Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I–V Relations and Critical Potentials. Part I: Analysis

In this work, we analyze a one-dimensional steady-state Poisson–Nernst–Planck type model 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, and only the hard sphere component of the excess (beyond the ideal) electrochemical potential is included. The 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. Our 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 and, treating the sizes as small parameters, we also derive an approximation of the I–V (current–voltage) relation. For this relatively simple situation, it is found that the ion size effect on the I–V relation can go either way—enhance or reduce the current. More precisely, there is a critical potential value Vc so that, if V > Vc, then the ion size enhances the current; if V < Vc, it reduces the current. There is another critical potential value Vc so that, if V > Vc, the current is increasing with respect to λ =  r2/r1 where r1 and r2 are, respectively, the radii of the positively and negatively charged ions; if V < Vc, the current is decreasing in λ. To our knowledge, the existence of these two critical values for the potential was not previously identified.