Two Approximate Solutions of the Ornstein-Zernike (OZ) Integral Equation

TL;DR

This thesis reviews liquid-state theories based on the Ornstein-Zernike equation, deriving analytical solutions for hard-sphere models using approximations like PY and MSA, and providing explicit thermodynamic expressions.

Contribution

It offers a comprehensive derivation of analytical solutions to the OZ equation for hard-sphere systems, enhancing clarity and completeness in the literature.

Findings

Analytical solutions for the OZ equation under PY and MSA approximations.

Explicit expressions for thermodynamic properties like the equation of state.

Enhanced mathematical clarity in deriving correlation functions and thermodynamics.

Abstract

This thesis explores the evolution of liquid-state theories based on the Ornstein-Zernike (OZ) equation, summarizing the foundational methods developed by Baxter, Lebowitz, Wertheim, and others. A unifying feature of these approaches is their shared analytical strategy: by introducing an intermediate function with specific mathematical properties, they effectively decouple the total correlation function and the direct correlation function. This allows the OZ equation to be solved within specific spatial intervals by exploiting regions where either the total or direct correlation function is known. Furthermore, this work presents a comprehensive derivation of analytical solutions to the OZ integral equation under the hard-sphere model. This includes applications of the Percus-Yevick (PY) approximation for both single- and multi-component systems, as well as the Mean Spherical Approximation (MSA) for systems of charged hard spheres. Building upon these analytical solutions, explicit expressions for macroscopic thermodynamic properties, such as the equation of state and activity coefficients, are rigorously derived. These derivations extensively employ advanced mathematical techniques, including Fourier transforms, complex analysis, and integral equation theory. Notably, many of the intermediate analytical steps and thermodynamic derivations presented herein offer a level of clarity and completeness previously absent from the existing literature.