An analytical expression for the direct correlation function of a hard-sphere liquid derived from the extended scaled particle theory

Abstract

Integral equation methods based on the Ornstein–Zernike equation have been widely used for calculations of the radial distribution functions, g(r), of simple liquids. The closure that connects a direct correlation function, c(r), with a total correlation function, h(r), is necessary for solving the integral equation. In this work, we propose an analytical expression of the direct correlation function for a hard-sphere liquid through the solvation free energy for a pair of hard-spheres as a solute derived from the extended version of scaled particle theory (XSPT) instead of using iterative calculations with some closure. The solvation free energy has an analytical expression given by the XSPT that treats solvation of an arbitrary shaped particle into a hard-sphere liquid. In the case of a hard-sphere liquid, our new expression for the direct correlation function gives an analytical expression of the radial distribution function in Fourier Transform k-space. In addition, the radial distribution function in real-space quantitatively agrees with the Monte Carlo simulation results better than the Percus–Yevick equation method under equivalent numerical conditions. Our method with the proposed direct correlation function in this work is expected to be applied to estimation of the contributions from hydrophobic hydration in the hydration free energy of two associated molecules, which is a key to understanding the mechanism for association of proteins.