Flexoelectricity, strain gradients, and singularities in ferroelectric nanostructures

The effect of flexoelectricity on the formation and evolution of domain structures in ferroelectric materials is developed by integrating strain gradient theory into a finite element phase field model. Length scales associated with elastic strain gradients and the corresponding polarization gradient across a domain wall are integrated into the time- dependent Ginzburg- Landau theory and numerical simulated. Theoretical relations of a shear strain gradient along an electrode/ dielectric interface are first solved and verified numerically using the finite element model. A singularity in the strain gradient- induced polarization is shown to occur as the elastic strain gradient length scale approaches a flexoelectric length scale. The theory and finite element modeling is then extended to quantify strain gradient electromechanics near 180 and 90 tetragonal phase domain structures. It is shown that the strain gradient length scale ( l1) strongly influences changes in strain across the domain walls but has a negligible effect on the polarization. Strain gradient effects become negligible when l1 ' 0: 1lP, where lP is the polarization domain wall length scale.