l to be m o s t i n c l u d e the R D M F 0.1M. T h e N a , K , C s , M , C l " , B r - , a n d I+ solutions + a c c u r a t e at Hence, w e have repeated solutions concentration is c o n s t a n t a polarizable water-like model dilution, while o f the aqueous have been c o n s i d e r e d . expect + moment, m with 1 that w i t h i n the S C M F t h e o r y , a s d e s c r i b e d in C h a p t e r only model c o n s i d e r e d in s e c t i o n s 2 , 3 a n d 4 o f incorporate a n d at l o w c o n c e n t r a t i o n . + dipole s y s t e m s w i t h i n the S C M F t h e o r y point would have been f o r a non-polarizable solvent in the bulk solutions solutions o f the R D M F solutions chapter incorporate e x a m i n e the e f f e c t s up t o t h i s in this e q u a l t o the e f f e c t i v e local electric are e f f e c t i v e electrolyte which W e have C o n s e q u e n t l y , the model chapter reported only of NaCl and C s l were (i.e., f o r c ^ O . I M ) . It w a s f o u n d calculations for for concentrations ions were all studied investigated that infinite at o r b e l o w at infinite at f i n i t e at the l o w (but f i n i t e ) - concentrations 0.1M) effect done the in all pure e x a m i n e d , the - inclusion of u p o n the average previously reported tetrahedral 239 solvent local the fields RDMF of the c a l c u l a t i o n s , the was used in all had a n e g l i g i b l e bulk. T h e r e f o r e , as effective model (~0.1% at dipole calculations was moment of reported in this section. First w e u^g( r ) , values as determined of u r obtained two-body from the additional u s i n g e q . (4.29). given by ^g( ) with those the shall consider from problem the a single Figure 51 that the interaction which C l e a r l y , the RDMF theory are m u c h lateral solvent a Na W e recall the low predicts smaller fields for polarizable an i o n , a n d h e n c e r e p r e s e n t s interaction In F i g u r e 51 w e RDMF theory e q . (4.1c). of ion-solvent that than density those are h a v i n g infinite the dilution e q . (4.1c) c o n s i d e r s solvent values have c o m p a r e d i o n at + term, particle limit. for the It a distance c a n be s e e n additional determined a very at from large only r from ion-solvent e q . (4.1c). effect upon r), X & even the at smaller RDMF large). larger result From separations. In C h a p t e r for is e s s e n t i a l l y u^( r) l s Figure 51 w e than u^g( r ) a l l the ions at A p ( r = d ^ ) =0.425D. The We s We note contact find that r) shown that Na value that have 1/9 o b s e r v e that f o r contact. investigated. IV w e of u u^p( r ) + a similar of that range e is a b o u t times result rapidly long ( r ) (when qp u^g( r ) f o r drops at six was Na + from is found for corresponds contact, to almost r e a c h i n g z e r o at a r e d u c e d s e p a r a t i o n o f 0 . 5 d , then i n c r e a s e s s l i g h t l y t o a s m a l l p e a k at r - d = 0 . 9 d „ . N o n e o f the o t h e r i o n s e x a m i n e d here s h o w e d is S ^ g such a peak, although range 0.4d„ < r - d - ° rapidly S was to electrostatic its all t h e s e B e y o n d one solvent long-range limiting attraction between be relatively terms virtually S additional found due t o <d„. 1S approach The for of the small at u did ig( ) r diameter r) an i o n and a s o l v e n t range w h e n potential. flatten in was found due t o Very l i t t l e c h a n g e in ion potentials respectively. of We mean find solvation. This w o u l d interaction has b e e n force have at s h o w n the infinite dilution t h a t , in g e n e r a l , the seem consistent increased slightly. effects for addition w i t h the of NaCl of fact For NaCl w e the A and r slight unlike MBr, RDMF that the 9jg( ) upon u P(r) RDMF the a very A 52 and 53 w e to the compared with u P( r ) w a s o b s e r v e d , a l t h o u g h c l o s e i n s p e c t i o n r e v e a l s 1 s i n c r e a s e in g.:_(r) at a l m o s t all s e p a r a t i o n s . In F i g u r e the IS behaviour. short ion-solvent ions improves ion ion-solvent o b s e r v e that w ^ j ( r ) has - 240 Figure 5 1 . Additional ion-solvent dilution. The solid line interaction term represents u due t o (r) Ap(r) as g i v e n for by AD dashed line is u ig( ) r obtained using the RDMF theory. a N a * ion at e q . (4.1c), w h i l e infinite the - 242 Figure Effect NaCl of the in the RDMF u p o n w. .(r) tetrahedral 1 solvent at A represents the result when u-Hr) for 52. N a C l . The s o l i d AD line is /3w. .(r) for 1 2 5 ° C w h e n u . „ ( r ) = 0 , w h i l e the is is g i v e n b y the R D M F t h e o r y . dashed line - 244 - Figure Effect for of the R D M F u^(r)=0, while upon w ^j(r) 53. f o r M B r . A s in Figure 5 2 , the s o l i d the d a s h e d line is the R D M F result. line is - 246 - become slightly although more it t h e n positive near becomes more contact negative b e y o n d , a s c a n be s e e n in F i g u r e 52. w^j(r) to more Figures 52 a n d 53 w e d i s c o v e r that larger effect work positive upon w^j(r) that o f <j>) are e x t r e m e l y Thus, w e would NaCl expect these two solutions The e f f e c t s infinite dilution Here w e f i n d observe all f o r N a and M + small separations, indicating the proposed hydrophobic interaction would of model of improve addition A of u P ( r ) are s h o w n u ions ^g( ) to grow [82] h a v e aqueous electrolyte of these potentials opposite w^ ^( r ) solvation. the s o l v a t i o n . w. . ( r ) o f their very interaction deeper. This In C h a p t e r 3 effects. to more For M positive A t very a p p r o a c h their makes i s in f a c t electrolyte v a l u e s at with ion-solvent e f f e c t s , and F o r N a w e s e e in F i g u r e 54 that the negative the n e g a t i v e v a l u e s at a l l s e p a r a t i o n s . wells interaction what in r) with the s o l v e n t . T h u s , if expect the w e l l in w ^ ( r ) that u iAt g( ) w i " contribute r The l o w concentration l o w concentration limiting limiting affect solutions slope. the l i m i t i n g which behaviour contribution, of C j g for model that o u r n u m e r i c a l It i s o b v i o u s t o the l i m i t i n g Clearly then, u from behaviour A p (r) ion-solvent results do F i g u r e 56 that of C j g , almost c a n be e x p e c t e d t o of all thermodynamic depend upon t o C j g (cf. t h e R D M F h a v e b e e n c o m p a r e d in w e find laws. laws for small is o b s e r v e d in Figure 54. with and without respective the t o t a l significantly we + strong an a p p r e c i a b l e c o n t r i b u t i o n doubling + is consistent i o n s ; the i n c r e a s e d is i n c r e a s e d w e w o u l d slope. and C s l solutions 56. of o f m e a n f o r c e at This behaviour large to more IV w e h a v e s h o w n S ^ * , t o the l i m i t i n g Figure s o l u t i o n s (in coefficients e q . (4.98)) a n d h a v e d e r i v e d an e x p r e s s i o n (cf. e q . (4.99)) f o r i t s NaCl shown in F i g u r e s 54 a n d 55, r e s p e c t i v e l y . show shifts r nature shifts recent IX are a r e s u l t ion-solvent M B r . In v e r y and Rossky the l i k e - i o n In s e c t i o n 3 w e h a v e p r o p o s e d that the case c h a n g e s in the a c t i v i t y upon + improved 1 5 ions C o m p a r i n g the r e s u l t s in b e e x p e c t e d t o r e d u c e the h y d r o p h o b i c consequently shifts c h a n g e s in w. . ( r ) . AD due t o u - „ ( r ) o b s e r v e d here f o r is in w. . ( r ) l} a n d large that the a d d i t i o n of u ^ ( r ) concentration. o f the R D M F that i o n s , in t h i s in s i g n i f i c a n t at f i n i t e r sensitive to small the s h i f t s and M B r to result ^g( ), u appears t o have a relatively theory, Pettitt properties of separated distance and For M B r the addition the R D M F f o r larger some thermodynamic particular at the s o l v e n t v a l u e s at a l l s e p a r a t i o n s . using M c M i l l a n - M a y e r level that w i t h the a d d i t i o n properties correlations. of W e remark - 247 - Figure Effect Na + ion of the in the RDMF upon tetrahedral w ^ ( ) f° r solvent r at N a . The s o l i d + 25°C A line r e p r e s e n t s the result when u 54. ig( ) r , s when line is 0w^( r) u ^ ? ( r ) = 0 , w h i l e the is g i v e n b y the RDMF t h e o r y . for a dashed - 248 - - 249 - Figure Effect for of the u^(r)=0, l S RDMF while upon A r) for the d a s h e d line 55. M . As + is the in F i g u r e 5 4 , the RDMF result, s o l i d line is - 251 Figure Effect of difference the RDMF upon C j g . For ease between the infinite and s o l i d triangles are RHNC results respectively, when is g i v e n and t r i a n g l e s line is the T h e d a s h e d line by e q . (4.99). limiting u ^(r) sum is of S dots are r e s u l t s given c the the solutions, slope, S , determined represents plotted itself. The C s l and N a C l solutions, respectively, when solid have v a l u e , C j g , and C j g circles theory. The latter comparison we u ^ g ( r ) = 0 . The open RDMF the of model C s l and N a C l e=88.3. 56. for model (3.48) and dilution - c by the using and S ^ * for 3 eq. where - that the l i m i t i n g v a l u e s , C ^ g , are a l s o Therefore, even though ion-solvent is pair thermodynamic As RDMF solutions Quantities virtually great concentrations (become smaller NaCl sensitivity large which that effect it upon included the these the e a r l i e r u ^ ( r ) = 0 and K were our earlier also only results. slightly d i s c u s s i o n , the d e r i v a t i v e s of o f the R D M F , e v e n at the l o w F o r C s l the d e r i v a t i v e b y about in the The average energies of t o the a d d i t i o n in m a g n i t u d e ) term solutions. and C s l s o l u t i o n s constant from small d e p e n d e n c e a n d the f a c t electrolyte unchanged from expect examined. 1/r by the R D M F . it t o h a v e a r e l a t i v e l y at l o w c o n c e n t r a t i o n . A s w e would showed + of s u c h a s the d i e l e c t r i c affected. lny allow above, model studied remained term properties mentioned were appears as a relatively 4 u ^ ( r) symmetric - increased dramatically p o t e n t i a l , its l o n g - r a n g e a spherically many 253 w a s found to 2 0 % at 0 . 1 M , w h i l e increase for NaCl it d e c r e a s e d b y a p p r o x i m a t e l y 10%. From partial o u r d i s c u s s i o n in C h a p t e r molecular volume inclusion o f the R D M F . For IV it w o u l d o f the s o l u t e Their infinite w i t h the R D M F F i g u r e 3 6 ) is a l m o s t even if w e t o t a l l y determining turned dilution doubled ignore o n . W e find difference in t h e c o m p r e s s i b i l i t i e s T h u s , it w o u l d a p p e a r that that the R D M F t h e o r y r—>°°. What is not clear is w h y t h i s at l o n g r a n g e et al. [167] f o r the s p e c i a l c a s e w h e n known. would (cf. e q . (4.96b)) result is e q u i v a l e n t the s o l v e n t results but n o n - p o l a r i z a b l e . available O n e o f the m o s t s e e m t o be that aspect likely of (see in e x c e e d s the f o r the a n d real an e x a c t water. result and p —^ u a n c 2 In C h a p t e r f o r the a v e r a g e is p o l a r i z a b l e but ' IV w e local t o the e x p r e s s i o n o f Pollock non-polar. f o r the case when the H o w e v e r , s i n c e there f o r the latter 2 Unfortunately, correcting s h o u l d be the c a s e . the R D M F slope of V in the l i m i t s e x p r e s s i o n (cf. e q . (4.97)) w a s o b t a i n e d is p o l a r simulation dilution 7° =44.9 slope still solvent approximate, even field similar at i n f i n i t e only shown solvent that of three, even after have very limiting o f the t e t r a h e d r a l dependence. and use only c e q . (4.99) d o e s n o t r e p r e s e n t is s t i l l while c v a factor strong i n c r e a s e in S _ to S that the s e n s i t i v e to the that the l i m i t i n g S , w e d i s c o v e r that t h e t h e o r e t i c a l than show u^(r)=0, the H N C c o n t r i b u t i o n by more not values due t o a s i m i l a r real r e s u l t A s h o u l d be p a r t i c u l a r l y e x a m p l e , f o r C s l V ^ =75.6 c c / m o l e w h e n cc/mole seem obvious are n o computer s y s t e m , the a c c u r a c y o f e q . (4.97) is s o u r c e s o f error our m e a n f i e l d in the R D M F approach theory in w h i c h w e - 254 - assume that only be c o n s i d e r e d . the a v e r a g e One might dipole-dipole-ion) to be important M o r e o v e r , since u^P( r ) small errors in the a v e r a g e upon C T moments also expect field. influence dipole the results tetrahedral permanent moment different all solvent dipole excess local these upon with the gas phase value has a dielectric polarizable polarizable two solutions mean force long-range for have asymptotic constants. nonpolarizable differ employing earlier This of in this the polarizability as being tetrahedral being Closer limits. Thus, most comes much shifted (for the m o s t more r e v e a l s that about their o f the d i s s i m i l a r i t y through the d i f f e r e n c e expect to have part) f r o m the polarizable tetrahedral model reported solvent. f° KCI in r in F i g u r e for o f the i o n s negative values the p o t e n t i a l s of respective s e e n in between solutions thermodynamic those f o r the included solvation is solvent C l e a r l y , the r e s u l t s better behaviour Nevertheless, we would solvent r Also to much inspection oscillatory models force. jj( ) w of 1.855D, w h i c h in C h a p t e r V ) , w h i l e solvents. essentially chapter. nonpolarizable mean three section we will ignoring moment W e find quadrupole employing In t h i s using effective also studied, with of totally similar tetrahedral dramatically solutions ^j( ) r solvent. very were tetrahedral potentials w large obtained square solutions dilution. dipole [118]. markedly. solvent, the t w o s o l v e n t dielectric the differ 2 1 an In F i g u r e 5 7 w e h a v e c o m p a r e d model <AE (R)> , sections were o f 28.4 (as r e p o r t e d e =97.4. the n o n p o l a r i z a b l e reported the e f f e c t and nonpolarizable in the p o l a r i z a b l e for those local slope. in C h a p t e r V ) w i t h models its permanent constant model four o u t at i n f i n i t e for water 57 are the p r i m i t i v e the (described we shall examine and taking excess Solvents solvent results solvent its limiting aqueous electrolyte carried (e.g., have a r e l a t i v e l y e being the the a v e r a g e will the i o n n e e d correlations m = 2 . 6 0 5 D a n d an e f f e c t i v e but c l o s e l y r e l a t e d First the model Model g field in the p r e v i o u s moment 0 =2.57B. calculations compare reported towards (and hence C j g ) d e p e n d s u p o n 6. Results Obtained Employing D i f f e r e n t All the t h r e e - b o d y in d e t e r m i n i n g , and consequently C directed w ij( ) r their utilizing properties in s e c t i o n which 3 for 255 Figure Potentials of nonpolarizable for w ij( ) r mean force tetrahedral determined for at infinite dilution polarizable dotted primitive model obtained constants given in the 57. for s o l v e n t s . The s o l i d m o d e l s , r e s p e c t i v e l y . The functions - text. and KCI in and d a s h e d and nonpolarizable dash-dot from lines polarizable lines are R H N C tetrahedral represent e q . (2.99) u s i n g the and their results solvent respective dielectric - 256 - - 257 - We tensor have also of water investigated in the solvent the model. been replaced with an e f f e c t i v e Chapter V). different performed dipole the no longer of the were the negative the in the tetrahedral all solvent. appears to Finally, we the moment of water examine its effect We note the C receiving ion-ion the At cases) to those of the the upon C 2 ion aqueous electrolyte little influence model effects of the solvent. The C dipole 2 moment 94.9, both values being very of F i r s t , let 58 w e in the 2 v octupole have plotted 9 j , ( ) s (r) tetrahedral solvent. in the of a Na In the C occurrence of i o n s , as indicated separation for N r ion + 2 y 0.65d . a + a n c (or in the l tetrahedral This moment by I- less the quadrupole tensor of upon solutions. of the octupole we shall solvent those to properties (differing particular properties of of (as d i s c u s s e d the dielectric tetrahedral ions were F" ' o ion-solvent n s m e t r , equivalently, for octupole in Figure 58 b y about c l o s e to great all studied at solvent. changes f° r g- octupole addition + us e x a m i n e the along with increase C being a e T h e L i , N a , E q , K , C s , F-, E q - , C h and dilution generalization m = 2 . 6 3 4 D and a solvent. + changes i o n s o l v a t i o n and octupole v of infinite the s o l v a t i o n and upon t h e r m o d y n a m i c solutions. + slight effect the In constant + Only similar upon was + preference because aqueous electrolyte quadrupole in C h a p t e r V ) has an e f f e c t i v e + and N a / F " ) thermodynamic of are solvation same solution with s m a l l zz c o m p o n e n t v e x p e c t , the the 0 . 5 M the also solvent due t o of was quadrupole s o l v e n t , this addition has several effective tensor. structure water-like c o n s i d e r the for that the a slight be very of v it (as d i s c u s s e d in + found to properties 2 model computation would were have relatively to of solvent dilution single quadrupole and the T h u s , the will A quadrupole in c h a r g e (e.g., E q / E q - ions observed from KCI s o l u t i o n thermodynamic model of full moment infinite A s we m o d e l , as d i s c u s s e d b e l o w . model u s i n g the tetrahedral solvent. opposite of at solvent. ion-solvent that out constant positive moment 1% in a l m o s t water tetrahedral but In the KCI s o l u t i o n . dielectric zz c o m p o n e n t quadrupole solvent carried a model in s i z e s m a l l e r than than the the of square quadrupole quadrupole v symmetric, with noticed deal 2 and the ions equal the C 0.5M for s a m e as f o r the of at in the moment of of Calculations were ions effects the is not solvent both packing of C 2 structure. In octupole v Figure solvent a F" i o n ) in the ions slight the show solvent i n c r e a s e in 9 ^ ( ) r s a surprising result a t a a about the reduced s i n c e the addition - 258 Figure Ion-solvent and C the 2 v radial octupole tetrahedral distribution functions s o l v e n t s . The s o l i d at line - 58. infinite is 9i ( ) s o l v e n t . T h e d a s h e d and d o t t e d and a F~ i o n , r e s p e c t i v e l y , in the C 9 dilution octupole r s f° r for a Na lines represent solvent. the + (or tetrahedral F-) i o n results for in a Na + - 259 - - of the octupole packing within g^ (r) for s N a ) ion C second 2 larger for interaction In average in the decrease indicating presence of observe that have the solvent the oscillatory solvation for the the that of oscillatory a Na ions of The <cost9. _(r)> It In the of we by C of the Na solvent the from significantly narrower slightly octupole e q . (2.91)), w e smaller distribution from its moment same size. observed moment for about of angles. value F i g u r e 59 that the particles the octupole for s structure. The p o s i t i o n s F-, are o b v i o u s l y the model. contact solvent standard in the All larger more the results for of is a g a i n f o u n d to for C F- the 2 y solvent. orientation of specific reported here for and a n i o n s , a l t h o u g h decrease with for the ; s o l v e n t , indicating influence size. a increases effect of least near for of the the cations and F- i o n s w e r e increasing ion + d e v i a t i o n , o , is a n i o n s than + the we F- and N a s o l v e n t , at Na by g E v i d e n t l y , the the and s h a p e s <cost9^ (r)> both actually + + affected of standard Na , a Na , cost9. (r) as 1S octupole of a general F r o m F i g u r e 59 values deviation case tetrahedral directionally effects cations that In the around < c o s t ^ ( r ) > , particularly 20%) in the is s u c h that the c o n t a c t , has b e c o m e the (by octupole the find in the find is in V determined anion and F~ i o n s + presence of octupole y solvent the of the solvent, we 2 tetrahedral examine of for the solvent H o w e v e r , if equal size c a n be s e e n f r o m octupole the octupole y ion-dipole). for to 2 be ion-octupole s relative appears to C C remains s the influenced octupole in 9 ^ ( ) influence nature <cos#^ ( ) > , particularly v in the r in the ion + the short-range r 2 a Na decrease with d e c r e a s e in d i p o l a r C for in (or to moments moment ion + a F- o b s e r v e that structure peak in s o l v a t i o n , is f o u n d the structure for c a n be m a d e f o r shell structure cation. solvents. dipole contact is s h a r p e r than s e c o n d peak w e in g e n e r a l f o r over model. in turn observation have c o m p a r e d of in the dropped ions. to is s t r o n g l y in the asymmetry an a p p a r e n t peaks similar that solvated solvent in the is s h a r p e r than i o n than - and t e t r a h e d r a l ions octupole of because of orientation small the size a F tetrahedral s e e that the T h u s , the find F i g u r e 59 w e moment of for (with respect octupole these Na . defined ion A stabilize In F i g u r e 58 w e B e y o n d the + and h e n c e the increasing v for preferentially moment, v amplitude Moreover, we always 2 solvent. the 2 in C h a p t e r V t o solvent, which 9^J< '- F~ than clearly C r solvent solvent. C tetrahedral in found solvent. in the octupole v was pure a F- i o n peaks octupole more the in the + in the moment - 260 of also octupole 2 y - 261 Figure <cos0^ (r)> s The lines at infinite are as d e f i n e d dilution for in F i g u r e the 58. - 59. tetrahedral and C 2 y octupole solvents. - 263 Figure Potentials of mean force tetrahedral s o l v e n t s . The tetrahedral and C 9 at infinite solid octupole - 60. dilution and d a s h e d solvents, for lines L i F in the C 2 y are R H N C r e s u l t s respectively. octupole for the and - Let us n o w presence mean of force solvents C of w 2 at octupole infinite more octupole v jj( ) separation of structure for (in then, w e expect dramatically different replaced the larger for by 2 ij(r) observe the at like-ion contact that the behaviour Figure 2 v the much of For L i huge of at that ion octupole well h a v e the Once again, we It solvents + it for the C 2 shown We that thermodynamic Clearly to show tetrahedral octupole y solvent solvent. was s h o u l d a l s o be p o i n t e d the in minimum solutions. the is r a reduced octupole v in w h i c h we solvent out that is for and t e t r a h e d r a l d e f i n e d the the As the solvation w results that the small upon effects ions. solvent ^j( ) grows the of the octupole s we moment to repulsive solvent In F i g u r e 61 w e is h e l d t o see the is). that octupole moment for it structure, also evident behaviour degree ion and Thus, for deeper, while short-range the b e c o m e s sharper e m p h a s i z e here the the Here more tetrahedral shell structure in 9 ^ ( ) ion-solvent in Figure 6 1 . be c l o s e l y r e l a t e d t o r r in b o t h d i s c u s s e d a b o v e , the peak However, we influence the solvent. F\ tightly in of found with of contact well a n d F- i o n s + b e c o m e s much r) octupole v case (i.e., h o w pronounced for 2 Li addition range s e e m s t o c h a n g e s in the note well C in the is true. largest the f i n d that in the attractive for have been s h o w n due t o short solvent relationship which 2 ^j( ) w maximum) at even NaCl) solution mean force attractive ^( r ) how [82] C dependence when between effects shallower of in the LiF second is n o w of octupole v first s e e that the aqueous electrolyte solvent. potentials c o n v e r s e i s true long-range are m o s t octupole and the 2 for large c h a n g e s in s o m e a LiF (or concentration dramatic 5 8 ) and the opposite for + ions and R o s s k y in and C increased preference of the The potentials o b s e r v e that minimum to smaller. and the solvation C lny model. peak Its an model difference model. consequently the <j>) o f and t e t r a h e d r a l quite solvent becomes of v is m u c h octupole v 2 i o n p a i r s the The C C We contact inward. can result particular might the s m a l l pair Pettitt c h a n g e s in w ^ j ( r ) properties w of structure L i F in b o t h t e t r a h e d r a l indicating this ion-ion solvent in F i g u r e 6 0 . further 0.75d again r e c a l l the w o r k small for the F r o m F i g u r e 60 w e still about of in the (including solvent. is s h i f t e d r bridging dilution negative - sensitivity moment have been c o m p a r e d generally the the c o n s i d e r the 265 Li may F upon in (see + the well in F i g u r e of - be 58, r ) . - 266 Figure Like-ion potentials octupole for LiVLi of mean and t e t r a h e d r a l + and force at s o l v e n t s . The 61. infinite solid F / F " , r e s p e c t i v e l y , in the dash-dot lines represent octupole solvent. /3w^( r) for - Li dilution and d o t t e d tetrahedral + for Li + lines and F- in the are s o l v e n t . The RHNC C 2 results dashed and F", r e s p e c t i v e l y , in the C and 2 v 268 - TABLE X. 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 per i o n at i n f i n i t e d i l u t i o n . R e s u l t s f o r the C 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 are c o m p a r e d . The v a l u e s g i v e n are in k T u n i t s a n d are t h o s e o f the effective systems. 2 v ION Tetrahedral Solvent Li* Na Eq* K Cs FEqci- -424.1 -363.8 -316.2 -296.4 -255.8 -363.8 -316.2 -278.7 -230.5 + + + 1- We some will n o w examine properties dilution. The average in T a b l e X , where octupole solvents have b e e n octupole moment t o the s o l v e n t average ion-solvent ion-solvent XO in the C 2 v of the converse than If w e e x a m i n e we again solvent respect find some octupole expect large from has upon infinite dilution are g i v e n and C „ 2v o f the effect u p o n the our previous in m a g n i t u d e ) for a cation and o p p o s i t e more is l a r g e s t interesting energy in t h e t e t r a h e d r a l H e n c e , if w e c o n s i d e r t w o i o n s in c h a r g e strongly W e also observe the i n d i v i d u a l the i o n - o c t u p o l e t o their has a fairly {i.e., s m a l l e r interacts t o the i o n - s o l v e n t ion-quadrupole at the t e t r a h e d r a l f o r the s a m e c a t i o n are e q u a l d o e s the c a t i o n . in s o l v a t i o n contributions in b o t h is true f o r a n i o n s . + asymmetry solutions W e f i n d that the a d d i t i o n A s we would N a / F ~ ) , w e f i n d that the a n i o n than ions model positive solvent the s a m e s i z e w h i c h solvent moment X octupole solvent, while that the o c t u p o l e e n e r g i e s p e r i o n at i n f i n i t e included. d i s c u s s i o n s , U- _ / N . i s m o r e Octupole Solvent y of electrolyte for several energies. 2 -393.5 -342.1 -301.7 -284.7 -249.6 -394.7 -336.5 -291.7 -235.1 the influence o f the t h e r m o d y n a m i c results C from f o r the s m a l l e s t (e.g., E q / E q - o r + with the C T a b l e X that results. f o r a n y o f the i o n s For anions is negative in the t e t r a h e d r a l in the C and ion-octupole listed 2 y in T a b l e X , octupole a n d the i o n - d i p o l e a n d solvent. 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 octupole this e n e r g i e s h a v e d e c r e a s e d (i.e., i n c r e a s e d in m a g n i t u d e ) values v ions. ion-dipole, ion-quadrupole energy 2 energy For cations is p o s i t i v e with in the C 2 (indicating y a net 269 - - average repulsion between ion-dipole when the and ion-quadrupole a C2 most the octupole V favourable favourable note the i o n and the octupole orientations ion-solvent s h o w n that terms upon the partial dilution (3.64a) w e i m m e d i a t e l y energy ionic listed at f i n i t e - P C° ) s partial the C recorded is and tetrahedral solvents. in T a b l e X I a l o n g w i t h immediately are m u c h obvious larger than from those have reported five X,^ times larger sets o f experiment. These corrected very large much than can be clearly effect smaller and those seen from {i.e., m o r e holds and o p p o s i t e between their always volume. Then ionic f o r ions in b o t h been i n real w a t e r . in t h e p r e s e n t F o r the C a more 2 v compressibility partial results r e s p e c t i v e v a l u e s o f V? in t h e C 2 v comparison both corrected o f real w a t e r in e q . in T a b l e X I . moment volumes. solvent, while o r N a / F - ) there solvent i n x^) b e t w e e n in t h e C + solvent. solvent is octupole the octupole molar It e q . (6.4), w e s e e reasonable 2 W e f i n d that V . v octupole in s i z e is a dramatic octupole has a solvent the converse M o r e o v e r , f o r t w o ions equal value dilution x£ f o r t h e t e t r a h e d r a l T a b l e X I that in c h a r g e (i.e., E q * / E q - u s i n g e q . (6.4) o f experiment, w e compute in the tetrahedral h a v i n g t h e larger Comparing eqs. If w e r e - e x a m i n e negative) f o r a cation f o r anions. III w e into t w o independent results to allow the isothermal individual 5% of (6.4) v a l u e s f o r V.° a r e a l s o g i v e n f o r the same cation relationship anion upon In C h a p t e r i o n s at i n f i n i t e at 2 5 ° C . T h u s , in o r d e r results than c o m p r e s s i b i l i t y , x£, o f t h e pure of water f o r V.° u s i n g o n l y It volume. does not depend upon the differences of theoretical values (6.4). still. less We in T a b l e X . T a b l e X I that t h e t r e n d s (in s e c t i o n 3 ) that than that i s 5 0 % larger {i.e., o n e w h i c h is earlier respect t o the i o n . T h e s e v a l u e s f o r V.° h a v e experimental that V.° d e p e n d s u p o n t h e i s o t h e r m a l We correspond to concentration). molecular i octupole i o n , none o f , is h a v e c a l c u l a t e d v a l u e s f o r V.° f o r s e v e r a l v Clearly, have we 2 negative. is a l w a y s 2 = kTxJd V.° is an individual with molar and the a positive v" c a n b e s p l i t is not the case V? where moment f o r all the ions at i n f i n i t e with less orientations of the ion-octupole energy (of c o u r s e this (3.12) and interacts and quadrupole Finally, let us focus have particle of the octupole that t h e m a g n i t u d e total energies have b e c o m e solvent dipole moment) and equal difference solvent, with the (i.e., a p p e a r i n g a s t h o u g h it i s larger in 0 - 270 - TABLE XI. I n d i v i d u a l i o n i c p a r t i a l m o l a r v o l u m e s at i n f i n i t e d i l u t i o n . R e s u l t s f o r the t e t r a h e d r a l a n d 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 t h t h o s e f o r real w a t e r at 2 5 ° C . T h e e x p e r i m e n t a l partial molar v o l u m e s are f r o m T a b l e 6 o n page 376 o f R e f . 5. The v a l u e g i v e n in p a r e n t h e s e s a r e c o r r e c t e d r e s u l t s a s d i s c u s s e d in t h e t e x t . A l l v a l u e s f o r V.° are g i v e n in c c / m o l e . ? ION Tetrahedral Solvent -103.1 -62.7 -29.8 -14.5 22.6 -62.7 -29.8 0.3 53.0 Li* Na* Eq* K* Cs* FEq- ci|- solution). totally (that due t o d i f f e r e n c e s is t o the total presence solvent the of C l e a r l y , f o r ions C (-20.0) (-12.2) (-5.8) (-2.8) (-12.2) (-5.8) (0.1) (1°- ) 3 o f the s a m e s i z e volume e x p e c t that One view effect just currently then interaction expect e n e r g y , the g r e a t e r occurs very solvation tightly (i.e., shell. that smaller Table value due t o the has a slightly 2 y larger of the i o n and electrostriction, d e d u c t i o n is univalent octupole than in t h e C would ions approach. o f the usually implies 2 y is held in m o r e is the first solvation numbers f o r the N a * octupole of solvents c o n c l u d e that t h e a n i o n , in t h i s o f V . ° , but a g a i n t h i s most solvent coordination number of the F i g u r e 58 w e o b s e r v e that t h e electrostriction) e q . (5.2)), w e f i n d that Again, w e would [5] o f t h e s o l v e n t t h e larger t h e m a g n i t u d e [5] s u g g e s t s that If w e r e - e x a m i n e e x p e r i e n c e s greater (using 0 between the d e g r e e for small s h e l l o f a F- i o n in t h e C F- i o n w h i c h shell. the an i o n , w h i c h a N a * i o n . F u r t h e r m o r e , if w e c o m p u t e and F- i o n s the near in NA m u s t b e in T a b l e s X a n d X I . L e t us t r y a n o t h e r held for electrostriction the f i r s t ' s o l v a t i o n first of b y the values — 23.7 41.4 the e l e c t r o s t r i c t i o n and h e n c e t h e s m a l l e r V.° s h o u l d b e . U n f o r t u n a t e l y , t h i s contradicted — 3.4 15.5 3.3 any dissimilarity to the average Consequently, w e would ion-solvent -11.2 -7.4 change or c o m p r e s s i o n o f the s o l v e n t should be proportional the average (-29.3) (-22.9) (-17.6) (-15.1) (-8.8) (-0.3) (5.9) (11.4) (20.5) apparent electrostriction of the ion). N o w o n e might solvent. Expt. Octupole Solvent v -226.5 -177.5 -136.1 -116.6 -68.3 -2.0 46.0 87.9 158.5 (4.4) in their 2 solvent in i t s f i r s t case F-, should be contrary it is solvation have t o t h e r e s u l t s in X I . In o u r d i s c u s s i o n o f Figure 5 8 w e h a d m e n t i o n e d apparent shell - differences solvent. these in Q ^ g f r ) T o g e t h e r , the changes ions which XI for (apart from molar to of the We dependence is longer above size. be an i m p o r t a n t least V.°, F- and N a packing of the would factor for the correlations aqueous octupole y that it is solvent around in V.° s e e n in ion-solvent which the C, suggest the long-range in d e t e r m i n i n g model of in the dissimilarities C l e a r l y t h e n , the ion-dipole ions strongly structure gross + give values electrolyte S ) y ionic solutions Table structure rise to of the partial being study. conclude this of for observations most long-range in t h i s range long-range for equal v o l u m e s , at considered solvent in the accounts ions appears a t 271 discussion by as s h o w n consistent w i t h the in T a b l e noting X I , for experimental that the ions results ionic in the given. charge C 2 y and octupole size - 272 - CHAPTER VII CONCLUSIONS In t h i s dielectric thesis we properties i n c l u d e the properties solvent of the investigated. known We of h a v e e x a m i n e d the model aqueous electrolyte as a m o l e c u l a r water-like species. models crystal water-like radii were solvent models were m e a s u r e d v a l u e s of of water were Our the model theory RHNC t h e o r y , it symmetry. [68] had to and the generate all the the [104] and in o r d e r the program O Z and R H N C which into alkali The which and p o l a r i z a b i l i t y integral equation tensor in the uses m e t h o d s , the In o r d e r t o presented in t h i s apply a multi-component system h a v e at general f o r m s t h e s i s , both for the system. In simplifications of the least the e q u a t i o n s w a s w r i t t e n a n d then study to we have exploited properties e x p r e s s i o n s c a n not conditions equations and taking obtain applicable to of formalism real the indeterminate k-dependent appropriate determinate of C 2 y multipole used to pure s o l v e n t K i r k w o o d and integrals solutions. be a p p l i e d d i r e c t l y By defining exact electrolyte leads to Kirkwood-Buff a b l e to the derive general relationships between quantities. directly parameters. and systems. charge neutrality were adjustable h a v e e x a m i n e d the present thermodynamic we incorporating hard-sphere diameters. generalized for species certain thermodynamic Kirkwood-Buff ones also a s c h a r g e d hard s p h e r e s . exclusively. O Z equation w e all the results solution In t h i s structural f o c u s i n g u p o n the moments studied using be f i r s t the computer potential for explicitly were a s hard s p h e r e s multipole being e m p l o y e d almost when A low-order systems were our g e n e r a l i z a t i o n o f result no f r e e l y the also treated study simple ions simply determine and and included. RHNC which u s e d to with ions, primarily h a l i d e s , and h e n c e h a v e m o d e l l e d t h e s e Ionic in this used were microscopic (molecular) properties univalent solutions which The dielectric solvents employed T h e ion and s o l v e n t have considered only structural, thermodynamic Buff over h ^( fl T h e usual because application results analogs of for the k—>0 l i m i t s system we of the analytically, expressions. These relationships s y s t e m s s i n c e the r) have considered are - 273 - incorporates reported have the solvent results used is The be a p p l i e d t o more complicated limiting M o r e o v e r , the microscopic the approximation for macroscopic this the of electrolyte of involving present the R an aqueous for a from the found s electrolyte exact be the the limiting theory. functionally HNC is rather solutions results molecular to However, when HNC theory systems. macroscopic is c o m p a r e d w i t h the overestimate ion a n d the was shown properties solvent have which investigation to exact inaccurate we have slope for local considered solvent RDMF other could the We rise to for considered, by about an it. an e f f e c t upon the e m p h a s i z e d that systems present More be e x a m i n e d , e.g., complicated correlations at low general where short-range model b o t h the to SCMF an potential thermodynamic concentration. ease solvent and between formalisms for the a distance developed in the aqueous electrolyte models potentials at spherical relative of (like the those the electrolyte us a solvent of be u s e d w i t h b e s i d e s the study. allows approach of have a second laws most The moment spherical potential limiting which problem present dipole Furthermore, this ion-solvent and c o u l d the experienced by an e f f e c t i v e a thesis we theory, which field examined. to have a l s o d e v e l o p e d is a m e a n around In t h i s with many-body fluctuations) average total theory be the applied to RDMF particles may in w h i c h potential. field theory depend upon in the more electric [67] c a n be determine levels of solvent ignoring additive f o r m a l i s m , the are g e n e r a l of (by approximation s h o u l d a g a i n be thesis approximation pairwise of The described two a polarizable is r e d u c e d and g i v e s to have concentration. average ion. we SCMF salt level approximation) possess law w we thermodynamic s u r p r i s i n g l y , the extracted that the containing in o r d e r finite examine the model SCMF model detailed in t h i s discover study is the the at It was + slope of polarization how from Not the an e f f e c t i v e solution more [6]. expression. solutions of solvent lny limiting our and c o m p a r e d w i t h the limiting appears to these problem shown for of method magnitude. In the first limiting behaviour macroscopic F o r the HNC theory order law result, w e quantity. have and c a n e a s i l y Debye-Hiickel theory to we o n l y , the Debye-Hiickel equivalent Although system also examined through species. salt/solvent concentration was molecular a two-component general low expressions obtained for as a true b o t h the and the and h a v e higher solutions ions and ions multipole - moments. charges RHNC S o l u t i o n s of could theory presented Several in t h i s system theory. hard-sphere models with liquid water were the agreement investigated able to over obtain found functions of a spherical potential addition of this properties relatively have of potential our w a t e r - l i k e little to of effect a somewhat also the effect reported of range o f higher low water hard-sphere the was systems for and The are agree RHNC in rather poor However, we 25°C model. low The o c t u p o l e models constants which fluids at study for Model temperature. the improved these temperature. liquid any in t h i s temperatures our w a t e r - l i k e the potentials. [71,110] quadrupoles. at at pair dependence for for model. upon larger to form examined have dielectric values, particularly addition soft solvents were a large correct s t r u c t u r e different general angle-dependent linear experiment, particularly the several that the previously d i p o l e s and distribution with out ions of c o u l d , in p r i n c i p l e , be a p p l i e d t o that These systems were for point thesis similar to with experimental with The HNC b a s i s set be quite results We characterized by o b s e r v e d to well one salt c l o s e l y related w a t e r - l i k e RHNC pressures. than be e a s i l y s t u d i e d . multi-component u s i n g the more 274 through was dielectric constant, although u p o n the structure the simple M o r e o v e r , the temperature moment were it w i t h i n the dielectric shown was to have found model to fluid systems. Virtually all of the aqueous electrolyte solutions we investigated at concentration employed a polarizable solvent only and square quadrupole moments. For these s o l u t i o n s the dipole SCMF a p p r o x i m a t i o n , the s h o w n to average finite model be essentially total dipole average independent moment of the local electric of salt dielectric constants obtained qualitative agreement w i t h the experimental of aqueous electrolyte point out that study were Our confirmed relatively demonstrated hard-sphere found to all the by our simple a remarkable diameters be e x t r e m e l y w a s taken to for these of numerical model diversity the limiting ions. for of model values for solutions, particularly microscopic in the at and law was low dielectric smaller of in constants in We also this solutions simply varying i o n s , such as N a C l , larger The concentration. behaviour through soluble, whereas pairs are concentrations. aqueous electrolyte of the be a c o n s t a n t . expressions derived at with within bulk solutions the higher results Pairs model c o n c e n t r a t i o n , and h e n c e solvent equilibrium field have the were i o n s , such as M ' l , were - relatively insoluble. properties were salt. ion The M o r e o v e r , ion s o l v a t i o n shown size mean consistent the solutions of experimental were s y s t e m , the for ion-solvent were the in t h i s by we to larger disrupt clearly defined ion pairs. large ions. structure found The on the most dependence was Finally, we The e f f e c t s aqueous volumes is solutions, provide NaCl e.g., for solutions case some of the insight solutions. mean activity NaCl into We find coefficient, c a n h a v e a large electrolyte dilution of the by our only size of were of both influence, even in the the remark that were found RDMF theory at infinite by be were of Smaller greater at higher a r o u n d an Near c o n t a c t was both either very dependence of RLHNC the at higher have a like and small the in rather correlations results poor examined for and at low both strong seen to for like-ion dilution ion d e c r e a s e s , although very unlike or very ion-ion screening e f f e c t s , although previous to ion structure ionic a much shell, particularly concentration simple to dependence, yet case ion-ion, solutions. structure solvent ion-ion the became obvious observed. the of model structure solvation demonstrated solutions investigation within effect nature p a r t , the counter-ion results. may second solvation e s p e c i a l l y true dominated infinite this short-range was present in the little c o u n t e r - i o n in the correlations at However, even a detailed as the molecular F o r the models molar concentration model structure effects showed This was concentrations. for + a was observed our e x p e c t , the and c o u n t e r - i o n influence lny low of observed disagreed with solvent-solvent i o n s , although becomes strong behaviour structure the might concentrations. partial behaviour very s u c h as the have made A s we dependence w a s asymmetry quantities aqueous electrolyte ion-ion concentrations. ion-solvent size concentration. found concentration ion aqueous electrolyte study other properties short-range study more the values for reported the c h a n g e s i g n at and s o l v e n t - s o l v e n t d e g r e e than real C s l and K C I . exhibited low In t h i s to by S o m e of of thermodynamic relatively ions found those and S o m e of demonstrated results ion-solvent at coefficient. ions. unusual b e h a v i o u r that isothermal r e s u l t s , e.g., l e s s than the c o m p r e s s i b i l i t y , the was large sensitive to the that v^ be q u i t e thermodynamic several thermodynamic activity with s l o p e of to and many dependence of examined, including and the 275 at some higher u s i n g the agreement several same with the model concentration. The - electric field by lateral the RDMF due solvent had o n l y although quantities long-range RDMF electrolyte the thermodynamic impact. upon g ^ ^( r ) . The F u t h e r m o r e , the packing structure Several or in earlier by our Our Nevertheless, we water-like was ionic sensitivity results is very tensor found partial study thesis. ions clearly influence these where results solutions systems the In this only presented requires o f the low study at 2 5 ° C . models would in this thesis a s s u m p t i o n that o n l y a pairwise McMillan-Mayer theory level f o r real structural and study o f was addition shown of how the more octupole an ion additive ion-solvent immediately behaviour of water at r e l a t i v e l y exhibited upon i o n b o t h t o higher Ions w i t h moments temperature higher (e.g., C N ) s h o u l d aqueous dependence o f high temperatures much better. T h e easily be used t o test the potential in o r d e r need be e m p l o y e d in t o study above investigation. w e have c o n s i d e r e d m o d e l can very dilution is solvated. detailed concentration. C l e a r l y , the effect values o f V.°. moment multipole large long-range needs t o be extended order o f the t o h a v e a large at i n f i n i t e interesting be expected to work [25,28] insensitive t o the have been made should be examined, particularly present in t h e contribution O b v i o u s l y , further volumes The v e r y properties (e.g., d i v a l e n t ) a n d w i t h model in d e t e r m i n i n g (e.g., h e x a d e c a p o l e ) and t o f i n i t e electrolyte the s u g g e s t that the moments be investigated. RDMF o f w a t e r , the molar and t h e r m o d y n a m i c also for some t o have a particularly solvation charges some have been able t o clearly details strongly important o f larger o f the upon inaccurate, even t o be relatively t o the d i s c u s s i o n s in t h i s examination influence in determining solvent s u g g e s t i o n s f o r further solutions correlations. A s m a c r o s c o p i c result solutions. found quadrupole moment present known structure, is warranted. individual extreme on ion-ion F o r e x a m p l e , the o f polarization o f the t o our ion-solvent t o be relatively reduced C o n s e q u e n t l y , the H o w e v e r , at l e a s t 2 limits. ion s o l v a t i o n w a s The o c t u p o l e demonstrated out V . of electrolyte effects moment impact e x c e e d s the solutions. zz component octupole for properties Whereas separations. short-range notably turns importance these polarization small a larger concentration slope shown t o be substantially o b s e r v e d t o h a v e a large theory and l o w demonstrate u p o n the quantities, most limiting aqueous charge w a s effect RDMF was the - f i e l d s , e v e n at s m a l l a small thermodynamic the ionic it a p p e a r e d t o h a v e e x p e c t e d , the to t o the 276 primitive model 277 - electrolyte Another solutions. extension examination model outlined thesis. for present solutions in t h i s conclude develop investigation colloidal realistic solutions. models the more We to to large electrolyte hopefully not of This by R a t h e r , its these again stating results main purpose systems and t o might ion solvation help the measurable thermodynamic here. also understand to is a l s o be the to quantities. in p r o g r e s s double purpose model of layers out [187]. [133] present the same but u s i n g the the is in the complicated investigated methods study was of aqueous electrolyte systematically examine simple learn w h a t how carried Finally, more an e x a c t was in o r d e r being and e l e c t r i c that for currently which particles models and report is study considered - details is i m p o r t a n t of the ion in determining solvation affect - 278 - LIST OF REFERENCES 1. M u r r e l l , J . N . , and Boucher, E.A., Properties W i l e y , N e w Y o r k , 1982. 2. F r a n k s , F., W a t e r . T h e R o y a l 3. Helper, L.G., and S m i t h , W.L., P r i n c i p l e s York, 1975. 4. F r a n k s , F „ e d . , W a t e r : A C o m p r e h e n s i v e T r e a t i s e . V o l . 1: T h e P h y s i c s and P h y s i c a l C h e m i s t r y o f Water. Plenum P r e s s , N e w Y o r k , 1971. 5. C o n w a y , B.E., Ionic H y d r a t i o n in C h e m i s t r y a n d B i o p h y s i c s . E l s e v i e r S c i e n t i f i c P u b l i s h i n g C o . , A m s t e r d a m , 1981. 6. H a r n e d , H.S., a n d O w e n s , B.B., P h y s i c a l C h e m i s t r y o f E l e c t r o l y t e S o l u t i o n s . 3rd ed., Reinhold Publishing Corp., N e w York, 1959. 7. R o b i n s o n , R.A., a n d S t o k e s , R.H., E l e c t r o l y t e P r e s s , N e w Y o r k , 1959. 8. G u g g e n h e i m , E . A . , a n d S t o k e s , R.H., E q u i l i b r i u m P r o p e r t i e s o f A q u e o u s Solutions of Single Strong Electrolytes. Pergamon Press, Toronto, 1969. 9. C o n w a y , B.E., a n d B a r r a d s , R.G., e d s . , C h e m i c a l S o l u t i o n s . W i l e y , N e w Y o r k , 1966. Society of Liquids and Solutions. of Chemistry, L o n d o n , 1983. of Chemistry. MacMillan, New Solutions. 2nd ed., A c a d e m i c Physics of Ionic Ann. Rev. Phys. Chem., 3 1 , 3 4 5 (1980). 10. W o l y n e s , P.G., 11. Hamer, W . J . , ed., The Structure York, 1959. 12. F r i e d m a n , H.L., Chem. Sci., 2 5 , 4 2 (1985). 13. Enderby, J.E., and N e i l s o n , G.W., 14. E n d e r b y , J . E . , H o w e l l s , W . S . , a n d H o w e , R.A., (1973). 15. N a r t e n , A . H . , V a s l o w , F., a n d L e v y , H.A., 16. Enderby, J.E., 17. C l a u s i u s , R., 18. van't 19. P l a n c k , M . , Z. Physik. Chem., 1, 577 (1887). 20. Arrhenius, of Electrolyte Solutions. Wiley, N e w Rep. Prog. Phys., 44, 5 9 3 (1981). Chem. Phys. Lett., 2 1 , 109 J. Chem. Phys., 5 8 , 5017 Ann. Rev. Phys. Chem., 34, 155 (1983). Poggendorffs Anna/en, 101, 3 3 8 (1857). H o f f , J.H., Z . S., Z . Physik. Chem., 1, 481 (1887). Physik. Chem., 1, 631 (1887). (1973). - 279 21. F a l k e n h a g e n , H., E l e c t r o l y t e s , t r a n s : 1934. R.P. B e l l , C l a r e n d o n Press, 22. D e b y e , P . , a n d H u c k e l . E . , Physik. Z., 24, 185, 3 0 5 (1923). 23. O n s a g e r , L., Physik. Z., 28, 2 7 7 (1927) 24. Outhwaite, C.W., A Specialist Periodical Report: V o l . 2 , T h e C h e m i c a l S o c i e t y , L o n d o n , 1975. 25. Anderson, H.C, Modern Aspects of Electrochemistry, and J . O ' M . B o c k r i s , V o l . 1 1 , 1975. 26. M c M i l l a n , W . G . , a n d M a y e r , J . E . , J. Chem. Phys. 13 , 2 7 6 (1945). 27. H a n s e n , J . P . , a n d M c D o n a l d , I.R., T h e o r y P r e s s , L o n d o n , 1976. 28. F r i e d m a n , H.L., Ann. Rev. Phys. Chem., 32, 179 (1981). 29. R a m a n a t h a n , P . S . , a n d F r i e d m a n , H.L., J. Chem. Phys., 54, 1086 30. R a s a i a h , J . C . , J. Sol. Chem., 2, 301 (1973). 31. Barker, J.A., and Watts, 32. Rahman, A . , and Stillinger, 33. B a r k e r , J A . , a n d H e n d e r s o n , D., Rev. Mod. Phys., 48, 5 8 7 (1976). 34. L e v e s q u e , D., W e i s s , J . J . , a n d H a n s e n , J . P . , A p p l i c a t i o n s o f t h e M o n t e C a r l o M e t h o d in S t a t i s t i c a l P h y s i c s , e d . , K. B i n d e r , S p r i n g e r - V e r l a g , N e w Y o r k , 1984. 35. R e i m e r s , J.R., a n d W a t t s , R.O., Chem. Phys., 91, 201 36. R e i m e r s , J.R., W a t t s , 37. B a r n e s , P . , F i n n e y , J.L., N i c h o l a s , J . D . , a n d Q u i n n , J . E . , Nature, 282, 4 5 9 (1979). 38. Wojcik, 39. J o r g e n s e n , W . L . , J. Am. Chem. Soc, 103, 3 3 5 (1981). 40. K u h a r s k y , R . A . , a n d R o s s k y , P . J . , J. Chem. Phys., 82, 5 1 6 4 41. J o r g e n s e n , W . L . , C h a n d r a s e k h a r , J . , M a d u r a , J . D . , I m p e y , R.W., a n d K l e i n , M.L., J. Chem. Phys., 79, 9 2 6 (1983). 42. Mezei, M., Swaminathan, 3366 (1979). Statistical of Simple Oxford, Mechanics. e d s . B.E. C o n w a y Liquids. Academic (1971). R.O., Chem. Phys. Lett., 3, 144 (1969). F.H., J. Chem. Phys., 55, 3 3 3 6 (1971). (1984). R.O., a n d K l e i n , M.L., Chem. Phys., 64, 9 5 (1982). M . , a n d C l e m e n t i , E., J. Chem. Phys., 84, 5970 (1986). (1984). S . , a n d B e v e r i d g e , D.L., J. Chem. Phys., 7 1 , 280 - 43. L i e , G . C , C l e m e n t i , E . , a n d Y a s h i m i n e , M . , J. Chem. Phys., 6 4 , 2 3 1 5 (1976). 44. Stillinger, 45. N e u m a n n , M . , J. Chem. Phys., 8 2 , 5663 46. N e u m a n n , M . , J. Chem. Phys., 8 5 , 1567 47. W o o d , D.W., W a t e r : A C o m p r e h e n s i v e Recent A d v a n c e s . Plenum P r e s s , N e w 48. S c e a t s , M . G . , S t a v o l a , M . , a n d R i c e , S . A . , J. Chem. Phys., 7 0 , 3927 (1979). 49. M a t s u o k a , O . , Y o s h i m i n e , M . , a n d C l e m e n t i , E., J. Chem. Phys., 6 4 , 1351 (1976). 50. B o u n d s , D.G., Molec. Phys., 5 4 , 1355 51. Chandrasekhar, J . , Spellmeyer, Soc, 106, 9 0 3 (1984). 52. M i l l s , M.F., R e i m e r s , J.R., a n d W a t t s , 53. Mezei 54. I m p e y , R.W., M a d d e n , P . A . , a n d M c D o n a l d , I.R., J. Chem. Phys., 8 7 , 5071 (1983). 55. Heinzinger, F.H., a n d R a h m a n , A . , J. Chem. Phys., 6 0 , 1545 (1974). (1985). (1986). T r e a t i s e , e d . F. F r a n k s , V o l . 6 : Y o r k , 1979. (1985). D.C., a n d J o r g e n s e n , W . L . , J. Am. Chem. R.O., Molec. Phys., 5 7 , 7 7 7 (1986). M . , a n d B e v e r i d g e , D.L., J. Chem. Phys., 74, 6902 (1981). K., a n d V o g e l , P . C . , Z . Naturforsch., 3 1 A , 4 6 3 (1976). 5 6 . ° S z a s z , G.I., H e i n z i n g e r , K., a n d R e i d e , W . O . , Z. Naturforsch., 3 6 A , 1067 (1981). 57. C l e m e n t i , E . , a n d B a r s o t t i , R., Chem. Phys. Lett., 5 9 , 21 (1978). 58. H e i n z i n g e r , K., Pure Appl. Chem., 5 7 , 1031 59. R o s s k y , P . J . , Ann. Rev. Phys. Chem., 3 6 , 321 (1985). 60. W a t t s , R.O., A S p e c i a l i s t P e r i o d i c a l C h e m i c a l S o c i e t y , L o n d o n , 1973. 61. S t e l l , G . , P a t e y , G . N . , a n d H o y e , J . S . , Adv. Chem. Phys., 38, 183 (1981). 62. P a t e y , G . N . , Molec. Phys., 3 4 , 4 2 7 (1978). 63. P a t e y , G . N . , Molec. Phys., 3 5 , 1413 64. P a t e y , G . N . , L e v e s q u e , D., a n d W e i s , J . J . , Molec. Phys., 3 8 , 1635 65. P a t e y , G . N . , L e v e s q u e , D., a n d W e i s , J . J . , Molec. Phys., 3 8 , 2 1 9 (1979). Report: (1985). Statistical M e c h a n i c s . V o l . 1, (1978). (1979). - 281 - 66. Wertheim, M . S . , J. Chem. Phys., 55, 429I (1971). 67. C a r n i e , S . L . , a n d P a t e y , G . N . , Molec. Phys., 47, 1129 (1982). 68. F r i e s , P.H., a n d P a t e y , G . N . , J. Chem. Phys., 82, 4 2 9 (1985). 69. L e v e s q u e , D., W e i s , J . J . , a n d P a t e y , G . N . , Molec. Phys., 5 1 , 3 3 3 (1984). 70. L e e , L.Y., F r i e s , P.H., a n d P a t e y , G . N . , Molec. Phys., 55, 751 (1985). 71. P e r k y n s , J . S . , F r i e s , P.H., a n d P a t e y , G . N . , Molec. Phys., 57, 5 2 9 (1986). 72. C a r n i e , S . L . , C h a n , D.Y.C., a n d W a l k e r , G.R., Molec. Phys., 43, 1115 (1981). 73. C h a n , D.Y.C., M i t c h e l l , D . J . , a n d N i n h a m , B.W., J. Chem. Phys., 70, 2946 (1979) . 74. L e v e s q u e , D., W e i s , J . J . , a n d P a t e y , G . N . , J. Chem. Phys., 72, 1887 (1980) . 75. L e v e s q u e , D., W e i s , J . J . , a n d P a t e y , G . N . , Phys. Lett., 66A, 115 (1978). 76. Outhwaite, 77. A d e l m a n , S . A . , a n d C h e n , J . H . , J. Chem. Phys., 70, 4 2 9 1 (1979). 78. A d e l m a n , S . A . , a n d D e u t c h , J . M . , J. Chem. Phys., 60, 3 9 3 5 79. Patey, G.N., and Carnie 80. K u s a l i k , P . G . , a n d P a t e y , G . N . , J. Chem. Phys., 79, 4 4 6 8 81. Kusalik, P.G., " A Theoretical Study of Dilute A q u e o u s Electrolyte S o l u t i o n s " , M a s t e r ' s T h e s i s , U n i v e r s i t y o f British C o l u m b i a , 1984. 82. P e t t i t t , B . M . , a n d R o s s k y , P . J . , J. Chem. Phys., 84, 5836 83. K u s a l i k , P . G . , a n d P a t e y , G . N . , t o be published. 84. K u s a l i k , P . G . , a n d P a t e y , G . N . , t o be published. 85. K u s a l i k , P . G . , a n d P a t e y , G . N . , a c c e p t e d , J. Chem. Phys. 86. Kusalik, P.G., and Patey, G.N., submitted, 87. F r i e d m a n , H.L., A C o u r s e in S t a t i s t i c a l E n g l e w o o d C l i f f s , N e w J e r s e y , 1985. 88. McQuarrie, D A . , Statistical 89. H i l l , T.L., A n Introduction to S t a t i s t i c a l Thermodynamics. A d d i s o n - W e s l e y , R e a d i n g , M a s s a c h u s e t t s , 1960. C . W . , Molec. Phys., 37, 1229 (1977). (1974). S.L., J. Chem. Phys., 78, 5 1 8 3 (1983). (1983). (1986). J. Chem. P h y s . Mechanics. M e c h a n i c s . Harper Prentice-Hall, and R o w , N e w York, 1976. - 282 - 90. O r n s t e i n , L . S . , and Z e r n i k e , F., Proc. Acad. Sci. Amsterdam, 17, 7 9 3 (1914). 91. L e b o w i t z , J . L . , and P e r c u s , J.K., Phys. Rev., 144, 251 (1966). 92. P e r c u s , J . K . , a n d Y e v i c k , G . J . , Phys. Rev., 110, 1 (1958). 93. V a n L e e u w e n , J . M . , G r o e n e v e l d , M . J . , a n d De B o e r , J . , Physica, 25, 7 9 2 (1959). 94. M e e r o n , E., J. Math. Phys., 1, 192 (1960). 95. M o r i t a , T., Prog. Theor. Phys., 23, 8 2 9 (1960). 96. R u s h b r o o k e , G . S . , Physica, 26, 2 5 9 (1960). 97. V e r l e t , L., Nuovo Cimento, 18, 7 7 (1960). 98. P e r e r a , A . , K u s a l i k , P . G . , a n d P a t e y , G . N . , Mol. Phys., In P r e s s . 99. P e r e r a , A . , K u s a l i k , P . G . , a n d P a t e y , G . N . , s u b m i t t e d , J. Chem. Phys. 100. P e r e r a , A . , K u s a l i k , P . G . , a n d P a t e y , G . N . , t o be p u b l i s h e d . 101. B l u m , L., a n d T o r r u e l l a , A . J . , J. Chem. Phys., 56, 3 0 3 (1971). 102. B l u m , L., J. Chem. Phys., 57, 1862 (1972). 103. B l u m , L., J. Chem. Phys., 58, 3 2 9 5 104. K i r k w o o d , J . G . , a n d B u f f , F.P., J. Chem. Phys., 19, 7 7 4 (1951). 105. B o r n , M . , a n d O p p e n h e i m e r , J.R., Ann. Phys. (Leipz.l, 84, 4 5 7 (1927). 106. H i r s c h f e l d e r , J . O . , C u r t i s s , C.F., a n d B i r d , R.B., M o l e c u l a r and L i q u i d s . W i l e y , N e w Y o r k , 1 9 5 4 . 107. Kielich, S., A Specialist Periodical Report: Dielectric and Related M o l e c u l a r P r o c e s s e s . V o l . 1, T h e C h e m i c a l S o c i e t y , L o n d o n , 1 9 7 2 . 108. M e s s i a h , A . , Quantum 109. R o t e n b e r g , M . , B e v i n s , R „ M e t r o p o l i s , N . , and W o o t e n , N . , T h e 3 - i a n d 6-i S y m b o l s . M a s s a c h u s e t t s Institute o f T e c h n o l o g y , C a m b r i d g e , 1959. 110. P e r k y n s , J . S . , " T h e S o l u t i o n t o the R e f e r e n c e H y p e r n e t t e d C h a i n A p p r o x i m a t i o n f o r Fluids of Hard S p h e r e s W i t h D i p o l e s and Quadrupoles with Application to Liquid A m m o n i a " , Master's Thesis, University of British C o l u m b i a , 1985. 111. W a l k e r , G.R., " F l u i d s w i t h A n g l e - D e p e n d e n t A u s t r a l i a n N a t i o n a l U n i v e r s i t y , .1983. (1973). Theory of Gases Mechanics. W i l e y , N e w York, 1958. Potentials", Ph.D. Dissertation, - 283 - 112. B u c k i n g h a m , A . D . , Quart. Rev. Chem. Soc. Lond., 1 3 , 183 (1959). 113. P r i c e , S.L., S t o n e , A . J . , a n d A l d e r t o n , M . , Molec. Phys., 5 2 , 9 8 7 (1984). 114. S t o g r y n , D.E., a n d S t o g r y n , A . P . , Molec. Phys., 1 1 , 371 (1966). 115. J o h n , I.G., B a c s k a y , G . B . , a n d H u s h , N . S . , Chem. Phys., 5 1 , 4 9 (1980). 116. N e w m a n n , D., a n d M o s k o v i t z , J . W . , J. Chem. Phys., 4 9 , 2 0 5 6 117. H u i s z o o n , C . , Molec. Phys., 5 8 , 8 6 5 , 118. D y k e , T.R., a n d M u e n t e r , J . S . , J. Chem. Phys., 5 9 , 3 1 2 5 119. V e r h o e v e n , J . , a n d D y m a n u y s , A . , J.Chem. Phys., 5 2 , 3222 120. C i c c a r i e l l o , S . , a n d D o m e n i c o , G . , J. Chem. Soc, Faraday, 8 1 , 1163 (1968). (1986). (1973). (1970). (1985) . 121. N a r t e n , A . H . , a n d L e v y , H A . , J. Chem. Phys., 5 5 , 2 2 6 3 (1971). 122. S o p e r , A . K . , a n d P h i l l i p s , M . G . , Chem. Phys., 1 0 7 , 4 7 (1986). 123. P a u l i n g , L., T h e N a t u r e o f t h e C h e m i c a l P r e s s , I t h a c a , N.Y., 1960. 124. M o r r i s , D.F.C., Struct. Bonding, 4 , 6 3 (1968). 125. A d e l m a n , S . A . , J. Chem. Phys., 6 4 , 7 2 4 (1976). 126. B r i g h a m , E.O., a n d M o r r o w , R.E., T h e S p e c t r u m , N e w Y o r k , 1967. 127. L a d o , F., J. Comp. Phys., 8 , 4 1 7 (1971). 128. A b r a m o w i t z , M . , a n d S t e g u n , I.A., e d s . , H a n d b o o k F u n c t i o n s . D o v e r , N e w Y o r k , 1970. 129. L a d o , F., Molec. Phys., 3 1 , 1117 130. L e e , L.L., a n d L e v e s q u e , D., Molec. Phys., 2 6 , 1351 131. V e r l e t , L., a n d W e i s , J . J . , Phys. Rev. A., 5 , 9 3 9 (1972). 132. P e r k y n s , J . S . , K u s a l i k , P . G . , a n d P a t e y , G . N . , Chem. Phys. Lett., 1 2 9 , 2 5 8 (1986) . 133. Torrie, G.M., Kusalik, P.G., and Patey, G.N., to be published. 134. G e r a l d , C.F., A p p l i e d N u m e r i c a l M i l l s , O n t a r i o , 1978. 135. C h a n , D.Y.C., M i t c h e l l , D . J . , N i n h a m , B.W., a n d P a i l t h o r p e , B A . , J. Chem. Phys., 6 9 , 691 (1978). Bond. 3rd ed., Cornell Fast Fourier Transform. of University IEEE Mathematical (1976). (1973). A n a l y s i s . 2nd ed., A d d i s o n - W e s l e y , D o n - 284 - 136. M e n d e n h a l l , W . , Introduction to P r o b a b i l i t y and S t a t i s t i c s . 4th e d . , Duxbury P r e s s , North Scltuate, M a s s a c h u s e t t s , 1975. 137. Hasted, J.B., Aqueous 138. F r o l i c h , H., T h e o r y O x f o r d , 1958. 139. B o t t c h e r , C.J.F., T h e o r y o f E l e c t r i c S c i e n t i f i c , A m s t e r d a m , 1973. 140. W e i d n e r , R.T., a n d S e l l s , R.L., E l e m e n t a r y and B a c o n , B o s t o n , 1973. 141. H u b b a r d , J . B . , C o l o n o m o s , P., a n d W o l n e s , P . G . , J. Chem. Phys., 7 1 , 2652 (1979). 142. K i r k w o o d , J . G . , J. Chem. Phys., 4, 5 9 2 (1936). 143. Hcfye, J . S . , a n d S t e l l , G . , J. Chem. Phys., 6 4 , 1952 (1976). 144. Stillinger, 145. F r i e d m a n , H.L., a n d R a m a n a t h a n , P . S . , J. Chem. Phys., 7 4 , 3756 146. B e e b y , J . L . , J. Phys. Chem., 6, 2262 147. H a l l , D . G . , J. Chem. Soc, Faraday Trans., 6 7 , 2 5 1 6 (1971). 148. R a s a i a h , J . C . , a n d F r i e d m a n , H.L., J. Chem. Phys., 5 0 , 3 9 6 5 149. F r i e d m a n , H.L., K r i s h m a n , C . V . , a n d J o l i c o e u r , C . , Ann. New York Academy Sci., 2 0 4 , 19 (1973). 150. Castellan, G.W., Physical Chemistry. A d d i s o n Massachusetts, 1971. 151. R a s a i a h , J . C . , a n d F r i e d m a n , H.L., J. Chem. Phys., 4 8 , 2742 152. Hcfye, J . S . , a n d S t e l l , G . , J. Chem. Phys., 7 1 , 1985 (1979). 153. Rasaiah, J.C., Isbister, (1981). 154. R a s a i a h , J . C . , J. Chem. Phys., 7 7 , 5711 (1982). 155. P a t e y , G . N . , a n d V a l l e a u , J . P . , Chem. Phys. Lett., 5 8 , 157 (1978). 156. Wertheim, 157. V e n k a t a s u b r a m a n i a n , V . , G u b b i n s , K.E., G r a y , C . G . , a n d J o s l i n , C . G . , Molec. Phys., 52, 1441 (1984). Dielectrics. Chapman and Hall, L o n d o n , 1973. of Dielectrics. 2nd ed., Oxford F A . , and Lovett, University Press, P o l a r i z a t i o n . 2nd e d . , Elsevier Classical Physics. V o l . 2, Allyn R., J. Chem. Phys., 4 9 , 1991 (1968). (1970). (1973). (1969). Wesley, Reading, (1968). D . J . , a n d S t e l l , G . , J. Chem. Phys., 7 5 , 4 7 0 7 M . S . , Ann. Rev. Phys. Chem., 3 0 , 471 (1979). - 285 - 158. C a i l l a l , J . M . , L e v e s q u e , D., W e i s , J . J . , K u s a l i k , P . G . , and Molec. Phys., 5 5 , 6 5 (1985). 159. Patey, G.N., Torrie, 160. V e s l e y , F . J . , Chem. Phys. Lett., 5 6 , 390 161. P o l l o c k , E.L., A l d e r , B . J . , and 162. M u r a d , S . , Molec. Phys., 5 1 , 5 2 5 163. P a t e y , G . N . , L e v e s q u e , D., and 164. Murphy, 165. S p i e g e l , M.R., M a t h e m a t i c a l H a n d b o o k M c G r a w - H i l l , N e w Y o r k , 1968. 166. Steinhauser, (1981). 167. P o l l o c k , E.L., A l d e r , B . J . , and 49 (1980). 168. K u s a l i k , P . G . , and 169. Weast, R . C , ed., CRC Handbook P r e s s , C l e v e l a n d , O h i o , 1975. 170. J a n s o o n , V . M . , and (1972). 171. E i s e n b e r g , D., and K a u z m a n n , W . , The C l a r e n d o n , O x f o r d , 1969. 172. C o u l s o n , C . A . , and 173. P e r e r a , A . , K u s a l i k , P . G . , and 174. Washburn, 1926. 175. C o n w a y , B.E., and 176. B e h r e t , H., S c h m i t h a l s , 9 6 , 7 3 (1975). 177. H a r r i s , F.E., and 178. G e i s e , K., K a a t z e , U., a n d 179. Haggis, G.H., Hasted, J.B., and (1952). G . M . , and V a l l e a u , J . P . , J. Chem. Phys., 7 1 , 96 (1979). (1978). P a t e y , G . N . , Physica A., 1 0 8 , 14 (1981). (1984). W e i s , J . J . , Molec. Phys., 5 7 , 337 W . F . , J. Chem. Phys., 6 7 , 5 8 7 7 O., and Patey, G.N., Bertagnolli, (1986). (1977). o f Formulas and Tables. H., Ber. Bunsenges. Phys. Chem., 8 5 , 4 5 P r a t t , L.R., Proc. Natl. Acad. Sci. U.S.A., 7 7 , Patey, G.N., unpublished work. of Chemistry a n d P h y s i c s . 56th e d . , C R C F r a n c k , E . U . , Ber. Bunsenges. Phys. Chem., 7 6 , 9 4 3 Structure and Properties E i s e n b e r g , D., Proc. R. Soc. A., 2 9 1 , 4 4 5 Water. (1966). Patey, G.N., to be published. E.W., e d . . I n t e r n a t i o n a l Critical Tables. McGraw-Hill, New V e r r a l l , R.E., J. Phys. Chem., 7 0 , 3 9 5 2 F., and of York, (1966). B a r t h e l , J . , Z. Phys. Chem. Neue. Folge., O ' K o n s k i , C.T., J. Phys. Chem. 6 1 , 310 (1957). P o t t e l , R., J. Phys. Chem., 7 4 , 3718 (1970). B u c h a n a n , T . J . , J. Chem. Phys., 2 0 , 1452 - 286 180. B a r t h e l , J . , K r i i g e r , J . , a n d S c h a l l m e y e r , E., Z. Phys. Chem. Neue. 1 0 4 , 5 9 (1977). 181. G u c k e r , F.T., C h e r n i c k , C.L., a n d R o y - C h o w d b u r y , P., Proc. Natl. U.S.A., 55, 12 (1966). 182. W e n , W . I . , a n d S a i t o , S . , J. 183. M i l l e r o , F . J . , Chem. Rev., 184. F r a n k s , F., a n d S m i t h , H.T., Trans. Faraday Soc, 63, 2 5 8 6 185. H a m e r , W . J . , a n d W u , Y . C . , J. Phys. Chem. Ref. Data, 1, 1047 (1972). 186. L i n d e n b a u m , S . , a n d B o y d , G . E . , J. Phys. 187. P a t e y , G.N., and Kusalik, P.G., to be published. 188. S p r i n g e r , J . F . , P r o h r a n t , M . A . , a n d S t e v e n s , F.A., J. Chem. Phys., (1973). 189. L a d o , F., Molec. 190. P a t e y , G . N . , J. Chem. Phys., Phys., Phys. Chem., 68, 2639 Folge., Acad. Sci. (1964). 7 1 , 147 (1971). Chem., 68, 9 1 1 (1964). 3 1 , 1117 (1976). 7 2 , 5763 (1967). (1980). 58, 4 8 6 3 - 287 - APPENDIX A TREATMENT OF POTENTIAL TERMS IN c(12) In g e n e r a l , care m u s t mnl ( ) uv,a(i r a n c j be t a k e n J ^(r) 'uvjaP when numerically performing ' ^ nn multipolar long-range contributions c b e i n g c o n s i d e r e d in t h i s due to / 3 u ^ ( l 2 ) that a a s r—>°>. a forward and (cf. e q s . (2.32), (2.34) a n d (2.46), (2.47)). fluids ^12)—>-Bu ( 1 2) ap ap both a backward Hankel transforms hard-sphere in h a n d l i n g the l o n g - r a n g e t a i l s in W e find F o r the s t u d y , it is the are o f p r i m a r y it c o n v e n i e n t concern, since t o d e f i n e the short-range f u n c t i o n s C where mnl;S/_\ Mv;a0 = ( r ) c mnl / ,;a/3 ( \ r ) . + mnl / \x,;a/3 > /3u „(r), uv, a/3 ' m n l X m n l „(r) = 0, ' and d £ f l is the h a r d - s p h e r e contact defined below. multipolar First distance. are s h o r t - r a n g e d , a n d t h e r e f o r e ( r>d . a/3 « \ ' A 1 a ) r<d „ a/3 W e note that all potential • s y s t e m w e need to consider only (A.1b) 000 \)Q. ^ ^ ' r a terms a r e n o t i n c l u d e d in X that m n s are n o t ^"„(r). uv,ap let u s c o n s i d e r a s y s t e m w h i c h terms contains no charged species. For for which 1 ^ 2 . It f o l l o w s from n o n e o f the H a n k e l t r a n s f o r m s , cu u . af£k), will ~ 1 1 2 b e h a v i o u r at s m a l l k. In f a c t , a l l but ^ Q Q . ^ k ) w i l l g o t o e q s . (2.10b) a n d (3.38a) that divergent for for In e q . ( A . 1 b ) w e a s s u m e that 3 have /» > for m n l ^ O H this \ ( r ) M - zero a s k—>0. F u r t h e r m o r e , t h e integral c a n be performed X^"^ ^(r) a r^d / which j f full a ™ * terms [61,135] is only moment). and t r a n s f o r m transformed namely that c | u a t e s t o constant a o f the short-range 00-<x/3^ "^ r to term c to obtain species, t w o additional a n c transforms Using C r k) w i l l diverge ^OOIQ.^") 1 c numerically; = A / r C A N Then inserting e q . ( A . 2 ) into j3 h a s ' 3E that ( moment terms. transforms ' 2 n t a s 1/k ( o f o f Fourier e q s . (2.10b) a n d ( A . 1 b ) w e w r i t e r) P°' e associated with these a d v a n t a g e o f the l i n e a r i t y the short-range ^ ' 00^ap^ ^" a has a charge a n d particle o f particles ' A 2 ) a and e q . (2.34b) a n d e v a l u a t i n g t h e yields k ) = 7 % a/3 i=/-T. W e n o t e behaviour. that a y *oJ!w where v depend upon the charge a n d dipole /3, r e s p e c t i v e l y . integral e and dipolar the case w h e n particle analytically. A will makes no contribution a s k—>0 C Q Q ] $0;aff where ionic n o integral W e take only both treatment, are normally It c a n be s h o w n a dipole j a/3' uv;a,p containing special that there c o u r s e , this ^mnl , r b e a d d e d t o t h e hat t r a n s f o r m a system require o term ^ [103] .that f o r „(r). 1 For F o f the functions a n a l y t i c a l l y , a n d it c a n be s h o w n ' a/3 c a n then transforms Hence, the potential uv;ap uv;aP out „(r)=0. „ X ap -mnl the 288 - ^QO'a/3^^ F r o m the d e f i n i t i o n *?00 • a/3^ ^^ behaviour that will w '" ' a s oc '' ver ^ of ^ ^ ( 1 2 ) 9 e pose no numerical e q . (2.46b)) b e c a u s e t h e d i v e r g e n t a s ^ a s problems terms s i n ( k v * { a - 3 ) k a st ' [ ^5] s m a l l k iec o r r e c t 1 (cf. e q . (2.38)) ^— ®- F ° > r t u n a it c l e a r l y t e l y , this follows divergent in t h e back H a n k e l t r a n s f o r m in t h e i n t e g r a l will cancel (cf. e x a c t l y at - small 289 - k. When both CQQ^ ^(k) particles diverges a a short-range 1/k /3 are c h a r g e d , it as k—>0. 2 Thus, we r w n Following previous workers X c a n be s h o w n again split ( r ) v _ " s transform e [188-190], we a ($ , . "T~ q a 1 6 analytically to ' n t ° numerically, c a n be derived define (A.4a) ' ] that r -ar, _ q ( P c a n be F o u r i e r t r a n s f o r m e d o [61,135] oo^a// ^ c c a n be e a s i l y F o u r i e r t r a n s f o r m e d function, ^ o o ^ a / / ^ ' >000 . 00;a/5 which and function, which and a l o n g - r a n g e analytically. as a give 2 - *"v%[77?777] miaif" where q^ and q ^ are the c h a r g e s o n the chosen with c a r e s o a s not but also allow r. it It must is a g a i n the behaviour must we to that as c22^ ( k): UU;ap note its short-range the an i o n i c system '" n a at v - < A must a , a r 9 at e same small not r, of divergent concentrations will 4 B ) be some reasonable value the e finite become t C > Fourier t r a n s f o r m h29^ J r) UU;ap contribute. Thus, function ~000 c a n be F o u r i e r t r a n s f o r m e d For w that at ~000;S/- x which /Jq^q^/r k be s c r e e n e d , and h e n c e define r *?QtVa// ^ we 0 The constant cause ^ Q o ^ a p ^ ^ A g Q / ^ C r ) —> case ions. • at * numerically infinite rOOO without d i l u t i o n , the K * difficulty. fact that the functions m n "1 ^uv a^ ' certain for FFT a T the r e n ° functions. k=0 [126] point ) of l o n 9 e s c r e e n e d r e q u i r e s that r F i r s t , the long-range which was *?§0'aj? *k A tail ignored ' S ° ' t h e of in the l o n 9- r special attention h()(Va// ^ r numerical a n 9 e t a i l o f be p a i d must be integration h 00'a// ^' r to corrected {i.e., W h i c by h the - has a 1/r d e p e n d e n c e at large r 290 - [61], w i l l affect b o t h QQ . ^ C ^ R A ' ANC 022 CQQ, ^(r). a Within HNC theory, both [ h.QQ ] r) ] numerical c a l c u l a t i o n (both 2 t r a n s f o r m ) , but derived. large r. (except significant functions The resulting in the 1/r* for that t h e s e when tails have a dependence upon are t r u n c a t e d a c l o s u r e c a l c u l a t i o n and analytical expressions which W e remark unimportant quite at the correct correction terms determining larger c h a r g e s . C ^ ) for s for in the this Hankel truncation are s m a l l a n d univalent during c a n be relatively i o n s , but can b e c o m e - 291 - APPENDIX B REPRESENTATIVE EXAMPLES OF EXPONENTIAL INTEGRALS First w e will consider F where interest a is s o m e here. It an i n t e g r a l (1+zcr)^ = a H constant. 'dr L It is t h e b e h a v i o u r is c o n v e n i e n t F o f the g e n e r a l to write = a [ l + I, n + form , (B.1) o f F a s K—> 0 that is o f e q . (B.1) in t h e f o r m I ] , (B.2a) 2 where 0 u ~ 2 r J d (B.2b) z -2nr •dr (B.2c) 2 -7KT I, = K j e dr d (B.2d) d r and 0 0 z Then using standard tables of integrals [165], it is p o s s i b l e t o s h o w that -2/cr -2/cd - 2K S d dr (B.3a) and I* = We c a n then insert (B.3b) e e q s . (B.2c) a n d (B.3) into F and K -2/cd 2 -2KA = a expanding the exponential yields e q . (B.2a) t o obtain (B.4) - F = a|g-|/c T h e r e f o r e , in the limit F = a We general will now we K—>0. - + /c d----J . 2 have (B.5) that [a " I"] ' examine the (B.6) K—>0 d e p e n d e n c e of an integral of the form R+d -/cr F = a f ( 1+/cr)R-d where 292 again a is some constant. (r +R -d ) 2 2 Equation 2 - 2 (2rR) (B.8) c a n then F = a 2 dr , be w r i t t e n -J • (B.7) as (B.8a) where 3 -/cr J r e dr , R-d R I, = K 6 + J R+d Io ^ = d / 2 R-d r" e (B.8b) K t -KT (B.8c) dr , , R+d _ I. = -2 (R^+d^) / re R-d 9 r K r dr (B.8d) dr , (B.8e) K 1 0 I O R + - d = -2(R^+d ) J e R-d Z Q 7 9 0 ~ = *(R -d ) / £ R-d R I_ rK r 2 1 2 + d Kr P 2 dr (B.8f) r and I_ Using tables standard [165], w e forms .2 2 for j 2 , 2 R + d = (R*-d*r J R-d the can rearrange integrals in e - K r ^ - 2 — dr . (B.8g) r e q s . ( B . 8 b - e ) and e q . (B.8a) t o obtain (B.8g) as g i v e n in - F = 3 -Kt\ a ( 4 2 8r 8 x K K L + 293 - R+d 2 ( R + d ) (v+-) 2 l ( R - d ) - 2 2 K If we then evaluate its = a e „ K ./cd [ ^r 8 e 3 /C L which we F can w r i t e -KR = a e By is the expanding 8 3 K K—> 0 the that J J 8d 2 K 8Rd' (B.10) K J as + 8 R 2 K • e /cd - N o w , it find _ 8R 2 K _ I jR-d 8Rd] L - (B.9) 2 L limits, we - K d Hr~J _ 8 R _ 8 d I 3 2 2 /C K K -KR F e q . (B.9) at 2 limiting /cd - e - /cd e ] [8d 8Rdjj- Kd + behaviour exponentials, we e - -/cd e of + e -Kdj e q . (B.11) w e can s h o w that 2/cd. + -j/c at small d (B.11) wish to determine. K (B.12a) and e Inserting /cd , - K d + e e q s . (B.12) i n t o F 2 « d 2 + e q . (B.11) and s i m p l i f y i n g = (l+/cR)e 3d J k R (B.12b) 2 yields (B.13) - 294 - APPENDIX C TRANSFORMATION OF THE ROTATIONAL INVARIANT <1> (12) 123 As in t h e t e x t , w e w i l l Bertagnolli of make u s e o f the n o t a t i o n [166] in o u r d i s c u s s i o n . R e f . 1 6 6 , it i s e a s y t o s h o w Using the rotation that of Steinhauser and matrix in the (x, y , z ) f r a m e given b y e q . (39) of reference (see F i g u r e 6(b)) x 2 = cosa2cos^COS72 - s i n a 2 c o s ^ COS72 + -sin/3 2 - c o s c ^ c o s ^ sin72 _ -sina cos/? si 72 + 2 (C.1a) cosa2Sin72 cos7 n *2 " sina2sin72 2 2 sina.2COS72 cosa cos7 2 (C.1b) 2 s i n ^2 s i n 72 sinp^cosc^ z sin/L, s i n a ^ 2 = cos/3 In the i o n r e f e r e n c e given frame (C.1c) 0 w e take advantage o f the r e s t r i c t i o n on as b y e q . (4.66a) a n d w r i t e -sina -I X 2 = I 2 sin72 sin7 cosa^ I I 2 (C.2a) -cos 2 7 - s i n a ^ cos72^ *2 cosc^ C0S72^ s i n ?2 (C.2b) - 295 - cos< z z In order to reference rotation take to the matrix the (C.2c) sini 1 2 unit vectors , (x, y, z ) r e f e r e n c e R, as g i v e n by d a r | from 4 frame, we multiply e q . (4.69), w h i c h the x~ i n cosa^ = -sina -sina I 2 2 sin7 cos7 cosa *2 cos7 2 -sincuf c o s 7 cos by 2 sin7 sinco 2 (C.3b) 2 sin7 COSCJ 2 cosw °2 (C.3c) sm< cos We can n o w a s s o c i a t e d w i t h the of the two forms obtain two for x "2 sina) expressions relating different y 2 > the cos7 cos6j sincj 2 vectors (C.3a) cosu 2 of 2 sinco - I 2 unit frame sincj 7 sin7 ion yields - s i n c u s i n 7 ^ cosco + c o s °2 2 *2 S the 2 frames and z . 2 of the two reference From sets by of Euler equating e q s . (C.1c) and angles components (C.3c) w e have that cost s i n p\, COSCJ = sin< sin coscu, and cos/3 Similarly, we use the 2 = °2 I c o s c ^ sincj z-components (C.3a) and (C.1b), (C.3b), to (C.4a) obtain of x 2 (C.4b) and y 2 > as g i v e n by e q s . (C.1a), - r cos7 2 296 - COS7 j = sint^ s i n 7 L sino + 2 COSC0 -i ? sina = (C.5a) 2 and sin7 Then j r 2 = sint^ - c o s 7 L using the trigonometric 2 shown c o s a ) -i = sinc^ . -I (C.5b) 2 cos27 = cos7 - cos27 = 2sin7COS7 that 2 i d e n t i t i e s [165] 2 and t h e r e q u i r e m e n t sin7 sinco + sin27 2 sin 7 , (C.6a) , (C.6b) = 0 , which f o l l o w s from e q . (4.66b), it c a n be that cos27 2 = cos27 I 2 . 2 s i n C O S 2 5—f- -1 . 2 s i nw -I 0 O) - sin _ (C.7a) N and sin2 , 7 = -2cos27 I 9 s i n o [sino;cosJ| 2 9 ^ L sineIf o n e s u b s t i t u t e s e q s . (C.4) a n d (C.7) into # ( c > ? b ) -I e q . (4.64b) a n d t a k e s =0, one can o b t a i n t h e r e s u l t $ 1 2 3 (12) 2 2 1 = V%3 c o s / 3 ( c o s CJ - s i n 1 + 2sin/3 c o s 2 7 1 I T 2 I s i n to)cos27 2 I . cosc^ cosc^ sinw 2 L - From I I . o2 "1 sinc^sinc^sinco 3 1 1 1 ^ 2 j — c o s co Lsinc^ 1 + 2 s i n cu, sincocosco | (C.8) e q . (C.4a) w e have the i d e n t i t i e s sincucoscu cosc^ and o . 2 p = j sina^ : COSCJ (C.9a) - 297 - coseu sincip sincu = ^ COSCI2 Using these relationships, together we <^ simplify (12) e q . (C.8) w h i c h given — cosco with simple trigonometric can eventually in e q . (4.70). . be w r i t t e n a s the (C.9b) identities [165], expression for
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- The structural, thermodynamic and dielectric properties...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
The structural, thermodynamic and dielectric properties of electrolyte solutions : a theoretical study Kusalik, Peter Gerard 1987
pdf
Page Metadata
Item Metadata
Title | The structural, thermodynamic and dielectric properties of electrolyte solutions : a theoretical study |
Creator |
Kusalik, Peter Gerard |
Publisher | University of British Columbia |
Date Issued | 1987 |
Description | In traditional theories for electrolyte solutions the solvent is treated only as a dielectric continuum. A more complete theoretical picture of electrolyte solutions 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 solutions which explicitly include a water-like molecular solvent. The ions are modelled simply as charged hard spheres and only univalent ions are considered. The water-like solvent is also treated as a hard sphere into which the low-order multipole moments and polarizability tensor of water are included. The reference hypernetted-chain theory is used to study the model systems. The formalism of Kirkwood and Buff is employed to obtain general expressions relating the microscopic correlation functions and the thermodynamic properties of electrolyte solutions without restricting the nature of the solvent. The low concentration limiting behaviour of these expressions is examined and compared with the macroscopic results determined through Debye-Hückel theory. The influence of solvent polarizability is examined at two theoretical levels. The more detailed approach, the R-dependent mean field theory, allows us to consider the average local electric field 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 well with experiment. Model aqueous electrolyte solutions are studied both at infinite dilution and at finite 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 simply 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 detail, revealing a wealth of structural information. Ionic solvation is generally found to be very sensitive to the details of the interactions within the system. |
Subject |
Electrolyte solutions |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-08-13 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0060479 |
URI | http://hdl.handle.net/2429/27365 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemistry |
Affiliation |
Science, Faculty of Chemistry, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1987_A1 K87.pdf [ 11.23MB ]
- Metadata
- JSON: 831-1.0060479.json
- JSON-LD: 831-1.0060479-ld.json
- RDF/XML (Pretty): 831-1.0060479-rdf.xml
- RDF/JSON: 831-1.0060479-rdf.json
- Turtle: 831-1.0060479-turtle.txt
- N-Triples: 831-1.0060479-rdf-ntriples.txt
- Original Record: 831-1.0060479-source.json
- Full Text
- 831-1.0060479-fulltext.txt
- Citation
- 831-1.0060479.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0060479/manifest