Transport in Porous Media

, Volume 88, Issue 2, pp 249–270 | Cite as

Derivation of Micro/Macro Lithium Battery Models from Homogenization

  • Francesco CiucciEmail author
  • Wei Lai


In this article, we develop a micro–macroscopic coupled model aimed at studying the interplay between electrokinetics and transport in lithium ion batteries. The system studied consists of a solid (electrode material) and a liquid phase (electrolyte) with periodic microscopic features. In this work, homogenization of generalized Poisson–Nernst–Planck (PNP) equation set leads to a micro/macro formulation similar in nature to the one developed in Newman’s model for lithium batteries. Underlying conservation equations are derived for each phase using asymptotic expansions and mathematical tools from homogenization theory, starting from a PNP micromodel, and in particular Newman’s model is obtained as a corollary of the micro/macro approach developed here. The advantage of homogenization lies in the fact that effective parameters can be derived directly from the analysis of the periodic microstructure and from the application of the theory developed in this article. In addition, the advantages of using homogenization in Lithium ion battery modeling are outlined. Lastly, this work is a necessary step toward more general homogenized models and toward mathematical proofs, and it is also needed preliminary analysis for multiscale computational schemes.


Newman’s model Porous electrode Lithium batteries Poisson–Nernst–Planck Homogenization 

List of symbols



e in the solid and X in the liquid


Normal of Γ oriented from the liquid towards the solid


Scale separation = l micro/l MACRO


Approximation of f at order k in δ


Interface between liquid and solid phase oriented from the liquid towards the solid


Thermodynamic factor \({=\gamma=1+\frac{\partial \log f(c)}{\partial \log c}}\)


Dimensionless electrochemical potential \({=\frac{e}{U_{\rm T}}\mu + z \phi}\)


Effective dimensionless diffusivity matrix


Approximation of f at order 0 in δ


Current density


Macroscopic coordinate


Microscopic coordinate = x/δ

\({\mathcal L}\)

Liquid phase

\({\mathcal S}\)

Solid phase


Chemical potential = \({\mu = k_{\rm B} T \log\left(f\pm(c)\frac{c}{\left(c\right)^0}\right)}\)


Gradient operator with respect to the dimensional coordinates


Gradient operator with respect to the macro coordinates


Gradient operator with respect to the micro coordinates


Electrical potential


Order 0 approximation of the reduced electrochemical potential of e in \({\mathcal S = \Phi_1 =-\frac{\left(\tilde \mu_-^{\rm s}\right)_0}{e}}\)


Order 0 approximation of the reduced electrochemical potential of Li+ in \({\mathcal L = \Phi_ 2=\frac{\left(\tilde \mu_+^{\rm l}\right)_0}{e}}\)


Effective dimensionless conductivity tensor


Conductivity \({=Dc \frac{e^2}{k_{\rm B}T}}\)


Tortuosity factor

\({\tilde \mu}\)

Electrochemical potential \({= \mu + ez\phi}\)

\({\tilde \mu^\star}\)

Reduced electrochemical potential \({=\frac{\hat{\mu}}{z e}}\)

\({\tilde \nabla}\)

Gradient operator with respect to the dimensionless coordinates

\({\tilde{{\bf x}}}\)

Dimensionless macroscopic coordinate

\({\tilde{{\bf y}}}\)

Dimensionless microscopic coordinate \({=\tilde{\bf x}/\delta}\)




Permittivity of vacuum


Relative dielectric constant






Boltzmann’s constant




Macroscopic lengthscale


Microscopic lengthscale






Thermal voltage \({=\frac{k_{\rm B} T}{e}}\)


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Allaire G., Brizzi R.: A multiscale finite element method for numerical homogenization. Multiscale Model. Simul. 4(3), 790–812 (2005)CrossRefGoogle Scholar
  2. Amirat Y., Shelukhin V.: Electroosmosis law via homogenization of electrolyte flow equations in porous media. J. Math. Anal. Appl. 342(2), 1227–1245 (2008)CrossRefGoogle Scholar
  3. Armand M., Tarascon J.-M.: Building better batteries. Nature 451(7179), 652–657 (2008)CrossRefGoogle Scholar
  4. Balke N., Jesse S., Morozovska A.N., Eliseev E., Chung D.W., Kim Y., Adamczyk L., Garcia R.E., Dudney N., Kalinin S.V.: Nanoscale mapping of ion diffusion in a lithium-ion battery cathode. Nat. Nanotechnol. 5(10), 749–754 (2010)CrossRefGoogle Scholar
  5. Bard A.J., Faulkner L.R.: Electrochemical Methods: Fundamentals and Applications, 2nd edn. Wiley, New York (2000)Google Scholar
  6. Bensoussan A., Lion J.L., Papanicolaou G.: Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications. North Holland, Amsterdam (1978)Google Scholar
  7. Bohm M., Showalter R.E.: Diffusion in fissured media. SIAM J. Math. Anal. 16(3), 500–509 (1985a)CrossRefGoogle Scholar
  8. Bohm M., Showalter R.E.: A nonlinear pseudoparabolic diffusion equation. SIAM J. Math. Anal. 16(5), 980–999 (1985b)CrossRefGoogle Scholar
  9. Canon, É., Jäger, W.: Homogenization of Nonlinear Adsorption-Diffusion Processes in Porous Media, p. 123. SFB-Preprints, Heidelberg (1993)Google Scholar
  10. Cervera J., Schiedt B., Ramírez P.: A Poisson/Nernst-Planck model for ionic transport through synthetic conical nanopores. Europhys. Lett. 71(1), 35–41 (2005)CrossRefGoogle Scholar
  11. Chen Y.H., Wang C.W., Zhang X., Sastry A.M.: Porous cathode optimization for lithium cells: Ionic and electronic conductivity, capacity, and selection of materials. J. Power Sources 195(9), 2851–2862 (2010)CrossRefGoogle Scholar
  12. Christensen J.: Modeling diffusion-induced stress in Li-ion cells with porous electrodes. J. Electrochem. Soc. 157(3), A366–A380 (2010)CrossRefGoogle Scholar
  13. Ciucci F., Hao Y., Goodwin D.G.: Impedance spectra of mixed conductors: a 2D study of ceria. Phys. Chem. Chem. Phys. 11, 11243–11257 (2009)CrossRefGoogle Scholar
  14. Cushman J.H., Bennethum L.S., Hu B.X.: A primer on upscaling tools for porous media. Adv. Water Resour. 25(8–12), 1043–1067 (2002)CrossRefGoogle Scholar
  15. Darling R., Newman J.: Modeling side reactions in composite Liy Mn2O4 electrodes. J. Electrochem. Soc. 145(3), 990–998 (1998)CrossRefGoogle Scholar
  16. Dezanneau G., Morata A., Taranón A., Peiró F., Morante J.R.: Effect of grain size distribution on the grain boundary electrical response of 2d and 3d polycrystals. Solid State Ion. 177(35–36), 3117–3121 (2006)CrossRefGoogle Scholar
  17. Doyle, M.: Design and Simulation of Lithium Rechargeable Batteries. Dissertation, PhD thesis, University of California Berkeley (1995)Google Scholar
  18. Doyle M., Newman J.: Analysis of capacity-rate data for lithium batteries using simplified models of the discharge process. J. Appl. Electrochem. 27(7), 846–856 (1997)CrossRefGoogle Scholar
  19. Doyle M., Newman J., Gozdz A.S., Schmutz C.N., Tarascon J.-M.: Comparison of modeling predictions with experimental data from plastic lithium ion cells. J. Electrochem. Soc. 143(6), 1890–1903 (1996)CrossRefGoogle Scholar
  20. Doyle M., Meyers J.P., Newman J.: Computer simulations of the impedance response of lithium rechargeable batteries. J. Electrochem. Soc. 147(1), 99–110 (2000)CrossRefGoogle Scholar
  21. Efendiev Y., Hou T.Y.: Multiscale Finite Element Methods: Theory and Applications. Surveys and Tutorials in the Applied Mathematical Sciences. Springer-Verlag, Berlin (2009)Google Scholar
  22. Fang W., Kwon O.J., Wang C.-Y.: Electrochemical-thermal modeling of automotive li-ion batteries and experimental validation using a three-electrode cell. Int. J. Energy Res. 34(2), 107–115 (2010)CrossRefGoogle Scholar
  23. Fleig J.: The grain boundary impedance of random microstructures: numerical simulations and implications for the analysis of experimental data. Solid State Ion. 150(1–2), 181–193 (2002)CrossRefGoogle Scholar
  24. Edwin Garcia R., Chiang Y.-M.: Spatially resolved modeling of microstructurally complex battery architectures. J. Electrochem. Soc. 154(9), A856–A864 (2007)CrossRefGoogle Scholar
  25. Garcia R.E., Chiang Y.-M., Carter W.C., Limthongkul P., Bishop C.M.: Microstructural modeling and design of rechargeable lithium-ion batteries. J. Electrochem. Soc. 152(1), A255–A263 (2005)CrossRefGoogle Scholar
  26. Golmon S., Maute K., Dunn M.L.: Numerical modeling of electrochemical-mechanical interactions in lithium polymer batteries. Comput. Struct. 87(23–24), 1567–1579 (2009)CrossRefGoogle Scholar
  27. Gomadam P.M., Weidner J.W., Dougal R.A., White R.E.: Mathematical modeling of lithium-ion and nickel battery systems. J. Power Sources 110(2), 267–284 (2002)CrossRefGoogle Scholar
  28. Gu W.B., Wang C.Y.: Thermal-electrochemical modeling of battery systems. J. Electrochem. Soc. 147(8), 2910–2922 (2000)CrossRefGoogle Scholar
  29. Hornung, U. (ed.): Homogenization and Porous Media. Interdisciplinary Applied Mathematics. Springer, New York (1996)Google Scholar
  30. Hornung U., Showalter R.E.: Diffusion models for fractured media. J. Math. Anal. Appl. 147(1), 69–80 (1990)CrossRefGoogle Scholar
  31. Jamnik J., Maier J.: Generalised equivalent circuits for mass and charge transport: chemical capacitance and its implications. Phys. Chem. Chem. Phys. 3(9), 1668–1678 (2001)CrossRefGoogle Scholar
  32. Jikov V.V., Kozlov S.M., Oleinik O.A.: Homogenization of Differential Operators and Integral Functionals, 1st edn. Springer, New York (1994)Google Scholar
  33. Kuwata N., Iwagami N., Tanji Y., Matsuda Y., Kawamura J.: Characterization of thin-film lithium batteries with stable thin-film Li3PO4 solid electrolytes fabricated by arf excimer laser deposition. J. Electrochem. Soc. 157(4), A521–A527 (2010)CrossRefGoogle Scholar
  34. Lai, W., Ciucci, F.: Mathematical modeling of porous battery electrodes—revisit of newman’s model. Electrochim. Acta (2010) doi: 10.1016/j.electacta.2011.01.012
  35. Lai W., Ciucci F.: Thermodynamics and kinetics of phase transformation in intercalation battery electrodes—phenomenological modeling. Electrochim. Acta 56(1), 531–542 (2010)CrossRefGoogle Scholar
  36. Lai W., Ciucci F.: Small-signal apparent diffusion impedance of intercalation battery electrodes. J. Electrochem. Soc. 158(2), A115–A121 (2011)CrossRefGoogle Scholar
  37. Lai W., Haile S.M.: Impedance spectroscopy as a tool for chemical and electrochemical analysis of mixed conductors: a case study of ceria. J. Am. Ceram. Soc. 88(11), 2979–2997 (2005)CrossRefGoogle Scholar
  38. Looker J., Carnie S.: Homogenization of the ionic transport equations in periodic porous media. Transp. Porous Med. 65(1), 107–131 (2006)CrossRefGoogle Scholar
  39. Moyne C., Murad M.: Macroscopic behavior of swelling porous media derived from micromechanical analysis. Transp. Porous Med. 50(1), 127–151 (2003)CrossRefGoogle Scholar
  40. Newman J., Tiedemann W.: Porous-electrode theory with battery applications. AIChE J. 21(1), 25–41 (1975)CrossRefGoogle Scholar
  41. Nolen J., Papanicolaou G., Pironneau O.: A framework for adaptive multiscale methods for elliptic problems. Multiscale Model. Simulation 7(1), 171–196 (2008)CrossRefGoogle Scholar
  42. Pavliotis G.A., Stuart A.M.: Multiscale Methods: Averaging and Homogenization. Texts in Applied Mathematics, vol. 53. Springer, Berlin (2008)Google Scholar
  43. Ramadass P., Haran B., White R., Popov B.N.: Mathematical modeling of the capacity fade of Li-ion cells. J. Power Sources 123(2), 230–240 (2003)CrossRefGoogle Scholar
  44. Samson E., Marchand J., Beaudoin J.J.: Describing ion diffusion mechanisms in cement-based materials using the homogenization technique. Cem. Concr. Res. 29(8), 1341–1345 (1999)CrossRefGoogle Scholar
  45. Schuss Z., Nadler B., Eisenberg R.S.: Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model. Phys. Rev. E 64(2–3), 036116 (2001)CrossRefGoogle Scholar
  46. Sepúlveda, M., Postel, M.: Identification of the effective diffusion for a transport equation modeling chromatography. IWR/SFB-Preprints, Abstract 96-06 (1996)Google Scholar
  47. Shearing P.R., Howard L.E., Jorgensen P.S., Brandon N.P., Harris S.J.: Characterization of the 3-dimensional microstructure of a graphite negative electrode from a Li-ion battery. Electrochem. Commun. 12(3), 374–377 (2010)CrossRefGoogle Scholar
  48. Showalter R.E.: Diffusion models with microstructure. Transp. Porous Med. 6(5), 567–580 (1991)CrossRefGoogle Scholar
  49. Showalter R.E., Walkington N.J.: Diffusion of fluid in a fissured medium with microstructure. SIAM J. Math. Anal. 22(6), 1702–1722 (1991)CrossRefGoogle Scholar
  50. Thomas K., Newman J., Darling R.: Mathematical Modeling of Lithium Batteries, pp. 345–392. Kluwer Academic Publisher, New York (2002)Google Scholar
  51. Vogt, Ch.: A Homogenization Theorem Leading to a Volterra Integro-Differential Equation for Permeation Chromotography. SFB 123, Preprint No 155. SFB-Preprints, Heidelberg (1982)Google Scholar
  52. Wakihara M.: Recent developments in lithium ion batteries. Mater. Sci. Eng. 33(4), 109–134 (2001)CrossRefGoogle Scholar
  53. Wang C.-W., Sastry A.M.: Mesoscale modeling of a Li-ion polymer cell. J. Electrochem. Soc. 154(11), A1035–A1047 (2007)CrossRefGoogle Scholar
  54. Whitaker S.: The Method of Volume Averaging (Theory and Applications of Transport Porous Media), 1st edn. Springer, Berlin (1998)Google Scholar
  55. Wilson J.R., Cronin J.S., Barnett S.A., Harris S.J.: Measurement of three-dimensional microstructure in a LiCoO2 positive electrode. J. Power Sources 196(7), 3443–3447 (2011)CrossRefGoogle Scholar
  56. Zhang Q., Guo Q., White R.E.: Semi-empirical modeling of charge and discharge profiles for a LiCoO2 electrode. J. Power Sources 165(1), 427–435 (2007)CrossRefGoogle Scholar
  57. Zhang X., Sastry A. M., Shyy W.: Intercalation-induced stress and heat generation within single lithium-ion battery cathode particles. J. Electrochem. Soc. 155(7), A542–A552 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences, Interdisciplinary Center for Scientific ComputingUniversity of HeidelbergHeidelbergGermany
  2. 2.Department of Chemical Engineering and Materials ScienceMichigan State UniversityEast LansingUSA

Personalised recommendations