Multiphase-field modeling of spinodal decomposition during intercalation in an Allen-Cahn framework

Abstract

Owing to its unique thermodynamical description, phase separation has largely been modeled in the Cahn-Hilliard framework. In the present work, as a computationally efficient alternative, a multicomponent, multiphase-field model operating in Allen-Cahn framework is presented and subsequently employed to simulate spinodal decomposition. Stability analysis shows that the formulation can cover the effect of phase separation while simplifying the extension to multiple phases and incorporation of additional driving forces. Computational efficiency of the proposed approach is compared with the conventional technique by modeling intercalation in a representative one-dimensional domain. Moreover, intercalation within a multigrain system involving multiparticle interaction is studied. Our results suggest initiation of phase transformation at higher order junctions as well as a grain-by-grain intercalation behavior in a two-phase cathode material such as the well-studied LiFePO 4 .

Modeling intercalation in cathode materials with phase-field methods: Assumptions and implications using the example of LiFePO4

“…1a. Within the multi-phase approach, the regular solution is replaced by an interpolation of phasedependent chemical free energy contributions [6]. These functions can be fitted such that the resulting open circuit potential matches experimental measurements close to the thermodynamic equilibrium as shown in 3.8…”

“…For the sake of investigating polycrystalline samples, we furthermore employ a multiphase-field model based on the formulation presented in [6]. Following the notation in [14] we express the free energy functional in dependence of N phases with corresponding order parameters φ = {φ 1 , ..., φ α , ..., φ N } T and K chemical species with molar fractions c = {c 1 , ..., ci , ..., cK } T as…”

“…( 19) is employed. In our previous work [6], we outlined how conventional parameters such as interfacial energies and width can be related and, additionally, showed that the instability limits for nucleation of new phases can be matched. However, the two modeling approaches exhibit different kinetics with regard to nucleation and phase separation which will partly be discussed in the remainder of the paper.…”

“…The multiphase-field method is a suitable tool to investigate the coupled effects of electro-chemo-mechanical driving forces on the transport processes of intercalated ions [5,6]. Although, the phase-field approach originally emerged to study the microstructure evolution during solidification, it has been applied to a broad variety of fields up to date [7][8][9].…”

“…This framework has been sucessfully combined with elastic driving forces based on jump conditions [31,32] to model martensitic transformations [12], crack propagation [33][34][35] and the evolution of electric field to study electromigration [36]. It has further been shown that grain boundary diffusion [37] as well as the instability leading to phase separation in the miscibility gap [6] can be effectively modelled. The aim of this work is to show the potentials of both approaches for the numerical screening of promising intercalation electrode materials while also discussing their complementary range of applicability and their limitations due to modeling assumptions.…”