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Understanding size-dependent migration of a two-phase lithiation front coupled to stress

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Abstract

Recent in-situ experiments show that stress-driven migration of the phase interface during two-phase lithiation in nanosized particles exhibits self-limiting and size-dependent behaviors wherein the mechanism remains unclear. In the reaction-limited regime, we develop a mechano-kinetic coupling model with a nonlinear kinetic law to study the size effect of such phase boundary movement accounting for possible sources of stresses by chemical lithiation, concurrent plasticity, surface/interface elasticity, and elastic softening of the lithiated phase. We show that both hydrostatic and non-hydrostatic stresses contribute to the driving force for the phase interface movement and result in the size-dependent slowing down behavior of the phase interface. The obtained results reveal why the interface movement slows down more dramatically in the smaller particle, and there are similar lithiation time scales in nanoparticles of different sizes observed in experiments.

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References

  1. Tarascon, J.M., Armand, M.: Issues and challenges facing rechargeable lithium batteries. Nature 414(6861), 359–367 (2001). https://doi.org/10.1038/35104644

    Article  Google Scholar 

  2. Chan, C.K., Peng, H., Liu, G., Mcllwrath, K., Zhang, X.F., Huggins, R.A., Cui, Y.: High-performance lithium battery anodes using silicon nanowires. Nat. Nanotech. 3(1), 31–35 (2008). https://doi.org/10.1038/nnano.2007.411

    Article  Google Scholar 

  3. Mukhopadhyay, A., Sheldon, B.W.: Deformation and stress in electrode materials for Li-ion batteries. Prog. Mater. Sci. 63, 58–116 (2014). https://doi.org/10.1016/j.pmatsci.2014.02.001

    Article  Google Scholar 

  4. Zhang, L., Song, Y., He, L., Ni, Y.: Variations of boundary reaction rate and particle size on the diffusion-induced stress in a phase separating electrode. J. Appl. Phys. 116(14), 143506 (2014). https://doi.org/10.1063/1.4897459

    Article  Google Scholar 

  5. Zhao, K., Pharr, M., Vlassak, J.J., Suo, Z.: Fracture of electrodes in lithium-ion batteries caused by fast charging. J. Appl. Phys. 108(7), 073517 (2010). https://doi.org/10.1063/1.3492617

    Article  Google Scholar 

  6. Tang, M., Huang, H.Y., Meethong, N., Kao, Y.H., Carter, W.C., Chiang, Y.M.: Model for the particle size, overpotential, and strain dependence of phase transition pathways in storage electrodes: application to nanoscale olivines. Chem. Mater. 21(8), 1557–1571 (2009). https://doi.org/10.1021/Cm803172s

    Article  Google Scholar 

  7. Li, D., Zhou, H.: Two-phase transition of Li-intercalation compounds in Li-ion batteries. Mater. Today 17(9), 451–463 (2014). https://doi.org/10.1016/j.mattod.2014.06.002

    Article  Google Scholar 

  8. Chang, L., Lu, Y., He, L., Ni, Y.: Phase field model for two-phase lithiation in an arbitrarily shaped elastoplastic electrode particle under galvanostatic and potentiostatic operations. Int. J. Solids Struct. 143, 73–83 (2018). https://doi.org/10.1016/j.ijsolstr.2018.02.033

    Article  Google Scholar 

  9. Lu, Y.Y., Ni, Y.: Stress-mediated lithiation in nanoscale phase transformation electrodes. Acta Mech. Solida Sin. 30(3), 248–253 (2017). https://doi.org/10.1016/j.camss.2017.05.004

    Article  Google Scholar 

  10. Liu, X.H., Fan, F., Yang, H., Zhang, S., Huang, J.Y., Zhu, T.: Self-limiting lithiation in silicon nanowires. ACS Nano 7(2), 1495–1503 (2013). https://doi.org/10.1021/nn305282d

    Article  Google Scholar 

  11. McDowell, M.T., Ryu, I., Lee, S.W., Wang, C., Nix, W.D., Cui, Y.: Studying the kinetics of crystalline silicon nanoparticle lithiation with in situ transmission electron microscopy. Adv. Mater. 24(45), 6034–6041 (2012). https://doi.org/10.1002/adma.201202744

    Article  Google Scholar 

  12. Drozdov, A.D., Sommer-Larsen, P., deClaville Christiansen, J.: Self-limiting lithiation of electrode nanoparticles in Li-ion batteries. J. Appl. Phys. 114(22), 223514 (2013). https://doi.org/10.1063/1.4844535

    Article  Google Scholar 

  13. Bazant, M.Z.: Theory of chemical kinetics and charge transfer based on nonequilibrium thermodynamics. Acc. Chem. Res. 46(5), 1144–1160 (2013). https://doi.org/10.1021/ar300145c

    Article  Google Scholar 

  14. Cui, Z., Gao, F., Qu, J.: Two-phase versus two-stage versus multi-phase lithiation kinetics in silicon. Appl. Phys. Lett. 103(14), 143901 (2013). https://doi.org/10.1063/1.4824064

    Article  Google Scholar 

  15. Zhao, K., Pharr, M., Wan, Q., Wang, W.L., Kaxiras, E., Vlassak, J.J., Suo, Z.: Concurrent reaction and plasticity during initial lithiation of crystalline silicon in lithium-ion batteries. J. Electrochem. Soc. 159(3), A238–A243 (2012). https://doi.org/10.1149/2.020203jes

    Article  Google Scholar 

  16. Jia, Z., Li, T.: Stress-modulated driving force for lithiation reaction in hollow nano-anodes. J. Power Sources 275, 866–876 (2015). https://doi.org/10.1016/j.jpowsour.2014.11.081

    Article  Google Scholar 

  17. Bower, A.F., Guduru, P.R., Chason, E.: Analytical solutions for composition and stress in spherical elastic–plastic lithium-ion electrode particles containing a propagating phase boundary. Int. J. Solids Struct. 69–70, 328–342 (2015). https://doi.org/10.1016/j.ijsolstr.2015.05.018

    Article  Google Scholar 

  18. Lu, Y., Chang, L., Yao, H., He, L., Ni, Y.: Transition from deceleration to acceleration of lithiation front movement in hollow phase transformation electrodes. J. Electrochem. Soc. 164(13), A3371–A3379 (2017). https://doi.org/10.1149/2.0781713jes

    Article  Google Scholar 

  19. Gao, F., Hong, W.: Phase-field model for the two-phase lithiation of silicon. J. Mech. Phys. Solids 94, 18–32 (2016). https://doi.org/10.1016/j.jmps.2016.04.020

    Article  MathSciNet  Google Scholar 

  20. Bai, P., Cogswell, D.A., Bazant, M.Z.: Suppression of phase separation in LiFePO\(_4\) nanoparticles during battery discharge. Nano Lett. 11(11), 4890–4896 (2011). https://doi.org/10.1021/nl202764f

    Article  Google Scholar 

  21. Liang, L., Qi, Y., Xue, F., Bhattacharya, S., Harris, S.J., Chen, L.Q.: Nonlinear phase-field model for electrode–electrolyte interface evolution. Phys. Rev. E 86(5 Pt 1), 051609 (2012). https://doi.org/10.1103/PhysRevE.86.051609

    Article  Google Scholar 

  22. Deshpande, R., Qi, Y., Cheng, Y.-T.: Effects of concentration-dependent elastic modulus on diffusion-induced stresses for battery applications. J. Electrochem. Soc. 157(8), A967 (2010). https://doi.org/10.1149/1.3454762

    Article  Google Scholar 

  23. Cui, Z., Gao, F., Cui, Z., Qu, J.: A second nearest-neighbor embedded atom method interatomic potential for Li–Si alloys. J. Power Sources 207, 150–159 (2012). https://doi.org/10.1016/j.jpowsour.2012.01.145

    Article  Google Scholar 

  24. Huang, S., Fan, F., Li, J., Zhang, S., Zhu, T.: Stress generation during lithiation of high-capacity electrode particles in lithium ion batteries. Acta Mater. 61(12), 4354–4364 (2013). https://doi.org/10.1016/j.actamat.2013.04.007

    Article  Google Scholar 

  25. Lu, Y.Y., Ni, Y.: Effects of particle shape and concurrent plasticity on stress generation during lithiation in particulate Li-ion battery electrodes. Mech. Mater. 91, 372–381 (2015). https://doi.org/10.1016/j.mechmat.2015.03.010

    Article  Google Scholar 

  26. Cheng, Y.-T., Verbrugge, M.W.: The influence of surface mechanics on diffusion induced stresses within spherical nanoparticles. J. Appl. Phys. 104(8), 083521 (2008). https://doi.org/10.1063/1.3000442

    Article  Google Scholar 

  27. Liu, Y., Lv, P., Ma, J., Bai, R., Duan, H.L.: Stress fields in hollow core–shell spherical electrodes of lithium ion batteries. Proc. R. Soc. A 470(2172), 20140299–20140299 (2014). https://doi.org/10.1098/rspa.2014.0299

    Article  Google Scholar 

  28. Hao, F., Fang, D.: Diffusion-induced stresses of spherical core-shell electrodes in lithium-ion batteries: the effects of the shell and surface/interface stress. J. Electrochem. Soc. 160(4), A595–A600 (2013). https://doi.org/10.1149/2.054304jes

    Article  Google Scholar 

  29. Yu, H.H., Suo, Z.: Stress-dependent surface reactions and implications for a stress measurement technique. J. Appl. Phys. 87(3), 1211–1218 (2000). https://doi.org/10.1063/1.371999

    Article  Google Scholar 

  30. Cui, Z.W., Gao, F., Qu, J.M.: Interface-reaction controlled diffusion in binary solids with applications to lithiation of silicon in lithium-ion batteries. J. Mech. Phys. Solids 61(2), 293–310 (2013). https://doi.org/10.1016/j.jmps.2012.11.001

    Article  MathSciNet  Google Scholar 

  31. Asaro, R., Lubarda, V.: Mechanics of Solids and Materials. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  32. Shenoy, V.B., Johari, P., Qi, Y.: Elastic softening of amorphous and crystalline Li–Si phases with increasing Li concentration: a first-principles study. J. Power Sources 195(19), 6825–6830 (2010). https://doi.org/10.1016/j.jpowsour.2010.04.044

    Article  Google Scholar 

  33. Fischer, F.D., Svoboda, J.: Stresses in hollow nanoparticles. Int. J. Solids Struct. 47(20), 2799–2805 (2010). https://doi.org/10.1016/j.ijsolstr.2010.06.008

    Article  MATH  Google Scholar 

  34. Gurtin, M.E., Murdoch, A.I.: Continuum theory of elastic-material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)

    Article  MathSciNet  Google Scholar 

  35. Suo, Z.: Motions of microscopic surfaces in materials. Adv. Appl. Mech. 33(33), 193–294 (1997). https://doi.org/10.1016/S0065-2156(08)70387-9

    Article  MathSciNet  MATH  Google Scholar 

  36. Freund, L.B., Suresh, S.: Thin film materials: stress, defect formation and surface evolution. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  37. Zhang, X., Lee, S.W., Lee, H.-W., Cui, Y., Linder, C.: A reaction-controlled diffusion model for the lithiation of silicon in lithium-ion batteries. Extreme Mech. Lett. 4, 61–75 (2015). https://doi.org/10.1016/j.eml.2015.04.005

    Article  Google Scholar 

  38. Cui, Z., Gao, F., Qu, J.: A finite deformation stress-dependent chemical potential and its applications to lithium ion batteries. J. Mech. Phys. Solids 60(7), 1280–1295 (2012). https://doi.org/10.1016/j.jmps.2012.03.008

    Article  MathSciNet  Google Scholar 

  39. Xu, Y.H., Yin, G.P., Zuo, P.J.: Geometric and electronic studies of Li\(_{15}\)Si\(_{4}\) for silicon anode. Electrochim. Acta 54(2), 341–345 (2008). https://doi.org/10.1016/j.electacta.2008.07.083

    Article  Google Scholar 

  40. Liu, X.H., Zheng, H., Zhong, L., Huang, S., Karki, K., Zhang, L.Q., Liu, Y., Kushima, A., Liang, W.T., Wang, J.W., Cho, J.H., Epstein, E., Dayeh, S.A., Picraux, S.T., Zhu, T., Li, J., Sullivan, J.P., Cumings, J., Wang, C., Mao, S.X., Ye, Z.Z., Zhang, S., Huang, J.Y.: Anisotropic swelling and fracture of silicon nanowires during lithiation. Nano Lett. 11(8), 3312–3318 (2011). https://doi.org/10.1021/nl201684d

    Article  Google Scholar 

  41. Sethuraman, V.A., Chon, M.J., Shimshak, M., Srinivasan, V., Guduru, P.R.: In situ measurements of stress evolution in silicon thin films during electrochemical lithiation and delithiation. J. Power Sources 195(15), 5062–5066 (2010). https://doi.org/10.1016/j.jpowsour.2010.02.013

    Article  Google Scholar 

  42. Zang, J.-L., Zhao, Y.-P.: A diffusion and curvature dependent surface elastic model with application to stress analysis of anode in lithium ion battery. Int. J. Eng. Sci. 61, 156–170 (2012). https://doi.org/10.1016/j.ijengsci.2012.06.018

    Article  MathSciNet  MATH  Google Scholar 

  43. Bernstein, N., Aziz, M.J., Kaxiras, E.: Amorphous-crystal interface in silicon: a tight-binding simulation. Phys. Rev. B 58(8), 4579–4583 (1998). https://doi.org/10.1103/PhysRevB.58.4579

    Article  Google Scholar 

  44. Deshpande, R., Cheng, Y.T., Verbrugge, M.W., Timmons, A.: Diffusion induced stresses and strain energy in a phase-transforming spherical electrode particle. J. Electrochem. Soc. 158(6), A718–A724 (2011). https://doi.org/10.1149/1.3565183

    Article  Google Scholar 

  45. Christensen, J., Newman, J.: A mathematical model of stress generation and fracture in lithium manganese oxide. J. Electrochem. Soc. 153(6), A1019–A1030 (2006). https://doi.org/10.1149/1.2185287

    Article  Google Scholar 

  46. 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). https://doi.org/10.1016/j.jmps.2014.05.001

    Article  MathSciNet  MATH  Google Scholar 

  47. Gösele, U., Tu, K.N.: “Critical thickness” of amorphous phase formation in binary diffusion couples. J. Appl. Phys. 66(6), 2619 (1989). https://doi.org/10.1063/1.344229

    Article  Google Scholar 

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Acknowledgements

We would like to thank A. F. Bower for helpful discussions and valuable suggestions. Y. Ni was supported by the National Natural Science Foundation of China (Grant No. 11472262), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB22040502), the Collaborative Innovation Center of Suzhou Nano Science and Technology, and the Fundamental Research Funds for the Central Universities. A.K. Soh was supported by the Advanced Engineering Programme, Monash University Malaysia.

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Lu, Y., Soh, A.K., Ni, Y. et al. Understanding size-dependent migration of a two-phase lithiation front coupled to stress. Acta Mech 230, 303–317 (2019). https://doi.org/10.1007/s00707-018-2303-3

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  • DOI: https://doi.org/10.1007/s00707-018-2303-3

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