Skip to main content
SpringerLink
Log in

Analytical modeling and simulation of porous electrodes: Li-ion distribution and diffusion-induced stress

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

A new model of porous electrodes based on the Gibbs free energy is developed, in which lithium-ion (Li-ion) diffusion, diffusion-induced stress (DIS), Butler–Volmer (BV) reaction kinetics, and size polydispersity of electrode particles are considered. The influence of BV reaction kinetics and concentration-dependent exchange current density (ECD) on concentration profile and DIS evolution are numerically investigated. BV reaction kinetics leads to a decrease in Li-ion concentration and DIS. In addition, concentration-dependent ECD results in a decrease in Li-ion concentration and an increase in DIS. Size polydispersity of electrode particles significantly affects the concentration profile and DIS. Optimal macroscopic state of charge (SOC) should consider the influence of the microscopic SOC values and mass fractions of differently sized particles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Dunn, B., Kamath, H., Tarascon, J.M.: Electrical energy storage for the grid: a battery of choices. Science 334, 928–935 (2011)

    Article  Google Scholar 

  2. Palacin, M.R., de Guibert, A.: Why do batteries fail? Science 351, 1253292 (2016)

    Article  Google Scholar 

  3. Shpigel, N., Levi, M.D., Sigalov, S., et al.: In situ hydrodynamic spectroscopy for structure characterization of porous energy storage electrodes. Nat. Mater. 15, 570–575 (2016)

    Article  Google Scholar 

  4. Liu, H., Strobridge, F.C., Borkiewiez, O.J., et al.: Capturing metastable structures during high-rate cycling of LiFePO\(_4\) nanoparticle electrodes. Science 344, 1252817 (2014)

    Article  Google Scholar 

  5. Banerjee, J., Dutta, K.: Materials for electrodes of Li-ion batteries: issues related to stress development. Crit. Rev. Solid State 42, 218–238 (2017)

    Article  Google Scholar 

  6. Prussin, S.: Generation and distribution of dislocations by solute diffusion. J. Appl. Phys. 32, 1876–1881 (1961)

    Article  Google Scholar 

  7. Malavé, V., Berger, J.R., Zhu, H., et al.: A computational model of the mechanical behavior within reconstructed Li\(_{x}\)CoO\(_{2}\) Li-ion battery cathode particles. Electrochim. Acta 130, 707–717 (2014)

    Article  Google Scholar 

  8. Verbrugge, M.W., Cheng, Y.T.: Stress and strain-energy distributions within diffusion-controlled insertion-electrode particles subjected to periodic potential excitations. J. Electrochem. Soc. 156, A927–A937 (2009)

    Article  Google Scholar 

  9. Li, Y., Zhang, K., Zheng, B.: Interaction between diffusion and stresses in composition-gradient electrodes. Solid State Ion. 283, 103–108 (2015)

    Article  Google Scholar 

  10. Di Leo, C.V., Rejovitzky, E., Anand, L.: A Cahn–Hilliard-type phase-field theory for species diffusion coupled with large elastic deformations: application to phase-separating Li-ion electrode materials. J. Mech. Phys. Solids 70, 1–29 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Stein, P., Xu, B.: 3D Isogeometric analysis of intercalation-induced stresses in Li-ion battery electrode particles. Comput. Methods Appl. Mech. 268, 225–244 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zuo, P., Zhao, Y.P.: A phase field model coupling lithium diffusion and stress evolution with crack propagation and application in lithium ion batteries. Phys. Chem. Chem. Phys. 17, 287–297 (2015)

    Article  Google Scholar 

  13. Zuo, P., Zhao, Y.P.: Phase field modeling of lithium diffusion, finite deformation, stress evolution and crack propagation in lithium ion battery. Extreme Mech. Lett. 9, 467–479 (2016)

    Article  Google Scholar 

  14. Woodford, W.H., Chiang, Y.M., Carter, W.C.: “Electrochemical shock” of intercalation electrodes: a fracture mechanics analysis. J. Electrochem. Soc. 157, A1052–A1059 (2010)

    Article  Google Scholar 

  15. Woodford, W.H., Carter, W.C., Chiang, Y.M.: Strategies to avert electrochemical shock and their demonstration in spinels. J. Electrochem. Soc. 161, F3005–F3009 (2014)

    Article  Google Scholar 

  16. Vanimisetti, S.K., Ramakrishnan, N.: Effect of the electrode particle shape in Li-ion battery on the mechanical degradation during charge–discharge cycling. J. Mech. Eng. Sci. 226, 2192–2213 (2011)

    Article  Google Scholar 

  17. Zhang, T., Guo, Z.: Effects of electrode properties and fabricated pressure on Li ion diffusion and diffusion-induced stresses in cylindrical Li-ion batteries. Model. Simul. Mater. Sci. 22, 025016 (2014)

    Article  Google Scholar 

  18. Deshpande, R., Cheng, Y.T., Verbrugge, M.W., et al.: Diffusion induced stresses and strain energy in a phase-transforming spherical electrode particle. J. Electrochem. Soc. 158, A718–A724 (2011)

    Article  Google Scholar 

  19. Lim, C., Yan, B., Yin, L.: Simulation of diffusion-induced stress using reconstructed electrodes particle structures generated by micro/nano-CT. Electrochim. Acta 75, 279–287 (2012)

    Article  Google Scholar 

  20. Heubner, C., Schneider, M., Michaelis, A.: Investigation of charge transfer kinetics of Li-Intercalation in LiFePO\(_{4}\). J. Power Sources 288, 115–120 (2015)

    Article  Google Scholar 

  21. Guo, Y., Smith, R.B., Yu, Z., et al.: Li intercalation into graphite: direct optical imaging and Cahn–Hilliard reaction dynamics. J. Phys. Chem. Lett. 7, 2151–2156 (2016)

    Article  Google Scholar 

  22. Kuzmin, R.N., Maximov, D.S., Savenkova, N.P., et al.: Mathematical modeling of hysteresis in porous electrodes. J. Math. Chem. 50, 2471–2477 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Golmon, S., Maute, K., Lee, S.H., et al.: Stress generation in silicon particles during lithium insertion. Appl. Phys. Lett. 97, 033111 (2010)

    Article  Google Scholar 

  24. Jagannathan, M., Chandran, K.S.R.: Analytical modeling and simulation of electrochemical charge/discharge behavior of Si thin film negative electrodes in Li-ion cells. J. Power Sources 247, 667–675 (2014)

    Article  Google Scholar 

  25. Swamy, T., Chiang, Y.M.: Electrochemical charge transfer reaction kinetics at the silicon–liquid electrolyte interface. J. Electrochem. Soc. 162, A7129–A7134 (2015)

    Article  Google Scholar 

  26. Richardson, G., Denuault, G., Please, C.P.: Multiscale modelling and analysis of lithium-ion battery charge and discharge. J. Eng. Math. 72, 41–72 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, S., Jiang, J., Shi, W., et al.: Butler–Volmer-equation-based electrical model for high-power lithium titanate batteries used in electric vehicles. IEEE. Trans. Ind. Electron. 62, 7557–7568 (2015)

    Article  Google Scholar 

  28. Hess, A., Roode-Gutzmer, Q., Heubner, C., et al.: Determination of state of charge-dependent asymmetric Butler–Volmer kinetics for LixCoO\(_{2}\) electrode using GITT measurements. J. Power Sources 299, 156–161 (2015)

    Article  Google Scholar 

  29. Gwak, Y., Moon, J., Cho, M.: Multi-scale analysis of an electrochemical model including coupled diffusion, stress, and nonideal solution in a silicon thin film anode. J. Power Sources 307, 856–865 (2016)

    Article  Google Scholar 

  30. Zhang, Q., Guo, Q., White, R.E.: A new kinetic equation for intercalation electrodes. J. Electrochem. Soc. 153, A301–A309 (2006)

    Article  Google Scholar 

  31. Latz, A., Zausch, J.: Thermodynamic derivation of a Butler-Volmer model for intercalation in Li-ion batteries. Electrochim. Acta 110, 358–362 (2013)

    Article  Google Scholar 

  32. Ferguson, T.R., Bazant, M.Z.: Nonequilibrium thermodynamics of porous electrodes. J. Electrochem. Soc. 159, A1967–A1985 (2012)

    Article  Google Scholar 

  33. Lee, J.K., Yoon, W.Y., Kim, B.K.: Kinetics of reaction products of silicon monoxide with controlled amount of Li-ion insertion at various current densities for Li-ion batteries. J. Electrochem. Soc. 161, A927–A933 (2014)

    Article  Google Scholar 

  34. Bazant, M.Z.: Theory of chemical kinetics and charge transfer based on nonequilibrium thermodynamics. Accounts Chem. Res. 46, 1144–1160 (2013)

    Article  Google Scholar 

  35. Levi, M.D., Aurbach, D.: Kinetics of electrochemically induced phase transitions in ion-insertion electrodes and the chemical diffusion coefficient. J. Solid State Electr. 12, 409–420 (2007)

    Article  Google Scholar 

  36. Bates, A., Mukherjee, S., Schuppert, N., et al.: Modeling and simulation of 2D lithium-ion solid state battery. Int. J. Energy Res. 39, 1505–1518 (2015)

    Article  Google Scholar 

  37. Okajima, Y., Shibuta, Y., Suzuki, T.: A phase-field model for electrode reactions with Butler–Volmer kinetics. Comput. Mater. Sci. 50, 118–124 (2010)

    Article  Google Scholar 

  38. Zhao, Y., Xu, B.X., Stein, P., et al.: Phase-field study of electrochemical reactions at exterior and interior interfaces in Li-ion battery electrode particles. Comput. Methods Appl. Mech. Eng. 312, 428–446 (2016)

    Article  MathSciNet  Google Scholar 

  39. Xu, B.X., Zhao, Y., Stein, P.: Phase field modeling of electrochemically induced fracture in Li-ion battery with large deformation and phase segregation. GAMM-Mitt. 39, 92–109 (2016)

    Article  MathSciNet  Google Scholar 

  40. Newman, J., Tobias, C.: Theoretical analysis of current distribution in porous electrodes. J. Electrochem. Soc. 109, 1183–1191 (1962)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  43. Lai, W., Ciucci, F.: Mathematical modeling of porous battery electrodes—Revisit of Newman’s model. Electrochim. Acta 56, 4369–4377 (2011)

    Article  Google Scholar 

  44. Landstorfer, M., Jacob, T.: Mathematical modeling of intercalation batteries at the cell level and beyond. Chem. Soc. Rev. 42, 3234–3252 (2013)

    Article  Google Scholar 

  45. Zheng, J.P., Crain, D.J., Roy, D.: Kinetic aspects of Li intercalation in mechano-chemically processed cathode materials for lithium ion batteries: electrochemical characterization of ball-milled LiMn\(_{2}\)O\(_{4}\). Solid State Ion. 196, 48–58 (2011)

    Article  Google Scholar 

  46. Chung, D.W., Shearing, P.R., Brandon, N.P., et al.: Particle size polydispersity in Li-ion bateries. J. Electrochem. Soc. 161, A422–A430 (2014)

    Article  Google Scholar 

  47. Wu, W., Xiao, X., Wang, M., et al.: A microstructural resolved model for the stress analysis of lithium-ion batteries. J. Electrochem. Soc. 161, A803–A813 (2014)

    Article  Google Scholar 

  48. Jokar, A., Rajabloo, B., Désilets, M., et al.: Review of simplified pseudo-two-dimensional models of lithium-ion batteries. J. Power Sources 327, 44–55 (2016)

    Article  Google Scholar 

  49. Aberg, S.: A simple model for the measurement of charge transfer resistances of reactions with slow kinetics. J. Electroanal. Chem. 439, 63–71 (1997)

  50. Zhang, X., Shyy, W., Sastry, A.M.: Numerical simulation of intercalation-induced stress in Li-ion battery electrode particles. J. Electrochem. Soc. 154, A910–A916 (2007)

    Article  Google Scholar 

  51. Zhang, X., Sastry, A.M., Shyy, W.: Intercalation-induced stress and heat generation within single lithium-ion battery cathode particles. J. Electrochem. Soc. 155, A542–A552 (2008)

    Article  Google Scholar 

  52. Guo, Z., Zhu, J., Feng, J., et al.: Direct in situ observation and explanation of lithium dendrite of commercial graphite electrodes. RSC Adv. 5, 69514–69521 (2015)

    Article  Google Scholar 

  53. Yang, L., Yan, Y., Ran, Z., et al.: A new method for generating random fibre distributions for fibre reinforced composites. Compos. Sci. Technol. 76, 14–20 (2013)

    Article  Google Scholar 

  54. Takahashi, K., Srinivasan, V.: Examination of graphite particle cracking as a failure mode in lithium-ion batteries: a model-experimental study. J. Electrochem. Soc. 162, A635–A645 (2015)

    Article  Google Scholar 

  55. Takahashi, K., Higa, K., Mair, S., et al.: Mechanical degradation of graphite/PVDF composite electrodes: a model-experimental study. J. Electrochem. Soc. 163, A385–A395 (2016)

    Article  Google Scholar 

  56. Orvananos, B., Malik, R., Yu, H.C., et al.: Architecture dependence on the dynamics of nano-LiFePO\(_4\) electrodes. Electrochim. Acta 137, 245–257 (2014)

    Article  Google Scholar 

  57. Elul, S., Cohen, Y., Aurbach, D.: The influence of geometry in 2D simulation on the charge/discharge processes in Li-ion batteries. J. Electroanal. Chem. 682, 53–65 (2012)

    Article  Google Scholar 

  58. Jung, S., Jung, H.Y.: Charge/discharge characteristics of Li-ion batteries with two-phase active materials: a comparative study of LiFePO\(_{4}\) and LiCoO\(_{2}\) cells. Int. J. Energy Res. 40, 1541–1555 (2016)

    Article  Google Scholar 

  59. Lim, C., Yan, B., Kang, H., et al.: Analysis of geometric and electrochemical characteristics of lithium cobalt oxide electrode with different packing densities. J. Power Sources 328, 46–55 (2016)

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the financial support by the National Natural Science Foundation of China (Grants 11472165, 11332005).

Author information

Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, China

    Liang Ji & Zhansheng Guo

  2. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai, 200072, China

    Zhansheng Guo

Authors
  1. Liang Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhansheng Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhansheng Guo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ji, L., Guo, Z. Analytical modeling and simulation of porous electrodes: Li-ion distribution and diffusion-induced stress. Acta Mech. Sin. 34, 187–198 (2018). https://doi.org/10.1007/s10409-017-0704-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-017-0704-5

Keywords

Search

Navigation