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This work examines the geometric mean of concentrations and its behavior invarious situations, as well as the reversal permanent charge problem, thecharge sharing seen in x-ray diffraction. Observations are obtained fromanalytical results established using geometric singular perturbation analysisof 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 unequaldiffusion constants to acquire a system for determining the reversal potentialand reversal permanent charge. They studied the reversal potential problem andits dependence on diffusion coefficients, membrane potential, membraneconcentrations, etc. Here we use the same approach to study the dual problem ofreversal permanent charges and its dependence on other conditions. We considertwo ion species with positive and negative charges, say Ca$^+$ and Cl$^-$, todetermine the specific conditions under which the permanent charge is unique.Furthermore, we investigate the behavior of geometric mean of concentrationsfor various values of transmembrane potential and permanent charge.

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