Simple and accurate expressions for radial distribution functions of hard disk and hard sphere fluids

Hongqin Liu

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arXiv · 2310.12297

Simple and accurate expressions for radial distribution functions of hard disk and hard sphere fluids

Hongqin Liu

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Abstract

Analytical expressions for radial distribution function (RDF) are of critical importance for various applications, such as development of the perturbation theories for equilibrium properties. Theoretically, RDF expressions for odd-dimensional fluids can be obtained by solving the Percus-Yevick integral equations. But for even-dimensional cases, such as the hard disk (2D) fluid, analytical expressions are infeasible. The only 2D RDF is a heuristic expression proposed by Yuste et al. (J. Chem. Phys. 99, 2020,1993), which approximates the 2D RDF with an interpolation of the RDFs for hard-rod (1D) and hard-sphere (3D) fluids and provides acceptable estimations for an intermediate and low density range. In this work, we employ a simple and empirical expression for the 2D RDF and the 3D RDF based on the approach proposed by Trokhymchuk et al. for the 3D RDF (J. Chem. Phys., 123, 024501, 2005). The parameters are determined in such a way that the final RDF expressions are thermodynamically consistent, namely the pressure constraint and the isothermal compressibility constraint are both satisfied. The new RDFs for the 2D and 3D hard spheres are highly accurate for the entire density range up to the first-order phase transition points. The predictions of the first coordination numbers are consistent with simulation results for 3D fluid. Finally, by using the 2D RDF with a primitive second-order perturbation theory, the pressure-volume-temperature relation and vapor-liquid equilibrium are calculated for the 2D Lennard-Jones fluid. Comparisons with the simulation data show promising results.

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Hongqin Liu. 2023-10-18. Simple and accurate expressions for radial distribution functions of hard disk and hard sphere fluids. https://arxiv.org/abs/2310.12297

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