** Summary **

The general objective of our project is to develop, analyze and numerically approximate new
models for multi-scale multi-phase solidification phenomena in real metallic alloy
systems and to describe the interaction of solidification effects occuring on different
length and time scales. We mainly focus on growth structures
such as dendrites, eutectics and grains appearing on a mesoscopic length scale.
The microstructure formation and the type of growth morphology
is influenced by the macroscopic solidification conditions such as the temperature field.
Interactions of effects occur on different length scales as e.g. temperature
diffusion length, mass diffusion length and capillary length.
We aim to determine new growth laws bridging the evolution on larger and smaller scales.

A non-isothermal phase-field model for systems with arbitrary numbers of components and phases
has been developed in a general and thermodynamically consistent way
including a variety of different anisotropy expressions and free energy densities. A
relation to a classical model with moving boundaries has been established and there are
existence and uniqueness results of weak solutions. A 3D parallel simulator
has been developed and applied for
extensive numerical simulations of dendritic and eutectic microstructure formations in 2D and 3D.
The simulations validate the phase-field model and illustrate the large range of
applications to multiscale solidification phenomena that can be describe with our new methods
as e.g. the formation of lamellar or rod like microstructures in eutectic alloys.

For the third application period we intend to improve the order of convergence to the
related sharp interface model in order to allow for small kinetic undercooling as well as
to make numerical computations more efficient.
The results obtained for two phase models can be generalized to multi-phase systems.
Another part of our work will be application of homogenization methods on eutectic growth
during directional solidification. We want to describe the motion of the effective solid
liquid interface which is coupled to the temperature distribution. We will investigate
general analytical as well as numerical strategies to obtain macro fluxes from the
evolution on the microscopic scale. Further challenges will lie in the relaxation of
the assumptions we made to get the existence and uniqueness results.

The 3D parallel simulator will be further optimized with respect to memory usage and
computing time by improving the parallelization algorithms and by developing an adaptive
grid generator. A main emphasis lies in numerical simulations
of anisotropic crystal growth, of dendritic and eutectic growth morphologies in 2D and 3D and
in binary and ternary alloy systems. In numerical simulations, characteristic multiscale
microstructure formations will be studied resulting from phase transformations driven by
thermal and solutal diffusion. We also plan to continue the application of the model
simulator to real material systems (e.g. Ni, Ni-Cu, NiAl-X, Al-Cu).
The description of more complex microstructures such as the solidification process of
eutectic colonies and the coupled growth of dendrites with interdendritic eutectic substructures
is a challenging aim. This aim integrates our experiences with a number of single phenomena
of dendritic and eutectic growth.
In the following, the goals of our project are specified in more detail:

__Analysis of the phase field equations__

For the existence and uniqueness results that we already obtained we note that the
conditions on the occuring potentials are too strong for the cases we are interested
in, namely, a dependence of the concentration on chemical potential (difference) of the
form
and a linear dependence of the internal energy on the temperature.
We have first ideas how to handle the degenerating terms due to the mentioned dependences.
Afterwards we intend to investigate the uniqueness and the regularity problem.
__Second order asymptotics__

The approximation of the sharp interface model to second order in the small interface
thickness is challenging for multi phase-field models. Particulary the
second order asymptotics at triple junctions where several phases meet needs to be
studied. One first task for the next application period will be to show uniqueness
of the solution of the correction problem that
we have already derived and to show a decay behaviour of solutions.
As a next step we will investigate the problem of getting
second order accurate angle conditions at the external
boundary. This should also give a hint for the
situation at triple junctions which we want to study after.

__Application of homogenization methods__

To determine the profile, the position and the velocity of the effective solid liquid front
during eutectic solidification we will make use of homogenization methods. Figure 1
shows the situation and motivates the scales that should enter our analysis. We expect that on
the macroscopic scale of the temperature diffusion length we will have to describe the motion of
the effective solid liquid interface which is coupled to the distribution of the temperature.
Effective velocities and energy fluxes should depend on the evolution on the microscopic scale
which is the mass diffusion length.
On this scale, the theory by Jackson and Hunt establishes relations between the interfacial
undercooling, the growth velocity and the typical size of the microstructures. We want to
furnish a mathematically more rigorous basis.
The two scale approach is insufficient to recover their theory. As a third,
even smaller scale, the height deviations of the solid-liquid interface as indicated in Figure
1 (right image) should be taken into account. As a first step, we will investigate a
two scale approach with the mass diffusion length and this even smaller scale to recover the
results by Jackson and Hunt. At a later stage, the thermal diffusion length will be reincluded
as a third scale into our considerations. To couple effects on a microscopic scale to a model on a
macroscopic scale numerically, we want to investigate the Heterogeneous Multiscale Method
(HMM) of E and Enquist.

The 3D parallel simulator will be further optimized in order to treat multiscale solidification phenomena in larger domains. We plan to further develop the parallelization algorithm to be applicable on shared as well as distributed memory network of workstations. Further, we will develop an adaptive grid generator for each of the three types of equations and we will consider dynamic time stepping. Another important aspect is the continuation of our already ongoing discussion with groups using other numerical methods. We will compare results of our 3D simulator with simulations performed using multigrid methods by Kornhuber and Krause or using FE toolboxes such as DEAL II and ALBERT.

The goal is to investigate the effect of anisotropy on the evolution of interfaces and phase boundaries. In particular, we will consider crystal growth in 3D (Fig. 2 a)), dendrites and eutectic grains as a result of anisotropic growth (Fig. 2 b)) and the formation of spiral structures (Fig. 2 c)).

__Formation of dendritic networks__

Dendritic arrays with dendrite networks of different crystallographic orientation
will be computed in 2D and 3D. Between the differently oriented dendrites, grain boundaries
are formed (Fig. 3 a)).
Another focus will be the investigation of characteristic length
scales during growth in a narrow channel such as tip radius, channel
width, diffusion length, thermal field and growth velocity.
We will report on the extent of side branch formation depending
on the orientation of the crystal. __Binary eutectic solidification__

A main focus of the project is the study of the solidification
of eutectic structures. By taking the specific phase diagrams of real alloys, we will
compute solidification at eutectic and off-eutectic
compositions. The solid-liquid interface profile and the curvature for different
process conditions will be evalutated and compared with theoretical results.
The appearance of regular oscillations along the Solid-Solid interface is
led by the path of the triple junctions. We plan to systematically measure
the amplitude and the wave length of the oscillations depending
on the solidification conditions (see Fig. 3 b)).
It is planned to develop new laws for moving triple junctions under the influence
of thermal and solutal diffusion fields.
The stability of eutectic growth forms will be analyzed in 2D and 3D.

a) | b) |

For alloy systems with anisotropic phases, irregular growth structures
will be simulated and we examine typical minimum and maximum spacings of the facetted stripes.
During the growth of irregular eutectics, facets die as they approach
a minimum spacing, whereas new facets are born once the spacing
between two facetes reaches a maximum value.
In order to describe the birth of lamellae, a nucleation model has to be
incorporated in the phase-field formulation in a thermodynamically consistent way.
Furthermore, eutectic grain formations with lamellar eutectic grains
of different orientation will be computed on large domains. __Complex multiscale microstructures__

In ternary eutectic systems, there is a region in the phase diagram
below the ternary eutectic temperature, where the undercooled melt
transforms into three distinct solid phases (
).
We plan to compare 2D and 3D results of growth velocity, spacing etc. with our
generalization of the Jackson-Hunt analysis for ternary eutectics.
In regions of the phase diagram where one of the three alloy components
is of minor amount (), it acts as a ternary impurity. As
a result, the evolution of eutectic colonies is experimentally
observed (see Fig. 4, left image). There is a morphological
transition from a three phase eutectic solidification to a two phase
eutectic colony growth which we want to precisely determine.
This highly multiscale type of the eutectic colony structure is wide-spread
in alloy systems and therefore of great interest for our project.
The goal is to determine the influence
of the eutectic lamellar width (smaller scale) on the spacing between the
eutectic cells (larger scale).
Another important growth structure evolution or our consideration
is the solidification of a coarse primary dendritic network
with a fine interdendritic eutectic substructure (Fig. 4 right).

a) | b) |

- H. Garcke, B. Nestler and B. Stinner,
*A diffuse interface model for alloys with multiple components and phases*

SIAM J. Appl. Math. 64 (2004), pp. 775-779. - B. Nestler, Adam A. Wheeler and H. Garcke,
*Modelling of microstructure formation and interface dynamics*

Comp. Mater. Sci 26 (2003), pp. 111-119. - For
**further references**click