Exact Solution of Generalized Percus-Yevick Equation for a Mixture of Hard Spheres

TLDR: In this paper, the Percus-Yevick approximate equation for the radial distribution function of a fluid is generalized to an $m$-component mixture, which can be formulated by the method of functional Taylor expansion.

Abstract: The Percus-Yevick approximate equation for the radial distribution function of a fluid is generalized to an $m$-component mixture. This approximation which can be formulated by the method of functional Taylor expansion, consists in setting $\mathrm{exp}[\ensuremath{-}\ensuremath{\beta}{\ensuremath{\phi}}_{\mathrm{ij}}(r)]{C}_{\mathrm{ij}}(r)$ equal to ${g}_{\mathrm{ij}}(r)[{e}^{\ensuremath{-}\ensuremath{\beta}{\ensuremath{\phi}}_{\mathrm{ij}}(r)}\ensuremath{-}1]$, where ${C}_{\mathrm{ij}}$, ${g}_{\mathrm{ij}}$, and ${\ensuremath{\phi}}_{\mathrm{ij}}$ are the direct correlation function, the radial distribution function and the binary potential between a molecule of species $i$ and $a$ molecule of species $j$. The resulting equation for ${C}_{\mathrm{ij}}$ and ${g}_{\mathrm{ij}}$ is solved exactly for a mixture of hard spheres of diameters ${R}_{i}$. The equation of state obtained from ${C}_{\mathrm{ij}}(r)$ via a generalized Ornstein-Zernike compressibility relation has the form $\frac{p}{\mathrm{kT}}={[\ensuremath{\Sigma}{\ensuremath{\rho}}_{i}][1+\ensuremath{\xi}+{\ensuremath{\xi}}^{2}]\ensuremath{-}\frac{18}{\ensuremath{\pi}}\ensuremath{\Sigma}{ilj}^{}{\ensuremath{\eta}}_{i}{\ensuremath{\eta}}_{j}{({R}_{i}\ensuremath{-}{R}_{j})}^{2}\ifmmode\times\else\texttimes\fi{}[{R}_{i}+{R}_{j}+{R}_{i}{R}_{j}(\ensuremath{\Sigma}{\ensuremath{\eta}}_{l}R_{l}^{}{}_{}{}^{2})]}{(1\ensuremath{-}\ensuremath{\xi})}^{\ensuremath{-}3}$, where ${\ensuremath{\eta}}_{i}=\frac{\ensuremath{\pi}}{6}$ times the density of the $i\mathrm{th}$ component and $\ensuremath{\xi}=\ensuremath{\Sigma}{\ensuremath{\eta}}_{l}R_{l}^{}{}_{}{}^{3}$. This equation yields correctly the virial expansion of the pressure up to and including the third power in the densities and is in very good agreement with the available machine computations for a binary mixture. For a one-component system our solution for $C(r)$ and $g(r)$ reduces to that found previously by Wertheim and Thiele and the equation of state becomes identical with that found on the basis of different approximations by Reiss, Frisch, and Lebowitz.