On domain switching in deformable ferroelectrics, seen as continua with microstructure

Abstract

We obtain a three-dimensional continuum model for deformable ferroelectric bodies in their polar phase characterized by a spontaneous polarization. This is accomplished by assuming the body as comprised of a continuum with vectorial microstructure: in each point of the body therefore a gross and a fine structure are superposed, the gross structure representing a non linear polarizable elastic body and the vectorial fine structure describing the spontaneous polarization.¶Among the distinctive features of ferroelectric materials, the most interesting is represented by the organization of spontaneous polarization into a domain structure, which minimizes electrostatic energy and which can be modified by the application of electric and deformation fields. This process, called “polarization reversal” or “domain switching”, is associated with various hysteresis loops, the most typical being those between spontaneous polarization and electric field (dielectric hysteresis), and between strain and electric field (“butterfly” loop).¶From the balance laws of continua with vectorial microstructure and dissipation inequality we arrive at the evolution equation for the spontaneous polarization which allows for both inertial and dissipative terms and describes domain switching. We postulate a simple interaction mechanism between the spontaneous polarization and the pair electric field, deformation and arrive at, in the quasi-static case, to a minimization problem for a functional which reminds the micromagnetic energy of deformable ferromagnetics.¶For linearized kinematics we study, in the one-dimensional case, stable relative minimizers and give a simple justification for dielectric hysteresis and butterfly loops: under the hypothesis that the domain walls are sharp interfaces, the solutions we find explain the banded twin domains morphology which is typical of many ferroelectrics.