Poisson-Nernst-Planck systems for ion flow with density functional theory for hard-sphere potential: I-V relations and critical potentials. I: Analysis
type: Item
instance of: scholarly article
title: Poisson-Nernst-Planck systems for ion flow with density functional theory for hard-sphere potential: I-V relations and critical potentials. I: Analysis
zbMATH Open document ID: 1270.93011
author: Shuguan Ji
author: Weishi Liu
published in: Journal of Dynamics and Differential Equations
publication date: 2013-01-30T00:00:00Z
review text: In this paper, the one dimensional steady-state Poisson-Nernst-Planck (PNP) type model is analyzed for ionic flow through a membrane channel including ionic interactions modeled from the Density Functional Theory in a simple setting. Two oppositely charged ion species are involved with electroneutrality boundary conditions and with zero permanent charge. Only the hard sphere component of the excess (beyound the ideal) electrochemical potential is included. The mathematical model can be viewed as a singularly perturbed integro-differential system with a parameter resulting from a dimensionless scaling of the problem as the singular parameter. The performed analysis is a combination of geometric singular perturbation theory and functional analysis. The existence of a solution of the model problem for small ion sizes is established via a fixed-point problem. Further, treating the sizes as small parameters, it is derived an approximation of the I-V (current-voltage) relations. For this relatively simple situation, it is found that the ion size effect on the I-V relations can go either way - enhance or reduce the current. More precisely, it is obtained that there exists a critical potential value \(V_c\) so that, if \(V>V_c\), then the ion size enhances the current and if \(V<V_c\) it reduces the current. Also, it is shown that there exists another critical potential value \(V^c\) so that, if \(V>V^c\) the current is increasing with respect to \(\lambda=\frac{r_2}{r_1}\) (where \(r_1, r_2\) are respectively, the radii of the positively and negatively charged ions) and if \(V<V^c\), the current is decreasing in \(\lambda\). The understanding of the roles of \(V_c\) and \(V^c\) in this paper is at the very early stage and it is challenging for future research.
Mathematics Subject Classification ID: 34B16
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 35J57
Mathematics Subject Classification ID: 35R09
Mathematics Subject Classification ID: 78A37
Mathematics Subject Classification ID: 82B31
Mathematics Subject Classification ID: 92C35
zbMATH DE Number: 6132839
zbMATH Keywords: ion flow
zbMATH Keywords: PNP-DFT
zbMATH Keywords: hard-sphere
zbMATH Keywords: I-V relations
zbMATH Keywords: critical potentials
reviewed by: Vasile Dragan
MaRDI profile type: Publication
full work available at URL: s10884-012-9277-y
OpenAlex ID: W1988689739
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DOI: 10.1007/S10884-012-9277-Y
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description: scientific article; zbMATH DE number 6132839