**4. Conclusions**

We extended the phase-field model based on the Cahn-Hilliard equation to the lithium intercalation dynamics in a cathode with particles of distributed size. Considering the simplified phase-field intercalation in a single submicron spherical particle under galvanostatic condition, we established universal behavior observed for different sizes of electrode particles and a wide range of temperatures and C-rates. The universal rules were formulated for speed of front propagation and evolution of single-phase concentration. Two stages associated with single-phase and double-phase patterns are

typical for both charging and discharging processes. The single-phase stage takes approximately 10–15% of the process and plays an important role in the intercalation dynamics.

The particle size distribution causes the time-dependence of the interfacial flux density under galvanostatic conditions. The analytical results for a particle ensemble are presented for the case of the gammadistribution of particle sizes, and can be easily generalized to other PSDs. The universality of the established laws allowed us to formulate the boundary condition with time-dependent flux density for the Cahn-Hilliard equation. Numerical solutions of the corresponding boundary-value problem agree well with the obtained universal relations.

Thus, we obtained thegeneral analytical relations describing the intercalation process in ensemble of submicron electrode particle characterized by the arbitrary PSD. Other approaches based on the di ffusion equation for intercalating atoms can consider finite number of particle bins only and have higher computational cost.

The Cahn-Hilliard equation and obtained approximate relations on the phase-field intercalation can be used for modification of the well-known SPM and P2D models of LIBs.

**Author Contributions:** Conceptualization, P.L. and R.S.; methodology, P.L.; software, P.L.; validation, P.L. and R.S.; formal analysis, P.L.; writing—original draft preparation, P.L.; writing—review and editing, R.S.; visualization, P.L.; supervision, P.L. and R.S.; project administration, R.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors thank the Russian Science Foundation (project no. 19-71-10063) for financial support.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **List of Symbols**

