Perturbation theory of structure in mixtures near phase separation

A theory of the density-density correlation functions of classical binary mixtures is developed to treat the problem of phase separation. Particles in the mixture are assumed to interact through pairwise potentials, and the theory is thus appropriate both to insulating fluids and to metallic systems to the extent that these may be described, for structural purposes, by effective ion-ion pair interactions. Pair potentials are separated in the form $v_{\mathrm{ij}}={\overline{v}}_{\mathrm{ij}}+v_{\mathrm{ij}}^1$, where ${\overline{v}}_{\mathrm{ij}}$ is appropriate to a reference fluid and is so chosen that $v_{\mathrm{ij}}^1$ may be regarded as a perturbation on this reference fluid. The resulting perturbation theory can be framed in terms of functions $f_{\mathrm{ij}}$ closely related to the Ornstein-Zernike direct correlation functions. Exact equations for the $f_{\mathrm{ij}}$ are obtained by treating the densities of the mixture as basic variables in a linear-response problem. Approximate solutions to these equations (and hence to the structural problem) are given. To first order in $v_{\mathrm{ij}}^1$, these solutions are exact in the long-wavelength limit, the region of interest in the phase-separation problem. By way of application, we show the effect of these first-order corrections to the simple mean-field calculations previously applied to simple metallic mixtures.