It is the goal of this project to understand the interaction of different
length and time scales in phase separating systems with elastic misfit. If an
alloy is quenched below a certain critical temperature, a homogeneous mixture
of the alloy components will not be stable anymore. In a process called
spinodal decomposition - which happens on a very rapid time scale -
different phases appear which differ by the concentration of the alloy
components.
The Cahn-Hilliard equation and its extension with elasticity, the
Cahn-Larché model, have been originally introduced to model this phenomenon.
Later numerical simulations and formally matched asymptotic expansions showed
that the Cahn-Hilliard equation also models a process on an intermediate time
scale in which the regions occupied by the phases rearrange in order to
decrease their energy which at this stage is essentially given as the sum of
surface energy and elastic energy. This process is driven by the diffusion of
atoms. In the case where elastic energy contributions can be neglected this
leads to nearly spherical shapes.
On a third time scale the main remaining driving forces for energy reduction
is given by the interaction between different particles and interactions on a
larger length scale become important. On this large time scale one observes,
in the case that no elastic interactions are present, that small particles
shrink while large particles grow (Ostwald ripening). For the last two time
scales sharp interface models can also be used, i.e. in contrast to the
Cahn-Hilliard model where the interface between different phases is given by a
diffuse layer, now the interface is modeled as a sharp hypersurface.
The
influence of elastic interactions, e.g. through an elastic misfit due to
different lattice constants, can drastically influence the coarsening process.
In particular on the large time scale the elastic energy becomes comparable to
the surface energy and it might be possible to stabilize the coarsening
process (''inverse coarsening'').
Aims:
- We plan to complete the rigorous justification of the asymptotic limit of the
Cahn-Larché system.
- It is planned to rigorously derive scaling laws in the presence of
elastic interactions for some specific problems.
- We aim for a detailed comparison through numerical simulation
of the different models for phase separation
with respect to the qualitative behaviour as well as the statistical properties:
- the Cahn-Larché phase field model,
- the modified Mullin Sekerka sharp interface model,
- the reduced sharp interface model with rectangular particles, and
- the intermediate model by Blesgen/Luckhaus (in cooperation with the
SPP project ''Singular Limits of discrete models'' by Luckhaus and Blesgen)
- an extension of the model of Garcke and Nestler including elasticity
(see SPP project ''Analysis, modelling and simulation of multi-scale,
multi-phase solidification in alloy systems'').
Therefore, we have to consider efficient hierarchical solvers for the finite
element and boundary element approaches incorporated in these models and
to consider the convergence properties of the discretizations.
- The phase field approach for surface diffusion in the presence of
elastic interactions will be related to sharp interface models. We are in
particular interested in applications to epitaxial growth of thin solid
films and here it will be important to identify angle conditions given by
Young's law at the film-substrate-gas contact line. Here we plan to
cooperate with A. Voigt, who is applying for a new project within the SPP.
- We want to continue the study of solid, thin, elastic films in
cooperation with the project ''Multiscale folding patterns in
compressed elastic sheets'' of S. Conti within the SPP.
- One open problem is whether solutions of either Cahn-Hilliard or
Cahn-Larché systems converge to weak solutions
of the Mullins-Sekerka problem in the sense of Luckhaus and Sturzenhecker.
Here maybe ideas of
Röger and Hutchinson and Tonega might be helpful.