Electric double layer interaction of ionizable surfaces: Charge regulation for arbitrary potentials

Abstract

Electrostatic interactions between charged surfaces across an electrolyte solution are commonly described by boundary conditions of constant charge or constant surface potential. These two extremes are in general not appropriate for an equilibrium description of materials with ionizable surface groups, but do provide an upper and lower bound for the interaction energy. We propose a quantitative criterion which permits to evaluate the degree of charge regulation for surfaces with arbitrary electrostatic potential, as considered in the Poisson–Boltzmann theory. Our approach represents a generalization of the linearized regulation model proposed earlier for the framework of Debye–Hückel theory only. Like in the case of low surface potentials, the regulation behavior is generally determined by the competition between the capacities of the diffuse and the compact part of the electric double layer. Our results suggest a new way of using the limiting conditions of constant charge and constant potential to approximate the interaction free energy of charge regulating surfaces.