Poisson-Nernst-Planck systems for ion flow with density functional theory for hard-sphere potential: I-V relations and critical potentials. I: Analysis
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.
scientific article; zbMATH DE number 6132839
scientific article