Transport in Porous Media

, Volume 88, Issue 2, pp 249–270

# Derivation of Micro/Macro Lithium Battery Models from Homogenization

Article

## 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.

### Keywords

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

### List of symbols

+

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 $${=Dc \frac{e^2}{k_{\rm B}T}}$$

τ2

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

ε0

Permittivity of vacuum

εr

Relative dielectric constant

c

Concentration

D

Diffusivity

kB

Boltzmann’s constant

l

Liquid

lMACRO

Macroscopic lengthscale

lmicro

Microscopic lengthscale

s

Solid

T

Temperature

UT

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

## Preview

### References

1. Allaire G., Brizzi R.: A multiscale finite element method for numerical homogenization. Multiscale Model. Simul. 4(3), 790–812 (2005)
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)
3. Armand M., Tarascon J.-M.: Building better batteries. Nature 451(7179), 652–657 (2008)
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)
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)
8. Bohm M., Showalter R.E.: A nonlinear pseudoparabolic diffusion equation. SIAM J. Math. Anal. 16(5), 980–999 (1985b)
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)
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)
12. Christensen J.: Modeling diffusion-induced stress in Li-ion cells with porous electrodes. J. Electrochem. Soc. 157(3), A366–A380 (2010)
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)
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)
15. Darling R., Newman J.: Modeling side reactions in composite Liy Mn2O4 electrodes. J. Electrochem. Soc. 145(3), 990–998 (1998)
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)
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)
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)
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)
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)
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)
24. Edwin Garcia R., Chiang Y.-M.: Spatially resolved modeling of microstructurally complex battery architectures. J. Electrochem. Soc. 154(9), A856–A864 (2007)
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)
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)
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)
28. Gu W.B., Wang C.Y.: Thermal-electrochemical modeling of battery systems. J. Electrochem. Soc. 147(8), 2910–2922 (2000)
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)
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)
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)
34. Lai, W., Ciucci, F.: Mathematical modeling of porous battery electrodes—revisit of newman’s model. Electrochim. Acta (2010) doi:
35. Lai W., Ciucci F.: Thermodynamics and kinetics of phase transformation in intercalation battery electrodes—phenomenological modeling. Electrochim. Acta 56(1), 531–542 (2010)
36. Lai W., Ciucci F.: Small-signal apparent diffusion impedance of intercalation battery electrodes. J. Electrochem. Soc. 158(2), A115–A121 (2011)
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)
38. Looker J., Carnie S.: Homogenization of the ionic transport equations in periodic porous media. Transp. Porous Med. 65(1), 107–131 (2006)
39. Moyne C., Murad M.: Macroscopic behavior of swelling porous media derived from micromechanical analysis. Transp. Porous Med. 50(1), 127–151 (2003)
40. Newman J., Tiedemann W.: Porous-electrode theory with battery applications. AIChE J. 21(1), 25–41 (1975)
41. Nolen J., Papanicolaou G., Pironneau O.: A framework for adaptive multiscale methods for elliptic problems. Multiscale Model. Simulation 7(1), 171–196 (2008)
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)
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)
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)
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)
48. Showalter R.E.: Diffusion models with microstructure. Transp. Porous Med. 6(5), 567–580 (1991)
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)
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)
53. Wang C.-W., Sastry A.M.: Mesoscale modeling of a Li-ion polymer cell. J. Electrochem. Soc. 154(11), A1035–A1047 (2007)
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)
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)
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)