Explaining Key Properties of Lithiating TiO 2 -Anatase: A Phase Field Model for Materials with Multiple Phases

Phase field modelling has shaped our understanding regarding thermodynamic and kinetic material properties for lithium ion batteries 1-5 . Phase field models can capture phase separating systems upon lithiation assuming a regular solution model and a diffuse phase boundary that compete to lower the Gibbs free energy. In this way the essential physics of interface widths, phase morphology, phase separation stability, and variable solubility limits can be modelled. The phase field model simulates porous electrodes via a finite volume method by integrating charge transfer and non-equilibrium thermodynamic principles. When it comes to modelling materials with multiple phases, multiple regular solution models can be used that treat the material as a collection of lattices with different properties. The Gibbs free energy is the result of the linear addition of all contributions while interaction terms can be introduced to capture the effect of one lattice has on the other. TiO 2 anatase exhibits three phases corresponding to two phase transitions upon lithiation. The first biphasic transition gives rise to a voltage plateau of 1.84 V vs. Li/Li + where the Li 0.5 TiO 2 phase forms at the expense of the TiO 2 phase . The second biphasic transition typically appears as a sloping pseudo-plateau around 1.54 V marking the transformation from the Li 0.5 TiO 2 towards the Li 1 TiO 2 phase. The voltage profile strongly depends on the particle size, shape and the C-rate. From size and temperature dependent experiments the second phase transition is suggested to be kinetically limited due to the poor Li-ion diffusivity in the Li 1 TiO 2 phase 6 . Li-ion insertion properties in the TiO 2 - Li 0.5 TiO 2 range are vastly different from the Li 0.5 TiO 2 –Li 1 TiO 2 range due to huge repulsive interaction that plunge Li diffusivity in the Li 1 TiO 2 phase. In the Phase Field model we treated the two transitions as separate lattices described by two regular solution models with different order parameters. Lattice 1 describes the first phase transition (Li x TiO 2 , with 0