A Variational Model of Ferroelectric Microstructure

A simple model of the evolution of ferroelectric microstructure in single crystals is developed by using variational methods to minimize a functional based on free energy and dissipation. The thermodynamic variables in this formulation correspond to the positions of a set of domain walls within a single ferroelectric crystal. The total dissipation rate is taken to be a function of the velocities of the individual domain walls. With only two-degrees of freedom, a model of this type has sufficient richness to reproduce many of the observed features found in the single crystal response of a tetragonal ferroelectric. This includes the dielectric hysteresis, butterfly hysteresis, and the variation of the hysteresis loops under uniaxial compressive stress. Additional degrees of freedom can be used to capture multi-axiality. The model also highlights the marked differences in response that should occur when a single crystal is subjected to different boundary conditions.