Phase coexistence in colloidal suspensions: an analytic Poisson-Boltzmann treatment

We solve the linearized Poisson-Boltzmann equation analytically, subject to justifiable approximations, for a suspension containing a large number of identical spherical macroions under conditions of constant surface charge and zero added salt, in order to investigate the phase behavior of charge-stabilized colloidal suspensions. Our results for the electrostatic part of the Helmholtz free energy lead to an interaction which resembles the intermolecular interaction in the theory of molecular fluids. When combined with the ideal gas free energy of the counterions, this produces a van der Waals loop in the pV diagram, indicating coexistence between phases with different densities, for certain values of the macroion radius and charge. We also derive an expression for the surface potential of the macroions, and clarify the interpretation of the Poisson-Boltzmann equation.