David Schrade

On Phase Field Modeling of Ferroelectrics: Parameter Identification and Verification

This contribution introduces a thermodynamically consistent, fully electro-mechanically coupled micro-mechanical model for ferroelectric materials. Adopting a phase field concept, in which the spontaneous polarization is used as order parameter, a Ginzburg-Landau type theory is formulated for the evolution of the order parameter. The equations are discretized within the scope of the Finite Element Method, and implicit time integration is used to solve the non-linear evolution equation. Examples illustrate the physical meaning of phase field parameters and give an application to multi-axial switching in which experimental results are used for comparison.Copyright

Interaction of Cracks and Domain Structures in Thin Ferroelectric Films

The fracture behavior of ferroelectric materials is a complex problem that has been addressed in numerous experimental and theoretical studies. Several factors have been identified to play an important role, such as the applied electric field, the medium inside the crack, the electrical conditions on the crack faces, and polarization switching at or near the crack tip. In this investigation, a phase field model for ferroelectric domain evolution is used to calculate crack tip driving forces for mode-I cracks in barium titanate thin films. The driving forces are obtained by employing the theory of configurational forces, which is equivalent to considering the J-integral. Simulations are done for permeable, impermeable, semi-permeable, and energetically consistent crack face conditions with both air and water as crack medium. The finite element calculations are performed for films with thicknesses varying from 5 to 30 nm. The results show that the impermeable, semi-permeable and energetically consistent conditions lead to similar crack tip driving forces if air is used as crack medium. In the absence of mechanical loading, strong electric fields result in a closing crack tip driving force, while the use of water as crack medium leads to opposite driving forces. It can be confirmed that polarization switching at the crack tip has a significant effect on the driving force.

Modeling of Domain Structure Evolution in Ferroelectric Materials

A continuum phase field model is presented to simulate domain structures in ferroelectric single crystals. Special attention is given to domain structure evolution in the vicinity of crack tips. Using the spontaneous polarization as an order parameter the model is set up and implemented into a 2D finite element method. To evaluate crack driving forces the theory of configurational forces is extended to the phase field continuum and numerically realized within the finite element method. Simulations show the influence of boundary conditions and system parameters on the development of certain domain structures. Calculations with combined electric and mechanical loadings display the complex interaction of the loading with the domain structure and the crack driving force.

Domain wall pinning by point defects in ferroelectric materials

A continuum model for ferroelectric materials is presented where the spontaneous polarization is treated as an order parameter. The classic electric enthalpy consisting of elastic, dielectric and ferroelectric terms is extended by a phase separating potential and an interface energy which yields a phase field potential. The coupled material equations and the Ginzburg-Landau type evolution equation are derived from that phase field potential. The evolution equation as well as the mechanical and electro-static balance laws are solved using the Finite Element Method. The model is extended to allow for the simulation of point defects. Numerical examples are given for the defect-free case, and the influence of point defects is investigated.