New porous medium Poisson-Nernst-Planck equations for strongly oscillating electric potentials

TL;DR

This paper derives effective macroscopic equations for ionic transport in porous media with contrasting electric permittivities, accounting for strongly oscillating electric potentials through a novel asymptotic expansion, revealing new transport phenomena.

Contribution

It introduces a modified asymptotic multiple-scale expansion to derive new porous medium Poisson-Nernst-Planck equations that incorporate microscale effects into macroscale models.

Findings

Derivation of effective equations with a new transport term

Rigorous verification of solvability of the equations

Insights into microscale influence on macroscale ionic transport

Abstract

We consider the Poisson-Nernst-Planck system which is well-accepted for describing dilute electrolytes as well as transport of charged species in homogeneous environments. Here, we study these equations in porous media whose electric permittivities show a contrast compared to the electric permittivity of the electrolyte phase. Our main result is the derivation of convenient low-dimensional equations, that is, of effective macroscopic porous media Poisson-Nernst-Planck equations, which reliably describe ionic transport. The contrast in the electric permittivities between liquid and solid phase and the heterogeneity of the porous medium induce strongly oscillating electric potentials (fields). In order to account for this special physical scenario, we introduce a modified asymptotic multiple-scale expansion which takes advantage of the nonlinearly coupled structure of the ionic transport equations. This allows for a systematic upscaling resulting in a new effective porous medium formulation which shows a new transport term on the macroscale. Solvability of all arising equations is rigorously verified. This emergence of a new transport term indicates promising physical insights into the influence of the microscale material properties on the macroscale. Hence, systematic upscaling strategies provide a source and a prospective tool to capitalize intrinsic scale effects for scientific, engineering, and industrial applications.