Asymptotic Analysis on Dielectric Boundary Effects of Modified Poisson--Nernst--Planck Equations

The charge transport in an environment with inhomogeneous dielectric permittivity is ubiquitous in many areas such as electrochemical energy devices and biophysical systems. We theoretically study the equilibrium and dynamics of electrolytes between two blocking electrodes based on a modified Poisson--Nernst--Planck model with the dielectric boundary effect. Matched asymptotic analysis shows that a two-layer interfacial structure exists in the vicinity of the interfaces when the dielectric self-energy correction to the potential mean force is relatively weak. For this two-layer structured solution, the dielectric effect plays the dominate role in the first layer, while the solution in the second layer is mainly determined by the classical Poisson--Boltzmann equation. When the dielectric self energy becomes stronger, there is only one interfacial layer which is governed by the modified Poisson--Boltzmann equation with the dielectric self-energy correction in the Boltzmann factor. We perform a systematic in...