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The Poisson--Nernst--Planck (PNP) equations have been widely applied todescribe ionic transport in ion channels, nanofluidic devices, and manyelectrochemical systems. Despite their wide applications, the PNP equationsfail in predicting dynamics and equilibrium states of ionic concentrations inconfined environments, due to the ignorance of the excluded volume effect. Inthis work, a simple but effective modified PNP (MPNP) model with the excludedvolume effect is derived, based on a modification of diffusion coefficients ofions. At the steady state, a modified Poisson--Boltzmann (MPB) equation isobtained with the help of the Lambert-W special function. The existence anduniqueness of a weak solution to the MPB equation are established. Furtheranalysis on the limit of weak and strong electrostatic potential leads to twomodified Debye screening lengths, respectively. A numerical scheme thatconserves total ionic concentration and satisfies energy dissipation isdeveloped for the MPNP model. Numerical analysis is performed to prove that ourscheme respects ionic mass conservation and satisfies a corresponding discretefree energy dissipation law. Positivity of numerical solutions is alsodiscussed and numerically investigated. Numerical tests are conducted todemonstrate that the scheme is of second-order accurate in spatialdiscretization and has expected properties. Extensive numerical simulationsreveal that the excluded volume effect has pronounced impacts on the dynamicsof ionic concentration and flux. In addition, the effect of volume exclusion onthe timescales of charge diffusion is systematically investigated by studyingthe evolution of free energies and diffuse charges.

A Modified Poisson--Nernst--Planck Model with Excluded Volume Effect: Theory and Numerical Implementation