Gradient Elasticity Effects on the Two-Phase Lithiation of LIB Anodes

  • Ioannis Tsagrakis
  • Elias C. AifantisEmail author
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 90)


A coupled gradient chemoelasticity theory is employed to model the two-phase mechanism that occurs during lithiation of silicon nanoparticles used to fabricate next generation Li-ion battery (LIB) anodes. It is shown that the strain gradient length scale is able to predict the propagation of an interface front of nonzero thickness advancing from the lithiated to unlithiated region without necessarily including higher-order concentration gradients of the Li ions. Larger strain gradient coefficients (elastic internal lengths) induce more diffused interfaces and faster lithiation, which affect both internal strain and stress distributions in a similar way. Estimates for the migration velocity of the phase boundary are obtained and a range of values of the strain gradient length scale is shown to simulate the observed experimental results.


Internal Length Strain Gradient Full Lithiation Dimensionless Material Parameter Chemical Free Energy Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The input and discussions with Professor Katerina Aifantis of the University of Florida on the topic of LIBs were very useful and deeply appreciated. The support of the Ministry of Education and Science of Russian Federation under Mega-Grant No.14.Z50.31.0039 is also gratefully acknowledged.


  1. 1.
    Aifantis, K.E., Hackney, S.A.: An ideal elasticity problem for Li-batteries. J. Mech. Behav. Mater. 14(6), 413–427 (2003). Scholar
  2. 2.
    Aifantis, K.E., Dempsey, J.P.: Stable crack growth in nanostructured Li-batteries. J. Power Sources 143(1–2), 203–211 (2005). Scholar
  3. 3.
    Dimitrijevic, B.J., Aifantis, K.E., Hackl, K.: The influence of particle size and spacing on the fragmentation of nanocomposite anodes for Li batteries. J. Power Sources 206, 343–348 (2012). Scholar
  4. 4.
    Aifantis, K.E., Hackney, S.A., Kumar, V.R. (Eds.): High Energy Density Lithium Batteries: Materials, Engineering, Applications. Wiley-VCH, Weinheim (2010). Scholar
  5. 5.
    Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958). Scholar
  6. 6.
    Cahn, J.W.: On spinodal decomposition. Acta Metall. 9(9), 795–801 (1961). Scholar
  7. 7.
    Ryu, I., Choi, J.W., Cui, Y., Nix, W.D.: Size-dependent fracture of Si nanowire battery anodes. J. Mech. Phys. Solids 59(9), 1717–1730 (2011). Scholar
  8. 8.
    Bohn, E., Eckl, T., Kamlah, M., McMeeking, R.: A model for lithium diffusion and stress generation in an intercalation storage particle with phase change. J. Electrochem. Soc. 160(10), A1638–A1652 (2013). Scholar
  9. 9.
    Haftbaradaran, H., Song, J., Curtin, W.A., Gao, H.: Continuum and atomistic models of strongly coupled diffusion, stress, and solute concentration. J. Power Sources 196, 361–370 (2011). Scholar
  10. 10.
    Zhao, K., Pharr, M., Cai, S., Vlassak, J.J., Suo, Z.: Large plastic deformation in high-capacity lithium-ion batteries caused by charge and discharge. J. Am. Ceram. Soc. 94(S1), S226–S235 (2011). Scholar
  11. 11.
    Anand, L.: A Cahn–Hilliard-type theory for species diffusion coupled with large elastic-plastic deformations. J. Mech. Phys. Solids 60, 1983–2002 (2012). Scholar
  12. 12.
    Cogswell, D.A., Bazant, M.Z.: Coherency strain and the kinetics of phase separation in LiFePO4 nanoparticles. ACS Nano 6(3), 2215–2225 (2012). Scholar
  13. 13.
    Bagni, C., Askes, H., Aifantis, E.C.: Gradient-enriched finite element methodology for axisymmetric problems. Acta Mech. 228(4), 1423–1444 (2017). Scholar
  14. 14.
    Tsagrakis, I., Aifantis, E.C.: Thermodynamic coupling between gradient elasticity and a Cahn–Hilliard type of diffusion: size-dependent spinodal gaps. Contin. Mech. Thermodyn. (2017). Scholar
  15. 15.
    Tsagrakis, I., Aifantis, E.C.: Gradient and size effects on spinodal and miscibility gaps. Contin. Mech. Thermodyn. (submitted) (2017)Google Scholar
  16. 16.
    Liu, X.H., Wang, J.W., Huang, S., Fan, F., Huang, X., Liu, Y., Krylyuk, S., Yoo, J., Dayeh, S.A., Davydov, A.V., Mao, S.X., Picraux, S.T., Zhang, S., Li, J., Zhu, T., Huang, J.Y.: In situ atomic-scale imaging of electrochemical lithiation in silicon. Natl. Nanotechnol. 7, 749–756 (2012). Scholar
  17. 17.
    Wang, J.W., He, Y., Fan, F., Liu, X.H., Xia, S., Liu, Y., Harris, C.T., Li, H., Huang, J.Y., Mao, S.X., Zhu, T.: Two-phase electrochemical lithiation in amorphous silicon. Nano Lett. 13(2), 709–715 (2013). Scholar
  18. 18.
    Chen, L., Fan, F., Hong, L., Chen, J., Ji, Y.Z., Zhang, S.L., Zhu, T., Chen, L.Q.: A phase-field model coupled with large elasto-plastic deformation: application to lithiated silicon electrodes. J. Electrochem. Soc. 161(11), F3164–F3172 (2014). Scholar
  19. 19.
    Xie, Z., Ma, Z., Wang, Y., Zhou, Y., Lu, C.: A kinetic model for diffusion and chemical reaction of silicon anode lithiation in lithium ion batteries. RSC Adv. 6, 22383–22388 (2016). Scholar
  20. 20.
    Beaulieu, L.Y., Eberman, K.W., Turner, R.L., Krause, L.J., Dahna, J.R.: Colossal reversible volume changes in lithium alloys. Electrochem. Solid-State Lett. 4(9), A137–A140 (2001). Scholar
  21. 21.
    Berla, L.A., Lee, S.W., Cui, Y., Nix, W.D.: Mechanical behavior of electrochemically lithiated silicon. J. Power Sources 273, 41–51 (2015). Scholar
  22. 22.
    Aifantis, E.C., Serrin, J.B.: The mechanical theory of fluid interfaces and Maxwell’s rule. J. Colloid Interface Sci. 96(2), 517–529 (1983). Scholar
  23. 23.
    Aifantis, E.C., Serrin, J.B.: Equilibrium solutions in the mechanical theory of fluid microstructures. J. Colloid Interface Sci. 96(2), 530–547 (1983). Scholar
  24. 24.
    Burch, D., Bazant, M.Z.: Size-dependent spinodal and miscibility gaps for intercalation in nanoparticles. Nano Lett. 9(11), 3795–3800 (2009). Scholar
  25. 25.
    Bockris, J.O’M., Reddy, A.K.N., Gamboa-Aldeco, M.E.: Modern Electrochemistry 2A: Fundamentals of Electrodics, 2nd edn, p. 1213. Kluwer Academic Publishers (2002).
  26. 26.
    Purkayastha, R., McMeeking, R.: A parameter study of intercalation of lithium into storage particles in a lithium-ion battery. Comput. Mater. Sci. 80, 2–14 (2013). Scholar
  27. 27.
    Ding, N., Xu, J., Yao, Y.X., Wegner, G., Fang, X., Chen, C.H., Lieberwirth, I.: Determination of the diffusion coefficient of lithium ions in nano-Si. Solid State Ionics 180, 222–225 (2009). Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Aristotle University of ThessalonikiThessalonikiGreece
  2. 2.Michigan Technological UniversityHoughtonUSA
  3. 3.Beijing University of Civil Engineering and ArchitectureBeijingChina
  4. 4.ITMO UniversitySt. PetersburgRussia
  5. 5.Togliatti State UniversityTogliattiRussia

Personalised recommendations