Multipolar electrolyte solution models. IV. Thermodynamic perturbationtheoryJohn Eggebrecht and Pelin Ozler Citation: J. Chem. Phys. 98, 1552 (1993); doi: 10.1063/1.464272 View online: http://dx.doi.org/10.1063/1.464272 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v98/i2 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Multipolar electrolyte solution models. IV. Thermodynamic perturbation theory

John Eggebrechta) and Pelin Ozlerb)

Department of Chemical Engineering, Iowa State University, Ames, Iowa 50011

(Received 2 June 1992; accepted 1 September 1992)

Expressions for the equilibrium thermodynamic properties of a model electrolyte solution are obtained from the Taylor series expansion of the Helmholtz free energy in electrostatic interaction parameters. Rational polynomials, which are explicit in temperature, density, and composition, for ion-ion, ion-solvent, and solvent-solvent components of the free energy are used to extrapolate these series to low temperatures and finite ion concentrations. Comparisons of predicted internal energies, Helmholtz free energies, and compressibility factors are made with the results of Monte Carlo computer simulation. Predictions for the solutions are comparable in accuracy to those for the pure solvent. An expression for the solute concentration dependent dielectric response of the solution is developed and tested by comparison with simulation and experiment. This function introduces observed saturation with increasing ionic concentration to the ion-solvent and solvent-solvent contributions to the energies.

I. INTRODUCTION

Statistical mechanics provides a computational link, expressed in terms of few body distributions, between mo­lecular models and equilibrium thermodynamics. Very few models yield analytic distributions and, as the realism of the model increases, the numerical complexity of liquid­state iritegral equations increases more rapidly. One of the most useful concepts provided by the modern theory of liquids is that the average pair distribution of molecules in a homogeneous fluid is primarily determined by repulsive atomic interactions. 1 Thermodynamic perturbation theory exploits this to avoid a detailed account of the true fluid structure. The atomic distribution at the pair density level in a system composed of particles with only repulsive pair interactions is regarded as a reference fluid structure. An approximation for cohesive contributions to thermody­namic properties is obtained by summing the electrostatic interactions over this structure. For polar fluid models in which the intermolecular potentials can be expanded in a small set of spherical h~rmonics simple, explicit, expres­sions for the Helmholtz free energy are obtained.2

-4

In the A expansion this separation into reference and perturbation is made by dividing the pair potential as

u(12) =uo(12) +Av(12) (1)

with the perturbation parameter O<;;;A <;;; 1. Uo is the intermo­lecular potential of the reference fluid and is, in principle, dependent upon the relative orientational, as well as trans­lational, coordinates (12). In practice useful simplicity is achieved when the reference fluid interaction has spherical

a) Present address: Illinois Math and Science Academy, Aurora, Illinois 60506.

b)Present address: Department of Chemical Engineering, Middle East Technical University, Ankara, Turkey.

symmetry. Contributions to equilibrium thermodynamic properties from the perturbing potential, v( 12), are ob­tained by expanding the Helmholtz free energy in a Taylor series about A=O.

f3A =f3Ao- (a ~A Qc) A NpT,A=O

_~ (a2 Inpc) A2 _ ...

2 aA N pT,A=O ' (2)

where f3=l!kBT, kB is the Boltzmann constant, Qc is the configurational part of the partition function in the canon­ical ensemble, and Ao is the free energy of the reference fluid. Numerical evaluation of the coefficients of this ex­pansion beyond O(A3

) is difficult, and in application to molecular models with realistically strong perturbative in­teractions the series converges slowly. However, a Pade approximant can be used to extrapolate higher order terms in this alternating series.5 As a result, the free energy is reduced to a simple rational polynomial in density and inverse temperature which accurately predicts the free en­ergy, and its density and temperature derivatives, of point multipolar atoms2

-4 and simple polar fluids. 6 Applications to molecules with more asymmetric short-range interac­tions are less successful.

Strong electrolyte solutions are an important class of systems which, for many common solvents and solutes, have molecular constituents whose interactions at short range are to a good approximation spherically symmetric. The analysis of the relationship between the molecular and thermodynamic properties of these systems is complete only at high dilution where the solvent can be replaced by a dielectric continuum.7 The free energy in this limit is found to depend upon the molecular characteristics of sol­vent and solute species only through the parameter

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J. Eggebrecht and P. Ozier: Multipolar electrolyte solution models. IV 1553

(3)

the square of the reciprocal Debye length, as K3. Here, for simplicity, q is the charge and ()" the diameter of all ionic species. In a system with a total number of ions, Nb and volume, V, pi=Np 3 IV is the reduced ion density. Es is the equilibrium contribution to the dielectric response of the solution and is frequently replaced by the better character­ized dielectric constant of the pure solvent. We learn from McMillan and MayerS to view the solution as a vapor of ions interacting through a potential of mean force with a radial dependence e-Kr IrEs. This provides an expansion of the free energy in powers of density beyond K3.

Stell and Lebowitz (SL)9 obtained an expansion of the free energy of the dielectric continuum model in powers of K. Modifications of the Stell-Lebowitz expansion,1O an al­ternative free energy expansion due to Andersen and Chan­dler, II the hypernetted chain approximation,12 the Poisson equation in the nonlinear Debye-Hiickel approximation, 13 the Yvon-Born-Green equation with the superposition ap­proximation,14 and the mean spherical approximation 15,16 each give at least qualitative agreement with the exact ther­modynamic properties of the continuum solvent model. I7

Among these methods there is some variation in accuracy but much greater variation in numerical complexity. Free energy expansions are explicit in ion density and tempera­ture. The mean spherical approximation requires a numer­ical solution of a single nonlinear equation. The hypernet­ted chain approximation, which is the most accurate method, and Poisson equation methods, require numerical solutions of large systems of nonlinear equations.

Continuum solvent theories which lead to explicit state dependence are useful for the correlation of experimentally determined properties of electrolyte solutions. IS-20 The functional forms generated can be amended by treating model parameters, such as ion diameters, as state depen­dent to provide a useful representation of experimental data. These adjustable parameters, presumably, compen­sate for deficiencies in the solution model. However, in the preceding paper, hereafter referred to as Paper II, it was shown that the dimensionless group given by Eq. (3), with the usual replacement of Es with EO, does not properly rep­resent the state point of an electrolyte solution. Rather, the defining group is K5=~Es. Physically meaningful trends in fitting parameters from correlations based on the contin­uum solvent model could be obscured. As important is progress in the development of molecular descriptions of systems in which a bulk electrolyte solution is a compo­nent, such as electrode surfaces,colloidal dispersions, or macromolecule solvation. These require accurate predic­tive descriptions of the properties of model electrolyte so­lutions in which the solvent is considered explicitly. As it has become clear that the molecular nature of the solvent cannot be neglected, it has become possible, either through computer simulation21-23 or integral equation theory24-27 to return the solvent to solution models.

In this paper we consider the simplest solution model beyond the continuum solvent. Both ion and solvent spe-

cies are treated as hard spheres with equivalent diameters and embedded, ideal, multipoles. Solvent polarity beyond the dipole and electronic polarizability are neglected. The solution model and the coefficients of expansion (2) for this model are ,described in Sec. II.

The conceptual framework and mathematical tech­niques of the A. expansion are well developed for the solvent and these apply to the solution as well. However, some obstacles arise when ions are introduced. The success of the A. expansion for polar fluids is in part a consequence of the similarity of the distribution functions of hard sphere and hard sphere, multipolar, fluids at the same density. The ionic pair structure in an electrolyte solution is not similar to the hard sphere distribution function. This struc­tural dissimilarity appears in the form of divergent inte­grals in Eq. (2). The nonphysical divergence of the free energy is corrected for the continuum solvent model in Mayer's virial theory28 and in the Stell-Lebowitz theory9 by resummations involving terms of higher order in ion density and K, respectively. Henderson et al. 29 have consid­ered the problem for the dipolar molecular solvent. In Ap­pendix B the removal of divergent integrals from the KO

series is described. We find it necessary to evaluate two types of contributions to expansion (2) at O(A. 4) and these are provided in Appendix B.

As it is written, Eq. (2) provides only high tempera­ture limiting behavior. Extrapolation of this series is cen­tral to the success of thermodynamic perturbation theory for polar fluids. The Pade used in that application is not unique. There is little theoretical justification or guidance in the selection of an extrapolation procedure so that the development of a A. expansion for electrolyte solutions re­quires information derived from computer simulation. The necessary calculations were provided in Paper II. In Sec. III an assignment of terms in the A. expansion is made for the ion-ion, ion-solvent, and solvent-solvent components of the free energy by comparison with our Monte Carlo work and with expressions for the high temperature and infinite dilution limits. These limits have been obtained earlier from the asymptotic behavior of pair correlation function. 30,3! Here they are obtained, with corrections for short-range structure, from a summation of KO ordered terms in Eq. (2). Pade approximants which extrapolate these series to temperature and concentration regimes of practical importance are proposed in Sec. III. These are developed in terms of either explicit dependence on solvent polarity or the dielectric constant of the pure solvent. Only the latter recovers the Debye-Hiickel infinite dilution limit at low temperatures.

Comparisons are made of the predictions of these ap­proaches with the results of Monte Carlo simulation in Sec. IV. The accuracy of the predictions of the thermodynamic perturbation theory in terms of EO is found to be compara­ble to that of the A. expansion for the pure solvent.

The process of averaging over solvent orientations and positions by which the molecular solvent is simplified to a dielectric continuum is not well understood; if it were the state dependence of K would be known. However, it is clear that the averaging is to be done in the presence of the

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1554 J. Eggebrecht and P. Ozier: Multipolar electrolyte solution models. IV

solute. Therefore, it is the dielectric response of the solu­tion, and not the pure solvent, which screens the ion-ion interaction. In Sec. V we develop an expression for the static component of the dielectric response of the solution. Comparisons are made with the results of computer simu­lation and with experimental measurements on aqueous alkali halides. The solute concentration dependence of Es is accurately predicted in both cases. We show that an accu­rate representation of the solution dielectric substantially improves predictions of ion-solvent and solvent-solvent contributions to thermodynamic properties. However, the predictions of the A expansion for the total energies and compressibility factors for the molecular solvent model are only slightly affected by the replacement of EO with ES"

II. EXPANSION OF THE FREE ENERGY

The electrolyte solution model considered here is com­posed of N + positively and N _ negatively charged hard spheres in a solvent of Ns hard spheres with embedded ideal dipoles. Both solute and solvent. spheres have the same diameter, u. The mixture is electrically neutral and the ions are symmetrically charged, N + =N _. The re­duced charge is defined by q*2=lluknT. The Ns solvent species carry a reduced dipole moment f.t*2=/1,2lc?k nT. The thermodynamic state is defined by the temperature, T, and reduced densities, Pa = N au3 I V. Throughout the total ionic density is denoted by Pi= P + + P _.

Expansion (2) involves only the configurational par­tion function

Q .. 1 .. fdI ... fdN e-/3(Uo+AV). (4) c N!(41T)Ns

Here dl denotes integration over the translational coordi­nate dr, for ionic species and over the product dr, d(cos e,)d<p, for solvent species, where e, and <p, spec­ify the orientation of the dipolar vector of solvent molecule 1, and

(5)

{jay is the Kronecker delta and the prime on the last sum­mation requires that i:f= j when a = y. The greek indices here, and throughout unless otherwise noted, refer to mix­ture species.

With the nth derivative with respect to A of In Qc a sum of all n-fold products of perturbation potentials, vayCij), is generated. When the expansion parameter A is set equal to zero these derivatives are reduced to integrals over the reference fluid structure

For the ideal multipolar solution model considered here, the departure of the Helmholtz free energy from that of the hard sphere reference fluid, Ao, to 0(,,1,3) in expan­sion (2) becomes

f3(A_AO)=_~2 ~ PaPy f dr, f dr~0(r'2)(v~y(12» f33

+ 12 ~ PaPy f dr, f dr~O(r12) (v~y( 12) >

+~ ~ PaPyPo f dr, f dr2 f dr~o X (r'2,rI3,r32)(vay(12)vao( 13) voy(32». (7)

For a more general model with nonequivalent radii other terms arise. In particular, nonvanishing terms of order f3 appear, as well as terms involving the four-body reference fluid correlation function. For the restricted model of equivalent diameters these vanish as a consequence of elec­troneutrality

(8)

The angular integrations in Eq. (7) must now be per­formed,

1 I' 1211" (f(1 . .. n) ) =--n d cos e, d<p, ... .... .. (41T) -1 0

X I' d cos en (211" d<pn f(1··· n). -, Jo (9)

Since the reference potential is spherically symmetric an­gular integrations involve only products of perturbation potentials. Those considered here are electrostatic in origin and are pairwise additive. Ions interact via a bare Coulomb potential,

qllli vii(kl)=-.

rkl (10)

It is convenient to express the orientationally dependent part. of the ion-dipole, Viw and dipole-dipole, vJ.LW pertur­bation potentials as expansions in spherical harmonics, Y1m, whose arguments are the angles specifying the orien­tation of the symmetry axes of the dipole vector, Wk, and the intermolecular vector, wkl'

and

(12)

The summations are taken over all indices, {m}, which are allowed by the Clebsch-Gordan coefficients, C(l"i2,l;m"m2>m).32 Equations (11) and (12) are equiv­alent to Eqs. 01.4) and (II.S).

Angular integrations of the products of the perturba­tion potentials, in the form of Eqs. (11) and (12) can be performed analytically. The angular integrals vanish for

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J. Eggebrecht and P. Ozier: Multipolar electrolyte solution models. IV 1555

any products of spherical harmonics, Y1m, in a common orientational vector, cu, for which the sum of the principal indices, I, is odd.4 All of the nonvanishing angular integrals required here have been evaluated by Rasaiah and Stell.33

The remaining integrals over translational coordinates of the products of these angle averages with two- and three-body correlation functions, which we denote as I n and lars, respectively, are summarized in Appendix A. For the hard sphere reference fluid the I n and lars depend only on density. Numerical evaluations of these integrals have been performed3,l0,34 in the superposition approximation,

(13)

The integrals required here have been numerically ex­panded to fifth order in density with coefficients given in Appendix A.

Introducing the variables

(14)

and

(15)

Eq. (7) is written as follows:

/3(A-Ao) /3~sol K't/2 KrJiii 3J4K~ 9IiJ.LJ.LK~2 N N 161Tp + 481TP - 81TP + 81TP

3 ( 4J2 + /;;J.L) K;iy + 161Tp +S, (16)

where ~sol=~sol(P,Y) is the excess free energy of the pure solvent relative to that of the hard sphere reference at the same density and S is a sum containing divergent in­tegrals. The latter includes terms in Eq. (2) required to obtain a convergent series by resummation and are dis­played in Appendix B, Eq. (B2). With the exception of a single term in S, as described in Appendix B, these are integrands containing the coordinates of either some or no dipolar species whose integer indices form a complete cycle in which dipolar species interact directly only with ions. These so-called ring diagrams may be visualized as neck­laces in which no two dipolar beads are adjacent. The ex­ceptional term in S, which is O(A,4), involves two dipolar and two ionic species in which the dipoles interact directly.

The transformation ofEq. (7) to Eq. (16) requires the following integrals which are performed, in a manner which also defered to Appendix B, using bipolar coordi­nates:

and

f cos a3 41T dr3--Y--­

ri3'32 - rl2 .

(17)

(18)

A resummation of S in the continuum solvent model, in which terms involving solvent species are absent, was obtained by Stell and Lebowitz.9 A partial resummation

for the ion-dipole mixture was presented by Henderson et af. 29 with the result

/3 (A -Ao) /3~sol /3~SL 3J4/(~ 3K~ 9IiJ.LJ.LK~2 .. '4------+-+ .

N N N 81TP 81TP 81Tp

3J2K;iy . 3liiJ.LK;iy + 41Tp + 161Tp ;. (19)

Here /3~SL denotes the Stell-Lebowitz expansion in terms of KO

/3~SL Kl. J2K6 J 1K6 (lii/2 - 3Jo)Kg ~= -121Tp ~'161rp+ 81Tp + 241Tp -; (20)

Equation (19) was then modified by taking terms from the high temperature expansion of the mean spherical ap­proximation. 16 Chan35 has continued with this approach in a study of aqueous electrolytes for which a comparison with experimental data was not favorable. We consider Eq. ( 19) as written.

Equation (19) gives the Debye-Hiickel infinite dilu­tion result only in the limit as /3-.0. In this limit the lead­ing terms in the expansion of the dielectric constant of the pure dipolar solvent,36

Eo=I+3y+' . " (21)

can be used to write ion-dipole interactions in terms of the solvent dielectric. The Debye-Hiickel result,

. /3(A -AsoI) hm V Pi~O

K3

-:12-;';' (22)

is recovered for low temperatures from Eq. (19) by ex­panding 1/ Eo in a Taylor series in y,

1 -n=1-3ny+' . '. EO

(23)

A resummation similar to that performed by Hender­son et af. 29 is described in Appendix B. There ion-ion, ion­dipole, and dipole-dipole interactions are included in the analysis to obtain terms of O(K6) with the result,

(3(A-Ao)

N (24)

where the components of the free energy are assigned as

(3AAJ.LJ.L _ (EO-l)2 liJ.LJ.LK6 (EO-l)2 Kl .+ ... N - 8 - 11/2 , EO 1T P Eo 81T pEO

(25)

(26)

and

(3AAii Kl ~= 121TPEb12+'" . (27)

Terms of O(K~) appearing in Eqs. (25) through (27) are similar to the infinite dilution expressions obtained by oth-

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1556 J. Eggebrecht and P. Ozier: Multipolar electrolyte solution models. IV

ers30,37 from an evaluation of the energy e~quaiion using the ~ asymptotic behavior of pair correlations in an ion-dipole mixture. However, here corrections involving approximate short range structure, through the density dependent fac­tors Iillll and J4, appear. The O(Kg) terms in these equa­tions are identical to those obtained by Kusalik and patey26

from that approach. The assignment of remaining terms in Eq. (19) to

components of the free energy, expressed in terms of EO through the replacement of y from Eq. (23), have been made by comparisons with the results of our computer simulations. The ion-dipole free energy is expressed as

J 2 3

f3llAill=_(Eo-l) 4K

O +(Eo-l) KO 1/2 N EO 81Tp EO 61TpEO

(Eo-I) J2K6 + ----EO 41Tp·

(28)

Equation (BI5) provides an additional term in Eq. (19) which we assign as a dipole-dipole contribution. Also, although Eqs. (25) through (27) are the high tem­perature, low density limiting behaviors of the free energy expansion about the hard sphere reference, these equations do not recover the empirically established Debye-Hiickel limiting behavior. Again by selection among alternatives through comparison with simulation, we modify the coef­ficient of the term of O(Kg) in Eq. (25) so as to recover this limit,

f31lA1l1l_(EO-l)2IiIlIlK6 (Eo-I) K~ N - EO 81Tp - EO 121TpE6/2

+(Eo-l) IiillK6 .. ~ .(29) EO 161TpEo

Equations (19), which we will refer to as the A(Y) expan­sion, and Eqs. (28) and (29), which we will refer to as the MEo) expansion, constitute the main results of this section. We now turn to the extrapolation of these series through the use of Pade approximants.

III. PADE APPROXIMANTS

The expressions displayed in the preceding section are convergent only in the high temperature and infinite dilu­tion limits. For numerical values of KO and y which are of interest direct evaluation of either Eqs. (19) or (24) yield nonphysical values of thermodynamic properties. Equa­tions (20), (28), and (29) have been grouped as the lead­ing terms in alternating series in the components of the free energy. Higher order terms can be estimated by forming rational polynomials which recover the known terms by expansion. Extrapolations from the high temperature and infinite dilution limits are thus constructed.

In this manner the free energy of the pure solvent is accurately predicted by the [1,2] Pade approximant5

f31lA 27J~u2 sol OJ'

~=~= 81Tp J-'-6-+-I-Il-llpY-I-2· (30)

The properties of ionic solutions in the continuum sol­vent model are satisfactorily obtained from a [2,3] Pade38

which is written here in terms of KO

3 .. f3IlASL KO 1 +allKo

~=-121Tp 1+bfKo+b~K6' (31)

where

(32)

(33)

and

(34)

We now introduce Pade approximants for the mixture by a similar procedure. The Pade suggested in Ref. 29, while reducing to Eqs. (30) or (31) in the appropriate limits, is unacceptable since singularities appear at several ionic concentrations.

The assignment of individuals terms to free energy components is guided by the resummation of S. An assign­ment from Eq. (19) is more difficult. These terms remain ordered in f3 and we have seen in Paper II that the appro­priate ordering parameter is KO. The most successful of the many alternatives considered is considered in the following subsection.

The typical dependence on solute concentration of components of the configurational energy observed in com­puter simulations at fixed temperature and density is dis­played in Fig. 1. The simple linear dependence of the total energy is seen to combine a nearly linear contribution from ion-ion interactions with nonlinear, but compensating, contributions from ion-solvent and solvent-solvent inter­actions. These saturate as the concentration increases. We note that the Pade approximant given above for the ion­ion component is highly accurate. Pades considered in the following for the other components are also shown in Fig. 1.

A. A(y) expansion

We first consider Eq. (19) and separate the free energy into the components,

f3 flAllll f3flA sol 9Ii1l1lK'6J;2 f3flAsol

~+ N 81TP +~

27I!pY2

f3flAill 3IiIlK'6J; 3K~ 3(J2+IiiIl14)K;iy --=---.-+-+ .

N 81Tp 81Tp 41TP

2 3K(iY J4 - [1 + 2J4 (Jo+ Iull14 )KO]

81Tp 1'-:'2(Jo+IiiIl14)Ko

(35)

(36)

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J. Eggebrecht and P. Ozier: Multipolar electrolyte solution models. IV 1557

~ I·-l,-----,....---,-0.00 0.06 0.12

A

'<) 1

~ 1

/

I

/ I

p'~/M /

0.00

.- - ---e o

\ .... ~ \ \ \ \

0.06 0. 12

A

FIG. I. Acomparisonofthepredictionsofthe,1.P(Es), (-),,1.P(EO)' (---), and ,1.P(y) (_. - • ) expansions for the components of the internal energy with the typical dependence on solute concentration, at fixed temperature and total density, observed in Monte Carlo simulations. At the left are results for the total (e) and ion-ion (0) energies, and at the right the ion-dipole ce) and dipole-dipole CO) components are shown.

with (3llAsLIN given by Eq. (31). We will refer to Eqs. (31), (35), and (36) as the I!/(y) expansion. The assign­ment of contributions to the free energy of Eqs. (35) and (36) is consistent only at O(~) with the leading terms in the resummation in terms of Eo given by Eqs. (25) through (27). Also, Eqs. (35) and (36) are not consistent with the observation of Paper II that for large KO the free energy, relative to that of the pure solvent, should be independent of solvent polarity.

In Fig. 1 the incorrect approach to infinite dilution at low temperature of the I!/(y) expansion is apparent. How­ever, at higher concentrations the accuracy of the predic­tion of the total energy improves. Saturation of the ion­solvent and solvent-solvent components of the energy is present, but exaggerated. While the predictions of the total energies and free energies of this approach are in good agreement with simulation at high solute concentrations, predictions of the compressibility factor are quite poor.

B. i\.(EO) expansion

We now consider Eq. (24) and assign the components of the free energy as follows. The ion-ion contribution is written by simply modifying Eq. (31) to recover the lim­iting behavior of Eq. (27),

(3llA ii K~ Eo1l2+aiiKO -p;-- -121Tp 1 +bfKo+bfK6' (37)

where aii, by, and bq are given by Eqs. (32) through (34). The ion-dipole contribution is written

{3llA iJL=_ K6 (EO-I) J4-[9J~J~8+1]4KI3 .. N 81Tp EO 1-3J~KI2' (38)

where J2 = J2EO and K2=K6IEO' The dipole-dipole contribu­tion is written

{3llAf.'f.' = (3b.A so! + K6 (EO-I) N N 81TP EO

IIJLJL + [91iiJL1IJL/8 -1] 2KI3 X~~~~~~~~--

1 + 31iiJLK/4

where1lf.'f.' = l if.'f.'(Eo-l )/Eo'

(39)

Equations (37) through (39), which we refer to as the II/(EO) expansion, are consistent with Eqs. (25) through (27) and with the observed minor dependence of the free energy on polarity at large KO' Both {3b.A b• and (31lA be-

2 .,.. JLJL have, to a constant, as KO(EO-O/EO with changes in sol-vent polarity for large KO' The saturation of these compo­nents with increasing solute concentration, however, is absent. We will return to this effect in Sec. V with the introduction of a solute concentration dependent solution dielectric.

IV. COMPARISON WITH SIMULATION

The following comparisons are based on the Monte Carlo computer simulation results provided in Paper II. These calculations were made for fixed ratios of charge to dipolar strength parameters, X=q*2//-L*2, of either 64 or 40. The largest values of the reduced charge squared and reduced solute density examined was 160 and 0.12, respec­tively. Total reduced densities of 0.6786 and 0.72 were considered. For a nine molar aqueous solution of potas­sium flouride at 25 °C q*2~200, X~50, p~O.72, and Pi~O.12.

In the following calculations the pressure of the refer­ence fluid is taken from the equation of state for hard spheres due to Carnahan and Starling. 39 We use the per­turbation theory developed by Tani et al. 40 for the dielec­tric constant of the pure solvent. Predictions of this theory agree very well with the limited simulation data available for the dipolar hard sphere solvent.22

,4! Temperature and density derivatives of the dielectric constant of the pure solvent are neglected. We have noted the deficiencies of the I!/(y) expansion and consider it no further.

and

The total configurational energies and the components,

{3ilUii (3Ui=~ (40)

(41)

in the ,F(EO) expansion are compared with Monte Carlo computer simulation results from Paper II in Figs. 2 and 3. Note that the internal energy of the solvent is included in {3us' While the total energy is very accurately predicted, relative deviations from the correct values occur which are as large as 10% for {3ui and 20% for {3uS' as shown in Fig. 3.

Similar results are observed for the free energy. There is a cancellation of errors in the components which results

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1558 J. Eggebrecht and P. Ozier: Multipolar electrolyte solution models. IV

<::l-+--__ _

~ I

~ A q/'P~

0.02 0 tS~

~ o.o~ [:. tS~

0.08 + IN ~-I

0.12 x tS~

0.02 0 ~o

o.o~ * ~o 0.08 • ~o

~ 0.12 • -10

I 0 5 10 15

FIG. 2. A comparison of the predictions of the ilP( Eo) expansions for the total internal energy with the results of Monte Carlo simulation. The total reduced density is 0.6786.

in a very accurate prediction of the total free energy. In Fig. 4 a comparison is made of predictions of the ionic and solvent contributions from the Al(Eo) expansions with sim­ulation results at low and high ion density. The free ener­gies of the pure solvent have been removed from the sol­vent component and from the total free energy. Also shown in Fig. 4 are calculations employing the solution dielectric described in the following section.

The calculations of Fig. 4 have been made at the higher total density, p=O.72, whereas the calculations in Fig. 3

It) I

It) I

~ I

~ I

~ 0

"'" I ~U;/H

~ 11-----r---~----,_

o 5 10 15

FIG. 3. A comparison of the predictions of the components of the inter­nal energy from the il.P(EO) expansion with the results of Monte Carlo simulation. The total reduced density is 0.6786. The symbols correspond to the thermodynamic states displayed in Fig. 2.

<::l <::l

• ,~

'..:::.:--.:::.~

..... "-"-

'" • I ,

~ ~

\.. ~ ,

~'t \.. ' .. , o~ ," ,

~ , .. ~ .. ' .... , 0:--" I .. ,

\ o \

\ 0

a 0 .... b ~

0 50 100 0 100 200 A:Q8 A:Q8

FIG. 4. A comparison of ilP(Es ) and ilP(EO) expansions for the compo­nents of the Helmholtz free energy with the results of Monte Carlo sim­ulation. The total reduced density is 0.72 and the ratio of strength pa­rameters, X, is 40. Simulation results are labeled as follows: total (0), ion (A), and solvent (_). Perturbation theory results are shown for the ilP(Eo) expansion: total (---), ion (_._), and solvent (_ .. _); and ilP(Es )

expansion: total (-) and solvent (-- . -- ). The ionic contribution from the two approaches is indistinguishable on this scale. In (a) pi=0.0434 and in (b) PI=0.1194

are made at the lower total density. The density depen­dence appears to reliably portrayed, but more substantial evidence is provided in the evaluation of the pressure. In Fig. 5 compressiblity factors predicted by the Al(Eo) ex­pansion are compared with machine simulation results. Absolute errors in the predictions of the Al(Eo) expansion increase with increasing multipolar strength parameters, as is the case in applications to the pure solvent. However, in relative terms the deviation observed at the highest dipolar

~ ~'"

,0 50 /00 /50 200 250 Kg"

FIG. 5. A comparison of the predictions of the ilP( EO) expansion for the compressibility factor with the results of Monte Carlo simulation. The total reduced density is 0.6786. The symbols correspond to the thermo­dynamic states displayed in Fig. 2.

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J. Eggebrecht and P. OzIer: Multipolar electrolyte solution models. IV 1559

TABLE I. Numerical values of thermodynamic properties obtained by computer simulation (MC), the .:t(Eo) expansion [.:t(EO)]' and the .:t(Eo) expansion using the solution dielectric introduced in Sec. V [A.(Es)]. Uncertainty in the last digit in the Monte Carlo estimates is shown in parenthesis. The total reduced density is 0.6786.

-/3(at -aHS) -/3ut

fL*2 X Pi MC ,1.( .os) A.(Eo) MC A(Es) .1.(.00)

0.500 0.0 0.0000 0.14 0.154 0.154 0.279 0:283 0.283 0.500 64.0 0.217 O.S( 1) 0.656 0.656 0.97(2) 0.S70 0.S73 0.500 64.0 0.0434 1.39(2) 1.275 1.284 1.66(2) 1.553 1.571 0.500 64.0 0.0814 2.6(2) 2.412 2.464 2.92(2) 2.771 2.848 0.500 64.0 0.1194 3.7(4) 3.564 3.693 4.16(4) 3.985 4.158 1.750 0.0 0.0000 1.27 1.344 1.344 2.01 2.152 2.152 1.750 64.0 0.0217 3.8(3) 3.418 3.368 4.72(4) 4.530 4.496 1.750 64.0 0.0434 6.2(8) 5.950 5.844 7.38(4) 7.233 7.190 1.750 64.0 0.0814 10.6(9) 10.535 10.435 12.08(6) 11.998 12.069 1.750 64.0 0.1194 15.1(8) 15.130 15.1S2 16.8(1 ) 16.700 17.040 2.500 0.0 0.0000 2.21 2.343 2.343 3.33 3.545 3.545 2.500 64.0 0.0217 5.9(6) 5.393 5.281 7.30(4) 7.021 6.932 2.500 64.0 0.0434 9.48(9) 9.110 8.848 11.21(7) 10:970 10.805 2.500 64.0 0.0814 16.0(7) 15.848 15.449 18.06(9) 17.942 17.813 2.500 64.0 0.1194 22.4(4) 22.610 22.268 24.8(1) 24.840 24.951 0.800 0.0 0.0000 0.34 0.359 0.359 0.63 0.634 0.634 0.800 40.0 0.217 0.9(6) 0.845 0.841 1.31(3) 1.209 1.208 0.800 40.0 0.0434 1.65(6) 1.469 1.464 2.02(3) 1.900 1.905 0.800 40.0 0.0814 2.8(4) 2.637 2.646 3.27(2) 3.151 3.188 0.800 40.0 0.1194 3.9(5) 3.835 3.886 4.53(3) 4.408 4.510 2.400 0.0 0.0000 2.08 2.202 2.202 3.15 3.355 3.355 2.400 40.0 0.0217 4.2(2) 3.829 3.779 5.46(3 ) 5.254 5.211 2.400 40.0 0.0434 6.3(1) 5.911 5.772 7.78(5) 7.510 7.408 2.400 40.0 0.0814 10.1 (2) 9.790 9.515 11.77(7) 11.577 11.424 2.400 40.0 0.1194 13.8(3) 13.763 13.417 15.76(6) 15.660 15.539 4.000 0.0 0.0000 4.40 4.642 4.642 6.31 6.486 6.486 4.000 40.0 0.0217 8.1(3) 7.442 7.303 10.33(7) 9.731 9.596 4.000 40.0 0.0434 11.8(5) 11.012 10.614 14.24(6) 13.582 13.242 4.000 40.0 0.0814 18.2(7) 17.663 16.815 21.07(8) 20.531 19.909 4.000 40.0 0.1194 24.7(8) 24.478 23.272 27.92(S) 27.520 26.745

strength and largest value of KO is comparable to the error of the prediction of the it expansion for the pure solvent.

A quantitative assessment of the itP( Eo) is provided in Table I. Also shown are results which follow from this method by the replacement of the pure solvent dielectric with a solute concentration dependent dielectric intro­duced in the following section.

v. SOLUTION DIELECTRIC

Thermodynamic behavior at infinite dilution is well described by the continuum solvent model, to which Eq. (24) reduces in that limit. Useful extrapolations of the continuum solvent model have been constructed through an introduction of solute dependence in the properties of the continuum, primarily in terms of short-range structure. In this section an approximate expression for the solute concentration dependent dielectric constant is obtained and tested by comparison with computer simulation and with experimental values for aqueous alkali halides. This solution dielectric is then used in the definition of K, given by Eq. (3), and the Helmholtz free energy evaluated from the itP(EO) expansion with the replacement of EO by Es'

What is known theoretically from models containing a molecular solvent of the solution dielectric applies only in

/3p/p

MC A(Es) A(Es)

>5.20 5.172 5.172 5.03(4) 5.046 5.045 _ 4.91 (3) 4.920 4.916 4.68(3) 4.713 4.698 4.46(7) 4.518 4.489 4.17 4.068 4.068 3.70(6) 3.621 3.621 3.3(2) 3.210 3.199 2.6( 1) 2.549 2.503 1.9(3 ) 1.927 1.836 3.53 3.341 3.341 2.92(8) 2.708 2.714 2.3(2) 2.130 2.125 1:2( i) 1.201 1.150 0.2(1) 0.324 0.215 4.95 4.949 4.949 4.82(4) 4.830 4.829 4.66(5) 4.709 4.705 4.47(6) 4.510 4.499 4.26(5) 4.323 4.300 3.64 3.437 3.437 3.3 (1) 3.080 3.084 2.96(8) 2.741 2.744 2.4(2) 2.196 2.183 1.9(2) 1.689 1.647 2.29- 1.968 1.968 .1.7(1) 1.381 1.401 1.2(1 ) 0.821 0.855 0.2(2) -0.077 -0.058

-0.5(2) -0.917 -0.935

the approach to the infinite dilution limit. The low solute concentration expansion of Es has been considered by Friedman42 and by Patey and Carnie.43 The latter study provides the following result:

(42)

where

(43)

and

(44)

The preceding integrand contains Hankel transforms, h~n::(k), of the coefficients of the invariant expansion of the mixture correlation functions

hay(ij)=gayUj)-l= L h~n:/(rij)ifJlmn(ij), (45) Inm

where the ifJ are linear combinations of spherical harmon­ics.44

The integrand in Eq. (44) is dominated by small k 30

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1560 J. Eggebrecht and P. Ozier: Multipolar electrolyte solution models. IV

TABLE II. A comparison of the coefficient, 101, of the low ion density

expansion of the solution dielectric from Eq. (48) and from linearized hypernetted chain theory (CP) (Ref. 43) for three hard sphere multipo­lar solvents.

pf./, q*2 Y e l (48) el (CP)

0.600 40.0 2.094 312.1 350.5 0.732 200.24 7.735 1034.8 1073.9 0.800 200.24 2.513 9712.9 9794.1

- 3q*(E -1) 1· h?ll(k) =1· 0 1m 'I.t 1m * k-O k-O f-L pEok

(46)

Then, using the reference fluid structure for the spherically symmetric component of hp.p.(12) ,

. -000 hm hp.p. (k) = 41TJO , k-O

(47)

the integral I can be approximately replaced using the mean value theorem to obtain

(48)

Terms have been grouped on the right-hand side of Eq. ( 48) to indicate the weak dependence of El on temperature and density. .

In Table II a comparison is made of the values of E' predicted through the use of these approximations with those obtained from solutions of the Ornstein-Zernicke equation in the linearized hypernetted chain approxima­tion. The agreement between these two approaches im­proves with increasing density, as the accuracy of Eq. (47) increases, from roughly 10% at the lowest density to less than 1% at the highest density.

The introduction of a local dielectric function in con­tinuum solvent models of electrolyte solutions has been extensively examined. In this context one is concerned with nonlinear terms in an expansion of the dielectric constant in the polarizing external field per unit volume of a central ion. If orientational polarization of only the first solvent shell is considered, which is sufficient for the dipolar hard sphere solvent,22 the leading nonlinear term is proportional to ~. Booth45 derived an expression for the field depen­dence of the pure solvent dielectric as an extension of Kirk­wood's method for the evaluation of the local field in a dielectric sphere with an embedded charge. For the model considered here in which electronic polarizability is ne­glected this is

Es-l=3(Eo-1){ 1 21 } (49) 3aKQ sinh2[~(3a)1!2j .

Expansion of the sinh term to O(Kg) leads to

Eo-l Es-l=1+aK6· . (50)

This is one of a family of functional forms which Gra­hame46 had earlier considered, in the context of the Debye-

..... .;:.~

~t-----------------,---------------~ 0.00 -0.04 0.08

A

FIG. 6. A comparison of the predictions of Eqs. (50) and (51) with the results of computer simulations in which f./,*2=3.15, q*2=64, and p=0.6(e) and f./,*2=2.5, q*2= 160, and p=O.6786( .... ).

Huckel solution model, and found to be consistent with experimental measurements which were available at that time.

This model has since been further examined.47 How­ever, no satisfactory expression for the parameter a has, to our knowledge, yet been obtained. An approximate expres­sion may be obtained from the low concentration expan­sion ofEq. (50) by comparison with Eq. (42)

a (Eo-1) 3JO 271TpCol . (51)

In Fig. 6 Eqs. ( 50) and (51) are evaluated by com­parison with the results of computer simulation. As above, we use the perturbation theory developed by Tani et al. 40

for the dielectric constant of the pure solvent. Extremely long sampling chains are required for accuracte Monte Carlo estimates of Es- Consequently, the error bars for cal­culations presented in Ref. 22 are quite lftrge. In a later calculation, described in Paper I of this sequence, a suffi­ciently long sampling was performed to reduce the uncer­tainty (e in Fig. 6) and make a quantitative test of Eqs. (50}and (51).

A second test of Eqs. (50) and (51) is provided in Fig. 7 where comparisons are made with recent48 and earlier49

measurements of Es for aqueous solutions of three alkali halides at 25°C. The refractive index of water is intro­duced,

(52)

and is taken as 1.34. The experimental value of EO was taken as 79.2. Pauling radii50 were used for the ions with q*2 computed from the mean ionic diameter. A diameter of 2.8 A was used for water and the dipole moment was taken to be 2.63 D. The latter is an effective value, which includes electronic polarizability in an approximate manner, ob­tained in SCMF/HNC calculations by Kusa.lik and Patey.51 Equations (51) and (52) are in quantitative agree-

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J. Eggebrecht and P. Ozier: Multipolar electrolyte solution models. IV 1561

0.00 0.04

A 0.08

FIG. 7. A comparison of the predictions of Eqs. (50) and (51) with experimental values of the solution dielectric of three alkali halides: RbCI (-'-) and (X); CsCI (---) and (0); and LiCI r-), (.A.) (Ref. 48), and (0) (Ref. 49).

ment with the more recent measurements but disagree with the earlier values reported for LiCI solutions at higher con­centrations.

We now consider the replacement of co in the thermo­dynamic perturbation theory developed in the preceding sections with cs' An improvement in the predictions of ion-solvent and solvent-solvent components of the internal and free energies results as shown in Fig. 1. A partial sat­uration of these contributions is observed at high concen­trations. However, these contributions are found to ap­proximately cancel with the result that the total energies and compressibility factors are only very slightly altered. Numerical values are given in Table I for representative states at a total density of 0.6786. Comparable agreement is observed for the higher density states of Paper II.

VI. CONCLUDING REMARKS

Applications of the Il-expansion thermodynamic per­turbation theory have been made to the simplest electrolyte solution model beyond the continuum solvent. Expressions in solution polarity parameters have been obtained, and for the purely dipolar solvent, these have been reduced to rep­resentations in terms of the pure solvent dielectric con­stant. Pade approximants for the components of the Helm­holtz free energy have been used to predict Helmholtz free energies, internal energies, and pressure and these have been compared with the results of Monte Carlo computer simulation. At high ionic concentrations the predictions of internal energies in the IlP(y) expansion are reasonably ac­curate, but at low concentrations and temperatures this approach is unsuccessful. The most promising of the meth­ods presented here is the IlP(€o) expansion which gives accurate predictions of both energies and pressures over very broad composition and temperature ranges. At the two total densities for which simulation results are avail­able this theory performs well and in many applications this is the density regime of interest.

The important advantage of the methods described here is the avoidance of complications associated both with short-range contributions to the effective pair potential in

the continuum solvent mode152,53 and with orientationally

dependent pair potentials in the mean spherical and hyper­nefted chain theories. 54 Refinements of the solution model considered here, involving softening of the repulsive refer­ence potential, nonequivalent diameters, and polarizability can proceed by t~e application of well established integral equation and perturbation theory approaches for spheri­cally symmetric potentials. The rational polynomials for the thermodynamic properties of the model electrolyte so­lution generated by this application of thermodynamic per­tu~b~ti()Il. theory. ~a~~ ~ explicit. in "density, temperature, and concentration dependence and may prove useful in the rep­resentation of experimental data. While the complexity of the final expressions is somewhat greater than those con­structed from the continuum model, we have demonstrated that the essential physics is better captured by an expan­sion in KO which emerges from a molecular description of the solvent.

ACKNOWLEDGMENTS

The authors wish to express appreciation for support from the National Science Foundation (CBT 8811789) and the Fullbright Foundation for a Fellowship for Doc­toral study awarded to Pelin OzIer.

APPENDIX A: REFERENCE FLUID INTEGRALS

We define the following integrals of the reference fluid structure:

I n= 1'" dr"ri.-nho(r) 0<n<2,

I n= L"" dr ,z-ngo(r) n > 3,

where hoe 12) =go( 12)-1. Letting

IOP=~'J dri J dr3

and

(AI)

(A2)

(A3)

(A4)

(AS)

(A6)

where ho(123) =go(123) -go(12) -ho(3) -ho(32). In Eqs. (A4) through (A 7) ai is the angle at the ith vertex of a triangle formed by the three intermolecular: vectors along r 12' r 13' and f32'

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1562 J. Eggebrecht and P. Ozier: Multipolar electrolyte solution models. IV

TABLE III. Coefficients of density expansions of the indicated integrals of the hard sphere radial distri­bution function.

Co Cl C2 C3

Jo -0.33333 - 0.39217 "...0.445 12 0.403 13 J1 -0.50000 _ 0.515 24 ___ ~_ -0.600 90 0.53726 J2 -1.00000 0.45846 _~_0.29046 0.20023 J4 1.000 00- 0~309 13 _ -,;'0.242 28 0.02003 J6 0.33333 0.22649 . _0.04053 0.04109

lUi 1.500 00- -2.3840 _ 2,945 10 -2.90350

lUll- 1.83330 -0.704 82 0.467B4 -0,46023

lill-Il- 0.94685 0.26724 .0.87886 -1.69980

lll-Il-Il- 0.20833 0.24642 -=-O~102 03 -0.02408

When the reference system is taken to be the hard sphere fluid, as is the case here, values of these integrals depend only on density. These values have been fitted, us­ing the hard sphere radial distribution function in the Percus-Yevick approximation ss to polynomials of the form

S

_ '" a{3r i I a{3r- Lot c i p, i=O

(A8)

with a similar expression for the In- Coefficients of these expansions· are given in Table III. The leading coefficients for each integral can be evaluated analytically from the low density limit where goer) = 1 for r';?u. Values at nonzero densities were obtained by numerical integration.

As observed in Ref. 10, for polar fluids and for the continuum solvent solution model, improved agreement of

C4 Cs

-0.249_08 0.07155 :"-0.3319~ 0~095 40 -:::-0,10V.6 0,02981 ::-0.0).023 0.00298 -0.Q30 79 0.013 91

1.84200 -0.47399 0.34634 -0.11477 1.451-90 -0.49114

-0.02055 0.00745

the predictions of the A. expansion with simulation results is obtained with pair correlations in the Percus-Yevick approximation compared with those obtained using the empirically corrected V erlet-Weis functions, S6 despite the greater accuracy of the latter.

APPENDIX B: REMOVAL OF DIVERGENT INTEGRALS

S in Eq. (16) can be considered as a sum of two series

(B1)

where S3 contains divergent integrals with leading term O(f32) in which the reference fluid correlation functions, go( 1 . -.. n), take the asymptotic value of one. As in the case of the continuum solvent model, upon resummation these lead to a single term of O(f3312).

I

41T

9y2 00 (_1)n+lK5(n-2) 100 I f I I 1 +- L;- . dr12r12- dr3- .. - fdr +.

81Tp n=5 n-2. _ 0 41T r23 41T _ _ ~rn-2,n-lrn-l,1 (B2)

The three summations involve integrals over complete cycles in the integer indices, or rings, containing (n - m) ions and m dipoles in which no dipole-dipole interactions are present. Orientational averages over the dipolar vertices have been performed using Eqs. (11) and (18). For R3 the only terms present in f3(A-Ao) to O(f33) are those involving the ion-ion interaction. These have already been considered by Stell and Lebowitz9 and appear in Eq. (20). A contribution to R3 at OUf') is approximately obtained in the following subsection.

O(p4) terms

The third term in S3' which we denote sljJL, is O(p4) but must be included in our analysis to obtain the K~ contribution from the dipole-dipole interaction, as we shall see presently. The form taken by this term is derived from the following contribution to Eq. (2)

(B3)

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J. Eggebrecht and P. Ozier: Multipolar electrolyte solution models. IV 1563

The factors of 96 and 48 are the numbers of ways in which these terms, or their equivalents, arise in series (2). From Eqs. (11) and (12) and well known properties of spherical harmonics

~ N [ I (UilL(12)UILIL(23)UlLi(34)Uii(41)=271, 3 ~ L Ylm(l2)Y'I'm(34)- 3YIO(12)YIO(34) . 12r23 34r41 m

(B4)

The factor l/rl4 can also be expressed in terms of spherical harmonics and the integration over r4 performed

where aihas the meaning given in the text below Eq. (A7). Integration over r2 can now be performed by convert­

ing to bipolar coordintates, r=rI3, t=rI2' and s=r32, such that

~+f2-? cos a l

2rt (B6)

f2+?-~ cos a2

2st (B7)

and

~+?-f2 cos a3

2rs (B8)

Then dividing the domains of integration as described by Larsen et al.1O one finds

11" [Jr Ir+t I"" Jt+r =2 2 dt ds+ dt ds r I r-t r t-r

+ dt ds+ dt ds r 2s-4[s4 Ir II Ir+1 JI I r-I r-t r t-r

The second term in Eq. (B3) is similarily evaulated.

(UilL (12) ulLi(23) UilL (34) UlLi( 41) )

q4J.L4

and

9? J{ ~,.; cos a2 cos a4 12 23 34 41

(BlO)

(B5)

dr2 --=- dt r2 ds s-2 J cos a2 11" i"" Jr+t Yf2~3 r I max(1,1 r-ti)

411" X(f2+?-~)=-. (Bll)

rl3

From these angular and spatial integrations Eq. (B3), in terms of the variables K5 and y, becomes

(BI2)

In the preceding contributions to the free energy terms of O(tr) have been introduced and the necessary integrals evaluated in the low density limit where g( 1234) = 1. In

~ what follows we obtain an approximation for the remain­der of the second integral in Eq. (B3) by neglecting solvent-ion correlations for one of the dipolar particles

J cos a3 X dr4 ~? .

34 14 (B13)

The integration over r4 can now be carried out and the result simplified to obtain

., (B14)

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1564 J. Eggebrecht and P. Ozier: Multipolar electrolyte solution models. IV

We may now combine the first term in this result with the last term of Eq. (19) and the small y expansion of 1/ eO to obtain

(BI5)

which appears in Eq. (29). The second term in Eq. (BI4) can similarly be combined with the corresponding term in Eq. (19) to give

(BI6)

Although there is greater symmetry in Eq. (28) with this additional term we find that it is necessary to neglect it, along with all the other O(tr) terms which have not been considered, in the construction of the Pades for the free energy described in Sec. III.

Resummation of 53

Returning to Eq. (B2) we note that terms in K5n and ~n-ly can be combined when the derivative with respect to K5 is taken to give

xr~2 4~ J dr4 r14~43 ~ ••• ]. . (BI7)

The series can be summed in the {3--0 limit by expanding about y=O and expressing the result as a quadratic

1-3y+9y2- ...

1-6y+6y2 ( 3y)2 1+"2 + .... ~ (BI8)

As noted by Henderson et al., 29 where a similar resumma­tion is made, this replacement is consistent with a free energy expansion truncated at O(A3

). Taking Fourier transforms of the sum in· the preceding integral and then summing the resulting series in k space we obtain

K6(1-6y+6y2)

81Tp(1+3y/2)

K6(1- 6y+ 6y2)

81TP€6!2

where the small y expansions of to,

(BI9)

These terms are the ion-ion, ion-dipole, and dipole-dipole contributions, respectively, of O(K6) to the free energy.

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