Finite domain effects in steady-state solutions of Poisson-Nernst-Planck equations

TL;DR

This paper analyzes how finite domain sizes influence steady-state solutions of the Poisson-Nernst-Planck equations, revealing significant effects even in large domains and providing an asymptotic approximation with minimal error.

Contribution

It introduces an asymptotic matching method to approximate solutions of the PNP equations in large but finite domains, highlighting boundary layer sensitivities.

Findings

Finite domain effects are significant even for large domains.

Boundary layer structures are sensitive to domain size.

Approximate solutions with exponentially small errors are derived.

Abstract

Steady-state solutions of the Poisson-Nernst-Planck model are studied in the asymptotic limit of large, but finite domains. By using asymptotic matching for integrals, we derive an approximate solution for the steady-state equation with exponentially small error with respect to the domain size. The approximation is used to quantify the extent of finite domain effects over the full parameter space. Surprisingly, already for small applied voltages (several thermal voltages), we found that finite domain effects are significant even for large domains (on the scale of hundreds of Debye lengths). Namely, the solution near the boundary, i.e., the boundary layer (electric double layer) structure, is sensitive to the domain size even when the domain size is many times larger than the characteristic width of the boundary layer. We focus on this intermediate regime between confined domains and `essentially infinite' domains, and study how the domain size effects the solution properties. We conclude by providing an outlook to higher dimensions with applications to ion channels and porous electrodes.