Self-Consistent Ornstein-Zernike Approximation (SCOZA) and exact second virial coefficients and their relationship with critical temperature for colloidal or protein suspensions with short-ranged attractive interactions.

We focus on the second virial coefficient B2 of fluids with molecules interacting through hard-sphere potentials plus very short-ranged attractions, namely, with a range of attraction smaller than half hard-sphere diameter. This kind of interactions is found in colloidal or protein suspensions, while the interest in B2 stems from the relation between this quantity and some other properties of these fluid systems. Since the SCOZA (Self-Consistent Ornstein-Zernike Approximation) integral equation is known to yield accurate thermodynamic and structural predictions even near phase transitions and in the critical region, we investigate B2(SCOZA) and compare it with B2(exact), for some typical potential models. The aim of the paper is however twofold. First, by expanding in powers of density the condition of thermodynamic consistency included in the SCOZA integral equation, a general analytic expression for B2(SCOZA) is derived. For a given potential model, a comparison between B2(SCOZA) and B2(exact) may help to estimate the regimes where the SCOZA closure is reliable. Second, following the Vliegenthart-Lekkerkerker (VL) and Noro-Frenkel suggestions, the relationship between the critical B2 and the critical temperature Tc is discussed in detail for two prototype models: the square-well (SW) potential and the hard-sphere attractive Yukawa (HSY) one. The known simulation data for the SW model are revisited, while for the HSY model new SCOZA results have been generated. Although B2(HSY) at the critical temperature is found to be a slowly varying function of the range of Yukawa attraction ΔY over a wide interval of ΔY, it turns out to diverge as ΔY vanishes. For fluids with very short-ranged attractions, such a behavior contrasts with the VL assumption that B2 at the critical temperature should be nearly independent of the range of attraction. A very simple analytic representation is found for the available Monte Carlo data for Tc(HSY) and B2(HSY) as functions of the range of attraction, for ΔY smaller than half hard-sphere diameter.