On $\Gamma -$Convergence of a Variational Model for Lithium-Ion Batteries

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 $$I_{\epsilon }[u,c]:={\int }_{\Omega }\left(\frac{1}{\epsilon }f(c)+\epsilon \Vert \nabla c{\Vert }^2+\frac{1}{\epsilon }\mathbb{C}(e(u)-c{e}_0):(e(u)-c{e}_0)\right)dx,$$ 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 $\Omega \subset {\mathbb{R}}^2,$ it is proven that $\Gamma -{\lim }_{\epsilon \to 0}{I}_{\epsilon }=I_0,$ where $I_0$ is finite only for pairs $(u,c)$ such that $f(c)=0$ and the symmetrized gradient $e(u)=c{e}_0$ almost everywhere. Furthermore, $I_0$ is characterized as the integral of an anisotropic interfacial energy density over sharp interfaces given by the jumpset of $c.$

47 pages, 7 figures