Matheon-C14 Macroscopic models for precipitation in crystalline solids

Background.

Precipitation and subsequent growth of droplets of a new phase in a crystalline solid due to stress assisted diffusion can lead to significant deterioration of the desired properties of the crystal. One example is the formation of arsenic liquid droplets in a semi-isolating galliumarsenid crystal, which is used in the manufacturing of opto-electronic devices. The key question is how, by changing the model parameters, to either prevent precipitation or, if it occurs, to limit it to a homogeneous distribution of small precipitates. The latter is however difficult, since precipitation preferably occurs at point defects of dislocations.

In order to avoid many expensive experiments, the derivation of appropriate models and subsequent simulations are of large interest. The models to describe such processes are however complex: they consist of free boundary problems, given by the boundaries of the droplets. The motion of the free boundary is determined in a nonlinear way by the diffusion of the species in each phase and mechanical stresses, which arise due to point defects, dislocations and misfit between solid and liquid phase. Due to this complexity, the simulation of those models is limited to a few precipitates which is in contrast to the need in realistic situations.

However, due to a clear separation of length scales - the droplets have typically size much smaller than the distance between them - there is legitimate hope to consistently model and simulate the process by averaged models on a larger scale.

Research progress.

The aim of the project was to further develop the tools developed for basic models of phase transition [Nie99, NieO01, Nie03] to employ them in the more complex situations which are faced in experiments. Within the project C14 two new different macroscopic models for precipitation in GaAs have been developed and studied analytically.

On the one hand, based on a new diffusion model, including surface tension and bulk stresses [DreD04, DreD08], we derived rigorously a generalized mean field model for the radii evolution of liquid droplets in the crystal [Kim09]. This approach captures the main properties of the system and is suitable for numerics.

On the other hand, we proposed a new thermodynamically consistent Becker-Döring system [DreD06] and examined the analytical properties and long time behaviour of its solutions [Nal06, HerNN06].