A Dynamical Study of the Poisson-Nernst-Planck System for Three Ions

Abstract

Ionic channels and semiconductor devices use atomic scale structures to control macroscopic flows from one reservoir to another. The one-dimensional steady-state Poisson-Nernst-Planck (PNP) system is a useful representation of these devices, but experience shows that describing the reservoirs as boundary conditions is difficult. We study the PNP system for three types of ions with three regions of piecewise constant permanent charge. Reservoirs are represented by the outer regions with permanent charge zero. The PNP system can be treated as singularly perturbed system that has two limiting systems: inner and outer systems (termed fast and slow system in geometric singular perturbation theory). A complete set of integrals for the inner system is presented that provides information for boundary and internal layers. We will examine the effects of permanent charge on the fluxes of the ions through flux ratios, λ(Q) = J(Q)/J(0). J(Q) is the flux associated with nonzero permanent charge, Q, and J(0) will be the flux associated with zero permanent charge.