Nonlinear Electrostatics. The Poisson-Boltzmann Equation

TL;DR

This paper explores nonlinear electrostatics in conducting media, focusing on the Poisson-Boltzmann theory for electrolyte solutions near charged surfaces, providing analytical solutions and discussing inter-wall forces.

Contribution

It clarifies the appropriate free energy for Poisson-Boltzmann theory and derives analytical solutions for various electrolyte configurations.

Findings

Analytical solutions for the Poisson-Boltzmann equation in planar geometries.

Insights into free energy choices depending on boundary conditions.

Analysis of forces between charged walls with overlapping double layers.

Abstract

The description of a conducting medium in thermal equilibrium, such as an electrolyte solution or a plasma, involves nonlinear electrostatics, a subject rarely discussed in the standard electricity and magnetism textbooks. We consider in detail the case of the electrostatic double layer formed by an electrolyte solution near a uniformly charged wall, and we use mean-field or Poisson-Boltzmann (PB) theory to calculate the mean electrostatic potential and the mean ion concentrations, as functions of distance from the wall. PB theory is developed from the Gibbs variational principle for thermal equilibrium of minimizing the system free energy. We clarify the key issue of which free energy (Helmholtz, Gibbs, grand,...) should be used in the Gibbs principle; this turns out to depend not only on the specified conditions in the bulk electrolyte solution (e.g., fixed volume or fixed pressure), but also on the specified surface conditions, such as fixed surface charge or fixed surface potential. Despite its nonlinearity the PB equation for the mean electrostatic potential can be solved analytically for planar or wall geometry, and we present analytic solutions for both a full electrolyte, and for an ionic solution which contains only counterions, i.e. ions of sign opposite to that of the wall charge. This latter case has some novel features. We also use the free energy to discuss the inter-wall forces which arise when the two parallel charged walls are sufficiently close to permit their double layers to overlap. We consider situations where the two walls carry equal charges, and where they carry equal and opposite charges.