Polarization switching of ferroelectric ceramics with grain boundary effect: A simple continuum model

A one-dimensional continuum model is proposed to simulate the polarization switching of ferroelectric ceramics and investigate the effects of the presence of grain boundaries on the switching behavior. A ceramic bar is modeled as a composite system of grains and grain boundaries. A grain element consists of a material characterized by a Helmholtz free energy function and a grain boundary element consists of a linear dielectric material. A polarization switching in a grain element starts and proceeds according to a switching criterion and a kinetic relation, respectively. Electric conduction effects in both grain and grain boundary elements are included in the model. A system of differential equations is derived from the law of charge conservation and solved using the finite difference method. The predicted change of hysteresis loops with electrical loading rates is qualitatively consistent with experiments. The effects of the grain boundary on polarization switching are also calculated and discussed.