Criticality in the hard-sphere ionic fluid

A physically based mean-field theory of criticality and phase separation in the restricted primitive model of an electrolyte (hard spheres of diameter a carrying charges ± q) is developed on the basis of the Debye-Hückel (DH) approach. Simple DH theory yields a critical point at T∗ ≡ kBTa/q2 = 116, which is only about 15% above the best recent simulation estimates (Tc,sim∗ = 0.052–0.056) but a critical density (ϱc∗ ≡ ϱca3 = 164π ⋍ 0.005 that is much too small (ϱc,sim∗ = 0.023–0.035). Allowing for hard-core exclusion effects reduces these values slightly. However, correction of the DH linearization of the Poisson-Boltzmann equation by including pairing of + and − charges improves ϱc∗ significantly. Bjerrum's theory of the (required) association constant is revisited critically; Ebeling's reformulation is strongly endorsed but makes negligible numerical difference at criticality and below. The nature and size of the associated, dipolar ion pairs is examined quantitatively and their solvation free-energy in the residual fluid of free ions is calculated on the basis of DH theory. This contribution to the total free energy proves crucial and leads to a rather satisfactory description of the critical region. The temperature variation of the vapor pressure and of the density of neural dipolar pairs correlates fairly well with Gillan's numerical cluster analysis. Possible improvements to allow for larger ion clusters and to better represent the denser ionic liquid below criticality are discussed. Finally, the replacement of the DH approximation for the ionic free energy by the mean spherical approximation is studied. Reasonable critical densities are generated but the MSA critical temperatures are all 40–50% too high; in addition, the predicted density of neutral clusters seems much too low near criticality and, along with the vapor pressure, appears to decrease too rapidly by an exponential factor below Tc.