Non-localness of excess potentials and boundary value problems of Poisson-Nernst-Planck systems for ionic flow: a case study
In this paper, a form of natural boundary conditions is presented, so that the corresponding boundary value problems (BVP) of Poisson-Nernst-Planck (PNP) type systems with nonlocal excess potentials are generally well-posed. PNP type systems are basic primitive models for ionic flow through ion channels. Important properties of ion channels, such as current-voltage relations, permeation and selectivity, can be extracted from solutions of BVP of PNP type models. Many issues of BVP of PNP type systems with local excess potentials (including particularly classical PNP systems that treat ions as point-charges) are extensively examined analytically and numerically. On the other hand, for PNP type systems with nonlocal excess potentials, even the issue of well-posedness of BVP is poorly understood. In fact, the formulation of correct boundary conditions seems to be overlooked, even though complications of ionic behavior near the boundaries (locations of applied electrodes) have been long experienced in experiments and simulations. PNP type systems with nonlocal excess potentials can be viewed as functional differential systems and, for many approximation models of nonlocal excess potentials, as differential equations with both delays and advances. Thus, PNP type systems with nonlocal excess potentials have infinite degree of freedoms and BVP with the traditional two-point-boundary-conditions would be severely under determined. The mathematical theory for PNP with nonlocal excess potential would be significantly different from that for PNP with local excess potentials. This work, at an early stage toward a better understanding of related issues, provides some insights on interpretations of experimental designs of imposing boundary conditions and for correct formulations of numerical simulations.
Poisson-Nernst-Planck (PNP) type systems
scientific article; zbMATH DE number 6921348
scientific article