Asymptotic Preserving and Accurate scheme for Multiscale Poisson-Nernst-Planck (MPNP) system

TL;DR

This paper introduces a new asymptotic preserving scheme for a multiscale Poisson-Nernst-Planck system modeling the correlated motion of positive and negative ions near surface traps, validated through asymptotic analysis.

Contribution

It extends previous models to include both ion species simultaneously, capturing their Coulomb interaction and surface trap effects with an asymptotic preserving numerical scheme.

Findings

Validated the scheme through asymptotic analysis.

Accurately modeled ion interactions near surface traps.

Extended the model to include both positive and negative ions.

Abstract

In this paper, we propose and validate a two-species Multiscale model for a Poisson-Nernst-Planck (PNP) system, focusing on the correlated motion of positive and negative ions under the influence of a trap. Specifically, we aim to model surface traps whose attraction range, of length $\delta$, is much smaller then the scale of the problem. The physical setup we refer to is an anchored gas drop (bubble) surrounded by a flow of charged surfactants {(composed by positive and negative ions) that diffuses in water. When the diffusing surfactants reach the surface of the trap, the negative ions are adsorbed because of their hydrophobic tail that is attracted by the air bubble}. As in our previous works, the effect of the attractive potential is replaced by a suitable boundary condition derived by mass conservation and asymptotic analysis. The novelty of this work is the extension of the model proposed in \cite{astuto2023multiscale}, now incorporating the influence of both carriers -- positive and negative ions -- simultaneously, which is often neglected in traditional approaches that treat ion species independently. The two carriers interact through the Coulomb potential, that is computed by a Poisson equation. [...]