THEORY OF IONIC SOLUTIONS

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THEORY OF IONIC SOLUTIONS

J. Hansen

To cite this version:

J. Hansen. THEORY OF IONIC SOLUTIONS. Journal de Physique Colloques, 1984, 45 (C7), pp.C7-

97-C7-111. �10.1051/jphyscol:1984710�. �jpa-00224271�

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n09, Tome 45, septembre 1984 page C7-97

THEORY OF IONIC SOLUTIONS

J.P. Hansen

Laboratoire de Physique Théorique des ~iquides*, Université Pierre et Marie Curie, 75230 Paris Cedex 05, France

and Ecole Normale Supérieure de Saint-Cloud, 92211 Saint-Cloud, France Résumé - On passe en revue quelques progrès marquants en mécanique statisti- que des solutions ioniques. L'accent sera mis sur les modèles qui tiennent compte explicitement de la nature moléculaire du solvant polaire dans le calcul des proprié- tés thermodynamiques, structurales et diélectriques de solutions concentrées d'élec- trolytes simples.

Abstract - A review is given of some recent progress in the statistical mechanics of ionic solutions. Special emphasis is laid on simple models which take explicit account of the discrete molecular nature of the polar solvent to predict thermodynamic,structural and dielectric properties of concentrated solutions of simple electrolytes.

1

-

INTRODUCTION : THREE MEN IN A BOAT

In its simplest version an electrolyte solution comprises a solvent, usually water, and a single solute made up of positive cations and negative anions. This lecture deals with the Statistical Mechanics of such three-component systems, two of which carry electric charges, while the third is composed of highly polar, hydrogen-bonded molecules. The emphasis will be on semi-analytic theories for the structural, ther- modynamic and dielectric properties of simple ionic solutions.

The following notational conventions will be adopted : the solvent is made up of Po = No/V molecules per unit volume ; the solute is composed of pl = N1/V cations carrying a charge q l = Z, e, and pz = N2/V anions of charge q2= Z2 e, where e is the elementary (proton) charge. The total ionic number density will be denoted by

pi = p l

+

p2

.

Overall charge neutrality requires :

The total potential energy of the system is the sum of solvent-solvent, solvent - solute and solute-solute terms :

Except for polarizability effects, each of the three contributions is pairwise ad- ditive to a good approximation. As will become clear in section 3, the Statistical Mechanics of the pure polar solvent already represents a formidable challenge, so

that a full first principles description of the three-component problem has long been considered as untractable. For that reason attempts have been made long ago to reduce the complex three-component system to an effective two-component problem involving only solute coordinates. In a classic paper, McMillan and Mayer rl]showed how the grand-canonical partition function of the solvent-salute mixture can be transformed into an effective partition function for the solute species alone, by averaging over solvent coordinates. The solute ions interact then via an effective, solvent-averaged potential VNi,Lli ; the price to pay is that VN. N. is no longer

1' 1

* ~ ~ u i ~ e associée au CNRS

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984710

C7-98 JOURNAL

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PHYSIQUE

pairwise additive, but contains higher-than-pairwise terms, i.e. :

In the limit of vanishing solute concentration (infinite dilution limit), we expect that only the pair interaction survives and behaves as :

when the separation between ions i and j , r.. =

I:~

^- + ^r.

1,

goes to infinity ; in eq.(1 .4). denotes the dielectric constant o$'the pure salvent, describing the extent to which the bare Coulomb interaction between ions is shielded by the die- lectric solvent medium.

2 - THE PRIMITIVE MODEL : TWO MEN IN THE DRY.

The so - called "primitive model" (PM) neglects al1 higher than pairwise terms in eq.(1.3) and generalizes the asymptotic form (1.4) of the effective pair potential to finite solute concentrations and interionic distances. In its simplest version the mode1 is one of inpenetrable charged hard spheres;dropping the bar which sym- bolizes averaging over solvent coordinates, the pair potential of the PM reads :

where Ua denotes the core diameter of ionic species a(1

,<

a,b \( 2). The PM is a die- lectric continuum model since solvent effects come in only through the macroscopic dielectric shielding (1/c0 factor) of the Coulomb potential.

The interionic pair structure is characterized by three partial pair distribution functions gab(r). If h (r) = g (r)

-

1 denotes the

total

correlation functions

ab ab

(which vanish as r + C O ) , the so called direct correlation functions cab(r) are def ined by the three coupled 0rnst.ein-Zernike (OZ) relations [2] :

Another set of useful functions are the potentials of mean force wab defined by : ( 2 . 3 )

where f3 = l / k T is the inverse temperature.

B

A simple mean field calculation of the pair correlations isbased on the ansatz : (2.4) where @ (r) is the mean electrostatic potential at a distance r £rom an ion of spe-

cies a: This potential is related to the charge density around that ion by

Poisson's equation :

where the definition of the pair distribution functions as well as the charge neutra- lity condition (1.1) have been used. The set of eqs. (2.3-5) constitute Poisson - Boltzmann (PB) theory. Since the Coulomb potential is the Green's function of the Laplace operator, these eqs. can be rewritten in the compact integral form :

where

*

denotes the convolution product already encountered in eq. (2.2). Lineari- zation of the exponential in eq. (2.6) leads directly to the well-known Debye

-

Hückel limiting laws ; in particular the low concentration limit of the osmotic pressure P follows directly from the virial theorem :

where K is the inverse Debye screening length :

Correlations between ions inside the "polarization cloud" surrounding ion a can be approximatively taken into account by replacing the bare Coulomb potential vac(r) in the convolution product appearing in the PB equation (2.6) by the "renormalized potential" - kBT cac(r). Combining this substitution with the OZ relations (2.2) leads to the "hypernetted chain" (HNC) closure [:]2

Eqs.(2.2) and (2.9) form a closed set of non-linear integral equations which can be solved numerically (generally by an iterative procedure). The resulting pair distribution functions and thermodynamics turn out to be in excellent agreement with "exact" computer simulation data for the same model [.]3

In order to bypass the tedious numerical solutions of the non-linear HNC equations, one can aim at a fully analytic theory by linearizing the logarithm in the closure (2.9), when r

>

ORh

.

This leads to the widely used "means spherical approximation"

--

(MSA) []2 which supplements the OZ relation (2.2) by the closure :

~ q . (2.10a) is of course exact (inpenetrable ions), while eq. (2.10b), where c(S)(r) ab denotes the short-range (s) part of the direct correlation function, is only asymp- totically exact. The MSA for the restricted version (O ₁ = O2 = 0 . ; Z ₁ =

-

^{Z2 = 1)}

of the primitive model has indeed been solved analytically by Waisman and Lebowitz [4]. Since the restricted PM is symmetric with respect to charge conjugation, h (r) E h (r) (there are only two independent pair correlation functions), and

1 1 22

C7-100 JOURNAL DE PHYSIQUE

the OZ equations (2.2) are casily decoupled by the linear transformation :

-XI+' ^{(9 = [-Rji (y + A} ,, ^(4j-J

^{(2.1 la)}

The corresponding direct correlation functions c(')(r) and c(2)S(r) turn out to be polynomials of degree 3 for r < a

.

The excess interna1 energy :

is given by the physical solution of the second order equation :

which yields the correct Debye-Hückel limit at low concentrations.

The analytic solution of the MSA for the PM has been extended to arbitrary ionic mixtures (i.e. arbitrary charges and diameters) by Blum [5]. By adjusting the ion diameters

a-

these MSA results can be used to fit the ex~erimental osmotic pressures

a

of many salts over a wide range of concentrations [6]. It must however be kept in mind that the MSA is less accurate than the HNC approximation, particularly at high concentrations.

The ionic correlation functions hah(r) must obey two important sum rules derived by --

Stillinger and Lovett

c]

£rom a linear response argument, and recently proved rigo- rously by Martin and Gruber [8] :

Eq. (2.14a) expresses local charge neutrality, while (2.14b) is the "~erfect scree- mn'ng'' condition. Both G u l e s are characteristic of the infinite range of the bare Coulomb potential and follow in fact directly from the l/kL singularity of its Fourier transform ; they are model-independent and must also hold for the non - primitive models introduced in section 4.

The defects of the primitive mode1 are obvious : it ignores the molecular (discretel nature of the solvent, and hence ion-solvent correlations. This is clearly not justi- fied if ions and solvent molecules are of comparable size, but the continuum picture is probably a reasonable one for macroionic solutions like micelles [9] and charge - stabilized colloid suspensions [IO]. In the following sections the solvent is rein- troduced on an equal molecular footing with the solute.

3 - MOLECULAR MODELÇ FOR THE PURE SOLVENT : THE THIRD MAN

Solvent molecules are characterized by their permanent multipolar moments (dipole

+

^{t f 6}

moment p

,

quadrupole tensor Q

,

etc.) and by their polarizability tensor

a ,

which 6

'*

is isotropic, to a good approximation, (i.e.

a

=

a

1, with

a

a scalar polarizability) Water molecules in the gas phase have p = 1.85 x 10 -18 esu (=1.85 Debye) ;

0 3

Q ⁼ 2.5 x 10-26 esu and

a

= 1.44 A

.

Since their charge distribution is essential- T

ly tetrahedral, the quadrupole tensor is of the following form in a molecular £rame with the OZ axis along

*

fl

O

8

⁼

[ >

^{Q T} ("tetrahedral" quadrupole) (3.1)

O O O

On the other hand for linear molecules, the tensor takes the form :

- ?

1 QL ("1 inear" quadrupole)

The simplest mode1 for the pure solvent is one of dipolar hard spheres :( f O ; ft 8

Q

= a =

O ) characterized by the simple pair potential ;

where for any vector u

*

= u

û ,

and

d

l 2 is a "rotational invariantr1. The pair corre- lation functions h(1,2) and c(1,2) depend now on the molecular orientations as well

-f

as on the intermolecular separation vector r12 :

The OZ relation (2.2) must be generalized to include angular convolution :

where the normalization 0.= 4v for linear molecules (two polar angles) and fl = 8vL for non-linear molecules (three Euler angles). The general procedure to handle the

C7-102 J O U R N A L D E PHYSIQUE

the angle-dependence is to expand h and c in the complete set of rotational inva- riants

I [ 11

^:

where the first factor on the r.h.s. of eq. (3.7) denotes the 3-j symbol, familiar from the quantum theory of angular momentum, while the D are the Wigner matrix ele- ments (or generalized spherical harmonics). Molecular symmetry imposes restrictions on the terms appearing in eq. (3.6). In particular, for axially symmetric molecules,

JJ = ü = O, and :

where Y denotes the usual spherical harmonics. Taking the Fourier transform (with respect to rI2) of eq. (3.6) and using the standard Rayleigh expansion of exp{iz + };. leads to :

&p,,n,)= j 4 : ~ A . ' * ' . ^{( A ,} -Q~,-Q,)

with

Through the expansion (3.9) for h and c, the Fourier-transform of the OZ relation (3.5) is transformed into an infinite set of coupled algebraic equations relating the

hm'

and

;?'.

P U

For al1 practical purposes one must truncate the expansions (3.6) and specify a closure relation (MSA, HNC,

...

^{) .} This program was for the first time carried out by Wertheirri Cl21 ~ 1 1 0 lin~ited the expansion (3.6) to three terms in the case of dipolar hard spheres, interacting via the potential (3.3) ; since JJ = v = O in this case :

where Q O00 (1,2) = 1, 0'" (1.2) =

YI. Y2

and

ali2

(1,2) is defined in eq. (3.3).

The reason for restricting the inifite set of rotational invariants to the three appearing in eq. (3.11) is twofold. First, from a purely mathematical point of view, these three invariants form a closed set with respect to angular convolution : the convolution product of any two of these functions yields a -'0 belonging to the same set and no others. Secondly a knowledge of the ~rojections hooO, hl1' and h 112 suffices to determine the thermodynamic and dielectric properties of the model. In particular the isotherme1 compressibility

xT

and the excess interna1 energy are

given by : A

while the dielectric constant co follows directly £rom Kirkwood's formula [13] :

where y = 4 n ) ~ 2 po/(9kgT) and

gk is Kirkwood's g

-

factor equal to the mean square fluctuation of the total dipole moment * M of the sample :

Alternatively it can be shown that :

From the elementary properties of Hankel transforms it follows that :

C7-104 JOURNAL DE PHYSIQUE

which shows that the orientational correlations decay like the dipole-dipole poten- tial, i.e. there is no screening.

Starting from the truncated expansion (3.11) which leads to three coupled OZ rela- tions for the corresponding projections fimR(k) and CmR(k)

,

Wertheim Cl21 obtained an analytic solution with the MSA closure :

a

,a,, 4 0-

(3.18a)

The MSA turns out to underestimate considerably the dielectric constant & for 2 3

large reduced dipole moments y* = (y /O kB~)'I2. As in the case of ionicocorrela- tions in the primitive mode1,improvement may be sought from the HNC closure :

In order to obtain a tractable closure, the expansions of c and h in rotational invariants are again limited to 3 terms (cf. eq. (3.11)) and the logarithm in eq. (3.19) is linearized with respect to the projections hl1' and h1I2 ; this leads to the so-called linearized hypernetted chain (LHNC) approximation [14,15], a set of three closure relations between the three projections of h and c. These closure relations together with the corresponding projections of the OZ relation (3.5) must be solved numerically [15]. The spherically symmetric projections hooO and c O00 decouple £rom the other two projections ; this means that in the LHNC approximation

(as in the MSA), hooO(r) is not affected by the strength of the dipolar coupling (y*), clearly a defect of these closures. This implies in particular that the coor- dination number will be identical to that of the hard sphere fluid, i.e. close to

10 at high densities, so that LHNC (or MSA) theory applied to dipolar hard spheres is uncapable of describing a water-like structure with its characteristic tetrahe- dral coordination. This defect can in principle be overcome by expanding the loga- rithm in the HNC closure (3.19) to second order ; the resulting quadratic (QHNc) closure is however not self consistent since it would require the introduction of higher order rotational invariants [15].

The dielectric constant E calculated from LHNC or QHNC theory agrees reasonably well with computer simulation data. These results show that a purely dipolar mode1 for the solvent strongly overestimates orientational correlations, and hence at

.,. x

high couplings (y

&

2) typical of water.

The model must hence be refined by taking into account the quadrupole moment and the polarizability of the solvent molecules. Patey et al. [16] examined a model of hard spheres with embeded dipole and linear quadrupole. The pair potential is then of the form :

where

A r o o 0

(PL)

^{3 00}

j R - -

(dipole-dipole)

In this case the minimal basis with respect to angular convolution involves 10 rotational invariants,so thatlO terms must be retained in the expansion (3.6) of h(1,2) and c(1,2). Numerical solution of the LHNC and QHNC equations for this model shows that is strongly reduced by the presence of even a modest quadrupole moment. Physically quadrupolar interaction favours T - like molecular pair configu- rations, for which

cl . f i 2

⁹ 0 , leading to a strong reduction of the Kirkwood g -

-

factor (3.15), and hence of E

.

Similar conclusions have been reached by Carnie and Patey [17] for the case

09

a tetrahedral quadrupole, which is more appropriate for H O. In fact the reduction of E due to QT is too strong since with the H O gas

2 2

phase values of p and QT the model predicts Eo

"

25 for liquid water at room tempe- rature, way below the experimental value & 2 80. This apparent paradox can be resol- ved if in a final step the polarizability a of the molecules is taken into account.

The instantaneous dipole moment of molecule i is the sum of its permanent dipole and of the dipole induced by the local electric field

+

E i acting on that molecule :

-+ -+

where the sum runs over the No - 1 other molecules, and Tij =

ai

^{Vi ( 1 Ir. .) is the}

13

dipolar tensor. Clearly molecular polarizability introduces a complicated many body interaction which must be handled by iteration in a computer simulation of polarizable molecules 1181. These simulations, as well as theoretical considera- tions [14], show that to a good approximation the properties of a polarizable dipolar fluid can be related to those of a fluid with an effective permanent dipole:

p eff can be estimated from a mean field calculation [17] and turns out to be subs- tantially larger than the gas phase value in liquid water : p eff

"

2.56 Debye compared to p 2 1.85 D. Inserting this effective dipole moment, together with the quadrupole moment QT 2 2.5 x 1 0 - ~ ~ e s u in their LHWC calculation, Carnie and Patey [17] find the following values for in pure H20 at room temperature ( 0 = 2.8 AO;

C7-106 JOURNAL

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PHYSIQUE

2 5

Q;

⁼ (QT/o kg~)li2 = 0.9) : y = 1.85 D

,

y*

-

²

^,

^E

^'

²⁵

^,

^{p = p eff = 2.55 D}

^,

y K _ 2 . 7 , E - 8 0 2

.

~ ~ ~ ~

4 - DISCRETE SOLVENT MODELS OF IONIC SOLUTIONS : THE THREE MEN MEET.

In the last section it was shown how relatively simple models yields a reasonable microscopie description of the pure solvent. This section is devoted to similar models applied to the genuine three component system of solvent plus anions and cations, in which al1 three actors are treated on an equal footing. The simplest such model is a mixture of dipolar and charged hard spheres. The solvent is modeled by hard spheres of diameter ü and carrying a dipole moment p,while the symmetric + solute is made up of equal numbers of spheres of diameter 0; =

a ,

=

o7

and of charge

- + ' + - q1 = - q2 = q. The

The spherically symmetric term vap(r) includes the hard core repulsion and the bare Coulomb interaction qa qb/r between ions ; the second and third terms in eq. (4.1) represent the charge-dipole interaction (vlol(r) = pa qb/r ; vabl(r)

ab

= - q yb/r) while the last term is the dipole-dipole coupling (3.3). The ion-dipole

-

^A

coUpl?ng introduces tvo new rotational invariants @'O1(i,j) = pi

.

^r

@O1 1 ^A ij and

i , = pi

.

^r

^,

which, together with those appearing in eq. (3.ll),form a i i

closed set with respect to angular convolution. If the expansion (3.6) is limited to these 5 invariants, it is easy to verify that the pair structure of the symme- tric model is entirely described by 7 independent projections :

hOOO _{(r) : h22 (r)} O00 _; hOOO _{, 2 (r) ;} hOOO

_oo

_{(r) ;} ,0

o l

00 (r) h:?(r) ; hOil(r) =

-

^{h O1} 02 (r) ¹ ;

112 110 mnR

hO0 (r) ; hO0 (r). These are related to the corresponding cab (r) by 7 coupled OZ relations, which must be solved, subject to approximate closures.

For the MSA closure (2. IO), an analytic solution has been obtained [19,20] up to a set of coupled, non-linear algebraic equations. ~umerical solutions with the LHNC closure :

with the sum restricted to the 5 invariants enumerated above, have been obtained by Levesque et al. [21], who also examined the Q H N C closure. These calculations clearly

show that the ionic structure, as charecterized by hooO(r) and hy2°(r), difiers 1 1

markedly from the predictions of the primitive model of section 2, which invariably leads to monotonous hehaviour of these functions whit r. LHNC and Q H N C calculations indicate that both correlation functions exhibit much structure for r a 3 0 at high concentrations, and give strong evidence for the existence of anion-cation pairs separated by exactly one solvent molecule. Under certain physical conditions such

"trimers" may even be more probable than anion-cation "dimers".

In order to make contact with the primitive model one can define effective ion-ion potentials by reducing the three-component solvent-solute mixture to an effective two

-

component system involving only the ionic species [22]. Dropping the super- criptooO for the spherically symmetric ionic correlation functions, this program can be archived by introducing effective direct correlation functions related to the hab by generalized OZ equations :

Within the present mode1 of a mixture of charged and dipolar hard spheres, an ele- -ef f

mentary calculation leads to the following expression for the cab (k) [21,22] :

where

Eq. (4.4) expresses the fact that the effective ion-ion direct correlation function is determined by ion-ion, ion-solvent and solvent-solvent correlations in the full three-component mkture. Once czbf has been determined, an effective ion-ion poten- tial is derived by inverting the H N C closure :

(4.5)

For large separations, one expects :

C7-108 JOURNAL

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_PHYSIQUE

where E( f & ) is now a solute-dependent dielectric constant ; eq. (4.6), when generalized to al1 r

> ,

is just the primitive model result. LHNC calculations show that (r) differs markedly from the limit (4.6) for r< 30, reflecting the structure observed in gab(r). vab (r) should not be confused with the potential eff of mean force wab(r), defined by eq.(2.3) ; both potentials coïncide only in the limit of infinite dilution (p -t O). At finite ionic concentrations, the potential

i

of mean force between ions exhibits the usual Debye screening :

(n) ^{A-+@= Y. &IL} .- ^{x n}

More interestingly the ion-dipole and dipole-dipole potentials of mean force exhi- bit similar exponential screening, due to the presence of free charges [19,23]

.

It was shown in section 3 that the dipolar hard sphere fluid is not a good model for the solvent. For that reason Patey and Carnie [24] have recently refined the discrete solvent model of ionic solutions by solving the LHNC equation for a solu- tion of charged hard spheres in a "waterlike" solvent made up of spheres carrying an effective dipole moment (to account for polarizability effects) and a tetrahe- dral quadrupole. Their calculations show that the ion-ion potentials of mean

force are less structured and approach their continuum limit more rapidly than those obtained with a purely dipolar solvent yielding the same &

.

Ion pairs at contact are also less probable in the "waterlike" fluid which appears to be a "better" sol- vent than the dipolar fluid since it leads to stronger dissociation.

5 - SOLUTE-DEPENDENT DIELECTRIC PROPERTIES.

An important question is to which extent the solute modifies the dielectric constant of the solvent. Strictly speaking the dielectric constant of a conducting medium is infinite in the zero frequency limit, since

where o(w) is the a.c. conductivity which goes to a non-zero limit U (the d.c. con- ductivity) as w + O. Experimentally one can extract the residual (finite) part of the dielectric constant of the solution by taking the limit :

However E still contains kinetic contributions [25], and differs £rom the purely equilibrium part which describes the shielding of the Coulomb interaction between sol ions ; the latter quantity (E ) depends on the ion concentration and is not direc-

P,

tly accessible to experiment ;lit can however be extracted from a knowledge of the correlation functions of the solution in a number of different ways.

a) In the limit of infinite.dilution (p + O),

ED. goes over to the pure solvent i

value E and can be determined directly £rom the lpotential of mean force (2.5)

O'

between two isolated ions, acting as a

robe"

in the solvent :

b) At low, but finite ionic concentrations E deviates from E by aterm linear in pi [26] ; Pi

The slope

s,

can be expressed in terms of infinite dilution correlation functions The slope turns out to be negative, i.e. the dielectric constant of the solution decreases as solute is added. E~ depends however very sensitively on the model used to describe the solvent. Patey and Carnie [24] have estimated s1 from their LHNC calculations for the pure dipolar solvent model and for the "waterlike" model des- cribed in section 4, under physical conditions appropriate for water ; they find :

As

= E

-

^{E =}

-

28.4 ci ("waterlike" solvent)

Pi

O

AE

=

-

259 ci (dipolar solvent)

where ci is the solute concentration (in moles/%) ; the two slopes differ by an order of magnitude, but the slope calculated from the more realistic "waterlike"

solvent agrees reasonably well with indirect experimental determinations for alkali halide solutions.

c) For arbitrary solute concentrations there are three equivalent routes to & -

.

IJ

.

The first route is via Adelman's effective direct correlation functions and th;

associated effective ion-ion potentials (see eq. ( 4 . 5 ) . Examination of the asympto- tic behaviour of ceff (r) leads to the following expression [ 2 ? ] .

ab

with

-1 12 where

LU:)

is deiined in eq. (4.4b) and ;&)(k) = ;liO(k)

-

cOO ( k ) .

O0

The second route is via the Kirwood gK defined in eq. (3.15). In an ionic solution

E and gK are related by [21] : i

C7-11 O JOURNAL

DE

PHYSIQUE

which differs from eq. (3.14) valid in a pure solvent, due to the screening of the dipole-dipole interaction by the free charges (ions). Finally E can also be cal-

P .

culated £rom the perfect screening condition (2.14b), where K islgiven by eq. (2.8) with E replaced by E

.

P ¹

6 - CONCLUSIONS.

Discrete solvent models of ionic solutions represent a considerable progress over the more conventional primitive mode1 which ignores ion-solvent correlations. The former models yield a quantitavely reasonable description of thermodynamic and die-- lectric properties of simple ionic solutions. A detailed comparison with structural data from neutron diffraction experiments [27] is however meaningless, since the latter measure internuclear correlations, while the simple models described in this lecture refer to spherical molecules with point multipoles. Future improvenents will have to take into account the non-spherical

shape

(short-range anisotropy) of the solvent molecules, as well as extended charge distributions mjmicking more closely real water molecules. Preliminary work along these lines has been reported

~281

.

Extensions of the discrete solvent models to inhomogeneous situations (like an electric double layer near an electrode) have already been successfully formulated, in particular within the MSA [29]

.

ACKNOWLEDGEMENT :

The author is grateful to J.J.Weis for his advice during the preparation of this lecture.

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¹

]

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[

2

]

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-

[

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]

L.Blum, J. Chem. Phys.,

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]

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]

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]

M.S.Wertheim, Mol. Phys.

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-

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427 (1977) ;

2,

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108,

14 ( 1 9 8 1 ) . 1191 S.A.Adelman and J.M.Deutch, J. Chem. Phys.

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3935 ( 1 9 7 4 ) .

[20] L.Blum, Chem. Phys. Lett. 26, 200 (1974) ; J . Chem. Phys.

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2129 (1974) ; 3. Stat. Phys.

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451 ( 1 9 7 8 ) .

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2,

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[23] J.S.Hoye and G.Stel

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5183 ( 1 9 8 3 ) .

[25] J.B.Hubbard, P.Colonomos and P . G . W O ~ ~ ~ ~ S ; J. Chem. Phys.

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2652 ( 1 9 7 9 ) . [26] H.L.Friedman, J. Chem. Phys.

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1092 ( 1 9 8 2 ) .

[27] See G.Neilsonls

,

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1451 ( 1 9 8 2 ) . [29] L.Blum and D.Henderson, J. Chem. Phys.

E,

1902 ( 1 9 8 1 ) .