Geometric singular approach to Poisson-Nernst-Planck models with excess chemical potentials: Ion size effects on individual fluxes
We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck model for ionic flows through membrane channels. Excess chemical potentials are included in this work to account for finite ion size effects. This is the main difference from the classical Poisson-Nernst-Planck models, which treat ion species as point charges and neglect ion-to-ion interactions. Due to the fact that most experiments (with some exceptions) can only measure the total current while individual fluxes contain much more information on channel functions, our main focus is to study the qualitative properties of ionic flows in terms of individual fluxes under electroneutrality conditions. Our result shows that, in addition to ion sizes, the property depends on multiple physical parameters such as boundary concentrations and potentials, diffusion coe-cients, and ion valences. For the relatively simple setting and assumptions of the model in this paper, we are able to characterize, almost completely, the distinct effects of the nonlinear interplay between these physical parameters. The boundaries of different parameter regions are identified through a number of critical potential values that are explicitly expressed in terms of the physical parameters.Numerical simulations are performed to detect the critical potentials and investigate the quantitative properties of ionic flows over different potential regions. In particular, a special case is studied in Section 5 without the assumption of electroneutrality conditions.