Numerical Approach to Poisson–Nernst–Planck Models with Nonzero Permanent Charges and Steric Effects

We numerically study a quasi-one-dimensional classical Poisson–Nernst–Planck system with two oppositely charged ion species under different setups. Of particular interest is the following: (i) to numerically identify ten critical potentials, which play critical roles in characterizing the qualitative properties of individual fluxes; (ii) to numerically verify some interesting scaling laws related to boundary concentrations of the system; and (iii) to investigate the effects on the individual fluxes from finite ion sizes. Some numerical results support the analytical ones obtained in previous work and provide more intuitive illustrations of the abstract results, while some others provide important insights for future studies; particularly, the scaling law observed with both small permanent charges and finite ion sizes is quite different from previous results.