Under the influence of an electrical current, the uniform and stationary temperature of a sample results from the balance between the Joule heating and the dissipation through cooling. In a multiphase state this temperature can differ in different regions as the conductivity is usually phase dependent. The question is therefore, which phases are stable under which conditions, and how does a transition from one to another phase occurs, and under which conditions two-phase structures are possible. In general, this transition requires the formation of a nucleus and subsequent growth. For practical purposes the consideration of multiple nucleation sites and the discrimination between homogeneous and heterogeneous nucleation is important. We focus on the growth regime, which goes via the motion of a thermal wave, which is simultaneously the phase transformation front.
The motion of planar fronts has been studied, and the case of an alloy. Two-phase periodic structures are also possible under certain conditions. This already shows the complexity of different equilibrium and non-equilibrium patterns. Different external conditions like cooling, fixed-voltage or fixed-current setups, different electrode geometries etc. can be studied for the comparison with experiments. Apparently, this can lead to rather complex interfacial pattern formation processes.
The basic governing equations have been worked out and are applied there to planar front motion driven by Joule heating and its stability using analytical calculations. More complex geometries and boundary conditions, however, call for a numerical treatment. Therefore, the purpose of the present project is to apply a phase field model which is equivalent to a sharp interface description. As a more complex application in view of phase change materials we discuss filament growth, which has not yet been fully discussed on the level of sharp interface models, therefore showing the ability of the phase field model to easily generalize to various situations and applications.