Commun. Comput. Phys., 16 (2014), pp. 1298-1322.

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

Da Meng 1, Bin Zheng 1, Guang Lin 2*, Maria L. Sushko 1*

1 Pacific Northwest National Laboratory, Richland, WA 99352, USA.
2 Pacific Northwest National Laboratory, Richland, WA 99352, USA; Department of Mathematics, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA.

Received 4 September 2013; Accepted (in revised version) 12 May 2014
Available online 29 August 2014


We have developed efficient numerical algorithms for solving 3D steady-state 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(N\log 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 lithium-ion (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

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Key words: Poisson-Nernst-Planck equations, classical density functional theory, algebraic multigrid method, fast Fourier transform, Li-ion battery.

*Corresponding author.
Email: (D. Meng), (B. Zheng), (G. Lin), (M. L. Sushko)

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