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Radial Distribution Function in a Two Dimensional Core-Shoulder Particle System

Michael Wassermair, Gerhard Kahl, Andrew J Archer, Roland Roth

Abstract

An important quantity in liquid state theory is the radial distribution function $g(r)$. It can be calculated within the framework of classical density functional theory in two very distinct ways. In the test-particle route, one fixes a single fluid particle, turning it into an external potential in which the inhomogeneous structure of the fluid is calculated by minimising the functional. The second route to $g(r)$ in density functional theory employs the Ornstein-Zernike equation and the pair direct correlation function, that can be obtained from the second functional derivatives of the excess free energy functional. Since typically an approximate excess free energy functional is employed, one generally expects that the test-particle route, which requires only one functional derivative, to be more accurate than the Ornstein-Zernike route. Here we study a two dimensional core-shoulder particle system and present results that challenge this expectation. Our results show that in this system test-particle results for $g(r)$ are not always better than results obtained via the Ornstein-Zernike route.

Radial Distribution Function in a Two Dimensional Core-Shoulder Particle System

Abstract

An important quantity in liquid state theory is the radial distribution function . It can be calculated within the framework of classical density functional theory in two very distinct ways. In the test-particle route, one fixes a single fluid particle, turning it into an external potential in which the inhomogeneous structure of the fluid is calculated by minimising the functional. The second route to in density functional theory employs the Ornstein-Zernike equation and the pair direct correlation function, that can be obtained from the second functional derivatives of the excess free energy functional. Since typically an approximate excess free energy functional is employed, one generally expects that the test-particle route, which requires only one functional derivative, to be more accurate than the Ornstein-Zernike route. Here we study a two dimensional core-shoulder particle system and present results that challenge this expectation. Our results show that in this system test-particle results for are not always better than results obtained via the Ornstein-Zernike route.

Paper Structure

This paper contains 6 sections, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Theory
  3. Results
  4. Discussion and Summary
  5. OZ equation with the RPA closure
  6. The Percus test-particle route with the RPA DFT

Figures (2)

  • Figure 1: RDF $g(r)$ as obtained from the Ornstein-Zernike (OZ) route (blue lines), the test-particle (TP) route (red lines) and from GCMC simulations (black lines) for $\lambda=3.7$ and $T=0.7402$ and $\eta$-values as labelled. For this value of $\lambda$ we find that the results from the test-particle route agree better with the simulation results than those from the Ornstein-Zernike route.
  • Figure 2: RDF $g(r)$ as obtained from the Ornstein-Zernike (OZ) route (blue lines), the test-particle (TP) route (blue lines) and from GCMC simulations (black lines) for $\lambda=4.9$ and $T=1.2285$ and $\eta$-values as labelled. At low and intermediate densities the TP results are surprisingly bad, while the OZ results, except for the violation of the core condition, agree well with the simulation data for all densities considered here.