  In this paper, based on geometric singular perturbation analysis of a
quasi-one dimensional Poisson-Nernst-Planck model for ionic flows, we study the
problem of zero current condition for ionic flows through membrane channels
with a simple profile of permanent charges.
  For ionic mixtures of multiple ion species, under equal diffusion constant
condition, Eisenberg, et al [{\em Nonlinearity {\bf 28} (2015), 103-128}]
derived a system of two equations for determining the reversal potential and an
equation for the reversal permanent charge. The equal diffusion constant
condition is significantly degenerate from physical points of view. For unequal
diffusion coefficients, the analysis becomes extremely challenging. This work
will focus only on two ion species, one positively charged (cation) and one
negatively charged (anion), with two arbitrary diffusion coefficients.
Dependence of reversal potential on channel geometry and diffusion coefficients
has been investigated experimentally, numerically, and analytically in simple
setups, in many works. In this paper, we identify two governing equations for
the zero current, which enable one to mathematically analyze how the reversal
potential depends on the channel structure and diffusion coefficients. In
particular, we are able to show, with a number of concrete results, that the
possible different diffusion constants indeed make significant differences. The
inclusion of channel structures is also far beyond the situation where the
Goldman-Hodgkin-Katz (GHK) equation might be applicable. A comparison of our
result with the GHK equation is provided. The dual problem of reversal
permanent charges is briefly discussed too.
