Ab initio diffuse-interface model for lithiated electrode interface evolution

The study of chemical segregation at interfaces, and in particular the ability to predict the thickness of segregated layers via analytical expressions or computational modeling, is a fundamentally challenging topic in the design of novel heterostructured materials. This issue is particularly relevant for the phase-field (PF) methodology, which has become a prominent tool for describing phase transitions. These models rely on phenomenological parameters that pertain to the interfacial energy and thickness, quantities that cannot be experimentally measured. Instead of back-calculating these parameters from experimental data, here we combine a set of analytical expressions based on the Cahn-Hilliard approach with ab initio calculations to compute the gradient energy parameter κ and the thickness λ of the segregated Li layer at the ${\mathrm{Li}}_x\text{Si}$-Cu interface. With this bottom-up approach we calculate the thickness λ of the Li diffuse interface to be on the order of a few nm, in agreement with prior experimental secondary ion mass spectrometry observations. Our analysis indicates that Li segregation is primarily driven by solution thermodynamics, while the strain contribution in this system is relatively small. This combined scheme provides an essential first step in the systematic evaluation of the thermodynamic parameters of the PF methodology, and we believe that it can serve as a framework for the development of quantitative interface models in the field of Li-ion batteries.

Figure 1

Ab initio results for saturated and hypersaturated systems and the modeled concentration gradient at $y=0.9$. (a) Δ$G$ as a function of the state of charge $y$. Contributions from strain are presented in black open circles, and those from the chemical environment in red full circles. A blue dashed line bounds the unstable region where spinodal decomposition occurs. (b–d) Li-Si configurations at $y=0.9$ along with a schematic for the Li concentration gradient across the interface (shown with a dashed line) at $c$/2 to demonstrate uniform and nonuniform systems. Li: green atoms; Si: red atoms (total number of atoms $N_{\mathrm{at}}=160)$.

Figure 2

Analytical evaluation of the thermodynamic parameters κ and λ. (a) Energy density as a function of the molar fraction $y$. The points where the two curves meet represent solutions tο Eq. (4) for different values of κ. (b) Dependence of the Li mole fraction on the thickness of the segregated layer λ for various κ values. (c) Power law dependence of the segregated thickness λ on the thermodynamic parameter κ. Based on this expression and the ab initio models at $y=0.9$, the predicted thickness for the Li layer in Ref. [26] is calculated to be on the order of 7.24 nm.

Figure 3

SIMS measurement and experimental evaluation of the thickness. (a) SIMS depth profile of Li-Si ratio through the thickness of a lithiated Si film on Cu from Ref. [26]. (b) Schematic of bilayer approximation for the calculation of the thickness λ of the segregated layer. (c) Thickness λ of the segregated layer for two interface position scenarios.

Figure 4

Experimentally observed biaxial modulus and fitted curve.

Figure 5

(a) First cycle behavior of 50 nm thin film of Si. (b) Expanded view of voltage plateau observed during lithiation in first cycle.

Figure 6

Formation energy of $\mathrm{L}{\mathrm{i}}_x\mathrm{Si}$ showing DFT data points and spline fit.

Figure 7

Variation of interfacial energy Φ of $\mathrm{L}{\mathrm{i}}_x\mathrm{Si}\text{−}\mathrm{Cu}$ with Li concentration.

Figure 8

Li-Si configurations at $y=0.9$ with a gradient in the Li concentration across the interface at $c$/2 (shown with a green dashed line). Li: green atoms; Si: red atoms. Energies reported for (a) $N_{\mathrm{at}}=160$ atoms and (b) $N_{\mathrm{at}}=240$ atoms. Both models demonstrate similar behavior with the trends maintained. (c) The full length and cross section of area $A$ of the three 3D models used. Li: green atoms; Si: red atoms.