Some exact results and the application of the mean spherical approximation to charged hard spheres near a charged hard wall

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Abstract
An inconsistency within one of the Henderson–Abraham–Barker equations which fails to conserve local charge neutrality when applied to ionic mixtures near charged walls is discussed and is resolved by a careful analysis of the limit in which the wall particle is allowed to grow to infinite radius and zero curvature. This gives rise to a correction term which is needed only for the case of electrostatic interactions. The resulting equation is used to obtain exact asymptotic forms for the wall–fluid correlation functions at large positive and negative distances from the interface. Also, an exact relation for the contact value of the density profile is given. By means of a simple argument, results for both the total and direct wall–fluid correlation functions are given in the mean spherical approximation for charged hard spheres near a charged wall. These results are used to determine the capacitance of the interface in the mean spherical and exponential approximations. A modified form of the exponential approximation agrees closely with the Gouy–Chapman theory whereas the mean spherical approximation does not. On the basis of the exact expression for the contact value of the density profile it is argued that, for large charge on the wall, the Gouy–Chapman theory may give a reasonable capacitance even at high concentration.