Perturbation solutions for the nonlinear Poisson–Boltzmann equation with a high-order-accuracy Debye–Hückel approximation

Abstract

The Poisson-Boltzmann (P-B) equation is of fundamental importance in understanding solid-liquid electrolyte interfaces that are present in many fields. Due to the nonlinearity, it is usually challenging to find the explicit exact solutions of the P-B equation. The present work reports several perturbation solutions for the nonlinear P-B equation in the Cartesian and spherical coordinates. The new solutions contain a perturbation parameter from the high-order-accuracy approximation of the hyperbolic sine function and thus can apply to high zeta potential conditions. The comparison of the perturbation solutions with the traditional Debye-Hückel solutions and the full numerical solutions validates the robustness and accuracy of the perturbation solutions. The perturbation solutions are explicit and analytical and then can be used for a fast calculation of the EDL potential and interaction energy in versatile applications.