Phase field modeling of alloys
The modeling of alloy solidification strongly depends on the development of efficient phase field models. Quantitative predictions require a strict separation of the length scales in the problem, in particular of the phase field interface thickness from all the physical scales. Here we investigate in particular how models with a so called thin interface asymptotics are related to deep physical symmetry relations. This understanding is central to true quantitative modeling.
The kinetics of interfaces is inevitably very important for phase transformations. It causes in particular deviations from local equilibrium situations, as described by the bulk phase diagrams, at the interfaces between different phases. In particular for alloys it affects the entire microstructure formation and the concentrations in the growing phases. It can provoke the trapping of impurities in the growing phases and sometimes causes instabilitities of the growing fronts, which can lead to dendrites or banded structures.
In the bulk regions during isothermal solidification of alloys the dynamics is essentially described by diffusion equations. The interface, however, exhibits a much more complex behavior, as this is the region of the phase transformation additionally to diffusional exchange. In many cases the physical thickness of this interface region is small in comparison to the dimensions of the appearing patterns and can therefore be modeled as infinitely thin. This allows to link diffusional fluxes of the different atomic species to independent driving forces, which are related to the chemical potentials of the atoms. This link is established by Onsager relationships, which are strongly connected to deep physical symmetries.
In recent years, phase field models have become very popular for the modeling of alloy solidification. The most straightforward models are derived variationally from a free energy functional. However, these models typically suffer from limited accuracy of their predictions, as the phase field interface thickness, which is introduced as a numerical length scale parameter, needs to be small in comparison to the physical scales in order to provide quantitative results. For practical purposes, this is often hard to achieve due to the very large range of length scales present in the problem. The thin interface approach provides a way out of this dilemma, by leads to non-variational models. They, however, can lead to violations of the above Onsager symmetries.
At the present stage, the connection of the Onsager symmetries described on the level of sharp interface kinetics and phase field models is still fairly open. Physically, it is obvious, that all models should effectively reduce to the same predictions and therefore also to the same symmetry relations, but the connections are presently still unclear. In this project we therefore inspect the usual phase field models for the evolution of the phase field and concentration profiles in a generalized sense, such that also cross terms between the driving forces are taken into account. This allows for the first time to derive phase field models with the antitrapping current needed for quantitative modeling with thin interface asymptotics also in a variational sense. This leads to rigorous relationships between the phase field and sharp interface descriptions, and therefore for a deeper understanding of phase field models. This is an important step towards predicitive phase field models for phase transformations in alloys.