Mean Spherical Model Integral Equation for Charged Hard Spheres I. Method of Solution

The thermodynamic properties and radial distribution function of a ``primitive model electrolyte'' are calculated in the Mean Spherical Model (MSM) approximation. The system considered is formally described as a classical fluid of charged hard spheres (hard sheets in one dimension). The interaction potential between an ion of species i and an ion of species j, v_ij(r) is the sum of a Coulomb part and a hard core part q_ij(r)=∞ for ${\mathrm{r\; <\; R}}_{\mathrm{ij}}$ or 0 otherwise. The MSM approximation consists of supplementing the fact that the radial distribution function g_ij(r) is zero for ${\mathrm{r\; <\; R}}_{\mathrm{ij}}$ by equating the Ornstein—Zernike direct correlation function C_ij(r) to ‐βv_ij(r) for ${\mathrm{r\; >\; R}}_{\mathrm{ij}}\mathrm{,\hspace{0.17em}\beta }$ being the inverse temperature. In this paper, which is the first of two in this topic, we give the method of solution for this approximation and obtain C_ij(r) for ${\mathrm{r\; <\; R}}_{\mathrm{ij}}$ as polynomials in r, which have a structure similar to that found in the PY approximation for a mixture of uncharged hard spheres. We give the solution up to a set of algebraic equations in the polynomial coefficients. In the second paper we discuss the explicit solution and derived thermodynamic quantities.