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.  

TiO₂ 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_0.5TiO₂ phase forms at the expense of the TiO₂ phase . The second biphasic transition typically appears as a sloping pseudo-plateau around 1.54 V marking the transformation from  the Li_0.5TiO₂ towards the Li₁TiO₂ 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 Li₁TiO₂ phase⁶. 

Li-ion insertion properties in the TiO₂ - Li_0.5TiO₂ range are vastly different from the Li_0.5TiO₂ –Li₁TiO₂ range due to huge repulsive interaction that plunge Li diffusivity in the Li₁TiO₂ 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_xTiO₂, with 0<x<0.5), and lattice 2 describes the second phase transition (Li_xTiO₂, with 0.5<x<1) . The interaction was captured by the different parameters used to describe the kinetic and thermodynamic properties of the two phase transition reactions.  The large amount of experimental and theoretical results available in literature make it possible to avoid any fitting parameter in the current Phase Field model. All the thermodynamic and kinetic values we implemented were based on a careful selection from NMR, DFT, FF-MD, and electrochemical measurements.  The enthalpy of mixing was determined from DFT calculations, the phase boundary gradient penalty was calculated from the interface width determined by the anatase phase diagram constructed from neutron diffraction data.  Li-ion mobility from NMR experiments and FF-MD simulations was implemented, and the charge transfer coefficient was also obtained from NMR experiments. 

Our Phase Field model reproduces most of the experimentally observed phenomena, shedding light on the limitations and possibilities for anatase as an electrode material. C-rate and particle size behaviour shows good agreement with experiments both in terms of capacity and overpotential. Close to thermodynamic equilibrium, where kinetic limitations are eliminated either by cycling slow or by reducing the diffusion pathway, the slopy tail of the voltage profile becomes a flat plateau indicating a phase separation mechanism during the second phase transition. The role of the self-blocking layer of Li₁TiO₂phase is explained along with effects of surface area and preparation methods. Our simulations explain why the various experimental approaches increase capacities and cycling rates of anatase electrodes , and why anatase is not suitable for fast cycling. The anatase phase field model produces these results without any fitted parameters, showing the strong physical foundation of phase field modelling and its promise for modelling of other electrode materials. Finally, our work strengthens the background of simulating materials with multiple phases which pose a great challenge for conventional computational models. The simple and efficient thermodynamic treatment of separate lattices with no interaction terms can set the basis for modelling of  more complicated systems.

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2.         Ferguson, T. R.; Bazant, M. Z. J Electrochem Soc 2012, 159, (12), A1967-A1985.

3.         Cogswell, D. A.; Bazant, M. Z. Acs Nano 2012, 6, (3), 2215-2225.

4.         Bai, P.; Cogswell, D. A.; Bazant, M. Z. Nano Lett 2011, 11, (11), 4890-4896.

5.         Li, Y. Y. et al. Nat Mater 2014, 13, (12), 1149-1156.

6.         Borghols, W. J. H.; Lutzenkirchen-Hecht, D.; Haake, U.; van Eck, E. R. H.; Mulder, F. M.; Wagemaker, M., Phys Chem Chem Phys 2009, 11, (27), 5742-5748.