  Understanding time-dependent effects of charged vacancies on
electromechanical responses of materials is at the forefront of research for
designing materials exhibiting metal-insulator transition, and memresistive
behavior. A Rayleighian approach is used to develop a model for studying the
non-linear kinetics of reaction leading to generation of vacancies and
electrons by the dissociation of vacancy-electron pairs. Also, diffusion and
elastic effects of charged vacancies are considered to model
polarization-electric potential and strain-electric potential hysteresis loops.
The model captures multi-physics phenomena by introducing couplings among
polarization, electric potential, stress, strain, and concentrations of charged
(multivalent) vacancies and electrons (treated as classical negatively charged
particles), where the concentrations can vary due to association-dissociation
reactions. Derivation of coupled time-dependent equations based on the
Rayleighian approach is presented. Three limiting cases of the governing
equations are considered highlighting effects of 1) non-linear reaction
kinetics on the generation of charged vacancies and electrons, 2) the Vegard's
law (i.e., the concentration-dependent local strain) on asymmetric
strain-electric potential relations, and 3) coupling between a fast component
and the slow component of the net polarization on the polarization-electric
field relations. The Rayleighian approach discussed in this work should pave
the way for developing a multi-scale modeling framework in a thermodynamically
consistent manner while capturing multi-physics phenomena in ferroelectric
materials.
