Approximate analytic solution of the Poisson–Boltzmann equation for a spherical colloidal particle

Abstract

A novel perturbation scheme is developed to solve the Poisson–Boltzmann equation for the potential profile around a spherical colloidal particle of radius a in z-z electrolyte, suitable for values of κa in the intermediate regime 1 < κa < 15, where the Debye–Hückel linearized solution or the planar solution are not good approximations. The zero'th order solution is shown to be a better approximation to the exact numerical solution than the Debye–Hückel solution for κa > 1 for all values of the surface potential. The first order solution is also exhibited and the range of validity of the approximate solution extended into the small κa regime. The surface charge and Helmholtz free energy per unit area as functions of κa and surface potential are also derived and compared with the exact numerical results. The method developed herein is a general one capable of application to the case of unsymmetrical electrolytes and other geometries. As an example of this generality, the zero'th and first order solutions for the Poisson–Boltzmann equation in cylindrical geometry are displayed.