Title: Domain evolution in ferroelectrics: phase field and level set methods

Phase field modeling is widely used to model the evolution of domain patterns in various phase transformation problems. Existing phase field models for the evolution of domain structures in ferroelectrics are based on an Allen-Cahn type evolution law. Whereas the latter assumes a linear kinetic relation between the 
thermodynamic driving force acting on a domain wall and the domain wall velocity, experimental evidence shows that domain wall kinetics is actually nonlinear. For that reason, we develop a new phase field model for ferroelectrics [1] that incorporates nonlinearities in the kinetics of domain walls while fully accounting for electro-mechanical couplings. This model is a multi-phase-field generalization of the model of Alber & Zhu [2] (initially formulated for martensitic transformations) and incorporates the anisotropic dielectric, elastic, and piezoelectric properties of the different variants. Subsequently, we discuss the limitations faced by phase-field models (both by the classical Allen-Cahn type models and to a lesser extent by ours) when it comes to modeling quantitatively domain evolution in ferroelectrics. These are twofold: first, by construction, domain walls in phase-field models are endowed with a higher interfacial energy than the physical one, which ascribes undue weight to the curvature contribution to domain wall motion. Second, and this is related to the former, phase-field models fail to account for domain nucleation and nucleus growth at the levels of electric fields typical of experiments. In view of these limitations, we turn to the level set method and present results obtained with a state-of-the-art level set model for a hypothetical two-phase ferroelectric material. These results show that the level set method is a good candidate to address the above-mentioned limitations of the phase-field formulation. Generalization to multiple phases, as required to model real ferroelectric materials, remains an open question. It is interesting to note that models, limitations and questions presented in this talk extend beyond the modeling of ferroelectric domain evolution and apply in identical fashion to the modeling of stress-driven martensitic transformations.

References 

[1] L. Guin, D. M. Kochmann, 2023, A phase-field model for ferroelectrics with general kinetics. Part I: Model formulation. Journal of the Mechanics and Physics of Solids 
[2] H.-D. Alber, P. Zhu, 2013, Comparison of a rapidly converging phase field model for interfaces in solids with the Allen-Cahn model. Journal of Elasticity