Effect of Grain Size on Flexoelectricity

The grain-size effect of permittivity has been widely studied and its phenomenological theory is well established. In this work, we use such grain-size-effect theory to examine the grain-size effect of flexoelectricity, which has been overlooked so far. Two typical flexoelectric materials are employed, namely, ferroelectric ${\mathrm{Ba}\mathrm{Ti}\mathrm{O}}_{3}$ ceramics and paraelectric ${\mathrm{Sr}\mathrm{Ti}\mathrm{O}}_{3}$ ceramics. The results suggest that the variation of the intrinsic flexoelectric coefficient is completely synchronized with the variation of the grain-size-dependent permittivity for ${\mathrm{Sr}\mathrm{Ti}\mathrm{O}}_{3}$ ceramics. However, this synchronization is overturned in ${\mathrm{Ba}\mathrm{Ti}\mathrm{O}}_{3}$ ceramics, the flexoelectric coefficient of which increases with the grain size rather than dielectric permittivity. Under a flexoelectric field, it is proven that the polarization deflection of the 180\ifmmode^\circ\else\textdegree\fi{} domains (more likely to exist for large grain sizes) dominates the flexoelectricity of ferroelectric ${\mathrm{Ba}\mathrm{Ti}\mathrm{O}}_{3}$ ceramics. The observed flexoelectric grain-size effect is beneficial for the design of bulk materials with a high flexoelectric coefficient and provides a simple and effective way to improve the performance of actuators and sensors based on bulk flexoelectricity.