In this work, we formulate a unified framework of balance laws and thermodynamically-consistent constitutive equations which couple Cahn-Hilliard-type species diffusion with large elastic deformations of a body. The continuum framework was implemented in a finite-element program (using custom elements in Abaqus/Standard), where the fourth-order Cahn-Hillard-type diffusion equation was split into two second-order equations which are amenable to standard finite-element implementation.
We use our fully coupled theory and numerical implementation to study the combined effects of diffusion and stress on the lithiation of spheroidal-shaped particles. Importantly, we show that the morphology of lithiation is not necessarily “core-shell” type, as often assumed in the literature, but rather depends on the elastic deformation of the host cathode.
Di Leo, C.V., Rejovitzky, E. and Anand, L. (2014) A Cahn-Hilliard-type phase-field theory for species diffusion coupled with large elastic deformations: application to phase-separating Li-ion electrode materials. Journal of the Mechanics and Physics of Solids, 70, 1-29. [html]
Video: Diffusion only (no deformation) simulations to probe steady-state morphologies of the Cahn-Hilliard equation in 3D
Video: Lithiation of spheroidal particles with varying Young’s moduli. Curves show state-of-charge vs. time, top contours are of normalzied concentration, and bottom contours are of maximum principal stress.