Research Papers

Concentration-Dependent Chemical Expansion in Lithium-Ion Battery Cathode Particles

[+] Author and Article Information
Veruska Malavé

Department of Mechanical Engineering,
Colorado School of Mines,
Golden, CO 80401
e-mail: vmalaved@mines.edu

J. R. Berger

Department of Mechanical Engineering,
Colorado School of Mines,
Golden, CO 80401
e-mail: jberger@mines.edu

P. A. Martin

Department of Applied Mathematics
and Statistics,
Colorado School of Mines,
Golden, CO 80401
e-mail: pamartin@mines.edu

Crystal volumetric variations in other LIB electrode materials, such as LixM1∕6Mn2O4 derivative cathodes (M = Cr, Co, and Ni) and Li-alloy anodes, appear to experience linear volume changes with Li content [31,32].

The general spherically symmetric solution of Laplace's equation, ∇2Φ = 0, is Φ(r) = A + B/r.

For practical purposes, the saturation time is considered to be the time needed to lithiate the cathode particle until it reaches the maximum concentration of the compositional range studied (i.e., Li0.55CoO2).

Note that the terms C and x in LixCoO2 are equivalent.

Maximum discharge (at C=0.55) or charge (at C=0.37) is reached at t=0.5.

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 13, 2014; final manuscript received June 5, 2014; accepted manuscript posted June 11, 2014; published online June 23, 2014. Assoc. Editor: Pradeep Sharma.

J. Appl. Mech 81(9), 091005 (Jun 23, 2014) (9 pages) Paper No: JAM-14-1212; doi: 10.1115/1.4027833 History: Received May 13, 2014; Revised June 05, 2014

In this work, the effect of the concentration-dependent chemical-expansion coefficient, β, on the chemo-elastic field in lithium-ion cathode particles is examined. To accomplish this, an isotropic linear-elastic model is developed for a single idealistic particle subjected to potentiostatic-discharge and charge conditions. It is shown that β can be a key parameter in demarcating the chemo-stress–strain state of the cathode material undergoing nonlinear volumetric strains. As an example, such strains develop in the hexagonal-to-monoclinic-phase region of LixCoO2 (0.37 ≤ x ≤ 0.55) and, subsequently, the corresponding β is a linear function of concentration. Previous studies have assumed a constant value for β. Findings suggest that the composition-generated chemo-elastic field that is based on a linear-β dramatically affects both the interdiffusion and the mechanical behavior of the LixCoO2 cathode particle. Because the chemo-elastic phenomena emanate in a reciprocal fashion, the resulting linear β-based hydrostatic-stress gradients significantly aid the diffusion of lithium. Thus, diffusion is accelerated in either electrochemical process that the cathode material undergoes.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Malavé, V., Berger, J. R., Zhu, H., and Kee, R. J., 2014, “A Computational Model of the Mechanical Behavior Within Reconstructed LixCoO2 Li-Ion Battery Cathode Particles,” Electrochim. Acta, 130, pp. 707–717. [CrossRef]
Larché, F. C., and Cahn, J. W., 1985, “The Interactions of Composition and Stress in Crystalline Solids,” Acta Metall., 33(3), pp. 331–357. [CrossRef]
Bohn, E., Eckl, T., Kamlah, M., and McMeeking, R., 2013, “A Model for Lithium Diffusion and Stress Generation in an Intercalation Storage Particle With Phase Change,” J. Electrochem. Soc., 160(10), pp. A1638–A1652. [CrossRef]
Qi, Y., Guo, H., Hector, L. G., and Timmons, A., 2010, “Threefold Increase in the Young's Modulus of Graphite Negative Electrode During Lithium Intercalation,” J. Electrochem. Soc., 157(5), pp. A558–A566. [CrossRef]
Deshpande, R., Qi, Y., and Cheng, Y.-T., 2010, “Effects of Concentration-Dependent Elastic Modulus on Diffusion-Induced Stresses for Battery Applications,” J. Electrochem. Soc., 157(8), pp. A967–A971. [CrossRef]
Shenoy, V. B., Johari, P., and Qi, Y., 2010, “Elastic Softening of Amorphous and Crystalline Li-Si Phases With Increasing Li Concentration: A First-Principles Study,” J. Power Sources, 195(19), pp. 6825–6830. [CrossRef]
Gao, Y. F., and Zhou, M., 2011, “Strong Stress-Enhanced Diffusion in Amorphous Lithium Alloy Nanowire Electrodes,” J. Appl. Phys., 109(1), p. 014310. [CrossRef]
Yang, B., He, Y.-P., Irsa, J., Lundgren, C. A., Ratchford, J. B., and Zhao, Y.-P., 2012, “Effects of Composition-Dependent Modulus, Finite Concentration and Boundary Constraint on Li-Ion Diffusion and Stresses in a Bilayer Cu-Coated Si Nano-Anode,” J. Power Sources, 204, pp. 168–176. [CrossRef]
Stournara, M. E., Guduru, P. R., and Shenoy, V. B., 2012, “Elastic Behavior of Crystalline Li-Sn Phases With Increasing Li Concentration,” J. Power Sources, 208, pp. 165–169. [CrossRef]
He, Y.-L., Hu, H. J., Song, Y.-C., Guo, Z.-S., Liu, C., and Zhang, J.-Q., 2014, “Effects of Concentration-Dependent Elastic Modulus on the Diffusion of Lithium Ions and Diffusion Induced Stress in Layered Battery Electrodes,” J. Power Sources, 248, pp. 517–523. [CrossRef]
Lantelme, F., Groult, H., and Kumagai, N., 2000, “Study of the Concentration-Dependent Diffusion in Lithium Batteries,” Electrochim. Acta, 45(19), pp. 3171–3180. [CrossRef]
Renganathan, S., and White, R. E., 2011, “Semianalytical Method of Solution for Solid Phase Diffusion in Lithium Ion Battery Electrodes: Variable Diffusion Coefficient,” J. Power Sources, 196(1), pp. 442–448. [CrossRef]
Crank, J., 1975, The Mathematics of Diffusion, Oxford University, New York.
Purkayastha, R. T., and McMeeking, R. M., 2012, “An Integrated 2-D Model of a Lithium Ion Battery: The Effect of Material Parameters and Morphology on Storage Particle Stress,” Comput. Mech., 50(2), pp. 209–227. [CrossRef]
Darling, R., and Newman, J., 1998, “Diffusion in LiyMn2O4,” 193rd Meeting of the Electrochemical Society, San Diego, CA, May 3–8, Vol. 98–10, pp. 1–13.
Darling, R., and Newman, J., 1999, “Dynamic Monte Carlo Simulations of Diffusion in LiyMn2O4,” J. Electrochem. Soc., 146(10), pp. 3765–3772. [CrossRef]
Purkayastha, R., and McMeeking, R. M., 2012, “A Linearized Model for Lithium Ion Batteries and Maps for Their Performance and Failure,” ASME J. Appl. Mech., 79(3), p. 031021. [CrossRef]
Garcia, R. E., Chiang, Y.-M., Carter, W. C., Limthongkul, P., and Bishop, C. M., 2005, “Microstructural Modeling and Design of Rechargeable Lithium-Ion Batteries,” J. Electrochem. Soc., 152(1), pp. A255–A263. [CrossRef]
Zhang, X., Shyy, W., and Sastry, A. M., 2007, “Numerical Simulation of Intercalation-Induced Stress in Li-Ion Battery Electrode Particles,” J. Electrochem. Soc., 154(10), pp. A910–A916. [CrossRef]
Zhang, X., Sastry, A. M., and Shyy, W., 2008, “Intercalation-Induced Stress and Heat Generation Within Single Lithium-Ion Battery Cathode Particles,” J. Electrochem. Soc., 155(7), pp. A542–A552. [CrossRef]
Woodford, W. H., Chiang, Y.-M., and Carter, W. C., 2010, “Electrochemical Shock of Intercalation Electrodes: A Fracture Mechanics Analysis,” J. Electrochem. Soc., 157(10), pp. A1052–A1059. [CrossRef]
Chung, D.-W., Balke, N., Kalinin, S. V., and Garcia, R. E., 2011, “Virtual Electrochemical Strain Microscopy of Polycrystalline LiCoO2 Films,” J. Electrochem. Soc., 158(10), pp. A1083–A1089. [CrossRef]
Park, J., Lu, W., and Sastry, A. M., 2011, “Numerical Simulation of Stress Evolution in Lithium Manganese Dioxide Particles Due to Coupled Phase Transition and Intercalation,” J. Electrochem. Soc., 158(2), pp. A201–A206. [CrossRef]
Seo, J. H., Chung, M., Park, M., Han, S. W., Zhang, X., and Sastry, A. M., 2011, “Generation of Realistic Particle Structures and Simulations of Internal Stress: A Numerical/AFM Study of LiMn2O4 Particles,” J. Electrochem. Soc., 158(4), pp. A434–A442. [CrossRef]
Song, Y., Lu, B., Ji, X., and Zhang, J., 2012, “Diffusion Induced Stresses in Cylindrical Lithium-Ion Batteries: Analytical Solutions and Design Insights,” J. Electrochem. Soc., 159(12), pp. A2060–A2068. [CrossRef]
Lim, C., Yan, B., Yin, L., and Zhu, L., 2012, “Simulation of Diffusion-Induced Stress Using Reconstructed Electrodes Particle Structures Generated by Micro/Nano-CT,” Electrochim. Acta, 75, pp. 279–287. [CrossRef]
Yoshio, M., Brodd, R. J., and Kozawa, A., eds., 2010, Lithium-Ion Batteries: Science and Technology, Springer, New York.
Reimers, J. N., and Dahn, J. R., 1992, “Electrochemical and In Situ X-Ray Diffraction Studies of Lithium Intercalation in LixCoO2,” J. Electrochem. Soc., 139(8), pp. 2091–2097. [CrossRef]
Allen, J. L., Ding, M. S., Xu, K., Zhang, S., and Jow, T. R., 2004, “Li1+xFe1-xPO4: Electronically Conductive Lithium Iron Phospho-Olivines With Improved Electrochemical Performance,” 205th Electrochemical Society Fall Meeting, Honolulu, HI, October 3–8, Vol. 28, pp. 198–204.
Ohzuku, T., and Makimura, Y., 2006, “Formation of Solid Solution and Its Effect on Lithium Insertion Schemes for Advanced Lithium-Ion Batteries: X-Ray Absorption Spectroscopy and X-Ray Diffraction of LiCoO2, LiCo1∕2Ni1∕2O2, and LiNiO2,” Res. Chem. Intermed., 32(5), pp. 507–521. [CrossRef]
Huggins, R. A., 2009, Advanced Batteries: Materials Science Aspects, Springer, New York.
Wakihara, M., and Yamamoto, O., eds., 1998, Lithium Ion Batteries: Fundamentals and Performance, Wiley, Weinheim, Germany.
Mal, A. K., and Singh, S. J., 1991, Deformation of Elastic Solids, Prentice-Hall, Englewood Cliffs, NJ.
Yang, F., 2005, “Interaction Between Diffusion and Chemical Stresses,” Mater. Sci. Eng. A, 409(1–2), pp. 153–159. [CrossRef]
Cheng, Y.-T., and Verbrugge, M. W., 2009, “Evolution of Stress Within a Spherical Insertion Electrode Particle Under Potentiostatic and Galvanostatic Operation,” J. Power Sources, 190(2), pp. 453–460. [CrossRef]
Hao, F., Gao, X., and Fang, D., 2012, “Diffusion-Induced Stresses of Electrode Nanomaterials in Lithium-Ion Battery: The Effects of Surface Stress,” J. Appl. Phys., 112(10), p. 103507. [CrossRef]
Song, Y., Shao, X., Guo, Z., and Zhang, J., 2013, “Role of Material Properties and Mechanical Constraint on Stress-Assisted Diffusion in Plate Electrodes of Lithium Ion Batteries,” J. Phys. D: Appl. Phys., 46(10), p. 105307. [CrossRef]
Timoshenko, S., and Goodier, J. N., 1970, Theory of Elasticity, McGraw-Hill, New York.
Smith, G. D., 1985, Numerical Solution of Partial Differential Equations, 3rd ed., Oxford University, Oxford, UK.
Wiedemann, A. H., Goldin, G. M., Barnett, S. A., Zhu, H., and Kee, R. J., 2013, “Effects of Three-Dimensional Cathode Microstructure on the Performance of Lithium-Ion Battery Cathodes,” Electrochim. Acta, 88, pp. 580–588. [CrossRef]
Christensen, J., and Newman, J., 2006, “A Mathematical Model of Stress Generation and Fracture in Lithium Manganese Oxide,” J. Electrochem. Soc., 153(6), pp. A1019–A1030. [CrossRef]


Grahic Jump Location
Fig. 1

LixCoO2 crystal volume as a function of Li content (adapted from Reimers and Dahn [28]). The monoclinic phase is labeled as M1; two hexagonal phases are labeled as H1 and H2.

Grahic Jump Location
Fig. 2

Influence of hypothetical parameters of a linear β∧ on saturation time, t∧s, within the LixCoO2 cathode particle (0.37 ≤x≤ 0.55) at r∧ = 0.5: (a) ξ∧ effect at fixed η∧ = 1.0 and (b) η∧ effect at fixed ξ∧ = 1.0 during potentiostatic discharge conditions

Grahic Jump Location
Fig. 3

Influence of the β∧ parameter at r∧ = 0.5 for normalized values on the (a) Li diffusion and (b) hydrostatic stress during discharge-and-charge potentiostatic conditions (0.37 ≤x≤ 0.55)

Grahic Jump Location
Fig. 4

Evolution of the Li concentration profile across the LixCoO2 particle radius (a)–(d) and hydrostatic stress (e)–(f) under discharge potentiostatic conditions (0.37 ≤x≤ 0.55). Note that no steady state is reached by the reported t∧.

Grahic Jump Location
Fig. 5

Evolution of the Li concentration profile across the LixCoO2 particle radius (a)–(d) and hydrostatic stress (e)–(f) under charge potentiostatic conditions (0.55 ≥ x ≥ 0.37). Note that no steady state is reached by the reported t∧.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In