Solution to the Poisson–Boltzmann equation for particles covered by an ion-penetrable charged membrane

Abstract

The Poisson-Boltzmann equation governing the distribution of electrical potential of a planar, cylindrical or spherical particle covered by a charged, ion-penetrable membrane immersed in an a : b electrolyte solution has been analysed. Approximate analytical expressions for the distribution of electrical potential as well as its derivative have been derived. The former is readily applicable for the evaluation of the total free energy of the system under consideration. We have shown that in the limiting case, the present result reduces to that for a rigid, charged-surface model.