Evolution of compatible laminate domain structures in ferroelectric single crystals

The electromechanical properties and behaviour of ferroelectric single crystals are dominated by their domain structures. The domain structure can evolve according to the applied electrical and mechanical boundary conditions, but typically maintains a low energy state by adopting compatible configurations of microstructure in the form of multi-rank laminates. In this work, a model of domain structure evolution is developed, using a variational method to capture the dissipative nature of domain wall motion. The model describes the evolution of domain patterns with the constraint that they remain in low energy, compatible configurations. The electromechanical behaviour, such as microstructure evolution and hysteresis response, of periodic compatible laminate domain patterns in the tetragonal crystal system is studied. Estimates of the material response based on uniform field approximations are developed, and compared with a numerical model in which finite element analysis is used for accurate computation of the free energy. Microstructural transitions from one type of laminate domain pattern to another are included in the model by considering “pivot states”, which are the limiting states shared by more than one laminate pattern. The transition between distinct microstructural patterns at a pivot state is modelled as a bifurcation in which a material element notionally evolves along multiple paths simultaneously, representing sub-regions of the element evolving in different ways. The model is applied to study the hysteresis responses, such as dielectric hysteresis loops and butterfly strain loops, and switching mechanism of barium titanate (BaTiO₃) single crystals subjected to a variety of loads. The relationship between domain patterns and the behaviour of ferroelectric switching is discussed. The results show good general agreement with experimental data in the literature, reproducing several features such as the effect of stress on electrical hysteresis.