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

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 $\mathrm{r\; <\; R}$ , the hard core diameter, and a Yukawa potential ${\mathrm{-Ae}}^{\mathrm{-\kappa \; r}}\mathrm{/r\hspace{0.17em}}\mathrm{for}\mathrm{\hspace{0.17em}r\; >\; R}$ ; unlike particles interact via the same hard spheres potentials (same diameter) and an opposite Yukawa potential Ae ^−κr/r. We find that for $\text{A>0}$ , 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 ${\mathrm{\theta >\theta }}_c\mathrm{,\theta \equiv \beta \xi \epsilon }$ , where β is the reciprocal temperature, ξ the total reduced density of hard spheres and ${\mathrm{\epsilon \equiv \; Ae}}^{\mathrm{-\kappa \; R}}\mathrm{/R}$ . We interpret this to mean the existence of phase separation when θ is above its critical value θ_c We are able to calculate θ_c analytically as a function of A, κ, and R. We also find that when $\text{A<0}$ there are always homogenous solutions of the MSM. Finally when $\text{κ→ 0}$ and ${\mathrm{A=-e}}^2\mathrm{/D}$ , we recover the Waisman‐Lebowitz solution of the MSM for hard charged spheres of equal numbers of the $\mathrm{+e}$ and $\mathrm{-e}$ charged particles and dielectric constant D.