Kinematics, electromechanics, and kinetics of dielectric and piezoelectric crystals with lattice defects

A mathematical framework is formulated to address the electromechanical behavior of dielectric and piezoelectric solids containing lattice imperfections. The macroscopic displacement gradient encompasses recoverable elasticity, deviatoric plasticity arising from dislocation glide, and volumetric deformation attributed to point vacancies in the crystal. A linear connection on the spatial manifold of deformed lattice vectors describes gradients of stretch and rotation at the microscale caused by continuous distributions of various classes of crystal defects. It is shown that parallel transport of a lattice director vector with respect to this connection about a closed loop yields a discontinuity with contributions from the torsion of the connection (physically, from dislocations) and its curvature (physically, from rotational defects such as domain walls, and from gradients in vacancy concentration). Classical balance laws of electrostatics and mass and momentum conservation are invoked. A free energy function dependent upon lattice distortion, polarization, temperature, and defect densities is suggested. Thermodynamically consistent kinetic relations for dislocation glide and vacancy diffusion are then derived, with the chemical potential for the latter depending upon defect density, electric potential, hydrostatic pressure, and vacancy energy. The theory also explicitly considers mass rearrangement at the free surface of the substance. Two forms of the contribution of vacancies to the free energy are investigated in detail: a logarithmic function common in chemical mixing theory, and a quadratic function analogous to the convex strain energy used in continuum elasticity theory. For the latter case, the analytical solution of the diffusion equation in one spatial dimension, at steady state, illustrates the effects of defect charge, defect energy, and mechanical stress on the distribution of vacancies in a dielectric thin film. A specific example is given of how compressive residual stresses observed in experiments may be correlated with the equilibrium concentration of vacancies within grains of a polycrystalline dielectric thin film, influencing its electrical performance.