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We derive a set of modified Poisson-Nernst-Planck (mPNP) equations for iontransport from the variation of the free energy functional which includes themany-body Coulomb correlation in media of variable dielectric coefficient. Thecorrelation effects are considered through the Debye charging process in whichthe self energy of an ion is governed by the generalized Debye-H\"uckelequation. We develop the asymptotic expansions of the self energy taking theion radius as the small parameter such that the multiscale model can be solvedefficiently by numerical methods. It is shown that the variations of the energyfunctional give the self-energy-modified PNP equations which satisfy a properweak formulation. We present numerical results from different asymptoticexpansions with a semi-implicit conservative numerical method and investigatethe effect of the Coulomb correlation.

Modified Poisson-Nernst-Planck model with accurate Coulomb correlation in variable media