Direct numerical simulations of the modified Poisson-Nernst-Planck equations for the charging dynamics of cylindrical electrolyte-filled pores

TL;DR

This study uses finite element simulations of modified Poisson-Nernst-Planck equations to analyze charging timescales in cylindrical electrolyte pores, validating theoretical models across key dimensionless parameters.

Contribution

It provides a numerical validation of existing theoretical predictions for pore charging dynamics using detailed finite element simulations.

Findings

Charging timescales depend on applied potential, pore-to-reservoir resistance ratio, and pore radius to Debye length ratio.

The numerical results delineate the validity range of existing theories.

The study highlights the importance of specific dimensionless parameters in electrochemical pore charging.

Abstract

Understanding how electrolyte-filled porous electrodes respond to an applied potential is important to many electrochemical technologies. Here, we consider a model supercapacitor of two blocking cylindrical pores on either side of a cylindrical electrolyte reservoir. A stepwise potential difference $2\Phi$ between the pores drives ionic fluxes in the setup, which we study through the modified Poisson-Nernst-Planck equations, solved with finite elements. We focus our discussion on the dominant timescales with which the pores charge and how these timescales depend on three dimensionless numbers. Next to the dimensionless applied potential $\Phi$, we consider the ratio $R/R_b$ of the pore's resistance $R$ to the bulk reservoir resistance $R_b$ and the ratio $r_{p}/\lambda$ of the pore radius $r_p$ to the Debye length $\lambda$. We compare our data to theoretical predictions by Aslyamov and Janssen ($\Phi$), Posey and Morozumi ($R/R_b$), and Henrique, Zuk, and Gupta ($r_{p}/\lambda$). Through our numerical approach, we delineate the validity of these theories and the assumptions on which they were based.