Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment (Communications in Computational Physics, 2014)
We have developed efficient numerical algorithms for solving 3D steady-statePoisson-Nernst-Planck (PNP) equations with excess chemical potentials describedby the classical density functional theory (cDFT). The coupled PNP equations are discretizedby a finite difference scheme and solved iteratively using the Gummel methodwith relaxation. The Nernst-Planck equations are transformed into Laplace equationsthrough the Slotboom transformation. Then, the algebraic multigrid method isapplied to efficiently solve the Poisson equation and the transformed Nernst-Planckequations. A novel strategy for calculating excess chemical potentials through fastFourier transforms is proposed, which reduces computational complexity from $\mathcal{O}$($N^2$) to $\mathcal{O}$($NlogN$), where $N$ is the number of grid points. Integrals involving the Diracdelta function are evaluated directly by coordinate transformation, which yields moreaccurate results compared to applying numerical quadrature to an approximated deltafunction. Numerical results for ion and electron transport in solid electrolyte for lithium-ion(Li-ion) batteries are shown to be in good agreement with the experimental dataand the results from previous studies.