# 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

We have developed efficient numerical algorithms for the solution of 3D steady-state Poisson-Nernst-Planck equations (PNP) with excess chemical potentials described by the classical density functional theory (cDFT). The coupled PNP equations are discretized by finite difference scheme and solved iteratively by Gummel method with relaxation. The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation. Algebraic multigrid method is then 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(N2) to O(NlogN) where N is the number of grid points. Integrals involving Dirac delta function are evaluated directly by coordinate transformation which yields more accurate result compared to applying numerical quadrature to an approximated delta function. Numerical results for ion and electron transport in solid electrolyte for Li ion batteries are shown to be in good agreement with the experimental data and the results from previous studies.

- Authors:

- Publication Date:

- Research Org.:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1170083

- Report Number(s):
- PNNL-SA-98014

- DOE Contract Number:
- AC05-76RL01830

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Communications in Computational Physics, 16(5):1298-1322

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Meng, Da, Zheng, Bin, Lin, Guang, and Sushko, Maria L.
```*Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment*. United States: N. p., 2014.
Web.

```
Meng, Da, Zheng, Bin, Lin, Guang, & Sushko, Maria L.
```*Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment*. United States.

```
Meng, Da, Zheng, Bin, Lin, Guang, and Sushko, Maria L. Fri .
"Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment". United States.
doi:.
```

```
@article{osti_1170083,
```

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

author = {Meng, Da and Zheng, Bin and Lin, Guang and Sushko, Maria L.},

abstractNote = {We have developed efficient numerical algorithms for the solution of 3D steady-state Poisson-Nernst-Planck equations (PNP) with excess chemical potentials described by the classical density functional theory (cDFT). The coupled PNP equations are discretized by finite difference scheme and solved iteratively by Gummel method with relaxation. The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation. Algebraic multigrid method is then 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(N2) to O(NlogN) where N is the number of grid points. Integrals involving Dirac delta function are evaluated directly by coordinate transformation which yields more accurate result compared to applying numerical quadrature to an approximated delta function. Numerical results for ion and electron transport in solid electrolyte for Li ion batteries are shown to be in good agreement with the experimental data and the results from previous studies.},

doi = {},

journal = {Communications in Computational Physics, 16(5):1298-1322},

number = ,

volume = ,

place = {United States},

year = {Fri Aug 29 00:00:00 EDT 2014},

month = {Fri Aug 29 00:00:00 EDT 2014}

}