Publication 20-CNA-019

On Existence and Regularity for a Cahn-Hilliard Variational Model for Lithium-Ion Batteries

Kerrek Stinson
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
kstinson@andrew.cmu.edu

Abstract: 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 Neumann boundary condition $\partial_\nu \mu = R(c,\mu)$ for the chemical potential $\mu$, with $c$, the lithium-ion density. Importantly, $R$ depends exponentially on $\mu$. In arbitrary dimension, existence of a weak solution for the Cahn-Hilliard reaction model with elasticity is proven using a generalized gradient structure. This approach is, at present, restricted to polynomial growth in $R$. Working to remove this limitation, fixed point methods are applied to obtain existence of strong solutions of the Cahn-Hilliard reaction model without elasticity in dimensions $2$ and $3.$ This method is then extended to prove existence of higher regularity solutions in dimension $2$, allowing for recovery of exponential boundary conditions as in the physical application to lithium-ion batteries.

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