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-epsilon[u, c] := integral(Omega) (1/epsilon f (c) + epsilon parallel to del c parallel to(2) + 1/epsilon C(e(u) - ce(0)) : (e(u) - ce(0))dx, where f is a double well potential, 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 of R-2, it is proven that Gamma - lim(epsilon -> 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) = 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.