Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment

Abstract

Abstract.We have developed efficient numerical algorithms for solving 3D steadystate Poisson-Nernst-Planck (PNP) equations with excess chemical potentials described by the classical density functional theory (cDFT). The coupled PNP equations are discretized by a finite difference scheme and solved iteratively using the Gummel method with relaxation. The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation. Then, the algebraic multigrid method is applied to efficiently solve the Poisson equation and the transformed Nernst-Planck equations. A novel strategy for calculating excess chemical potentials through fast Fourier transforms is proposed, which reduces computational complexity from O(N 2 ) to O(Nlog N), where N is the number of grid points. Integrals involving the Dirac delta function are evaluated directly by coordinate transformation, which yields more accurate results compared to applying numerical quadrature to an approximated delta function. Numerical results for ion and electron transport in solid electrolyte for lithiumion (Li-ion) batteries are shown to be in good agreement with the experimental data and the results from previous studies. AMS subject classifications: 92C35, 35J47, 35J60, 35R09, 65M06, 65N55, 65T50

Low‐Defect and Low‐Porosity Hard Carbon with High Coulombic Efficiency and High Capacity for Practical Sodium Ion Battery Anode

“…[16] The simulations were performed using the model of a graphitic electrode embedded in electrolyte, 1 m NaClO 4 in EC/DEC (1:1 by volume) ( Figure S6, Supporting Information). [16] The simulations were performed using the model of a graphitic electrode embedded in electrolyte, 1 m NaClO 4 in EC/DEC (1:1 by volume) ( Figure S6, Supporting Information).…”

A Local Approximation of Fundamental Measure Theory Incorporated into Three Dimensional Poisson–Nernst–Planck Equations to Account for Hard Sphere Repulsion Among Ions

“…This integration is in the form of convolution. Gillespie et al have used fast Fourier transform to deal with the convolution [28], while Meng et al have used the definition of Dirac delta function and change of variables to transform these 3D integrals into 2D integrals on spheres and remove the singularity in the integrands [40]. Though both algorithms can solve the integro-differential equations, they cost lots of computer memory and time during calculation.…”

Multiscale model of metal alloy oxidation at grain boundaries

“…The PNP/cDFT simulations were done using code we have developed, which is parallelized by the multigrid approach and a linear scaling with respect to the number of grid points has been achieved. 19,28 This allows modeling alloy/oxide interfaces with the sizes up to a micrometer in 3D, so our model is able to directly connect with the length and time scales of the grain boundary oxidation experiments.…”

“…The overall approach stems from the coupled Poisson-Nernst-Planck/classical Density Functional Theory (PNP/cDFT) formalism informed by planewave DFT developed for ion and electron transport in metal oxides. [16][17][18][19] and the PNP based models of reactive transport developed in connection with enzymatic reactions. 20,21 In the present study, the PNP/cDFT model is adapted to the problem of IG oxidation dynamics in Ni-Cr and Ni-Al binary alloys.…”