Analysis of the hypernetted chain equation for ionic fluids

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Abstract
It is well known that the numerical solution of the hypernetted chain (HNC) equations yields satisfactory results for the pair correlation function of the primitive model of electrolytes and similar models of ionic particles over a considerable range of thermodynamic states. Despite this, it has become apparent that for low densities (or low ionic concentration in electrolytes) the numerical solution breaks down for temperatures well above the expected coexistence region between gas and liquid phases. Here we study the situation by analytic means, comparing it to a similar problem for sticky hard spheres in the Percus-Yevick (PY) approximation. On the basis of our analysis we conclude that the failure of the HNC is of the same nature and is connected to the existence of two possible solutions for low densities. When the temperature is lowered these solutions will merge into one at a particular temperature, below which a real solution is no longer possible. By extending our analysis to systems like the monoatomic Lennard-Jones fluid and comparing with previous results of Gallerani, Lo Vecchio and Reatto for two-Yukawa and Lennard-Jones systems in the PY approximation, we conclude that these are general common features of the HNC and PY approximations in the low-density regime. Numerically they appear to persist in the HNC closure at high densities as well (in contrast to the behaviour of the PY approximation) although our analysis is silent in this regard. Our analysis is consistent with the results of a recent comprehensive numerical study by Belloni.