  We propose a modified Poisson-Nernst-Planck (PNP) model to investigate charge
transport in electrolytes of inhomogeneous dielectric environment. The model
includes the ionic polarization due to the dielectric inhomogeneity and the
ion-ion correlation. This is achieved by the self energy of test ions through
solving a generalized Debye-H\"uckel (DH) equation. We develop numerical
methods for the system composed of the PNP and DH equations. Particularly,
towards the numerical challenge of solving the high-dimensional DH equation, we
developed an analytical WKB approximation and a numerical approach based on the
selective inversion of sparse matrices. The model and numerical methods are
validated by simulating the charge diffusion in electrolytes between two
electrodes, for which effects of dielectrics and correlation are investigated
by comparing the results with the prediction by the classical PNP theory. We
find that, at the length scale of the interface separation comparable to the
Bjerrum length, the results of the modified equations are significantly
different from the classical PNP predictions mostly due to the dielectric
effect. It is also shown that when the ion self energy is in weak or mediate
strength, the WKB approximation presents a high accuracy, compared to precise
finite-difference results.
