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

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 (over the ideal gas) free energy functional. Since typically an approximate excess free energy functional is employed, the test-particle route, which requires only one functional derivative, is more accurate than the Ornstein-Zernike route. Here we study a two dimensional core-shoulder particle system and find that in some circumstances the results from the Ornstein-Zernike route can be comparable in accuracy to the test-particle results for $r>σ$, the core diameter. We also examine in detail the asymptotic $r\to\infty$ decay of $g(r)$, finding a variety of possible decay wavelengths at different state points and state points where there is a crossover from one wavelength to a very different one. This behaviour is a signature pointing to the rich phase behaviour of the incipient solid phases.