Simple and Robust Solver for the Poisson-Boltzmann Equation

TL;DR

This paper introduces a simple, stable variational numerical method for solving the nonlinear Poisson-Boltzmann equation, validated against analytical solutions and applied to colloidal systems.

Contribution

It develops a new variational approach and an unconditionally stable algorithm for the Poisson-Boltzmann equation, enhancing robustness and simplicity over existing methods.

Findings

Algorithm is simple and unconditionally stable.

Validated with analytical planar solutions.

Applied successfully to colloidal sphere case.

Abstract

A variational approach is used to develop a robust numerical procedure for solving the nonlinear Poisson-Boltzmann equation. Following Maggs et al., we construct an appropriate constrained free energy functional, such that its Euler-Lagrange equations are equivalent to the Poisson-Boltzmann equation. We then develop, implement, and test an algorithm for its numerical minimization, which is quite simple and unconditionally stable. The analytic solution for planar geometry is used for validation. Furthermore, some results are presented for a charged colloidal sphere surrounded by counterions.