Implementation of the coupled two-mode phase field crystal model with Cahn–Hilliard for phase-separation in battery electrode particles

In this article, we present the behavior of two‐mode phase field crystal (2MPFC) method under a concentration dependent deformation. A mixed finite element formulation is proposed for the 2MPFC method that solves a 10th‐order parabolic equation. Lithium concentration diffusion in the electrode particle is captured by the Cahn–Hilliard (CH) equation and the host electrode material, Li$_{x}$Mn$_{2}$O$_{4}$ (LMO), which has a face‐centered cubic (fcc) lattice structure, is modeled using 2MPFC. The coupling between 2MPFC and CH models brings about the concentration dependent deformation in the polycrystalline LMO electrode particle. The atomistic dynamics is assumed to operate on a faster time‐scale compared to the diffusion of lithium, thereby both the 2MPFC and CH models evolve on two different time‐scales. The coupled 2MPFC–CH system models the diffusion induced grain boundary migration in LMO capturing the charging and discharging state of the battery.