The application of the generalized mean spherical approximation to the theory of the diffuse double layer

A system of charged hard spheres near a uniformly charged hard wall is considered. An approximation is established by postulating a closure for the Ornstein–Zernike (OZ) equations for this system. In this paper these OZ equations are solved for a closure in which the direct correlation functions are equal to the wall-ion potentials plus a sum of exponential functions. As a specific application of this solution we use one exponential and adjust two parameters to satisfy an approximate contact value theorem and give the same diffuse layer potential as is obtained using the hypernetted chain (HNC) approximation. Once this fit is made, the density, charge, and potential profiles can be easily calculated. The agreement with the corresponding HNC results is good. Comparison with the simpler Poisson–Boltzmann theory of Gouy and Chapman (GC) shows the GC theory to be better than one would expect. However, appreciable differences between the present results and the GC results for the diffuse layer potential are found.