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Geometric Mean of Concentrations and Reversal Permanent Charge in Zero-Current Ionic Flows via Poisson-Nernst-Planck Models

This work examines the geometric mean of concentrations and its behavior in various situations, as well as the reversal permanent charge problem, the charge sharing seen in x-ray diffraction. Observations are obtained from analytical results established using geometric singular perturbation analysis of classical Poisson-Nernst-Planck models. For ionic mixtures of multiple ion species Mofidi and Liu [{\em SIAM J. Appl. Math. {\bf 80} (2020), 1908-1935}] centered two ion species with unequal diffusion constants to acquire a system for determining the reversal potential and reversal permanent charge. They studied the reversal potential problem and its dependence on diffusion coefficients, membrane potential, membrane concentrations, etc. Here we use the same approach to study the dual problem of reversal permanent charges and its dependence on other conditions. We consider two ion species with positive and negative charges, say Ca$^+$ and Cl$^-$, to determine the specific conditions under which the permanent charge is unique. Furthermore, we investigate the behavior of geometric mean of concentrations for various values of transmembrane potential and permanent charge.