Effective charge of cylindrical and spherical colloids immersed in an electrolyte: the quasi-planar limit

TL;DR

This paper develops a new analytical expansion scheme for the non-linear Poisson-Boltzmann theory, providing exact results for the effective charge of cylindrical and spherical colloids in electrolytes when the macro-ion size is large compared to the Debye length.

Contribution

It introduces a hidden structure in the electrostatic potential and a resummation technique that yields novel exact results for colloid effective charge in the quasi-planar limit.

Findings

New expansion scheme for electrostatic potential

Exact effective charge results for large macro-ion radius

Agreement with numerical solutions

Abstract

We consider the non-linear Poisson-Boltzmann theory for a single cylindrical or spherical macro-ion in symmetric 1:1, together with asymmetric 1:2 and 2:1 electrolytes. We focus on the regime where $\kappa a $, the ratio of the macro-ion radius $a$ over the inverse Debye length in the bulk electrolyte, is large. Analyzing the structure of the analytical expansion emerging from a multiple scale analysis, we uncover a hidden structure for the electrostatic potential. This structure, which appears after a heuristic resummation, suggests a new and convenient expansion scheme that we present and work out in detail. We show that novel exact results can thereby be obtained, in particular pertaining to effective charge properties, in complete agreement with the direct numerical solution to the problem.