Calculated electrocapillary curve for a molten salt

If bulk properties of simple molten salts may be reasonably well understood in terms of the primitive model, the situation with respect to surface properties is less satisfactory. It has been shown that a simple model for the distributions at the free surface of a molten salt can give surface tension and surface energy in reasonable accord with experiment, provided that a factor guaranteeing local electroneutrality is introduced. In this model, properties are given in terms of bulk-salt distribution functions, for which the primitive model is used. The present work extends this model to the electrocapillary curve, i.e., variation of surface tension with surface charge density. The calculations are like those for the free surface, corresponding to the point of zero charge. The local electroneutrality correction, while extremely important for the magnitude of the surface tension, is much less important for its variation with surface charge, and hence the electrical capacitance. Our capacitances, derived from surface charges and potential drops derived from our model, are much too small, whereas the Gouy-Chapman model gives values which are much too large. The calculated variations of surface tension and potential drop with surface charge do not satisfy the thermodynamically derived Lippmann equations; neither does one obtain the same surface tension from different thermodynamically equivalent formulas. In order to understand the reasons and to improve the situation, we show how thermodynamic consistency may be restored to our model. Capacitances are still numerically much smaller than those reported experimentally. The rate constants of reaction of 2,4-dinitrofluorobenzene (DNF) with OH- in microemulsions of n-octane, tert-amyl alcohol, and cetyltrimethylammonium bromide (CTABr) and in micelles of CTABr and tert-amyl alcohol can be treated quantitatively by using a pseudophase ion-exchange model and the second-order rate constants in the microemulsion or micelle droplets are larger than that in water, but much smaller than those in moist tertiary alcohols. Reactions of DNF and 2,4-dinitrochlorobenene (DNC) in microemulsions or micelles containing primary alcohols


I. Introduction
Molten salts are important in batteries and other technological applications, but are also of great theoretical interest. They constitute the simplest electrolytes, involving only two charged species, with no uncharged species (e.g., the solvent in an electrolytic solution), and with both species describable by classical mechanics (unlike the conduction electrons of metals). Although the precise form of the forces between ions, whether they can be assumed pairwise additive,l and whether one needs to introduce a dielectric constant to represent their polarizability, is not completely settled, it is clear that treatment of the Coulombic attractions and repulsions constitutes a major part of the theoretical description of these systems. Indeed, many bulk properties of the molten salts can be understood in terms of the primitive model, which considers the ions as charged hard spheres in a dielectric medium.2-8 Understanding of surface properties is less advanced than understanding of bulk properties, although surface properties are of primary importance in electrochemistry. One of the basic concepts of modern electrochemistr?l2 is the polarizable electrode, a charged interface in which a change in surface charge density is accompanied by changes in the surface tension and in the potential drop across the interface, but not by current flow, so that it behaves like a capacitor. Such a system is usually exemplified by mercury in aqueous solution, but can be realized also by a metal in a molten salt.13-15 The Lippmann e q~a t i o n~J ' J~~' relates the variation of surface tension with potential drop (electrocapillary curve) to the surface charge. The surface tension can be measured directly and also the derivative of surface charge with potential drop, which is the capacitance (although the meaning of some of the measurements has been criti~ized'~). Measured properties are actually those of the interface as a whole, but conventional electrochemical wisdom is that the metal's contribution to certain properties is unimportant.18 For example, if one distinguishes between charged components of the salt and those of the metal, one shows" that the potential drop consish of salt and metal contributions (which are not necessarily the same as for free salt and metal surfaces), so that, if the metal's capacity is large, the capacity of the interface is essentially that of the salt. (This is emphatically not the case for the surface tension of the interface, which is dominated by that of the metal.) A number of approaches to learning about the polarizable molten salt-metal interface are being taken. Monte Carlo and molecular dynamics simulations for the liquidvapor interface of molten salts have been performed, giving information on the ionic distribution functions near the surface.lg These calculations have recently been extendeda to the interface at a charged repulsive wall (electrode) with surface tension and capacitance being calculated. Various statistical mechanical approaches are possible but have so far usually been tried in the electrolyte, rather than the molten salt, regime of the interface. The density profile, as well as the two-particle distribution functions needed for a calculation of surface tension, can be expressed in terms of the direct correlation functions c,, of the interface, and the short-range character of the ci, suggests that a reasonable approximation is to replace them by the corresponding functions for the bulk. This has been done21-25 by using a variety of approximations for the bulk functions, coupled with various approximations to the integral equation determining the wall-particle correlation functions (density profiles). Tests have been for the primitive model; indeed one flaw of this model, the replacement of the solvent by a continuous dielectric, becomes very important when the model is applied to the molten salt interface, since the dielectric constant cannot be separated from the ion density profile. The extension of density functional theories to systems involving Coulombic interactions2&% has also led to calculations of some properties of the liquid-vapor interface for molten s a l t~.~a By use of a number of approximations for the response functions in the surface, these theories give expressions which allow the use of properties of the bulk to calculate properties of the surface. Our own work3,-% has involved generating approximate distribution functions for the surface from those for the bulk and calculating surface tensions and surface energies for the free surface.
It is this approach that we extend, in the present paper, to calculation of the eiectrocapillary curve (surface tension as a function of surface charge) of the salt part of the interface. At the potential of zero charge (pzc), the model for the salt (in contact with a metal electrode) is identical with that for the salt in contact with its vapor (free surface). This is possible if the ion density profiles are so sharp that the repulsion due to the metal does not change their shape much. Some worker^'^^^^ have argued, on the basis of electrocapillary measurements, that the alkali halide surface in the electrode is not much changed, relative to the free surface, by the metal at the pzc. At potentials other than the pzc, it is difficult to perform calculations for the salt alone, since it carries a net charge. Thus the metal in the interface is represented in our calculations by a charged plane and calculated results for the globally neutral interface are compared to the corresponding properties for an ideal capacitor, formed from two charged planes. We will require knowledge of the bulk correlation functions of the salt, which are obtained from the generalized mean spherical approximation for equal sized charged hard spheres.8,36 The simpler mean spherical approximation, which has been extended to unequal ion sizes,7,37~38 has serious deficiencies in the pair distribution f~n c t i o n s .~~~~~~~~~~ Thus the salt is being considered in the context of the restricted primitive model. The dielectric constant is taken as unity, which is con-sistent7 with rigid ions; for a surface problem, there are difficulties, related to treatment of image forces, attached to using a value other than ~n i t y .~, ,~, Problems arising from the assumptions about the bulk salt, and from other aspects of the model, are discussed in section VII.
Section I1 presents our model for the surface and the formulas used for calculation of the surface tension of the charged interface. The Fowler model for the distribution functions, and our modification of it for charged-particle systems, are presented. The essence of the modification is satisfaction of a constraint of local electroneutrality; the method for accomlishing this is discussed in section 111, and results for the variational calculations of the "electroneutrality functions" are presented. The calculation of surface tensions is carried out in section IV. In section V, the results are presented, and the Lippmann equation is discussed.
One aim of statistical mechanical calculations like the present one must be to make connection with the thermodynamic description, which defines surface properties in a way different from ours; the thermodynamic definitions do not deal with the actual physical description of surface charges, etc. On the other hand, internal consistency is built in, whereas statistical mechanically calculated quantities do not necessarily satisfy thermodynamic relations, which become a test of the consistency of our assumptions. The Lippmann equation is one of these thermodynamic relations; as shown in section V, we find it not to be verified. A discussion points out a problem with the Born-Green-Yvon equilibrium condition, which leads to other surface tension formulas and redefined density profiles. None of our models give capacitances in accord with experiment. Section VI considers other ways in which the model may be improved and directions for future work.

Calculation of Surface Properties
The Kirkwood-Buff formula42 gives the surface tension in terms of the interparticle forces and the two-particle distributions p!;), where pi;) (?,, y2) Gl Gz gives the number of pairs of particles such that a particle of species i is in the volume element dr', at 7, and a particle of species J is in dF; at T2. It is assumed that only pairwise interactions are present. We write Pi;) = ~~l)(~l)~~l)(~z)g,~(~l,~~) (1) where P I ( ' ) is the one-particle density (density profile) of species i, depending only on the coordinate perpendicular to the interface, and the correlation function g,, depends on 7, and F; in the interface. In the bulk, p!" is, of course, a constant and g, can depend only on the interparticle distance rI2. The Fowler approximati~n~*-~ replaces the g, in (1) by the corresponding bulk functions so that, given the profiles, insertion of (1) into the Kirkwood-Buff formula allows the calculation of the surface tension in terms of the properties of the bulk fluid. (Modern theories of the interface make the same replacement for the direct correlation function45 which seems more reasonable because of the shorter range of this function.) While reasonable results can be obtained for some fluids by this procedure, molten salts are emphatically not among them. 31 The problem seems to be due to the Coulombic interactions, which impose4 a local electroneutrality condition on the correlation functions.
The local electroneutrality condition is that the net charge around an ion should be equal and opposite to the ion's charge. (It is not implied that the net charge density be zero everywhere; other authors have referred to this as "local electroneutrality".) This is not obeyed by the Fowler approximation because the surface, as expressed by the factor pi(z) in (l), truncates the two-particle di~tribution.~~ We have d i s c u s~e d~~~~~ modification of the Fowler approximation to guarantee local electroneutrality. Great improvements in surface tension and surface energy for the free (electrically neutral) surface of a molten salt have been obtained on introducing this m~dification.~'-~~ Croxton and M~Q u a r r i e~~ found imposition of a similar condition on their closure of the Born-Yvon-Green (BYG) equation for charged spheres at a charged surface led to improved results and suggested a similar modification should ameliorate theories which use bulk c.. for the surface. Local electroneutrality was also usedih in a theory for getting (bulk) gij of charged hard spheres in terms of gij for neutral spheres. In the present work, we extend the electroneutrality formulas to the salt surface in the presence of an external field. In this case, a separation of positive and negative charges in the salt leads to a double layer and a potential drop across the surface region. As for the free surface, our model yields a formula for surface tension which involves integrals over the bulk distribution functions; the function introduced to guarantee local electroneutrality is also determined by properties of the bulk distribution function (section 111). Our bulk salt is described by the restricted primitive model: anions and cations are oppositely charged hard spheres of equal size with t,he dielectric constant taken as unity. The distribution functions for the bulk are calculated according to the generalized mean spherical approximation (GMSA).36 Although the simpler mean spherical approximation (MSA) seems to give reasonable values for thermodynamic p r o p e r t i e~~,~~ and although the MSA has been solveds for hard spheres of different sizes we do not use it here because it does not describe well the detailed shapes6,36i37*39 of the gij which are central to our calculations. We note that other extensions of the MSA for the primitive model have been proposed and studied.5@54 While first developed and tested for electrolyte solutions, the GMSA seems to work better8,53 for higher concentrations and thus for molten salts (with dielectric constant unity).
As for the assumption of equal sizes for cation and anion cores, it seems not to be a bad one for bulk properties even  if the ions are actually of different sizes because, except where the disparity in ion sizes is very large (Li salts), ions of opposite charge are much more likely to approach to small distances than like-charged ions.3-5*55'60 The core size parameter we use in our calculations is in fact roughly appropriate for NaC1.5*6*32v33 However, the difference in ion sizes must be important in determining the structure of the surface, since this difference should produce a double layer and potential drop even for the free surface or for the interface at the potential of zero charge. The larger ions (usually the anions) will tend to protrude from the free surface, forming a layer of charge, and the absence of electric fields in the bulk implies an oppositely charged layer must exist below it. For ions against a hard wall (the metal of the electrode) the reverse should obtain. Sluckinm has recently discussed this effect using a perturbation theory applied to a treatment of these systems by the density functional formalism.Ba Since our model assumes identical density profiles for cations and anions in the absence of an external field, it seems most reasonable to say we are describing a fictitious salt for which anions and cations both have a hard-sphere diameter of 2.55 A, representing the averaging of anion and cation diameters for NaC1.
The radial distribution functions for a bulk liquid whose particles interact by Coulomb plus hard-sphere potentials are calculated according to the generalized mean spherical approximation (GMSA) with the formulas of Stell and c o -w~r k e r s~~~~~~~ which generate gD and gs where (2) (3) (superscript b refers to bulk). The distribution functions we obtained are given in Figure 1. In the absence of external field, i.e., at the potential of zero charge, the  Flgure 2. Ion and metal charge densities for use in two-particle distributions, eq 1. expression for the surface tension does not involve 2, so it was not calculated in our previous work. The local electroneutrality conditions will be imposed by modifying gD, leaving g' unchanged. Away from the potential of zero charge, where positive and negative ion distributions are different, g, __ can differ from g-+, and g, + from g--. The quantities g+ --g+ + and g-+ -g--can thus be different in the surface region, and two functions will be determined to guarantee local electroneutrality. We write g+ -(?1?2) + g+ +(F;?J = 2gS(rlZ) and find f+ and f-from the electroneutrality conditions. The solution of the equations is discussed in the next section.
The one-particle densities remain to be specified in (1). The simplest assumption is to take them as stepfunctions as was done for some of our calculations of the free sur- The abrupt decrease of the density to zero is more appropriate for a fluid near a repulsive wall than for a free surface (for which an attempt to consider other profiles was madeM). It must be admitted that a condition exists on the contact density at a hard which is not satisfied by the assumption of stepfunctions. Furthermore, a fluid of particles with repulsive cores should have an oscillating density profile at a wall and molecular dynamics calculations28 for a molten salt show that the presence of Coulombic forces does not remove the oscillations (although charge ordering or layering seem to be absent, even for a charged wall). Of course, our pi:) do not satisfy the BYG equations with stepfunctions for the pi", and there is some ambiguity in the definition of surface tension when such an inconsistency is Our use of stepfunction profiles for surface tension calculation is based on a hope that details of the profiles will average out (see section VI for generation of oscillatory profiles from the model).
The charge distribution is shown in Figure 2 which implies that the centers of the first layer of ions of the salt lie in the plane z = 0. The displacement a is determined by the charge density according to related to the distance of closest approach between ions of the salt and the metal surface. If the ionic radius of the metal ions is R M and the charge density q M is supposed to lie on the first layer of ions, the estimate for d would be RM + lIzu, where u is the ionic core diameter. Finally, we note that all properties should be unchanged by a change in the sign of all charges, so the electrocapillary curve is symmetric about the pzc.

Imposing Local Electroneutrality
The local electroneutrality condition is that the net charge surrounding a positive (negative) ion at z should be equal to one negative (positive) charge. This includes the charges on the metal as well as the charges of other ions in the salt. In terms of the one-and two-particle distributions, the condition for a positive ion is Here, the charge density of the metal is and of course is uncorrelated (g = 1) with the ions of the salt. On introduction of the assumptions of eq 4 and 5, eq 7 becomes The corresponding equation for the charge around a negative ion is where only values of z1 I -a are of interest. If p d = 0 and p+ and p-are identical for the potential of zero charge, (10) becomes the same as (9) and gs does not appear.
Equations 9 and 10 may be simplified by writing gs as 1 + hs. The terms in 1 represent the total charge of the salt, which is equal and opposite to the total charge on the metal: Thus the term in p d disappears and the local electroneutrality equations do not involve the metal charge distribution. Equation 9 may be written, when stepfunctions are inserted for the density profiles, as  Figure 3. We may note that our correlation functions (eq 1,4, and 5) do not guarantee the symmetry i between the two-particle distributions. The symmetry would require g+ -= gs + gy to be identical with g-+ = gS + g? and hence f+[(zl + z2)/2] to equal f-[(zl + 2,)/2].
Since in fact f+ and fare only slightly different, the symmetry is effectively assured. An idea of how well one can satisfy the electroneutrality conditions is gained from the sum of the squares of the values of (11) or (12) evaluated for 100 values of zl. It must be noted that there is always a doubt63 whether a solution to an equation like (11) actually exists; in fact, the physics of the present situation mean that a mean-square solution, minimizing the meansquare deviation of the left side from zero, is what we actually should seek. The calculations we perform in the present paper are for a = 0 . 0 5~ and 0 . 1 0~. The smaller value corresponds to a charge per unit area )qM1 of 9634.3 esu/cm2 or 3.2136 pC/cm2.

IV. Calculation of Surface Tension
a multicomponent fluid is The Kirkwood-Buff formula for the surface tension of where i and j run over species and uij is the interaction potential between a particle of species i and a particle of species j , assumed to be a function of the interparticle separation only. We have three kinds of particles: positive ions, negative ions, and charges of the "metal" at d. There are no,correlations between particles of the third kind and either of the first two, or between particles of the third kind; in such cases, we write ~1 3 7~7~) = pi(zl)pj(zz), whereas pa!;) = pi(zl)pi(z,)g;j where correlations exist. The interaction potentials uij consist of the electrostatic interaction, which is present for all particles, and the hard-sphere repulsion, for particles of the molten salt only. Thus the surface tension has two parts The hard-sphere part, on introducing our assumptions corresponding to the two parts of uij.
for the gij, is, after a change of variables

(16)
Here /3 = l/kT, u = 1/2(21 + z 2 ) , w = z2 -zl, and we are using the usual treatmenP4 of the hard-sphere term, requiring an integration by parts in rI2. After some further algebra, we find  with va = 1/2(21 + z 2 ) and ci = a / a . We have assumed d C 1. These two contributions are given in Table I. It is seen that as ci moves away from zero, the charge due to & is overcompensated by the term in &. Since this term arises from interactions between particles of the salt, it is the same for -d as for ci.
The electrostatic contribution to the surface tension, using our assumptions for the pi:), is conveniently written where hij = gi, -1. The electric charge density pe is p + - for the present problem immediately yields the last expression of (20), which may be written in terms of q by using the formula (6) for a. The rest of YEL is rewritten by putting hi, = gi, -1, inserting our assumptions for the gij, and separating contributions of hS and gD. Thus   1/(4np) and the zeroth and  correction & makes the surface tension positive at the pzc. For large electrode charge one still gets negative values, but one must remember that the actual surface tension of the interface is that of the salt plus that of the metal. The latter can be several hundred dynes/centimeter at the pzc, and, as the interface is charged, additional contributions from the metal may arise. Our results for surface tensions and potential drops are given in Table V.

V. Surface Tensions and Lippmann Equation
The surface tension at the point of zero charge is what we calculate for the free surface, 97.23 dyn/cm. In comparing it to experimental values for the free surface of alkali halides, one should recall that we are assuming there is no double layer for the electroneutral surface, which would hold for equal anion and cation core sizes, whereas the actual cation-anion radius ratio for NaCl is far from 1. The ratios for NaC1, KC1, and RbCl are 0.52,0.73, and 0.82; surface tensions at 1128 K are 111.3, 92.8 and 86.2 dyn/cm, respectively.M The effect of the size asymmetry on surface tension is a matter for subsequent investigation. It may be noted that, if it is important in determining surface structure, it should produce oppositely directed double layers for the free surface (larger ions to the outside of the salt) and for the surface of the salt in the interface (larger ions away from the metal, i.e., toward bulk salt). However, surface energies and surface entropies may well be independent of the sign of the surface double layer.
Smirnov, Stepanov, and S h a r o~~~l~ have argued that the metal at the pzc does not alter the surface structure of the salt from that of the free surface, since -yfree and +ymeM-dt change in a parallel way as one goes from one alkali chloride to another. Defining the work of adhesion Wa as the surface tension of the metal alone plus the surface tension of the salt alone minus the interfacial tension of the metal-molten salt interface at the pzc, they find@ values for Wa of 139 dyn/cm2 for P b in alkali chlorides, 134 dyn/cm2 for In, and 105 dyn/cm2 for Bi (although  Wa is quite independent of the alkali cation, and only slightly dependent on the anion.67 On the other hand, Ukshe et al. 69 have interpreted their electrocapillary measurements in terms of a significant influence of the metal on the top layer of the salt structure. They conclude that each salt has a different structure in the double layer, according to the cation-cation radius ratio. The differential capacitance of the interface" is given by (dq/dV),,,,, where CL represents the bulk chemical potentials of the species, Le., bulk compositions is to be held c~n s t a n t . '~~~~ With stepfunctions for anion and cation densities, the electric field is 2?rep(z + Lia) for z between -6a and 0, 2irep6a for z between 0 and d, and zero elsewhere. Thus where Li is taken positive. These potentials are given in Table VI. With q = ape12 (the convention here is that q is the charge per unit area on the salt and V the electrical potential in bulk salt minus the electrical potential in bulk metal), we get Thus the capacitance at the point of zero charge or electrocapillary maximum (a = 0) is simply (47rd)-l, which is the value for an ideal capacitor. Capacitances for a # 0 are less than ideal value by a factor of d / ( d + lal).
The capacitance (4?rd)-', equal to 6.935,3.467, and 1.7337 pF/cm2 for d = 0 . 5~~ a, and 2a, respectively, is much smaller than any of the values measured by Ukshe et al. 69 for Pb-molten salt interfaces. For Pb-NaC1, at 1093 K, a value of 45 rF/cm2 is reported, which seems typical of values for molten salts, although there is some question of the importance of Faradaic contributions to the capacity,72 and of the contribution of the metal as well as of the salt. To obtain 45 pF/cm2 from our model, we would need d = 0 . 0 7 7~. It appears that the above calculation is not capable of accounting for the measured values. However, this calculation obtained q and V by assuming stepfunction densities for the ions; as discussed below, these are not the only one-particle densities one could use.
According to the Lippmann e q~a t i o n~' J~~~~*~~ the surface charge density can be obtained from the electrocapillary curve: Thus, we use the data of Table VI for d = 0 and f0.05 to write For the three values of d, we find b = 5.0057 X lo6, 2.7982 X lo6, and 1.47785 X lo6 cm-', respectively. The resulting capacitances at the point of zero charge are given by In mks units, the three values of b give capacitances of 11.124, 6.218, and 3.2844 pF/cm2, about twice the ideal values of '/,ad, and closer to measured values. Note that the electroneutrality correction y ' & , extremely important in getting a reasonable result for surface tension, is much less important for the electrical capacitance, given by the second derivative of the surface tension with potential. Apparently, the electroneutrality correction is relatively constant with electrode charge. If, using the data for d = 0 and h0.05, we write y as a parabolic function of a, the second derivative, for d/u = 1, is 23 464 dyn/cm without y& and 22 416 dyn/cm with this correction. The Lippmann equation is not satisfied by our model: the charge densities according to (26) and (27) approach twice the value obtained from the stepfunction charge densities. The reason for the discrepancy is found on examination of T&, which is the largest term in y which varies with charge. For d much larger than lal, it becomes y ! & ! = -ae2p2a2d = -4aq2d = -v2/4ad (28) Note that the field energy of an ideal capacitor is74 Goodisman and Amokrane with E the electric field, which differs from ygL in that the latter is a free energy, involving the work necessary to separate charges in forming the double layer, as discussed in Chapter 17 of ref 70 and elsewhere.75 Differeniation of (28) gives The violation of the Lippmann equation is related to the lack of mechanical e q~i l i b r i u m .~~~~~~~~ The Born-Green-Yvon equation, or mechanical equilibrium condition, is for a system in which all forces are central. It relates the one-and two-particle distributions and, by relating the corresponding contributions to the pressure, guarantees the constancy of the normal component of the pressure through the interface. When this constancy is used in conjunction with the formula (14) for the surface tension, which inv01ves~~J~ the difference between tangential ( x component) and normal (z component) pressure, a new formula is obtained. With P h representing the (isotropic) pressure of the homogeneous phase, we have (30) where gb. is the bulk correlation function, depending only on rlz. +his gives y in terms of "surface excess" densities and two-particle distributions, i.e.
The contribution of the electrostatic forces may be calculated as in eq 19-24. The term corresponding to y ! & is not expected to contribute much. In the remaining electrostatic terms -ne2p2 the first term, which is the largest, is just half of y&, so differentiation with respect to V will give the correct q. Unfortunately, consideration of the contribution of the hard-sphere potential to surface tension shows that (30) would give totally unreasonable surface tensions. Mechanical equilibrium and consistency among these surface tension expressions may be restored to our model by using, when surface charges and potentials are discussed, one-particle distributions PI' ) generated from the two-particle distributions pi:) of our model by eq 29. These pI1) are consistent with the p $ ) in the sense that (30) will lead to the same surface tension as (141, and the pressure normal to the interface will be independent of z. Since we have calculated y using (14), which requires only the pi;), the surface tensions will not be changed; if we use (30) to calculate the surface tension, we insert p!') calculated from (29) and p$' as before. There is still an inconsistency in that the two-particle ion-metal distributions, for which there is no correlation, are constructed with stepfunctions rather than the true one-particle distributions.
In order that the integration of (29) to give one-particle densities make sense, dpii)/dz must vanish for z --a. where dpl')/dz is given by (29); p(-) of course vanishes. The charge density at z = 0 is 6.65170 esu/cm2. The calculated value of q (charge on the salt) depends on x, the value of z at which we cut off the p?: To have Q = 9634.3 esu/cm2, we require W = 0.0343808, which corresponds to AV = 4.45903 X statvolts and y = -0.2147137 erg/cm2. Following our procedure of fitting the results for Q = 0 and *9634.3 esu/cm2 to a parabola, we have -m 0 = yo -1.079889 X 108AVL so dy/dV = 9630.5 esu/cm2 for AV = 4.45903 X the deviation from 9634.3 esujcm' being due to the fact that (37) is not precisely a parabola in W. The capacitance is 2.159778 X lo8 cm-l or 239.975 pF/cm2. More precisely, we use (38) to get the capacitance as dQ/dAV = K(cosh' W + sinh2 W) which at W = 0 is just K, or 239.881 /*F/cm2. The value is, as previously mentioned, much too high.

VII. Conclusion
The initial purpose of performing these calculations was to ascertain whether one could describe the charged interface of a molten salt (in contact with a charged wall representing the metal) by invoking the same simple assumption for the ion-ion distributions that gave reasonable surface tensions for the neutral surface. Additional assumptions are required for the salt-metal two-particle distribution, and we assumed no correlation. For the charged interface, one is interested in the potential difference between inside and outside the surface, and how it changes as the surface charge changes (electrocapillary curve). The capacities obtained are much too low. However, their calculation requires surface charges and potential drops which, unlike the surface tension as given by the Kirkwood-Buff formula, require the one-particle distributions p('). Since the pI1) assumed were inconsistent with the p $ , different theoretically equivalent surface tension formulas could give quite different results. A further inconsistency exists between the surface charges We may choose the value of x to make q equal to the value implicit in our pi?), i.e.
Here, we have put f* = 1 since the electroneutrality factors make only a small contribution to the change of surface tension with potential. The notation p+ and pis now used to distinguish the stepfunctions appearing in pi?) from the one-electron densities now being generated and used in q and V. In a product of two pi (i = + or -) the first factor is a function of z and the second of z'. The right side of (33) may be reduced, after considerable algebra, to an expression involving the moments and contact values of gS and gD. Then the solution of (33) leads to x = -0.39497~.
The potential drop across the entire interface, with contributions from the charge density of the metal as well as from the ions of the salt, may be written The value of the integral after insertion of (29) is 1.20318 X lo4 esufcm. Therefore, for d = 0.5u, u, and 2a, we have potential drops of 3.0556 X 4.5992 X and 7.6864 X statvolts (a = 0 . 0 5~) . Then the coefficient for the parabola of y in VL (eq 27) becomes, for the three cases, 1.4084 X lo6, 1.3247 X lo6, and 0.9775 X lo6 cm-l. The capacitances become 3.1298,2.9437, and 2.1725 pF/cm2, even lower than those previously calculated. The charge densities calculated from the y-V parabola are 8607, 12 186, and 15028 esu/cm2. The molecular dynamics calculations of He yes and Clarke,20 for charged hard spheres near a wall, model molten KCl. Although surface tensions were not reported, a value for capacity was derived from surface charges and potential drops, as computed from the charge densities. The value of 50-70 pF/cm2 is of the right size, although it was stated that the system is far from the pzc, and that errors in this quantity are large.
We may also compare our results with what one obtains from the Gouy-Chapman model. Apparently, the shortrange interionic forces and correlations, which are not considered, lead to corrections which cancel, as seems to occur in the same calculations on the related Debye-Huckel leading to satisfaction of the Lippmann equation. In the Gouy-Chapman model, the one-particle densities are assumed to vary according to a Boltzmann ~~ (77) S. N. Bagchi, Int. J. Math. Sci., 3, 607 (1980). derived from the Lippmann equation and those calculated from the pi1).
The first kind of inconsistency is removed by maintaining the @ and using the equation of mechanical equilibrium to derive pi1) from the p!:). Different surface tension formulas then give the same results, and mechanical equilibrium is assured in the region where pi') and pi?) are nonzero. Since pi1) must be truncated to assure the correct surface charge, a new inconsistency appears, as no repulsive interaction between metal and salt particles (which would make pjl) go to zero) is introduced into the calculations. Furthermore, if one wants to assume no correlation between particles of the metal m and ions of the salt i, the two-particle distribution pi$ should be pI1) X p i ) , with pi1) the same one-particle densities used for calculation of electrical properties. Inserting this into the mechanical equilibrium condition yields a more complicated integral equation for the pi1), solution of which has not been attempted here. The capacities calculated with pi1) derived from mechanical equilibrium are still quite small compared with those reported experimentally for the molten salt-metal surface. The Lippmann equation can be satisfied only for one choice of the distance of closest approach of salt ions to the metal. To help understand the origin of the problem, we considered the Gouy-Chapman model, which, ignoring nonelectrostatic interactions and interparticle correlations, satisfies the mechanical equilibrium condition as well as the Lippmann equation. For a molten salt in contact with a charged wall, this model gives capacities which are much too high. When a region free of ions (Stern layer) is introduced between the region of the salt ions and the charged surface (metal), the potential drop across the interface is increased with no change in the surface charge, thus reducing the capacity. However, the Lippmann equation is no longer satisfied. As in our model, we now have an ionic distribution which drops to zero within the region of the interface, implying a repulsive force, without inclusion of such a force in the surface tension calculation. Again we conclude that a consistent treatment of such forces is required in order for a model to satisfy the Lippmann equation.
Since this equation is easily demonstrated thermodynamically, a few words are appropriate about why we are interested in finding it in our model. The equation as generally stated involves the total surface charge density of the interface and the potential difference between homogeneous regions on either side of the interfacial region.
A model gives these quantities specific meaning by supplying information about the charge distributions, from which total density and potential drop are calculable. The charge distribution is also related through the mechanical equilibrium condition to the two-particle distribution, which may be used to calculate the surface tension. From this point of view, the satisfaction of the Lippmann equation is a necessary condition on the two-particle distributions, which describe the correlation due to the interparticle forces.