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Sharp-interface formation during lithium intercalation into silicon

Published online by Cambridge University Press:  28 March 2017

E. MECA ,
A. MÜNCH  and
B. WAGNER
E. MECA
Affiliation:
Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany email: meca@wias-berlin.de
A. MÜNCH
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GGUK email: muench@maths.ox.ac.uk
B. WAGNER
Affiliation:
Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany email: meca@wias-berlin.de Technische Universität Berlin, Institute of Mathematics, Straße des 17. Juni 136, 10623 Berlin, Germany email: wagnerb@wias-berlin.de
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Abstract

In this study, we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as amorphous silicon (a-Si). The governing equations couple a viscous Cahn–Hilliard-Reaction model with elasticity in the framework of the Cahn–Larché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the formation of phase boundaries having a sharp gradient in lithium ion concentration between the initial state of the solid layer and the intercalated region. We carry out a matched asymptotic analysis to derive the corresponding sharp-interface model that also takes into account the dynamics of triple points where the sharp interface intersects the free boundary of the Si layer. We numerically compare the interface motion predicted by the sharp-interface model with the long-time dynamics of the phase-field model.

Keywords

Asymptotic AnalysisPhase-Field ModelStability AnalysisInterface DynamicsNumerical Methods
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 1 , February 2018 , pp. 118 - 145
Copyright
Copyright © Cambridge University Press 2017 

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References

[1] Abels, H. & Schaubeck, S. (2015) Sharp interface limit for the Cahn-Larché system. Asymptotic Anal. 91 (3–4), 283–340.Google Scholar
[2] Bazant, M. Z. (2013) Theory of chemical kinetics and charge transfer based on nonequilibrium thermodynamics. Acc. Chem. Res. 46 (5), 1144–1160.CrossRefGoogle ScholarPubMed
[3] Bower, A. F., Guduru, P. R. & Sethuraman, V. A. (2011) A finite strain model of stress, diffusion, plastic flow, and electrochemical reactions in a lithium-ion half-cell. J. Mech. Phys. Solids 59 (4), 804–828.CrossRefGoogle Scholar
[4] Bronsard, L., Garcke, H. & Stoth, B. (1998) A multi-phase Mullins-Sekerka system: Matched asymptotic expansions and an implicit time discretisation or the geometric evolution problem. Proc. Roy. Soc. of Edingburgh: Sec. A Math. 128, 481–06.Google Scholar
[5] Bucci, G., Nadimpalli, S. P., Sethuraman, V. A., Bower, A. F. & Guduru, P. R. (2014) Measurement and modeling of the mechanical and electrochemical response of amorphous Si thin film electrodes during cyclic lithiation. J. Mech. Phys. Solids 62, 276–294.Google Scholar
[6] Burch, D. & Bazant, M. Z. (2009) Size-dependent spinodal and miscibility gaps for intercalation in nanoparticles. Nano Lett. 9 (11), 3795–3800, pMID: 19824617.Google Scholar
[7] Chakraborty, J., Please, C. P., Goriely, A. & Chapman, S. J. (2015) Combining mechanical and chemical effects in the deformation and failure of a cylindrical electrode particle in a Li-ion battery. International Journal of Solids and Structures 54, 66–81.Google Scholar
[8] Chakraborty, J., Please, C. P., Goriely, A. & Chapman, S. J. (2015) Influence of constraints on axial growth reduction of cylindrical Li-ion battery electrode particles. J. Power Sources 279, 746–758, 9th International Conference on Lead-Acid Batteries (LABAT) 2014.Google Scholar
[9] Cubuk, E. D. & Kaxiras, E. (2014) Theory of structural transformation in lithiated amorphous silicon. Nano Lett. 14 (7), 4065–4070.Google Scholar
[10] Cui, Z., Gao, F. & Qu, J. (2012) A finite deformation stress-dependent chemical potential and its applications to lithium ion batteries. Journal of the Mechanics and Physics of Solids 60 (7), 1280–1295.Google Scholar
[11] Fratzl, P., Penrose, O. & Lebowitz, J. L. (1999) Modeling of phase separation in alloys with coherent elastic misfit. J. Stat. Phys. 95 (5–6), 1429–1503.Google Scholar
[12] Fried, E. & Gurtin, M. E. (1994) Dynamic solid-solid transitions with phase characterized by an order parameter. Physica D: Nonlinear Phenom. 72 (4), 287–308.Google Scholar
[13] Garcke, H. (2003) On Cahn-Hilliard systems with elasticity. Proc. Edinburgh: Sect. A Math. 133 (02), 307–331.Google Scholar
[14] Garcke, H. & Kwak, D. J. C. (2006) On asymptotic limits of Cahn-Hilliard systems with elastic misfit. In: Mielke, A. (editor), Analysis, Modeling and Simulation of Multiscale Problems, Springer, Berlin, pp. 87–111.Google Scholar
[15] Huang, Y., Ngo, D. & Rosakis, A. (2005) Non-uniform, axisymmetric misfit strain: In thin films bonded on plate substrates/substrate systems: The relation between non-uniform film stresses and system curvatures. Acta Mech. Sin. 21 (4), 362–370.Google Scholar
[16] Leo, P., Lowengrub, J. & Jou, H.-J. (1998) A diffuse interface model for microstructural evolution in elastically stressed solids. Acta Mater. 46 (6), 2113–2130.Google Scholar
[17] Leo, P. & Sekerka, R. (1989) Overview no. 86: The effect of surface stress on crystal-melt and crystal-crystal equilibrium. Acta Metall. 37 (12), 3119–3138.Google Scholar
[18] Leo, P. H. & Sekerka, R. (1989) The effect of elastic fields on the morphological stability of a precipitate grown from solid solution. Acta Metall. 37 (12), 3139–3149.Google Scholar
[19] Levitas, V. I. & Attariani, H. (2014) Anisotropic compositional expansion in elastoplastic materials and corresponding chemical potential: Large-strain formulation and application to amorphous lithiated silicon. J. Mech. Phys. Solids 69, 84–111.CrossRefGoogle Scholar
[20] Liu, X. H., Fan, F., Yang, H., Zhang, S., Huang, J. Y. & Zhu, T. (2013) Self-limiting lithiation in silicon nanowires. ACS Nano 7 (2), 1495–1503.Google Scholar
[21] McDowell, M. T., Lee, S. W., Harris, J. T., Korgel, B. A., Wang, C., Nix, W. D. & Cui, Y. (2013) In situ tem of two-phase lithiation of amorphous silicon nanospheres. Nano Lett. 13 (2), 758–764.CrossRefGoogle ScholarPubMed
[22] McDowell, M. T., Lee, S. W., Nix, W. D. & Cui, Y. (2013) 25th anniversary article: Understanding the lithiation of silicon and other alloying anodes for lithium-ion batteries. Adv. Mater. 25 (36), 4966–4985.CrossRefGoogle ScholarPubMed
[23] Meca, E., Münch, A. & Wagner, B. (2016) Thin-film electrodes for high-capacity lithium-ion batteries: Influence of phase transformations on stress. Proc. R. Soc. A: Math. Phys. Eng. Sci. 472 (2193), 20160093.Google Scholar
[24] Meca, E., Shenoy, V. B. & Lowengrub, J. (2013) Phase-field modeling of two-dimensional crystal growth with anisotropic diffusion. Phys. Rev. E 88, 052409.CrossRefGoogle ScholarPubMed
[25] Ngo, D., Huang, Y., Rosakis, A. & Feng, X. (2006) Spatially non-uniform, isotropic misfit strain in thin films bonded on plate substrates: The relation between non-uniform film stresses and system curvatures. Thin Solid Films 515 (4), 2220–2229.CrossRefGoogle Scholar
[26] Novick-Cohen, A. (1988) On the viscous Cahn-Hilliard equation. Material instabilities in continuum mechanics, Edinburgh, 1985–1986, Oxford Sci. Publ., Oxford Univ. Press, New York, pp. 329–342.Google Scholar
[27] Owen, N., Rubinstein, J. & Sternberg, P. (1990) Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition. Proc. R. Soc. Lond. A 429, 505–532.Google Scholar
[28] Pego, R. L. (1989) Front migration in the nonlinear Cahn-Hilliard equation. Proc. R. Soc. London A: Math. Phys. Eng. Sci. 422 (1863), 261–278.Google Scholar
[29] Sethuraman, V. A., Chon, M. J., Shimshak, M., Srinivasan, V. & Guduru, P. R. (2010) In situ measurements of stress evolution in silicon thin films during electrochemical lithiation and delithiation. J. Power Sources 195 (15), 5062–5066.Google Scholar
[30] Shenoy, V., Johari, P. & 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), 6825–6830.Google Scholar
[31] Wang, J. W. et al. (2013) Two-phase electrochemical lithiation in amorphous silicon. Nano Lett. 13 (2), 709–715.CrossRefGoogle ScholarPubMed
[32] Wise, S., Kim, J. & Lowengrub, J. (2007) Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method. J. Comput. Phys. 226 (1), 414–446.CrossRefGoogle Scholar
[33] Zeng, Y. & Bazant, M. Z. (2014) Phase separation dynamics in isotropic ion-intercalation particles. SIAM J. Appl. Math. 74 (4), 980–1004.Google Scholar
[34] Zhao, K., Tritsaris, G. A., Pharr, M., Wang, W. L., Okeke, O., Suo, Z., Vlassak, J. J. & Kaxiras, E. (2012) Reactive flow in silicon electrodes assisted by the insertion of lithium. Nano Lett. 12 (8), 4397–4403.Google Scholar