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.
The microstructure of ferroelectric single crystals is a crucial factor that determines macroscopic properties and poling
behaviour. Recent models of domain configuration, (such as that of Li & Liu, Journal of Mechanics and Physics of
Solids, 2004) employ multi-rank laminate structures that satisfy compatibility in an average sense. In general, these
models result in high-rank structures, corresponding to fine microstructure. However, minimum energy structures may
be expected to have low rank and to satisfy compatibility requirements at every domain wall exactly. In this paper, the
criteria of exact compatibility and average compatibility are defined and then used to determine energy minimizing
microstructure in the tetragonal crystal system. In addition, the lowest rank construction of compatible laminate structure
for a given macroscopic state of strain and polarization is found. Based on this, poling paths from unpoled to the fully-poled
state in the tetragonal system are found, which allow the structure to stay in the lowest possible rank while
maintaining exact compatibility. The application of the theory to a broader class of crystal structures is discussed.