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

Abstract

The cathode material NaxFePO4 of sodium-ion batteries shows phase changes during intercalation. In this work, a phase-field model for NaxFePO4 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 NaxFePO4 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 NaxFePO4. The model captures the important feature of phase segregation into a sodium-poor phase FePO4 and a sodium-rich phase Na2/3FePO4. 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 NaxFePO4 and LixFePO4 of lithium-ion batteries to each other in terms of phase changes and stresses, and show that although the miscibility gap of NaxFePO4 is smaller than that of LixFePO4, the stresses in the cathode material NaxFePO4 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 NaxFePO4 compared to LixFePO4.