Отправлено: 17.07.15 15:13. Заголовок: Poisson–Nernst–Planck Systems for Ion Flow with Density

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

Received: 22 June 2012 / Revised: 20 September 2012 /

Abstract 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 V c 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.