Near- and far-field expansions for stationary solutions of Poisson--Nernst--Planck equations

TL;DR

This paper analyzes the stationary Poisson--Nernst--Planck equations with a large parameter, deriving asymptotic expansions and boundary phenomena to better understand ion distributions near charged surfaces.

Contribution

It introduces refined near- and far-field asymptotic expansions for solutions and compares a nonlocal Poisson--Boltzmann model with the classical model.

Findings

Solution blows up near boundary in thin region

Boundary concentration phenomenon confirmed

Comparison between Poisson--Nernst--Planck and Poisson--Boltzmann models

Abstract

This work is concerned with the stationary Poisson--Nernst--Planck equation with a large parameter which describes a huge number of ions occupying an electrolytic region. Firstly, we focus on the model with a single specie of positive charges in one-dimensional bounded domains due to the assumption that these ions are transported in the same direction along a tubular-like mircodomain. We show that the solution asymptotically blows up in a thin region attached to the boundary, and establish the refined "near-field" and "far-field" expansions for the solutions with respect to the parameter. Moreover, we obtain the boundary concentration phenomenon of the net charge density, which mathematically confirms the physical description that the non-neutral phenomenon occurs near the charged surface. In addition, we revisit a nonlocal Poisson--Boltzmann model for monovalent binary ions and establish a novel comparison for these two models.