Boundary Layer Solutions of Charge Conserving Poisson-Boltzmann Equations: One-Dimensional Case

TL;DR

This paper rigorously analyzes boundary layer solutions of charge conserving Poisson-Boltzmann equations in one dimension, revealing explicit formulas for electric potentials near boundaries and interior points, with applications to biological and physical systems.

Contribution

It provides the first rigorous asymptotic analysis of 1D boundary layer solutions for CCPB equations with explicit formulas for potential limits.

Findings

Asymptotic behaviors of solutions are rigorously proven.

Explicit nonlinear formulas determine potential gaps at boundaries.

Formulas are numerically solvable and differ from classical PB equations.

Abstract

For multispecies ions, we study boundary layer solutions of charge conserving Poisson-Boltzmann (CCPB) equations [50] (with a small parameter \k{o}) over a finite one-dimensional (1D) spatial domain, subjected to Robin type boundary conditions with variable coefficients. Hereafter, 1D boundary layer solutions mean that as \k{o} approaches zero, the profiles of solutions form boundary layers near boundary points and become flat in the interior domain. These solutions are related to electric double layers with many applications in biology and physics. We rigorously prove the asymptotic behaviors of 1D boundary layer solutions at interior and boundary points. The asymptotic limits of the solution values(electric potentials) at interior and boundary points with a potential gap (related to zeta potential) are uniquely determined by explicit nonlinear formulas (cannot be found in classical Poisson-Boltzmann equations) which are solvable by numerical computations.