Sharp interface limit of the Cahn–Hilliard reaction model for lithium-ion batteries

Abstract

We propose a weak solution theory for the sharp interface limit of the Cahn–Hilliard reaction model, a variational PDE for lithium-ion batteries. An essential feature of this model is the use of Butler–Volmer kinetics for lithium-ion insertion, which arises as a Robin-type boundary condition relating the flux of the chemical potential to the reaction rate, itself a nonlinear function of the chemical potential and the ion concentration. To pass through the nonlinearity as the interface width vanishes, we introduce solution concepts at the diffuse and sharp interface level describing dynamics principally in terms of an optimal dissipation inequality. Using this functional framework and under an energy convergence hypothesis, we show that solutions of the Cahn–Hilliard reaction model converge to a Mullins–Sekerka type geometric evolution equation as the width of the transition layer vanishes.