Analytic Solution of the Ornstein-Zernike Relation for Inhomogeneous Liquids

TL;DR

This paper develops an explicit analytical solution for the Ornstein-Zernike relation in inhomogeneous liquids, linking the direct correlation function to the pair distribution function, validated in one dimension and extended to three dimensions.

Contribution

It provides a novel explicit formulation of the Ornstein-Zernike equation for inhomogeneous liquids, connecting correlation functions with experimental observables.

Findings

Validated in 1D hard rod liquids

Extended formalism to 3D liquids

Demonstrated accuracy against exact solutions

Abstract

The properties of a classical simple liquid can be strongly affected by application of an external potential that supports inhomogeneity. To understand the nature of these property changes the equilibrium particle distribution functions of the liquid have, typically, been evaluated individually as functions of system control parameters, such as the packing fraction of a hard sphere liquid. In this study we focus attention on two distribution functions that characterize the inhomogeneous liquid: the pair direct correlation function $c(\mathbf{r}_1,\mathbf{r}_2)$ and the pair correlation function $g(\mathbf{r}_1,\mathbf{r}_2)$. We solve the Ornstein-Zernike equation for the inhomogeneous liquid to obtain $c(\mathbf{r}_1,\mathbf{r}_2)$ as an explicit function of $g(\mathbf{r}_1,\mathbf{r}_2)$, with the latter considered to be an experimental observable, using information about the well studied and resolved $g^0(\mathbf{r}_1-\mathbf{r}_2)$ and $c^0(\mathbf{r}_1-\mathbf{r}_2)$ for the parent homogeneous ($^0$) system. The result obtained with our formulation is tested against the exact solutions for the correlation and distribution functions of a one-dimensional inhomogeneous hard rod liquid. Following the success of that test the formalism is extended to obtain $c(\mathbf{r}_1,\mathbf{r}_2)$ as an explicit function of $g(\mathbf{r}_1,\mathbf{r}_2)$ in a three dimensional liquid.