Existence for a Cahn-Hilliard Model for Lithium-Ion Batteries with Exponential-Growth Boundary Conditions

TL;DR

This paper establishes the existence of regular solutions for a Cahn-Hilliard reaction model with exponential-growth boundary conditions, relevant for lithium-ion batteries, using fixed point methods in three dimensions.

Contribution

It introduces a mathematical framework for the Cahn-Hilliard model with exponential boundary conditions, advancing the analysis of lithium-ion battery models.

Findings

Proves existence of solutions in 3D for the model.

Handles exponential boundary conditions physically relevant to batteries.

Employs fixed point methods for nonlinear PDE analysis.

Abstract

The Cahn-Hilliard reaction model, a nonlinear, evolutionary PDE, was introduced to model phase separation in lithium-ion batteries. Using Butler-Volmer kinetics for electrochemical consistency, this model allows lithium-ions to enter the domain via a nonlinear Robin-type boundary condition $\partial_\nu \mu = R(c,\mu)$ for the chemical potential $\mu$, with $c$, the lithium-ion density. Importantly, $R$ depends exponentially on $\mu$. Fixed point methods are applied to obtain existence of regular solutions of the Cahn-Hilliard reaction model in dimension $3,$ allowing for recovery of exponential boundary conditions as in the physical application.