Numerical Solutions of the Convolution-Hypernetted Chain Integral Equation for the Pair Correlation Function of a Fluid. I. The Lennard-Jones (12, 6) Potential

Abstract

The integral equation of the convolution-hypernetted chain approximation is solved numerically at five temperatures for the Lennard-Jones (12, 6) potential. Due to an inconsistency in the approximation there are two equations of state associated with it. These are compared with each other, with other theories, and with the experimental argon equation of state. At high temperatures, the two equations of state agree both with experiment and with more a priori theory at low densities and bracket them at higher densities, their mean giving a good representation of both. The integral equation is found to be singular at certain temperature—density points, their locus forming a dome shaped curve in the T—ρ plane. These points are shown to correspond to the limits of metastability in the van der Waals gas, the highest temperature point being the critical point. The nonsingular points below the critical point correspond either to a homogeneous gas (low densities) or liquid (high densities) phase. The former are in good agreement with the three-term virial series, as expected, and the latter are a considerable improvement over it. The equation of state is a considerable improvement over that obtained from the Born—Green equation by Kirkwood, Lewinson, and Alder, the latter predicting negative pressures over most of the liquid range while these results show only positive pressures. The comparisons with experiment are obscured by the inadequacy of the (12, 6) potential function. The use of temperature ``compensated'' potential parameters (as determined by the second virial coefficient) indicates that the use of a more reasonable potential function in the theory will give a relatively good representation of the equation of state of a fluid over a wide range of temperatures and densities.