An ensemble of hard spheres

The Gouy-Chapman theory treats the electrolyte as consisting of point ions in a dielectric continuum. This is reasonable when the concentration of the ions is low, and the space charge is so far from the metal surface that the discrete molecular nature of the solution is not important. This is not true at higher electrolyte concentrations, and better models must be used in this case. Improvements on the Gouy-Chapman theory should explain the origin of the Helmholtz capacity. In the last section we have seen that the metal makes a contribution to the Helmholtz capacity other contributions are expected to arise from the molecular structure of the solution. 

The most important result is the existence of an extended boundary region, where the structure of solution differs significantly from the bulk, and where the potential deviates from the predictions of the Gouy-Chapman theory. In this model the interfacial capacity can be 

JDue to the polarizability of water, the effective value of the dipole moment for liquid water is expected to be somewhat higher than the value for the vapor so the calculated value is not bad. 

For water at room temperature, A 2.65. The natural interpretation of Eq. (17.20) is this The structure of the solution at the interface causes deviations from the Gouy-Chapman theory. The leading correction term is independent of the electrolyte concentration and therefore contributes to the Helmholtz capacity for water (s 3 A) one obtains a contribution of about 7.1 A (0.64 cm2//F-1). At very high concentrations terms of order k and higher become significant. These should cause deviations from a straight line in a Parsons and Zobel plot, which have indeed been observed [10]. 

If we combine these ideas with those developed in the previous section, we conclude that there are two contributions to the inverse Helmholtz capacity one from the structure of the solution, and one from the response of the surface electrons. It is natural to combine the 

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Figure 17.5 An ensemble of hard sphere ions and dipoles in contact with a metal.

Figure 17.6 Distribution of the electrostatic potential for an ensemble of hard sphere ions and dipoles in contact with a hard wall the straight line is the prediction of the Gouy-Chapman theory. Data taken from Ref. 8.

Camie and Chan30 treated an ensemble of hard-sphere ions and dipoles in contact with a hard wall. Specifically, they considered solvent molecules of radius rw with a point dipole at their center, and two kinds of ions, positive and negative, both with the same radius r,. One kind of ion can be adsorbed on the surface of the hard wall by a potential proportional to a Dirac delta function. Charge transfer between the metal (i.e. the hard wall) and the adsorbate was not... 

In principle, these correlation functions should be calculated from the interaction potentials c,y(l, 2) between the particles. However, even for an ensemble of hard spheres, this is an intractable problem, let alone for systems with realistic interactions. Therefore various schemes have been devised to calculate the correlation functions from approximate relations between these functions. For this purpose. 

The simplest molecular model for an electrolyte solution is an ensemble of hard spheres treated in the MSA. This can be combined with jellium to obtain a model for the whole interphase [46-48]. The hard sphere model has been solved at the PZC only, so the combined model is restricted to this point. It is natural to consider the jellium surface as a hard wall for the electrolyte and add the contributions of the hard-sphere electrolyte and jellium to... 

Another principle difficulty is caused by the structure of the electrochemical interface. The distribution of the particles and the electric potential has been investigated by numerous methods, starting from integral equations for hard-sphere ions and dipoles, molecular dynamics and Monte Carlo simulations based on force fields, and to a limited extent (short simulation times, small ensembles) by ab initio molecular dynamics. While the details depend on the system considered and the method employed, they all agree in an important point for the ionic concentrations usually... 

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