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.

Automated summary

The singularly perturbed phase field model is used to model lithium-ion batteries with chemical and elastic effects. The model's energy is based on a specific functional form suitable for strictly star-shaped Lipschitz domains.