Nonequilibrium thermodynamics of ferroelectric domain evolution

A ferroelectric crystal tends to form domains. Internal and external fields can cause domain walls to move, resulting in macroscopic phenomena such as aging, creep and fatigue. Domain evolution is a nonequilibrium thermodynamic process. The domain pattern, the state of the crystal, can be described with a set of generalized coordinates. The free energy of the crystal is a function of the generalized coordinates, to be pictured as a surface in the thermodynamic space, the free energy erecting and the generalized coordinates spanning. A point on this surface represents a (nonequilibrium) state of the crystal, the slopes of the tangent plane contacting the surface at the point are the generalized forces, and a curve on the surface is an evolution path. Thermodynamics requires that the path descend on the surface, but by itself does not determine the path. To complete this global picture of structural evolution, we formulate a variational principle that includes kinetics. The functional to be minimized consists of the free energy rate and a dissipation potential involving the domain wall mobility. The variational principle equips a viscosity matrix to every point on the thermodynamic surface. The approach results in a set of ordinary differential equations that govern the evolution of the generalized coordinates. We outline the operational aspects of this approach, make contact with the finite element method, and illustrate the approach by examples of one or two degrees of freedom.