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On \(\varGamma \)-Convergence of a Variational Model for Lithium-Ion Batteries

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Abstract

A singularly perturbed phase field model used to model lithium-ion batteries including chemical and elastic effects is considered. The underlying energy is given by

$$\begin{aligned} I_\varepsilon [u,c ] := \int _{\varOmega }\left( \frac{1}{\varepsilon }f(c)+\varepsilon \Vert \nabla c\Vert ^2+\frac{1}{\varepsilon }{\mathbb {C}}(e(u)-ce_0):(e(u)-ce_0)\right) \, \mathrm{d}x, \end{aligned}$$

where f is a double well potential, \({\mathbb {C}}\) is a symmetric positive definite fourth order tensor, c is the normalized lithium-ion density, and u is the material displacement. The integrand contains elements close to those in energy functionals arising in both the theory of fluid-fluid and solid-solid phase transitions. For a strictly star-shaped, Lipschitz domain \(\varOmega \subset {\mathbb {R}}^2,\) it is proven that \(\varGamma - \lim _{\varepsilon \rightarrow 0} I_\varepsilon = I_0,\) where \(I_0\) is finite only for pairs (uc) such that \(f(c) = 0\) and the symmetrized gradient \(e(u) = ce_0\) almost everywhere. Furthermore, \(I_0\) is characterized as the integral of an anisotropic interfacial energy density over sharp interfaces given by the jumpset of c.

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Acknowledgements

This paper is part of the author’s Ph.D. thesis at Carnegie Mellon University under the direction of Irene Fonseca and Giovanni Leoni. The author is deeply indebted to these two for their many hours spent watching the author scribble at a board and for expert guidance on many mathematical topics. Furthermore, the author is thankful for their many suggestions as to the organization of the paper and spotting a plethora of typos, which greatly improved the paper. The author was partially supported by National Science Foundation Grants DMS 1906238 and DMS 1714098.

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Correspondence to Kerrek Stinson.

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Stinson, K. On \(\varGamma \)-Convergence of a Variational Model for Lithium-Ion Batteries. Arch Rational Mech Anal 240, 1–50 (2021). https://doi.org/10.1007/s00205-020-01602-7

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