Solution of the mean spherical model for a mixture exhibiting phase separation

Abstract

We have solved analytically the Lebowitz-Percus Mean Spherical Model (MSM) approximation for a symmetric mixture of two species of particles, constrained to have an equal number of particles of species 1 and 2. In this mixture, like particles interact via a hard sphere potential for r < R, the hard core diameter, and a Yukawa potential −Ae−κ r/r for r > R; unlike particles interact via the same hard spheres potentials (same diameter) and an opposite Yukawa potential Ae −κr/r. We find that for A>0, corresponding to an attraction between particles of the same kind and a repulsion between particles of different kind, that there are no spatially homogeneous solutions of the MSM when θ>θc,θ≡βξε, where β is the reciprocal temperature, ξ the total reduced density of hard spheres and ε≡ Ae−κ R/R. We interpret this to mean the existence of phase separation when θ is above its critical value θc We are able to calculate θc analytically as a function of A, κ, and R. We also find that when A<0 there are always homogenous solutions of the MSM. Finally when κ→ 0 and A=−e2/D, we recover the Waisman-Lebowitz solution of the MSM for hard charged spheres of equal numbers of the +e and −e charged particles and dielectric constant D.