The role of electric field gradient in modeling elastic ferroelectrics

In this article a continuum model for simulating the microscopic behavior of ferroelectrics is proposed. The model is based on an alternative phase field formulation, where spontaneous polarization, mechanical strain, electric field and electric field gradient are used as constitutive variables. The electric field gradient is energetically conjugated to the quadrupole polarization, thus it is about a theory with electric quadrupoles. A variational principle has been used for the derivation of the field equations and the appropriate boundary conditions. Like polarization vector, it is assumed that electric quadrupole tensor is separated into a reversible and a spontaneous part, too. It is proved that spontaneous quadrupole tensor is responsible for the domain wall thickness. Computational examples concerning ferroelectric materials with charge and dipole defects are carried out with the proposed model. The additional boundary conditions provided by the variational principle seem to be useful especially to handle dipole defects.