A variational model of ferroelectric rank-2 laminate domain structures

Abstract

The equilibrium domain arrangements of ferroelectric single crystals are significantly affected by loads and boundary conditions. Domain structures evolve towards a minimum energy state. In this paper, a variational method, which minimizes a functional based on free energy and dissipation, is developed to model the evolution of several typical rank-2 laminate domain patterns in the tetragonal crystal system. Periodic laminates which satisfy compatibility at every domain wall are studied. These domain patterns include herringbone and vortex array structures. The unit cells for both types of domain pattern dictate a set of domain walls whose positions may vary while maintaining the same topology. The positions of domain walls are treated as thermodynamic variables in the formulation, and the total dissipation rate is then a function of the velocities of the domain walls. By using this model, many features normally observed in ferroelectric single crystals can be reproduced, such as the dielectric hysteresis loop and butterfly loop. The characteristics of the hysteresis loop for different topologies, as well as under different applied loads and boundary conditions are discussed. The model can readily be extended to higher rank laminate structures and other crystal systems.