Analytical solution to the Poisson-Nernst-Planck equations for the charging of a long electrolyte-filled slit pore

TL;DR

This paper provides an analytical solution to the Poisson-Nernst-Planck equations describing the charging dynamics of a long electrolyte-filled slit pore, revealing detailed behavior at different potential regimes and confirming prior numerical findings.

Contribution

It introduces an analytical model for the PNP equations in a long slit pore, explaining charge buildup dynamics at various potentials and validating the transmission line model predictions.

Findings

Analytical solution matches the transmission line model at small potentials.

Reproduces biexponential charge buildup at moderate to high potentials.

Provides insight into the slow charging dynamics at late times.

Abstract

We study the charging dynamics of a long electrolyte-filled slit pore in response to a suddenly applied potential. In particular, we analytically solve the Poisson-Nernst-Planck (PNP) equations for a pore for which $\lambda_D\ll H\ll L$, with $\lambda_D$ the Debye length and $H$ and $L$ the pore's width and length. For small applied potentials, we find the time-dependent potential drop between the pore's surface and its center to be in complete agreement with a prediction of the celebrated transmission line model. For moderate to high applied potentials, prior numerical work showed that charging slows down at late times; Our analytical model reproduces and explains such biexponential charge buildup.