Exact Solution to the Linearized Poisson–Boltzmann Equation for Spheroidal Surfaces

Abstract

The Poisson–Boltzmann equation governing the electrostatic potential distribution around charged spheroidal surfaces immersed in an electrolyte solution is derived. We show that if the surface potential is low, the linearized Poisson–Boltzmann equation can be solved exactly. Both constant surface potential and constant amount of surface charges are considered, the latter includes conductive and nonconductive surfaces. The results of numerical simulation reveal that for a spheroidal surface of constant potential the distance between the equal-potential surface to the charged surface is inversely proportional to the curvature of the latter. In other words, the thickness of the double layer is position dependent. For a nonconductive surface, the surface potential is inversely proportional to the curvature, and for a conductive surface, the charge density is proportional to the curvature of the surface. The present analysis is appropriate for a relatively thick double layer and/or a small focus.