Electric field based Poisson-Boltzmann: Treating mobile charge as polarization

TL;DR

This paper reformulates the Poisson-Boltzmann equation using an electric field-based approach that treats mobile charges as polarization, leading to a convex functional suitable for variational analysis of electrostatic interactions in electrolytes.

Contribution

It introduces a novel variational formulation of the Poisson-Boltzmann theory incorporating dielectric polarization and electric field energy density, enabling explicit treatment of screening effects.

Findings

The functional is convex, facilitating stable numerical solutions.

The approach explicitly accounts for ionic and solvent polarization effects.

Maxwell equations are integrated via a vector potential in the variational framework.

Abstract

Mobile charge in an electrolytic solution can in principle be represented as the divergence of ionic polarization. After adding explicit solvent polarization a finite volume of electrolyte can then be treated as a composite non-uniform dielectric body. Writing the electrostatic interactions as an integral over electric field energy density we show that the Poisson-Boltzmann functional in this formulation is convex and can be used to derive the equilibrium equations for electric potential and ion concentration by a variational procedure developed by Ericksen for dielectric continua (Arch. Rational Mech. Anal. 2007, 183, 299-313). The Maxwell field equations are enforced by extending the set of variational parameters by a vector potential representing the dielectric displacement which is fully transverse in a dielectric system without embedded external charge. The electric field energy density in this representation is a function of the vector potential and the sum of ionic and solvent polarization making the mutual screening explicit.