Poisson-Boltzmann thermodynamics of counter-ions confined by curved hard walls

TL;DR

This paper develops a theoretical framework using Poisson-Boltzmann theory to analyze the thermodynamics of counter-ions confined by curved hard walls in various geometries, including a sphere, providing exact solutions for contact densities.

Contribution

It introduces a novel differential equation approach to solve the 3D sphere problem, extending the understanding of counter-ion distributions in curved geometries within mean-field theory.

Findings

Exact series solutions for contact density at small and large surface charges.

Complete thermodynamic description of counter-ions inside a charged sphere.

Extension of solutions to geometries beyond parallel plates and cylinders.

Abstract

We consider a set of identical mobile point-like charges (counter-ions) confined to a domain with curved hard walls carrying a uniform fixed surface charge density, the system as a whole being electroneutral. Three domain geometries are considered: a pair of parallel plates, the cylinder and the sphere. The particle system in thermal equilibrium is assumed to be described by the nonlinear Poisson-Boltzmann theory. While the effectively 1D plates and the 2D cylinder have already been solved, the 3D sphere problem is not integrable. It is shown that the contact density of particles at the charged surface is determined by a first-order Abel differential equation of the second kind which is a counterpart of Enig's equation in the critical theory of gravitation and combustion/explosion. This equation enables us to construct the exact series solutions of the contact density in the regions of small and large surface charge densities. The formalism provides, within the mean-field Poisson-Boltzmann framework, the complete thermodynamics of counter-ions inside a charged sphere (salt-free system).