A Model for the Surface of a Molten Salt

A model is proposed for the two-particle distribution functions for the surface region of a system composed of two oppositely charged species with identical hard sphere repulsions. The distribution functions are formed from those for the bulk fluid by incorporating a cutoff corresponding to the surface and a multiplying factor defined so as to guarantee electroneutrality while maintaining the proper symmetry. Various methods for doing this are discussed. Good agreement is obtained for surface tension and surface energy. Density oscillations are predicted. The absorption spectra of various homologues and analogues of retinals, a total of 21, with varying number of double bonds (n) have been examined in detail under various conditions of solvent and temperature. Altogether six band systems have been identified and their oscillator strengths and transition energies are presented as functions of chain length. The origin of the transitions is discussed in the light of the results of semiempirical calculations available in the literature. The trend in the lower polyene systems (n = 2-4) where the l(n,a*) state is seen in absorption clearly indicates that this latter state is the lowest singlet state in these systems, and is close to the B, state in retinals and their analogues. The -280-nm band system in retinals and their analogues, heretofore not satisfactorily assigned, is traceable to a more intense band system in the lower homologues and is tentatively interpreted in terms of absorption of 6-s-trans conformers present in solution to the extent of -10% in equilibrium with distorted 6-s-cis conformers. The absorption spectra of retinones and c16 ketones indicate methyl-methyl and methyl-hydrogen steric interaction leading to geometric distortion of the polyene chain.


I. Introduction
We consider a system composed of two oppositely charged species, with the interaction potential between a particle of species i and one of species j given by a Coulombic potential, eLeJ/rLJ, plus a hard sphere repulsion, the hard sphere radii being identical for positively and negatively charged species (restricted primitive model).l This set of assumptions seems to give a good description of molten salts and other ionic systemsaZ By a model for the surface region, we mean a formula for the two-particle distribution functions pL,(2)(F;7z), where3 P,(~)(F;F;) d3rl d3r2 gives the average number of pairs of particles of species i and j , such that a particle of species i is found in a volume d3rl at F; and a particle of species j in a volume d3rz at Fz, From pL, ( Here z gives position on the direction normal to the surface, and B(zl) is the step function, equal to 1 for z1 > 0 and 0 for z1 I 0. The bulk distribution function depends only on r12, reflecting the isotropy and homogeneity of the fluid. For the surface region, pL,(2) may depend on z1 and the components of TlP Conventionally, one includes in the Fowler approximation a step-function formula for the one-particle distribution functions. However, the assumption for p(') is inconsistent with that for P (~) , as appears6p7 from the large and nonconstant normal pressure calculated using both assumptions. If p(') is derived from p@) using the Born-Yvon-Green equation, the calculated normal pressure is constant, as it should be.6-8 one can calculate the surface tension y and surface energy E,. When the generalized mean spherical approximation is used to obtain pUbulk, the results for y and E, for NaCl at 1128 K (see Table I) are so far from the experimental values7 as to indicate a serious problem with the approximation (1) for p,@), a problem which apparently is not present for one-component systems for which this model has been used. Use of a better interaction potential and pL?lk from Monte Carlo calculations does not improve the ~i t u a t i o n .~ The main source  of the difficulty seems to lie in the violation of the electroneutrality condition.
Calculated from ES = y -T dyl This is best expressed by writing where pibulk (= pjbdk) is the one-particle density of species i and defining gS(r12) = %k+-+ g++) ( 5 ) Since the bulk correlation function satisfies JOmgDr2 dr = 1/(47rp), we have electroneutrality when lzll is large, but not otherwise. The right side of (5) is plotted as a function of z1 in Figure 1A. Henceforth, we shall represent the combination of correlation functions given in (3b) by gB when the functions g+-and g++ are taken from bulk calculations without modification, and use gD for the corresponding function for the surface; gD depends on zl, z2, and F12, In our previous work, we corrected the problem by writing7 gD = fkl) gB(r12) (6) where and leaving gs unchanged. This makes (5) exactly equal to -e. Much improved values for surface tension and surface energy were obtained (see Table I). We pointed out, however, that a symmetry condition was violated. We should have where s122 = (xlx 2 ) 2 + (yl -y2)2, and this does not hold if gD of eq 6 is used. The present paper is concerned with correcting for electroneutrality while maintaining this symmetry.

A Model Satisfying the Symmetry Condition
the above choice is to write One reasonable way of obtaining a symmetrical g from gD(z122s12) = [fh) f(z2)11'2 gB(r12) (8) using eq 7 for f. The resultant gD is better than gB in assuring electroneutrality for small zl, as shown in Figure  1B. However, it is not an improvement elsewhere. Another method was suggested by us: one could put gD = f(zl) f(z2) gB(r12) and determine f by solving the integral equation (see eq 4) by an iterative method. Putting f0(z2) = 1, this gives for fi(zl), the first approximation for f(zl), just eq 7. Attempts to implement this procedure were unsuccessful: we were unable to get convergence to a reasonable f(z). Another solution to the symmetrization problem is to put gD = Lf(zl) + fWl gB(rlJ, leading to In this case, the integral equation is linear, but the iterative procedure again failed to converge. If it is assumed that f(z) = 1/2 (the asymptotic value) (see Figure 2). It appears that, as zo increases, f becomes more wildly oscillating, taking on larger positive and negative values. This probably explains our difficulties in getting solutions to the other integral equations by our iterative method.
After additional experimentation, we were led to try the physically reasonable assumption (10) This makes the multiplying factor on gB depend on the average of the z values of the two particles. We tried to use for f the function determined for the unsymmetrical model, according to eq 7, which uses the bulk correlation function gB(r) = l/z(g+--g++). Thus the same function was used as previously, but its argument was changed from z1 to 1/2(21 + z2). As shown in Figure lC, this choice of z leads to satisfactory charge neutrality (except when z1 is close to 0). It should be possible in principle to determine f so that electroneutrality obtained exactly when gD of (10) was employed, but we did not pursure this route. Instead we used the f of (7) to calculate properties of the surface, as we now discuss, with satisfactory results.

Calculated Properties
following form: For the purpose of these calculations, we wrote gD in the gD(z1,22,s12) = gBk12) This allows us to write the value for each property as the Fowler (superposition) value plus a correction. (Note that the correction is to the contribution of the electrostatic forces.) The function f (see eq 7) oscillates about unity as a function of its argument, the oscillations decreasing as the magnitude of its argument (which is always 10) increases. In fact, f is the reciprocal of the quantity plotted in Figure 1A. Calculated values off are given in Table I1 (see paragraph at end of text regarding supplementary material). For z -7.2u, we take f ( z ) = 1, so that the correction to the Fowler value of a property (superscript F) is an integral whose range is finite. Thus, the surface tension is written where j-= 1 4 . 4~ -z1,.r1; = z12 + slZ2, and z12 = -z21 = z2 -zl; cylindrical coordinates (z2,s12,q5) have been used in the integral and u is the hard sphere diameter.
As we have noted, our model of the interface is a model for the two-particle distribution functions P + + (~) (= P -(~) ) and p+A2) (= P -+ (~) ) , or, alternatively, their sum and difference ps and pD (see eq 2 and 3). We use the bulk-fluid distribution function for ps, as in eq 1, and modify the bulk-fluid function for pD. The one-particle distribution functions pi(1) are to be computed from the pjj(2) by the Born-Green-Yvon equation: Integrating (12) over all values of z1 and noting that pc!2) = 0 for z1 > 0, we obtain the difference between pi(0) and p;(-), which should be just the negative of the density of the anion or cation in the bulk fluid. As shown in our previous work,' use of the Fowler approximation (eq 1) gives the correct value, p/2, for this quantity, but, when we modify the distribution functions by multiplying pB by f(zl), this no longer holds true.
This deficiency in the previous model is remedied when the modification (9), which restores the symmetry of pD, is used. The effect of the correction is to add the following term to the integral of dpi(l)/dzl:

G(n+l)(lxl) (14)
where Gci)(y) is the ith moment of gB from y to 00, which we have previously tabulated. Equation 13 becomes In the first term, the integral over x vanishes because the integrand is odd; if L is large enough (>7.2 hard sphere diameters in the present case) so that f(u) = 1 for u < -L, the second term also vanishes.
For calculation of dpi(l)/dzl at a particular value of zl , the contribution additional to the Fowler contribution is obtained according to (see eq 13 and 14): Here, zo = -1 4 . 4~ -2z1. The results of this calculation are given in Figure 3. The density gradient dp(l)/dzl from the unmodified Fowler-Kirkwood-Buff model is plotted as well as dp(l)/dzl from our model (Fowler-Kirkwood-Buff results plus correction of eq 15). It is clear that our modification leads to more pronounced oscillations, ex-J. Goodisman and R. W. Pastor tending further into the bulk. With the step function of (eq l), dp/dzl vanishes for z1 > 0, while Figure 3 shows that it has its largest magnitude for z1 just below zero. The discontinuity in dp/dz1 could be removed by using a smooth function instead of 1 -B(zl) in eq 1.
From dpi(l)/dzl, one could compute the density profile, integrating inward from z1 = 0. Clearly, our model predicts a density profile which has strong oscillations is about ~m -~) . This may imply that such oscillations really exist in this system: they do not seem tos in other surface systems studied. The enhanced oscillations represent the influence of the electroneutrality constraint, which is a reflection of the long-range character of the Coulombic force, not present in systems considered in the past. In this system, we find, furthermore, attractive forces and repulsive forces of the same range. This would seem to create the possibility that dpi(l)/dzl take both positive and negative values in the surface region.
Returning to the surface tension, we find, by treating (11) like (13), the following additional contribution:

3~i22G'-2)(1zi21)1
For the surface energy, we find X (17) with u = 1/2(21 + z2), a! = 2(-7.2a -zl), and G(O) defined by (14). Since the moments of gB have been previously calculated, a double integral over z1 and z12 is required to evaluate the additional contributions to y and to Es. They are 178.94 and -619.44 dyn/cm, respectively. As shown in Table I, the total values for surface tension and surface energy are in reasonable agreement with experiment. As usual, the agreement is much better for the former.
We conclude that satisfaction of the electroneutrality and symmetry conditions on the two-particle distribution functions converts the Fowler-Kirkwood-Buff model of the surface region, untenable for this system, into one which could provide a reasonable description of the interface. What one requires now are additional constraints the predictions of the model. In particular, there is the fact that the improvement in calculated properties between the Fowler-Kirkwood-Buff and the present model is accompanied by a large increase in the oscillations in the one-particle distribution function or density.
One thermodynamic property which should be satisfied by the statistical mechanical model for the interface is the Gibbs-Helmholtz equation Verifying this equation is difficult; furthermore, it is likely that the sharp cutoffs in pij (eq 1) become worse approximations as the temperature increases. The fact that ES agrees much less well with experiment than y may be taken8 as an indication that the Gibbs-Helmholtz equation is not satisfied. Another property is expressed by the Gibbs-Lippmann equation where E, is the drop in electrical potential across the interface and q the electrical charge per unit area of the surface double layer.1° The present model of course has neither double layer nor potential drop. Modification of the assumptions, to produce asymmetries between anions and cations, would be interesting to study. Table 11, giving the function f which guarantees electroneutrality (3 pages). Ordering information is available on any current masthead page.

Supplementary Material Available:
The absorption spectra of various homologues and analogues of retinals, a total of 21, with varying number of double bonds (n) have been examined in detail under various conditions of solvent and temperature. Altogether six band systems have been identified and their oscillator strengths and transition energies are presented as functions of chain length. The origin of the transitions is discussed in the light of the results of semiempirical calculations available in the literature. The trend in the lower polyene systems (n = 2-4) where the l(n,a*) state is seen in absorption clearly indicates that this latter state is the lowest singlet state in these systems, and is close to the B, state in retinals and their analogues. The -280-nm band system in retinals and their analogues, heretofore not satisfactorily assigned, is traceable to a more intense band system in the lower homologues and is tentatively interpreted in terms of absorption of 6-s-trans conformers present in solution to the extent of -10% in equilibrium with distorted 6-s-cis conformers. The absorption spectra of retinones and c 1 6 ketones indicate methyl-methyl and methyl-hydrogen steric interaction leading to geometric distortion of the polyene chain.

I. Introduction
In recent years, a great deal of theoretical and experimental work has been done on retinals and related polyene sy~tems.l-~ However, a number of questions and controversies regarding the spectral properties of these systems still remain unresolved. Some of these concern the relative order of the three low-lying singlet states, l&-, lBU, and l(n,~*),~-lO the nature of the lowermost singlet state,11-14 the location of the cis band (lAg+ -lA, trans i t i~n ) ,~,~J J~ the assignment of the 280-300-nm band system (in r e t i n a l~) ,~J~~J ' and the absorption spectral behavior of the 14-methyl analogue of l l -~i s -r e t i n a l .~~J~ In the present investigation, we have studied 21 polyenals and polyenones that are related to retinals as analogues and homologues. The low-lying excited states 0022-3654/78/2082-208 1$0 1 .OO/O of photophysical and photochemical interest in many of these systems are expected to be comparatively sparsely located and to provide situations with relative state order different from that in retinals. Although there have been a few early spectroscopic studies2&22 on some of the polyene systems under examination, these were concerned with room temperature spectra and limited to band maxima (as routine work in the course of synthesis). No detailed systematic analysis has ever been undertaken with the object of understanding the existent problems concerning retinals, their homologues, and visual pigments.
In paper 1, we propose to present the absorption spectral data and attempt to interpret them in the light of the results of theoretical calculations available to date. In paper 2, we shall report the data on fluorescence, quantum