Interaction between Charged Surfaces on the Poisson−Boltzmann Level:  The Constant Regulation Approximation

Interaction forces between ionizable surfaces across an electrolyte solution on the Poisson−Boltzmann level are discussed within the constant regulation approximation. The chemical response of each surface is expressed in terms of two parameters, namely, the diffuse layer potential and the regulation parameter p. Both parameters are easily available because they arise naturally within classical equilibrium models for a single noninteracting surface. This approximation, thus, eliminates the need to treat the more intricate problem of two chemical adsorption equilibria coupled to the overlapping double layers between the surfaces. The ensuing simplicity makes this approach extremely versatile for the analysis of experimental data. The classical boundary condition of constant potential corresponds to p = 0, and that of constant charge corresponds to p = 1. While this approximation is rigorously correct at large separations, we find that it remains excellent down to contact in many realistic situations, such as in symmetric or asymmetric systems involving metal oxides or silica described by the 1-pK basic Stern model