DFG - Priority Research Program 1095
Analysis, Modeling and Simulation of Multiscale Problems

Research Project:
Analysis, modelling and simulation of multi-scale, multi-phase solidification in alloy systems

Research group Regensburg

Principal Investigator: Prof. Dr. Harald Garcke
Coworker: B. Stinner


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 $c = e^\mu/(1 + e^\mu)$ 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 $\varepsilon$ 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.

Figure 1: Different length scales involved during solidification. On the left the thickness of the temperature diffusion layer (largest scale). In the middle the mass diffusion layer is shown; at this scale the microstructures in the solid region can be seen. On the right the height deviations of the interface on the smallest scale.
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Numerical simulations and applications
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.

Anisotropic crystal growth
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)).

Figure 2: a) 3D crystal shape, b) eutectic grains and c) spiral growth structure.
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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.

Figure 3: a) Evolution of dendritic arrays with different orientations forming a dendritic grain boundary. b) Oscillatory eutectic growth indicating the scale of the amplitude and of the wave length.
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 ( $L \rightarrow S_1 + S_2 + S_3$). 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 ($<10\%$), 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).

Figure 4: a) Eutectic colony formation due to a ternary impurity in the system, b) primary dendritic growth with an interdendritic eutectic substructure, pictures Akamatsu and Faivre
a) b)

Some related references: