1. Materials Process Design and Control Laboratory MULTISCALE MODELING OF SOLIDIFICATION OF MULTICOMPONENT ALLOYS LIJIAN TAN Presentation for Thesis Defense (B-exam) Date: 22 May 2007 Sibley School of Mechanical and Aerospace Engineering Cornell University

2. Materials Process Design and Control Laboratory ACKNOWLEDGEMENTS SPECIAL COMMITTEE:  Prof. Nicholas Zabaras, M & A.E., Cornell University  Prof. Subrata Mukherjee, T & A.M., Cornell University  Prof. Stephen Vavasis, C.S., Cornell University  Prof. Doug James, C.S., Cornell University FUNDING SOURCES:  National Aeronautics and Space Administration (NASA), Department of Energy (DoE)  Sibley School of Mechanical & Aerospace Engineering  Cornell Theory Center (CTC) Materials Process Design and Control Laboratory (MPDC)

3. Materials Process Design and Control Laboratory OUTLINE OF THE PRESENTATION  Introduction – alloy solidification processes.  Micro-scale mathematical model  Applications  Interaction between multiple dendrites during solidification  Multi-scale modeling of solidification  Suggestions for future study

4. Materials Process Design and Control Laboratory Introduction and objectives of the current research

5. Materials Process Design and Control Laboratory Introduction Castings since 5500 BC…

6. Materials Process Design and Control Laboratory Will it break? Different microstructures Microstructure

7. Materials Process Design and Control Laboratory Alloy solidification 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

8. Materials Process Design and Control Laboratory Micro-scale mathematical model

9. Materials Process Design and Control Laboratory Two main difficulties Mathematical model  Applying boundary conditions on interface for heat transfer, fluid flow and solute transport.  Multiple moving interfaces (multiple phases/crystals).

10. Materials Process Design and Control Laboratory  Jump in temperature gradient governs interface motion  Gibbs-Thomson relation  No slip condition for flow  Solute rejection flux Complexity of the moving interface

11. Materials Process Design and Control Laboratory History: Devised by Sethian and Osher (1988) as a mathematical tool for computing interface propagation. Advantage is that we get extra information (distance to interface). This information helps to compute interfacial geometric quantities, define a novel model, doing adaptive meshing, and etc. Level Set Method We pay additional storage and extra computation time to maintain the above signed distance by solving Level set variable is simply distance to interface

12. Materials Process Design and Control Laboratory Assumption 1: Solidification occurs in a diffused zone of width 2w that is symmetric around the zero level set. A phase volume fraction can be defined accordingly. This assumption allows us to use the volume averaging technique. (N. Zabaras and D. Samanta, 2004) Present Model Don’t need to worry about boundary conditions of flow and solute any more!

13. Materials Process Design and Control Laboratory Unknown parameter k N. How will selection of k N affect the numerical solution? Assumption 2: The solid-liquid interface temperature is allowed to vary from the equilibrium 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!

14. Materials Process Design and Control Laboratory Effect of kN k N =0.001 k N =1k N =1000 Conclusion: Large k N converges to classical Stefan problem. T=-0.5 Ice T=-0.5 Water T=0 Ice Initial Steady state Numerical Solution For A Simple Problem If L=1, C=1

15. Materials Process Design and Control Laboratory In the simple case of fixed heat fluxes, interface temperature approaches equilibrium temperature exponentially. Stability requirement for this simple case is Although this is only for a very simple case, we find that selection of is stable for all problems we have considered. Stability Analysis

16. Materials Process Design and Control Laboratory Initial crystal shape Domain size Initial temperature Boundary conditions adiabatic With a grid of 64by64, we get Results using finer mesh are compared with results from literature in the next slide.  Benchmark problem Convergence Behavior

17. 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. Convergence Behavior Energy conserving makes the difference!

18. Materials Process Design and Control Laboratory Our diffused interface model with tracking of interface Phase field model without tracking of interface Computation Requirement Tracking interface makes the difference!

19. Materials Process Design and Control Laboratory Rotated surface tension Normal surface tension Mesh Anisotropy Study

20. Materials Process Design and Control Laboratory Crystal shape mainly determined by the anisotropy in surface tension not the initial perturbation. Mesh Anisotropy Study

21. Materials Process Design and Control Laboratory Applications 1.Pure material 2.Crystal growth with convection 3.Binary alloy 4.Multi-component alloy

22. Materials Process Design and Control Laboratory Effects of Undercooling (1) A small change in under-cooling will lead to a drastic change of tip velocity. (consistent with the solvability theory) (2) Increased undercooling leads to sharper dendrite shape and more obvious secondary dendrites.

23. Materials Process Design and Control Laboratory  Applicable to low under-cooling (at previously unreachable range using phase field method, Ref. Karma 2000) with a moderate grid. Extension to three dimension crystal growth

24. Materials Process Design and Control Laboratory Velocity of inlet flow at top: 0.035 Pr=23.1 Other Conditions are the same as the previous 2d diffusion benchmark problem. Crystal Growth with Convection

25. Materials Process Design and Control Laboratory  Similar to the 2D case, crystal tips will tilt in the upstream direction.  Distribute work and storage. (12 processors are used in the below example) Crystal Growth with Convection in 3D Thermal boundary layer

26. Materials Process Design and Control Laboratory  Difference between thermal boundary layer and solute boundary layer Alloy solidification  Tree type data structure for mesh refinement Coarsen Refine For alloys, uniform mesh doesn’t work very well due to the huge difference between thermal boundary layer and solute boundary layer.

27. Materials Process Design and Control Laboratory Initial crystal shape Domain size Initial temperature Boundary conditions no heat/solute flux Initial concentration Simple Adaptive Mesh Test Problem

28. 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) Results Using Adaptive Meshing

29. Materials Process Design and Control Laboratory Effects Of Refinement Criterion Interface position (curved interface) is the solved variable in this problem. Carefully choosing the refinement criterion leads to the same solution using a full grid.

30. Materials Process Design and Control Laboratory Crystal Growth of Alloy in 3D Ni-Cu alloy Copper concentration 0.40831 at.frac. Domain: a cube with side length 35  m Difficulties in this problem High under-cooling: 226 K High solidification speed High Lewis number: 14,860 Simulation of crystal growth of alloy in 3D is computationally very intensive. Our solution is to use both techniques of domain decomposition and adaptive meshing!

31. Materials Process Design and Control Laboratory Adaptive Domain Decomposition (Mesh Partition) 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Mesh Dual graph

32. Materials Process Design and Control Laboratory Technique Issues about Mesh Partition Efficient:  Require mesh partition very frequently (adaptive). Slow is unacceptable.  Maintain neighboring information using link list, e.g. for a node, there is a link list for its neighboring elements, and a link list for its neighboring edges.  Still linear in storage; greatly speed up the mesh partition procedure. Parallel:  Keep data distributed, work distributed. (Need to handle huge data)  Defined a global address (process id + pointer) Batch way:  (From + To) + Message Type + Message Length + Message content  Put all messages in a link list, and send them out together

33. Materials Process Design and Control Laboratory Colored by process id Demonstration of adaptive domain decomposition

34. Materials Process Design and Control Laboratory 3D CRYSTAL GROWH (Ni-Cu Alloy) 3 million elements (without adaptive meshing 200 million elements)

35. Materials Process Design and Control Laboratory 3D CRYSTAL GROWTH WITH CONVECTION Comparing with the pure material case, the growth for alloy is much more unstable due to the rejection of solution.

36. Materials Process Design and Control Laboratory Multi-component alloy system We use a signed distance function for each phase. Multi-phase system: one liquid phase + one or more than one solid phases. Relation between the signed distances: (1)Exactly one signed distance would be negative (2)The smallest positive signed distance has same absolute value of the negative signed distance

37. Materials Process Design and Control Laboratory  Stable growth with 4 seeds  Unstable growth with 2 seeds  Unstable to stable growth with 10 seeds Compute Eutectic Growth with Multiple Level Sets

38. Materials Process Design and Control Laboratory Solute concentration for peritectic growth of Fe – 0.3wt% C alloy at time 0.6s, 1.5s, 1.8s, and 2.4s. Compute Peritectic Growth with Multiple Level Sets

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

40. Materials Process Design and Control Laboratory Method 2: Markers to identify different region Method 1: A signed distance function for each phase. Each color (orientation of the crystal) is used as a marker. Efficient, appropriate for hundreds of crystals. Handle Multiple Interfaces

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

42. Materials Process Design and Control Laboratory As a feature of level set method, interface velocity must be evaluated at nodes near interface on both sides. Crystal orientation needs to be extended a certain distance away from the crystal to the liquid region. Extension of crystal orientation

43. Materials Process Design and Control Laboratory The purpose of this study is to verify the accuracy of using markers. Simple numerical study Growth of 9 initial seeds (circular shape) with different orientation.

44. 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).

45. 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.  We assume the nucleation sites fixed (do sampling first and then run the micro-scale model deterministically).

46. Materials Process Design and Control Laboratory Signed distance change due to nucleation We update the signed distance function at each node y, after a circular seed with radius R 0 is generated at location x i.

47. Materials Process Design and Control Laboratory CET transition study of Al-3%Cu alloy Follow conditions in Beckermann (2006). Relation between microstructure and processing parameters:

48. Materials Process Design and Control Laboratory Randomness effects

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

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

51. 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

52. 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)?

53. 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.

54. Materials Process Design and Control Laboratory Widely accepted assumptions Assumption 1: Without convection, macroscopic temperature can be modeled as Assumption 2: At a reasonably high solidification speed and without fluid flow, macroscopic concentration constant. Assumption 4: Volume fraction only depends on microstructure, and temperature. Assumption 3: Microstructure depends on macroscopic cooling history and thermal gradient history.

55. 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!

56. Materials Process Design and Control Laboratory 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

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

58. 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.

59. 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:

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

61. Materials Process Design and Control Laboratory Solve the previous problem Material properties: Boundary conditions: Initial condition: 010203040 0 10 20 30 40

62. 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

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

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

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

66. 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

67. Materials Process Design and Control Laboratory Liquid volume fraction 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

68. 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

69. Materials Process Design and Control Laboratory Solidification of Al-Cu alloy

70. Materials Process Design and Control Laboratory Step 1: Solution features of model M 1 2 3 4 56 7 8 9 10 11

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

72. Materials Process Design and Control Laboratory Periodic boundary condition for the sample problem Top half: results copied from below Bottom half: Computational domain Periodic boundary condition to minimize effects of boundary on directional solidification solution

73. Materials Process Design and Control Laboratory Lquid volume fraction for different microstructure features

74. Materials Process Design and Control Laboratory Iterative process for convergence Left half (black points): results after iter 0. Right half (green points): results after iter 3.

75. Materials Process Design and Control Laboratory Comparison with Lever rule (temperature at t=12.7s) Left: Lever rule Right: Database approach

76. Materials Process Design and Control Laboratory A B C D A (95mm,75mm) B (90mm,75mm) C (75mm,75mm) D (60mm,80mm) Microstructure in the domain 1 2 3 4 56 7 8 9 10 11 E F G H E (90mm,10mm) F (80mm,20mm) G (65mm,35mm) H (50mm,50mm) A B C D E F G H

77. Materials Process Design and Control Laboratory A B C D Fine columnar  coarse columnar  Equiaxed Microstructure from side to center A B C D

78. Materials Process Design and Control Laboratory Microstructure from corner to center E F G H Fine equiaxed  Coarse equiaxed E F G H

79. Materials Process Design and Control Laboratory Suggestions for future research

80. Materials Process Design and Control Laboratory Consider flow effects in the multi-scale model The computationally efficient model we used to identify relevant sample problems (with its analytical solution) is not applicable for problems with convection effects. Extension of the current technique or other techniques would be necessary to efficiently consider convection effects in a multi- scale framework.

81. Materials Process Design and Control Laboratory Modeling fluid structure interaction in micro-scale In our current micro-scale model, the crystal is assumed static after nucleation. In reality, the crystals would move, rotate, compact and break into fragments. Recently, there are lots of advances in fluid- structure interaction. These advances can be used to improve the micro-scale model.

82. Materials Process Design and Control Laboratory Atomic scale computation Our current micro-scale model relies on input from phase-diagram and a few parameters to mimic the crystal orientation anisotropy, surface tension effects, kinetic under-cooling effects and nucleation. Computation in the atomic scale (not continuum any more) and related multi-scale techniques to use atomic scale computation results are of great significance.

83. Materials Process Design and Control Laboratory Solid-Solid phase transformation In our current model, only liquid to solid phase transformation is considered. After this phase transformation, solid-solid transformation is also very crucial to the final microstructure. Modeling solid-solid phase transformation after solidification and study of the properties of the final microstructure is an open area.

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