Modeling of porous battery Electrodes with multiple phase transitions -- Part I: Modeling and homogenization

Abstract

We derive a thermodynamically consistent multiscale model for a porous intercalation battery in a half-cell configuration. Starting from microscopically resolved balance equations, the model rigorously couples cation and anion transport in the electrolyte with electron transport and solid- state diffusion in the active material through intercalation reactions. The derivation is based on non-equilibrium thermodynamics and periodic homogenization. The central novelty of this work lies in the systematic incorporation of multi-well free energy functions for intercalated cations into a homogenized DFN-type porous-electrode framework. This modeling choice leads to non-monotonic chemical potentials and enables a macroscopic descrip- tion of phase separation and multiple phase transitions within the electrode. While multi-well free energies are well established at the particle scale, their integration into homogenized porous- electrode models has so far been lacking. By extending the homogenization framework to include Cahn--Hilliard-type regularizations, phase-transition effects are retained at the electrode level. The resulting model exhibits an intrinsically coupled 3D+3D structure, in which macroscopic transport in the electrolyte is coupled to fully resolved microscopic diffusion within active parti- cles. This coupling naturally induces memory effects and time lags in the macroscopic voltage response, which cannot be captured by reduced single-scale models. Although the microscopic dynamics possess an underlying gradient-flow structure, we adopt a formal asymptotic approach to obtain a tractable DFN-type model suitable for practical simulations. This paper constitutes Part I of a three-part series and is devoted to the systematic derivation and mathematical formulation of the model. Numerical analysis, discretization strategies, simula- tion studies of transient cycling behavior, and experimental validation are deferred to Parts II and III. Part II focuses on finite C-rates, while Part III addresses open-circuit voltage conditions, where the predictive capabilities of the framework are investigated in detail.