Explaining key properties of lithiation in ${\mathrm{TiO}}_2$-anatase Li-ion battery electrodes using phase-field modeling

The improvement of Li-ion battery performance requires development of models that capture the essential physics and chemistry in Li-ion battery electrode materials. Phase-field modeling has recently been shown to have this ability, providing new opportunities to gain understanding of these complex systems. In this paper, a novel electrochemical phase-field model is presented that captures the thermodynamic and kinetic properties of lithium insertion in $\mathrm{Ti}{\mathrm{O}}_2$-anatase, a well-known and intensively studied Li-ion battery electrode material. Using a linear combination of two regular solution models, the two phase transitions during lithiation are described as lithiation of two separate lattices with different physical properties. Previous elaborate experimental work on lithiated anatase $\mathrm{Ti}{\mathrm{O}}_2$ provides all parameters necessary for the phase-field simulations, giving the opportunity to gain fundamental insight in the lithiation of anatase and validate this phase-field model. The phase-field model captures the essential experimentally observed phenomena, rationalizing the impact of C rate, particle size, surface area, and the memory effect on the performance of anatase as a Li-ion battery electrode. Thereby a comprehensive physical picture of the lithiation of anatase $\mathrm{Ti}{\mathrm{O}}_2$ is provided. The results of the simulations demonstrate that the performance of anatase is limited by the formation of the poor Li-ion diffusion in the ${\mathrm{Li}}_1{\mathrm{TiO}}_2$ phase at the surface of the particles. Unlike other electrode materials, the kinetic limitations of individual anatase particles limit the performance of full electrodes. Hence, rather than improving the ionic and electronic network in electrodes, improving the performance of anatase $\mathrm{Ti}{\mathrm{O}}_2$ electrodes requires preventing the formation of a blocking ${\mathrm{Li}}_1{\mathrm{TiO}}_2$ phase at the surface of particles. Additionally, the qualitative agreement of the phase-field model, containing only parameters from literature, with a broad spectrum of experiments demonstrates the capabilities of phase-field models for understanding Li-ion electrode materials, and its promise for guiding the design of electrodes through a thorough understanding of material properties and their interactions.

Figure 1

The crystal structure [23] and typical voltage profile of anatase during lithiation.

Figure 2

Schematic representation of the two lattice phase-field model for lithiation in anatase $\mathrm{Ti}{\mathrm{O}}_2$.

Figure 3

(a) Voltages profiles vs average concentration $\left(\tilde{X}\right)$ and (b) final concentration profiles in the particle at different C rates.

Figure 4

Li-ion concentration profiles in the electrolyte and in electrode particles at different electrode depths at different times during 2C discharge. The filling fraction in the displayed particles is the average of that in the simulated particles in each volume.

Figure 5

Voltage profiles vs average concentration $\tilde{X}$ for different increases in diffusivity at (a) 0.5C, and (b) 2C, and final concentration profiles inside the particles at (c) 0.5C and (d) 2C.

Figure 6

(a) Voltage profiles vs average concentration $\tilde{X}$ and (b) final concentration profiles of a spherical and a cylindrical particle at 0.5C and 2C.

Figure 7

Voltage profiles vs average concentration $\tilde{X}$ for particles with different radii at (a) 0.5C and (b) 2C.

Figure 8

Final concentration profiles at the end of a simulation at 0.5C and 2C for particles with a radius of (a) 5, (b) 20, and (c) 50 nm.

Figure 9

Concentration profiles during a 0.5C simulation at different lithiation stages for particles with a radius of (a) 5 and (b) 50 nm.

Figure 10

For different particle sizes and C rates the (a) final Li concentrations and (b) final thickness of the ${\mathrm{Li}}_1{\mathrm{TiO}}_2$ layer (defined as the depth at which ${\mathrm{Li}}_{0.6}{\mathrm{TiO}}_2$ occurs), the lines are a guide to the eye.

Figure 11

Overview of Li intercalation in anatase.