Liquid-liquid and liquid-solid phase separation and flocculation for a charged colloidal dispersion

We model the intercolloidal interaction by a hard-sphere Yukawa repulsion to which is added the long-range van der Waals attraction. In comparison with the Derjaguin-Landau-Verwey-Overbeek repulsion, the Yukawa repulsion explicitly incorporates the spatial correlations between colloids and small ions. As a result, the repulsive part can be expressed analytically and has a coupling strength depending on the colloidal volume fraction. By use of this two-body potential of mean force and in conjunction with a second-order thermodynamic perturbation theory, we construct the colloidal Helmholtz free energy and use it to calculate the thermodynamic quantities, pressure and chemical potential, needed in the determination of the liquid-liquid and liquid-solid phase diagrams. We examine, in an aqueous charged colloidal dispersion, the effects of the Hamaker constant and particle size on the conformation of a stable liquid-liquid phase transition calculated with respect to the liquid-solid coexistence phases. We find that there exists a threshold Hamaker constant or particle size whose value demarcates the stable liquid-liquid coexistence phases from their metastable counterparts. Applying the same technique and using the energetic criterion, we extend our calculations to study the flocculation phenomenon in aqueous charged colloids. Here, we pay due attention to determining the loci of a stability curve stipulated for a given temperature T₀, and obtain the parametric phase diagram of the Hamaker constant vs the coupling strength or, at given surface potential, the particle size. By imposing T₀ to be the critical temperature T_c, i.e., setting k_BT₀ (=k_BT_c) equal to a reasonable potential barrier, we arrive at the stability curve that marks the irreversible ⇆ reversible phase transition. The interesting result is that there occurs a minimum size for the colloidal particles below (above) which the colloidal dispersion is driven to an irreversible (reversible) phase transition.