Exact Solution of the Mean Spherical Model for Fluids of Hard Spheres with Permanent Electric Dipole Moments

Abstract

The mean spherical model is solved in closed form for a fluid of hard spheres with permanent electric dipole moments. Both the pair distribution function g(12) and the direct correlation function c(12) consist of a spherically symmetric term and two other terms with different dependences on the orientations of the two dipole moments. The spherically symmetric part is the solution of the Percus-Yevick equation for hard spheres. The angle-dependent terms satisfy two coupled integral equations, which can be decoupled by appropriate changes of the dependent variables. The solutions are expressed in terms of the solution of the Percus-Yevick equation for hard spheres for both positive and negative densities. The effect on g(12) of the finite size of the sample is calculated for the case of a sphere. The correction term in g(12) is found to be of order 1/𝒱, where 𝒱 is the volume of the sample. It is a function not only of the relative distance vector of the two molecules, but also of their positions in the sample. The contribution to the polarization is, nevertheless, constant throughout the sample, in agreement with classical electrostatics. The dielectric constant ε, calculated by Kirkwood's formula, is obtained in closed form. It is a function of a single variable which does not contain the hard sphere diameter.