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

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 $I_ε[u,c ] := \int_Ω\left( \frac{1}ε f(c) + ε\|\nabla c\|^2 + \frac{1}ε\mathbb{C} (e(u)-ce_0) : (e(u)-ce_0)\right) dx,$ 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 $Ω\subset \mathbb{R}^2,$ it is proven that $Γ- \lim_{ε\to 0} I_ε= I_0,$ where $I_0$ is finite only for pairs $(u,c)$ 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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