Approximate Analytic Expression of Surface Charge Density/Surface Potential Relationship for a Spherical Colloidal Particle Immersed in a General Electrolyte Solution

Abstract

We implement a variational iteration method for solving a Poisson–Boltzmann equation (PBE) describing a spherical colloid immersed in a general electrolyte solution. In this method, a general Lagrange multiplier is introduced to construct correction functional for the problem, and the multiplier is identified optimally via a variational method; a linearization solution of the original PBE is chosen as an initial approximation for the iteration. To proceed with the iteration analytically, we approximate the highly nonlinear term of the PBE by a polynomial of proper order. Due to presence of an exponent integral function in the first-order iteration solution, higher-order iteration solutions are analytically unavailable currently. Based on the first-order iteration solution, analytic expression for surface charge density/surface potential relationship is acquired. The present analytic solution contrasts sharply with previous ones by two striking features: (1) the present one surpasses well beyond previous approximate solutions in that general electrolyte type can be dealt with in a unified way, and (2) valid application scope of the present expression is in small κa domain and thus is complementary with those of previous ones.