Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T13:25:02.809Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  03 June 2015

Da Meng ,
Bin Zheng ,
Guang Lin  and
Maria L. Sushko
Da Meng*
Affiliation:
Pacific Northwest National Laboratory, Richland, WA 99352, USA
Bin Zheng*
Affiliation:
Pacific Northwest National Laboratory, Richland, WA 99352, USA
Guang Lin*
Affiliation:
Pacific Northwest National Laboratory, Richland, WA 99352, USA Department of Mathematics, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA
Maria L. Sushko*
Affiliation:
Pacific Northwest National Laboratory, Richland, WA 99352, USA
*
Email:da.meng@pnnl.gov
Email:bin.zheng@pnnl.gov
Corresponding author.Email:Guanglin@purdue.edu
Email:Maria.Sushko@pnnl.gov
Get access

Abstract

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(N2) to O(NlogN), 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.

Keywords

92C3535J4735J6035R0965M0665N5565T50Poisson-Nernst-Planck equationsclassical density functional theoryalgebraic multigrid methodfast Fourier transformLi-ion battery
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 5 , November 2014 , pp. 1298 - 1322
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Richardson, G., King, J., Time-dependent modelling and asymptotic analysis of electrochem-ical cells, J. Eng. Math. 59 (2007) 239–275.Google Scholar
[2]Bazant, M. Z., Kilic, M. S., Storey, B. D., Ajdari, A., Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions, Advances in Colloid and Interface Science 152 (2009) 48–88.Google Scholar
[3]Soestbergen, M. van, Biesheuvel, P., Bazant, M., Diffuse-charge effects on the transient response of electrochemical cells, Physical Review E 81 (2010) 021503.CrossRefGoogle ScholarPubMed
[4]Ciucci, F., Lai, W., Derivation of micro/macro lithium battery models from homogenization, Transp. Porous Med. 88 (2011) 249–270.Google Scholar
[5]Marcicki, J., Conlisk, A. T., Rizzoni, G., Comparison of limiting descriptions of the electrical double layer using a simplified lithium-ion battery model, ECS Transactions 41 (14) (2012) 9–21.Google Scholar
[6]Eisenberg, B., Ionic channels in biological membranes - electrostatic analysis of a natural nanotube, Contemp. Phys. 39 (6) (1998) 447–466.Google Scholar
[7]Kurnikova, M. G., Coalson, R. D., Graf, P., Nitzan, A., A lattice relaxation algorithm for three-dimensional Poisson-Nernst-Planck theory with application to ion transport through the gramicidin A channel, Biophys. J. 76 (1999) 642–656.Google Scholar
[8]Cardenas, A. E., Coalson, R. D., Kurnikova, M. G., Three-dimensional Poisson-Nernst-Planck theory studies: Influence of membrane electrostatics on Gramicidin A channel conductance, Biophys. J. 79 (2000) 80–93.Google Scholar
[9]Hollerbach, U., Chen, D. P., Busath, D. D., Eisenberg, B., Predicting function from structure using the Poisson-Nernst-Planck equations: Sodium current in the Gramicidin A channel, Langmuir 79 (13) (2000) 5509–5514.Google Scholar
[10]Coalson, R. D., Kurnikova, M. G., Poisson-Nernst-Planck theory approach to the calculation of current through biological ion channels, IEEE Transactions on Nanobioscience 4 (1) (2005) 81–93.Google Scholar
[11]Lu, B., Zhou, Y., Huber, G. A., Bond, S. D., Holst, M. J., McCammon, J. A., Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution, The Journal of Chemical Physics 127 (2007) 135102.Google Scholar
[12]Bolintineanu, D. S., Sayyed-Ahmad, A., Davis, H. T., Kaznessis, Y. N., Poisson-Nernst-Planck models of nonequilibrium ion electrodiffusion through a protegrin transmembrane pore, PLOS Computational Biology 5 (1) (2009) e1000277.Google Scholar
[13]Singer, A., Norbury, J., A Poisson-Nernst-Planck model for biological ion channels - an asymptotic analysis in a three-dimensional narrow funnel, SIAM J. Appl. Math. 70 (3) (2009) 949–968.Google Scholar
[14]Selberherr, S., Analysis and Simulation of Semiconductor Devices, Springer-Verlag/Wien, New York, 1984.Google Scholar
[15]Markowich, P., The Stationary Semiconductor Device Equation, Springer-Verlag/Wien, New York, 1986.Google Scholar
[16]Rouston, D., Bipolar Semiconductor Devices, McGraw-Hill, New York, 1990.Google Scholar
[17]Newman, J., Electrochemical Systems, Prentice Hall, 1991.Google Scholar
[18]Singh, Y., Density-functional theory of freezing and properties of the ordered phase, Physics Reports 207 (6) (1991) 351–444.Google Scholar
[19]Gillespie, D., Nonner, W., Eisenberg, R. S., Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux, J. Phys.: Condens. Matter 14 (2002) 12129–12145.Google Scholar
[20]Gillespie, D., Xu, L., Wang, Y., Meissner, G., (De)constructing the Ryanodine receptor: modeling ion permeation and selectivity of the Calcium release channel, J. Phys. Chem. B 109 (2005) 15598–15610.Google Scholar
[21]Gillespie, D., Energetics of divalent selectivity in a Calcium channel: the Ryanodine receptor case study, Biophys. J. 94 (2008) 1169–1184.Google Scholar
[22]Gillespie, D., Fill, M., Intracellular Calcium release channels mediate their own countercurrent: the Ryanodine receptor case study, Biophys. J. 95 (2008) 3706–3714.Google Scholar
[23]Sushko, M. L., Rosso, K. M., J. Liu, Size effects on Li/electron conductivity in TiÜ2 nanoparticles, Chem. Phys. Lett. 1 (13) (2010) 1967–1972.Google Scholar
[24]Cao, D., Wu, J., Microstructure of block copolymers near selective surfaces: theoretical predictions and configurational-bias Monte-Carlo simulation, Macromolecules 38 (2005) 971–978.Google Scholar
[25]Du, Y. A., Holzwarth, N. A. W., Li ion diffusion mechanisms in the crystalline electrolyte γ Li3PO4, Journal of the Electrochemical Society 154 (11) (2007) 999–1004.Google Scholar
[26]Sushko, M. L., Rosso, K. M., Zhang, J.-G. J., Liu, J., Multiscale simulations of Li ion conductivity in solid electrolyte, Chem. Phys. Lett. 2 (2011) 2352–2356.Google Scholar
[27]Sushko, M. L., Rosso, K. M., Liu, J., Mechanism of Li+/electron conductivity in rutile and anatase TiO2 nanoparticles, J. Phys. Chem. C 114 (2010) 20277–20283.Google Scholar
[28]Sushko, M. L., Liu, J., Structural rearrangements in self-assembled surfactant layers at surfaces, J. Phys. Chem. B 114 (2010) 3847–3854.Google Scholar
[29]Li, X., Qi, W., Mei, D., Sushko, M. L., Aksay, I., Liu, J., Functionalized graphene sheets as molecular templates for controlled nucleation and self-assembly of metal oxide-graphene nanocom-posites, Advanced Materials 24 (2012) 5136–5141.Google Scholar
[30]Hu, S., Li, Y., Rosso, K. M., Sushko, M. L., Mesoscale phase-field modeling of charge transport in nanocomposite electrodes for lithium-ion batteries, The Journal of Chemical Physics C 117(2013) 28–40.Google Scholar
[31]Golovnev, A., Trimper, S., Analytical solution of the Poisson-Nernst-Planck equations in the linear regime at an applied dc-voltage, The Journal of Chemical Physics 134 (2011) 154902.Google Scholar
[32]Ji, S., Liu, W., Poisson-Nernst-Planck systems for ion flow with density functional theory for hard-sphere potential: I-V relations and critical potentials. Part I: analysis, J. Dyn. Diff. Equat. 24 (2012) 955–983.Google Scholar
[33]Liu, W., Tu, X., Zhang, M., Poisson-Nernst-Planck systems for ion flow with density functional theory for hard-sphere potential: I-V relations and critical potentials. Part II: Numerics, J. Dyn. Diff. Equat. 24 (2012) 985–1004.CrossRefGoogle Scholar
[34]Wu, J., Srinivasan, V., Xu, J., Wang, C., Newton-Krylov-Multigrid algorithms for battery simulation, Journal of the Electrochemical Society 149 (10) (2002) A1342–A1348.Google Scholar
[35]Mathur, S. R., Murthy, J. Y., A multigrid method for the Poisson-Nernst-Planck equations, International Journal of Heat and Mass Transfer 52 (2009) 4031–4039.Google Scholar
[36]Coco, S., Gazzo, D., Laudani, A., Pollicino, G., A 3-D finite element Poisson-Nernst-Planck model for the analysis of ion transport across ionic channels, IEEE Transactions on Magnetics 43 (4) (2007) 1461–1464.CrossRefGoogle Scholar
[37]Lu, B., Holst, M. J., McCammon, J. A., Zhou, Y., Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: finite element solutions, Journal of Compu-tational Physics 229 (2010) 6979–6994.Google Scholar
[38]Zheng, Q., Chen, D., Wei, G.-W., Second-order Poisson-Nernst-Planck solver for ion transport, Journal of Computational Physics 230 (2011) 5239–5262.Google Scholar
[39]Hyon, Y., Eisenberg, B., Liu, C., A mathematical model for the hard sphere repulsion in ionic solutions, Commun. Math. Sci. 9 (2) (2011) 459–475.Google Scholar
[40]Falgout, R., Yang, U., hypre: a library of high performance preconditioners, in: Sloot, P., Tan, C., Dongarra, J., Hoekstra, A. (Eds.), Computational Science - ICCS 2002 Part III, Vol. 2331 of Lecture Notes in Computer Science, Springer-Verlag, 2002, pp. 632–641.Google Scholar
[41]Henson, V. E., Yang, U. M., BoomerAMG: a parallel algebraic multigrid solver and preconditioner, Applied Numerical Mathematics 41 (2002) 155–177.Google Scholar
[42]Pekurovsky, D., P3DFFT: A framework for parallel computations of Fourier transforms in three dimensions, SIAM J. Sci. Comput. 34 (4) (2012) C192–C209.Google Scholar
[43]Burger, M., Schlake, B., Wolfram, M.-T., Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries, Nonlinearity 25 (2012) 961–990.CrossRefGoogle Scholar
[44]Park, J.-H., Jerome, J., Qualitative properties of steady-state Poisson-Nernst-Planck systems: Mathematical study, SIAM J. Appl. Math. 57 (3) (1997) 609–630.Google Scholar
[45]Liu, W., Geometric singular perturbation approach to steady-state Poisson-Nernst-Planck systems, SIAM J. Appl. Math. 65 (3) (2005) 754–766.Google Scholar
[46]Jerome, J. W., Consistency of semiconductor modeling: an existence/stability analysis for the sationary Van Roosbroeck system, SIAM J. Appl. Math. 45 (4) (1985) 565–590.Google Scholar
[47]Rosenfeld, Y., Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing, Physical Review Letters 63 (9) (1989) 980.Google Scholar
[48]Rosenfeld, Y., Free energy model for inhomogeneous fluid mixtures: Yukawa-charged hard spheres, general interactions, and plasmas, J. Chem. Phys. 98 (10) (1993) 8126–8148.Google Scholar
[49]Hubbard, A. T. (Ed.), Encyclopedia of surface and colloid science, Vol. 3, CRC Press, 2002.Google Scholar
[50]Wang, K., Yu, Y.-X., Gao, G.-H., Density functional study on the structures and thermodynamic properties of small ions and around polyanionic DNA, Physical Review E 70 (2004) 011912.Google Scholar
[51]Butler, J. N., Ionic Equilibrium: Solubility and pH Calculations, Wiley-Interscience, 1998.Google Scholar
[52]Merkel, B. J., Planer-Friedrich, B., Groundwater Geochemistry: A Practical Guide to Modeling of Natural and Contaminated Aquatic Systems, 2nd Edition, Springer, 2008.Google Scholar
[53]Patra, C. N., Yethiraj, A., Density functional theory for the distribution of small ions around polyions, J. Phys. Chem. B 103 (1999) 6080–6087.Google Scholar
[54]Li, Z., Wu, J., Density-functional theory for the structures and thermodynamic properties of highly asymmetric electrolyte and neutral component mixtures, Physical Review E 70 (2004) 031109.Google Scholar
[55]Roth, R., Evans, R., Lang, A., Kahl, G., Fundamental measure theory for hard-sphere mixtures revisited: the White Bear version, J. Phys.: Condens. Matter 14 (2002) 12063–12078.Google Scholar
[56]Yu, Y.-X., Wu, J., Structures of hard-sphere fluids from a modified fundamental-measure the-ory, J. Chem. Phys. 117 (22) (2002) 10156–10164.Google Scholar
[57]Boda, D., Henderson, D., y Teran, L. Mier, Sokolowski, S., The application of density functional theory and the generalized mean spherical approximation to double layers containing strongly coupled ions, J Phys Condens Matter 14 (2002) 11945–11954.Google Scholar
[58]Gillespie, D., Hackbusch, W., Eisenberg, R. S., Density functiona theory of charged, hard-sphere fluids, Phys. Rev. E 68 (2003) 031503.Google Scholar
[59]Waisman, E., Lebowitz, J. L., Mean spherical model integral equation for charged hard spheres. II. Results, J. Chem. Phys. 57 (1972) 3093–3099.Google Scholar
[60]Mier-y-Teran, L., Suh, S., White, H., Davis, H., A nonlocal free-energy density-functional ap-proximation for the electrical double layer, J. Chem. Phys. 92 (8) (1990) 5087–5098.Google Scholar
[61]Yu, Y.-X., Wu, J., Gao, G.-H., Density-functional theory of spherical electric double layers and ζ potentials of colloidal particles in restricted-primitive-model electrolyte solutions, J. Chem. Phys. 120 (15) (2004) 7223–7233.Google Scholar
[62]Jerome, J., Analysis of Charge Transport: A Mathematical Study of Semiconductors, Springer-Verlag, Heidelberg, 1996.Google Scholar
[63]Im, W., Roux, B., Ion permeation and selectivity of OmpF Porin: a theoretical study based on molecular dynamics, Brownian dynamics, and continuum electrodiffusion theory, J. Mol. Biol. 322 (2002) 851–869.Google Scholar
[64]Hackbusch, W., Multi-Grid Methods and Applications, Springer, 1985.CrossRefGoogle Scholar
[65]Wesseling, P., An Introduction to Multigrid Methods, John Wiley & Sons, 1992.Google Scholar
[66]Bramble, J. H., Multigrid Methods, Chapman and Hall/CRC, 1993.Google Scholar
[67]Brandt, A., McCormick, S., Ruge, J., Algebraic multigrid for automatic multigrid solutions with application to geodetic computations, Tech. rep., Institute for Computational Studies, Fort Collins, Colorado (October 1982).Google Scholar
[68]Brandt, A., Algebraic multigrid theory: the symmetric case, Math. Comp. 19 (1986) 23–56.Google Scholar
[69]Cooley, J. W., Lewis, P. A. W., Welch, P. D., Application of the fast Fourier transform to computation of Fourier integrals, Fourier series, and convolution integrals, IEEE Transactions on Audio and Electroacoustics 15 (2) (1967) 79–84.Google Scholar
[70]Fu, Z.-W., Liu, W.-Y., Li, C.-L., Qin, Q.-Z., High-k lithium phosphorous oxynitride thin films, Applied Physics Letters 83 (24) (2003) 5008–5010.Google Scholar
[71]Wang, B., Kwak, B., Sales, B., Bates, J., Ionic conductivities and structure of lithium phosphorus oxynitride glasses, Journal of Non-Crystalline Solids 183 (1995) 297–306.Google Scholar
[72]Ribeiro, J., Sousa, R., Carmo, J., Goncalves, L., Silva, M., Silva, M., Correia, J., Enhanced solidstate electrolytes made of lithium phosphorous oxynitride films, Thin Solid Films 522 (2012) 8589.Google Scholar
[73]Johnson, O. W., One-dimensional diffusion of Li in rutile, Phys. Rev. A 136 (1964) 284–292.CrossRefGoogle Scholar
[74]Koudriachova, M. V., Harrison, N. M., de Leeuw, S. W., Diffusion of Li-ions in rutile. An ab initio study, Solid State Ionics 157 (2003) 35–38.Google Scholar
[75]Luo, W., Zhu, L., Zheng, X., Grain size effect on electrical conductivity and giant magnetore-sistance of bulk magnetic polycrystals, Chin. Phys. Lett. 26 (2009) 117502.Google Scholar