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Derivation of Micro/Macro Lithium Battery Models from Homogenization

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Abstract

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.

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Abbreviations

+:

Li+

−:

e− in the solid and X− in the liquid

ν :

Normal of Γ oriented from the liquid towards the solid

δ :

Scale separation = l micro/l MACRO

δk(f) k :

Approximation of f at order k in δ

Γ:

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

γ :

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

\({\hat{\mu}}\) :

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

Λ:

Effective dimensionless diffusivity matrix

(f)0 :

Approximation of f at order 0 in δ

j :

Current density

x :

Macroscopic coordinate

y :

Microscopic coordinate = x/δ

\({\mathcal L}\) :

Liquid phase

\({\mathcal S}\) :

Solid phase

μ :

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

\({\nabla}\) :

Gradient operator with respect to the dimensional coordinates

\({\nabla_x}\) :

Gradient operator with respect to the macro coordinates

\({\nabla_y}\) :

Gradient operator with respect to the micro coordinates

\({\phi}\) :

Electrical potential

Φ1 :

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

Φ2 :

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

Σ:

Effective dimensionless conductivity tensor

σ :

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

τ 2 :

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 \({=\tilde{\bf x}/\delta}\)

ε :

Porosity

ε 0 :

Permittivity of vacuum

ε r :

Relative dielectric constant

c :

Concentration

D :

Diffusivity

k B :

Boltzmann’s constant

l:

Liquid

l MACRO :

Macroscopic lengthscale

l micro :

Microscopic lengthscale

s:

Solid

T :

Temperature

U T :

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

References

  • Allaire G., Brizzi R.: A multiscale finite element method for numerical homogenization. Multiscale Model. Simul. 4(3), 790–812 (2005)

    Article  Google Scholar 

  • Amirat Y., Shelukhin V.: Electroosmosis law via homogenization of electrolyte flow equations in porous media. J. Math. Anal. Appl. 342(2), 1227–1245 (2008)

    Article  Google Scholar 

  • Armand M., Tarascon J.-M.: Building better batteries. Nature 451(7179), 652–657 (2008)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Bard A.J., Faulkner L.R.: Electrochemical Methods: Fundamentals and Applications, 2nd edn. Wiley, New York (2000)

    Google Scholar 

  • Bensoussan A., Lion J.L., Papanicolaou G.: Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications. North Holland, Amsterdam (1978)

    Google Scholar 

  • Bohm M., Showalter R.E.: Diffusion in fissured media. SIAM J. Math. Anal. 16(3), 500–509 (1985a)

    Article  Google Scholar 

  • Bohm M., Showalter R.E.: A nonlinear pseudoparabolic diffusion equation. SIAM J. Math. Anal. 16(5), 980–999 (1985b)

    Article  Google Scholar 

  • Canon, É., Jäger, W.: Homogenization of Nonlinear Adsorption-Diffusion Processes in Porous Media, p. 123. SFB-Preprints, Heidelberg (1993)

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Christensen J.: Modeling diffusion-induced stress in Li-ion cells with porous electrodes. J. Electrochem. Soc. 157(3), A366–A380 (2010)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Darling R., Newman J.: Modeling side reactions in composite Li y Mn2O4 electrodes. J. Electrochem. Soc. 145(3), 990–998 (1998)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Doyle, M.: Design and Simulation of Lithium Rechargeable Batteries. Dissertation, PhD thesis, University of California Berkeley (1995)

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Edwin Garcia R., Chiang Y.-M.: Spatially resolved modeling of microstructurally complex battery architectures. J. Electrochem. Soc. 154(9), A856–A864 (2007)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Gu W.B., Wang C.Y.: Thermal-electrochemical modeling of battery systems. J. Electrochem. Soc. 147(8), 2910–2922 (2000)

    Article  Google Scholar 

  • Hornung, U. (ed.): Homogenization and Porous Media. Interdisciplinary Applied Mathematics. Springer, New York (1996)

    Google Scholar 

  • Hornung U., Showalter R.E.: Diffusion models for fractured media. J. Math. Anal. Appl. 147(1), 69–80 (1990)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Jikov V.V., Kozlov S.M., Oleinik O.A.: Homogenization of Differential Operators and Integral Functionals, 1st edn. Springer, New York (1994)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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

  • Lai W., Ciucci F.: Thermodynamics and kinetics of phase transformation in intercalation battery electrodes—phenomenological modeling. Electrochim. Acta 56(1), 531–542 (2010)

    Article  Google Scholar 

  • Lai W., Ciucci F.: Small-signal apparent diffusion impedance of intercalation battery electrodes. J. Electrochem. Soc. 158(2), A115–A121 (2011)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Looker J., Carnie S.: Homogenization of the ionic transport equations in periodic porous media. Transp. Porous Med. 65(1), 107–131 (2006)

    Article  Google Scholar 

  • Moyne C., Murad M.: Macroscopic behavior of swelling porous media derived from micromechanical analysis. Transp. Porous Med. 50(1), 127–151 (2003)

    Article  Google Scholar 

  • Newman J., Tiedemann W.: Porous-electrode theory with battery applications. AIChE J. 21(1), 25–41 (1975)

    Article  Google Scholar 

  • Nolen J., Papanicolaou G., Pironneau O.: A framework for adaptive multiscale methods for elliptic problems. Multiscale Model. Simulation 7(1), 171–196 (2008)

    Article  Google Scholar 

  • Pavliotis G.A., Stuart A.M.: Multiscale Methods: Averaging and Homogenization. Texts in Applied Mathematics, vol. 53. Springer, Berlin (2008)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Sepúlveda, M., Postel, M.: Identification of the effective diffusion for a transport equation modeling chromatography. IWR/SFB-Preprints, Abstract 96-06 (1996)

  • 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)

    Article  Google Scholar 

  • Showalter R.E.: Diffusion models with microstructure. Transp. Porous Med. 6(5), 567–580 (1991)

    Article  Google Scholar 

  • Showalter R.E., Walkington N.J.: Diffusion of fluid in a fissured medium with microstructure. SIAM J. Math. Anal. 22(6), 1702–1722 (1991)

    Article  Google Scholar 

  • Thomas K., Newman J., Darling R.: Mathematical Modeling of Lithium Batteries, pp. 345–392. Kluwer Academic Publisher, New York (2002)

    Google Scholar 

  • Vogt, Ch.: A Homogenization Theorem Leading to a Volterra Integro-Differential Equation for Permeation Chromotography. SFB 123, Preprint No 155. SFB-Preprints, Heidelberg (1982)

  • Wakihara M.: Recent developments in lithium ion batteries. Mater. Sci. Eng. 33(4), 109–134 (2001)

    Article  Google Scholar 

  • Wang C.-W., Sastry A.M.: Mesoscale modeling of a Li-ion polymer cell. J. Electrochem. Soc. 154(11), A1035–A1047 (2007)

    Article  Google Scholar 

  • Whitaker S.: The Method of Volume Averaging (Theory and Applications of Transport Porous Media), 1st edn. Springer, Berlin (1998)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

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Authors and Affiliations

  1. Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences, Interdisciplinary Center for Scientific Computing, University of Heidelberg, Im Neuenheimer Feld 368, 69120, Heidelberg, Germany

    Francesco Ciucci

  2. Department of Chemical Engineering and Materials Science, Michigan State University, 3524 Engineering, East Lansing, MI, 48824, USA

    Wei Lai

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  1. Francesco Ciucci
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  2. Wei Lai
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Correspondence to Francesco Ciucci.

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Ciucci, F., Lai, W. Derivation of Micro/Macro Lithium Battery Models from Homogenization. Transp Porous Med 88, 249–270 (2011). https://doi.org/10.1007/s11242-011-9738-5

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