1. Materials Process Design and Control Laboratory MULTISCALE MODELING OF ALLOY SOLIDIFICATION LIJIAN TAN NICHOLAS ZABARAS Date: 24 July 2007 Sibley School of Mechanical and Aerospace Engineering Cornell University

2. Materials Process Design and Control Laboratory Alloy solidification is a multiscale process solid Mushy zone liquid ~10 -1 - 10 0 m (b) Microscopic scale ~ 10 -4 – 10 -5 m solid liquid (a) Macroscopic scale q os g Solidification system of the order meter, but micro-scale detail goes to the order 10^-5 m.

3. Materials Process Design and Control Laboratory  Jump in temperature gradient governs interface motion  Gibbs-Thomson relation  Solute rejection flux Complexity of the moving interface Growth of a single crystal is nontrivial due to the complexity of the moving crystal/liquid interface.

4. Materials Process Design and Control Laboratory Assumption 1: Solidification occurs in a diffused zone of width 2w that is symmetric around the interface. A phase volume fraction can be defined according the distance to interface, which is obtained from level set computation. This assumption allows us to use the volume averaging technique. (N. Zabaras and D. Samanta, 2004) Present micro-scale model

5. Materials Process Design and Control Laboratory Assumption 2: The solid-liquid interface temperature is allowed to vary from the Gibbs-Thomson temperature in a way governed by Gibbs-Thomson condition has to be satisfied (one of the major difficulties) Extended Stefan Condition Do not want to apply this directly, because any scheme with essential boundary condition is numerically not energy conserving. Introduce another assumption: Temperature boundary condition is automatically satisfied. Energy is numerically conserving!

6. Materials Process Design and Control Laboratory Our method Osher (1997) Different results obtained by researchers suggest that this problem is nontrivial. All the referred results are using sharp interface model. Triggavason (1996) Benchmark problem: Crystal growth with initial perturbation. Advantages of the energy conserving feature Energy conserving makes the difference!

7. Materials Process Design and Control Laboratory Le=10 (boundary layer differ by 10 times) Micro-segregation can be observed in the crystal; maximum liquid concentration about 0.05. (compares well with Ref Heinrich 2003) Adaptive meshing technique to speed up computation

8. Materials Process Design and Control Laboratory Each color represents one domain Domain decomposition technique to speed up computation

9. Materials Process Design and Control Laboratory Interaction between multiple crystals

10. Materials Process Design and Control Laboratory Different crystal orientation leads to different growth velocity. Crystal orientation

11. Materials Process Design and Control Laboratory Method 2: Markers to identify different crystals Method 1: A signed distance function for each crystal. For convenience, the physical meaning of “marker” is selected to be crystal orientation. Efficient, appropriate for hundreds of crystals. Handle Multiple Interfaces

12. Materials Process Design and Control Laboratory Comparison with method using multiple level sets Dashed line: method with multiple level set functions. Solid line: method with a single level set function (using markers).

13. Materials Process Design and Control Laboratory Nucleation model Crystals are not nucleated simultaneously. To simulate nucleation, we use the following model:  Nucleation sites: density ρ, location of each nucleation site totally random (uniformly distributed in the domain).  Orientation angle: orientation angle of each nucleation site totally random (uniformly distributed between 0 and 2π).  Each nucleation site becomes an actual seed iff the required undercooling is satisfied. The required undercooling is modeled to be a fixed value or as a random variable. Simplification: We assume the nucleation sites fixed (do sampling first and then run the micro-scale model deterministically).

14. Materials Process Design and Control Laboratory Randomness effects

15. Materials Process Design and Control Laboratory Multi-scale modeling

16. Materials Process Design and Control Laboratory An example which requires multi-scale modeling Material properties: Boundary conditions: Initial condition: 010203040 0 10 20 30 40

17. Materials Process Design and Control Laboratory Interaction between a large number of crystals

18. Materials Process Design and Control Laboratory Computational results using adaptive domain decomposition Computation time: 2 days with 8 nodes (16 CPUs). Cannot wait so long! Can we obtain results in a faster way (multi-scale modeling)?

19. Materials Process Design and Control Laboratory What we can expect from multi-scale modeling  Microstructure features are often of interest, e.g. 1 st /2 nd arm spacing, Heyn’s interception measure, etc. Let us denote these features as: Of course, we cannot expect microscopic details. But  We want to know macroscopic temperature, macroscopic concentration, liquid volume fraction.

20. Materials Process Design and Control Laboratory Macro-scale model Temperature Liquid volume fraction Microstructure features Unknown functions: First two equations coupled. Microstructure features determined as a post-processing process. Solve sample problems using the fully- resolved model (micro-scale model) to evaluate them!

21. Materials Process Design and Control Laboratory Multi-scale framework

22. Materials Process Design and Control Laboratory Model M for selecting relevant sample problems Infinite number of sample problems can be selected. How to select the ones related to our problem of interest is the key! Use a very simple model to find relevant sample problems. Model M: (1) treat material as pure material (sharp and stable interface) (2) do not model nucleation

23. Materials Process Design and Control Laboratory Solution features of model M Define solute features of model M to be the interface velocity and thermal gradient in the liquid at the time the interface passes through.

24. Materials Process Design and Control Laboratory Given any solution feature of model M, we can find a problem, such that features of model M for this problem equals to the given solution feature. Selection of sample problems Chose a domain (rectangle is used) with initial and boundary condition form the following analytical solution: Sample problem:

25. Materials Process Design and Control Laboratory Multi-scale framework

26. Materials Process Design and Control Laboratory Application of multi-scale model to the previous problem Material properties: Boundary conditions: Initial condition: 010203040 0 10 20 30 40

27. Materials Process Design and Control Laboratory Step 1: Get solution features of model M

28. Materials Process Design and Control Laboratory Step 1: Get solution features of model M Plot solution features of model M for all nodes in the feature spaces

29. Materials Process Design and Control Laboratory Step 2: Fully-resolved solutions of sample problems

30. Materials Process Design and Control Laboratory Step 2: Fully-resolved solutions of sample problems Database!

31. Materials Process Design and Control Laboratory Obtained liquid volume fraction

32. Materials Process Design and Control Laboratory Step 3: Use micro-scale information in macro-scale model

33. Materials Process Design and Control Laboratory Comparison of three involved models

34. Materials Process Design and Control Laboratory Use iterations to obtain temperature, volume fraction, microstructure features

35. Materials Process Design and Control Laboratory Temperature at time 130 Macro-scale model result with Lever rule Fully-resolved model results with different sampling of nucleation sites. Average Data-base approach result

36. Materials Process Design and Control Laboratory Mushy zone at time 130 Left: temperature field and volume fraction contours (0.95 and 0.05) Right: volume fraction contour on top of fully-resolved model interface position

37. Materials Process Design and Control Laboratory Predicted microstructure features Results in rectangle: predicted microstructure Results in the middle: fully-resolved model results Black solid line: predicted CET transition location

38. Materials Process Design and Control Laboratory THANK YOU FOR YOUR ATTENTION