Abstract
We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck model for ionic flows through membrane channels with fixed boundary ion concentrations and electric potentials. We consider two ion species, one positively charged and one negatively charged, and assume zero permanent charge. Bikerman's local hard-sphere potential is included in the model to account for ion size effects on the ionic flow. The model problem is treated as a boundary value problem of a singularly perturbed differential system. Our analysis is based on the geometric singular perturbation theory but, most importantly, on specific structures of this concrete model. The existence of solutions to the boundary value problem for small ion sizes is established and, treating the ion sizes as small parameters, we also derive approximations of individual fluxes and I-V (current-voltage) relations, from which qualitative properties of ionic flows related to ion sizes are studied. A detailed characterization of complicated interactions among multiple and physically crucial parameters for ionic flows, such as boundary concentrations and potentials, diffusion coefficients and ion sizes, is provided.
Highlights
We study the dynamics of ionic flows, the electrodiffusion of charges, through ion channels via a quasi-one-dimensional steady-state PoissonNernst-Planck (PNP) system
As a basic macroscopic model for electrodiffusion of charges, for ionic flows through ion channels ([8, 10, 15, 16, 17, 18, 19, 26, 27, 31, 38, 39, 62, 64, 72, 73, 74], etc.), under various reasonable conditions, PNP systems can be derived as reduced models from molecular dynamic models ([80]), from Boltzmann equations ([2]), and from variational principles ([34, 36, 37])
The main focus of this paper is to examine the qualitative properties of ion size effects on ionic flows via boundary value problem (BVP) (2.1)-(2.2) with local hard-sphere (LHS) model (2.4)
Summary
We study the dynamics of ionic flows, the electrodiffusion of charges, through ion channels via a quasi-one-dimensional steady-state PoissonNernst-Planck (PNP) system. PNP, Bikerman’s local hard-sphere potential, I-V relations, individual fluxes, ion size effects. This leads to our main interest studied, which contains four subsections.
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