# 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.

## Funding

### Variational Methods for Materials and Imaging Sciences

Directorate for Mathematical & Physical Sciences

Find out more...### Variational Methods for Materials Science and Mechanics

Directorate for Mathematical & Physical Sciences

Find out more...### Collaborative Research: GCR: Collective Behavior and Patterning of Topological Defects: From String Theory to Crystal Plasticity

Directorate for Mathematical & Physical Sciences

Find out more...## History

## Date

2021-05-22## Degree Type

- Dissertation

## Department

- Mathematical Sciences

## Degree Name

- Doctor of Philosophy (PhD)