Sodium Ion Batteries Particles: Phase-Field Modeling with Coupling of Cahn-Hilliard Equation and Finite Deformation Elasticity

The cathode material Na x FePO 4 of sodium-ion batteries shows phase changes during intercalation. In this work, a phase-field model for Na x FePO 4 is studied for the first time. The Cahn-Hilliard diffusion equation coupled to finite deformation elasticity is derived. Two finite deformation elasticity formulations based on elastic Green strain and logarithmic elastic strain, respectively, are compared. The material parameters for Na x FePO 4 are determined. We implemented the model in COMSOL Multiphysics for a spherically symmetric problem of sodium insertion into or extraction from a cathodic particle made of Na x FePO 4 . The model captures the important feature of phase segregation into a sodium-poor phase FePO 4 and a sodium-rich phase Na 2/3 FePO 4 . There is a visible difference for the concentration and stress between the small deformation theory and the finite deformation theories. Furthermore, we compare the two cathode materials Na x FePO 4 and Li x FePO 4 of lithium-ion batteries to each other in terms of phase changes and stresses, and show that although the miscibility gap of Na x FePO 4 is smaller than that of Li x FePO 4 , the stresses in the cathode material Na x FePO 4 are higher in the phase segregated state. As a result, the suppression of phase segregation by the elastic strain energy is more easily achieved in Na x FePO 4 compared to Li x FePO 4 .