Analysis of a Variational Model for Lithium-Ion Batteries

The Cahn-Hilliard reaction model, a nonlinear, evolutionary system of PDEs, was introduced to model phase separation in lithium-ion batteries. Using Butler-Volmer kinetics for electrochemical consistency, this model incorporates a nonlinear Robin-type boundary condition ∂νµ = R(c, µ) for the chemical potential µ, with c designating the lithium-ion density. Importantly, R depends exponentially on µ. For the static analysis, a singular perturbation of the underlying phase field energy with elasticity is studied. In dimension 2, it is proven that the energies have a sharp interface limit with energy finite only for locally laminate material displacements. In arbitrary dimension, existence of a weak solution for the evolutionary Cahn-Hilliard reaction model with elasticity is proven using a generalized gradient structure. This approach is, at present, restricted to polynomial growth in R. To remove this limitation, fixed point methods are applied to prove existence of higher regularity solutions in dimensions 2 and 3, allowing for recovery of exponential boundary conditions, as in the physical application to lithium-ion batteries.