Critical behavior of the hypernetted-chain equation for a Lennard-Jones mixture

The hypernetted-chain (HNC) approximation is applied to study the critical behavior of a Lennard-Jones mixture. The parameters of the model potential are appropriate to describe a He-Xe mixture in the thermodynamic region where ‘‘gas-gas phase separation’’ is known to occur. The locus of points in the temperature-concentration plane where the k→0 limit of the concentration-concentration structure factor $S_{\mathrm{cc}}$(k) diverges, i.e., the spinodal line, is determined through an extrapolation of the HNC results from the region where this integral equation is solvable, toward the phase stability boundary, where the algorithm becomes highly unstable and no solution can be found. The extrapolation is based on a power-law behavior of $S_{\mathrm{cc}}$(0) that is verified to hold in the realm of the available results. The critical temperature and concentration so obtained compare quite favorably with the experimental data.