1. Materials Process Design and Control Laboratory MULTISCALE MODELING OF ALLOY SOLIDIFICATION PROCESSES LIJIAN TAN Presentation for Admission to candidacy examination Date: 11 May 2006 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 FUNDING SOURCES:  National Aeronautics and Space Administration (NASA), Department of Energy (DoE), Aluminum Corporation of America (ALCOA)  Cornell Theory Center (CTC)  Sibley School of Mechanical & Aerospace Engineering Materials Process Design and Control Laboratory (MPDC)

3. Materials Process Design and Control Laboratory OUTLINE OF THE PRESENTATION  Brief review of alloy solidification process  Macro-scale Model  Meso-scale Model  Ongoing and future work

4. Materials Process Design and Control Laboratory Review of alloy solidification process

5. Materials Process Design and Control Laboratory Piping Micro shrinkage Macro shrinkage Shrinkage porosity (Ref. EPFL, Switzerland) DEFECTS DURING ALLOY SOLIDIFICATION Sub-surface liquation and crack formation (Ref. ALCOA corp.) Non-uniform growth and microstructure (Ref. ALCOA corp.) Freckle formation (Ref. Beckermann)

6. Materials Process Design and Control Laboratory MULTISCALE NATURE OF ALLOY SOLIDIFICATION solid Mushy zone liquid ~10 -1 - 10 0 m (b) Meso/micro scale ~ 10 -4 – 10 -5 m solid liquid (a) Macro scale (mushy zone is represented by volume fraction) q g

7. Materials Process Design and Control Laboratory Macro-scale solidification modeling

8. Materials Process Design and Control Laboratory VOLUME AVERAGING MODEL  Governing equations Legacy code developed by Deep Samanta (former MPDC graduate student) to solve this system based on stabilized FEM method.

9. Materials Process Design and Control Laboratory PREVIOUS WORK  L. Tan and N. Zabaras, "A thermomechanical study of the effects of mold topography on the solidification of Aluminum alloys", Materials Science and Engineering: A, Vol. 404, pp. 197-207, 2005  D. Samanta and N. Zabaras, "Macrosegregation in the solidification of Aluminum Alloys on uneven surfaces", International Journal of Heat and Mass Transfer, Vol. 48, pp. 4541-4556, 2005  D. Samanta and N. Zabaras, "Modeling melt convection in solidification processes with stabilized finite element techniques", International Journal for Numerical Methods in Engineering, Vol. 64, pp. 1769-1799, 2005  B. Ganapathysubramanian and N. Zabaras, "On the control of solidification of conducting materials using magnetic fields and magnetic field gradients", International Journal of Heat and Mass Transfer, Vol. 48, pp. 4174-4189, 2005  N. Zabaras and D. Samanta, "A stabilized volume-averaging finite element method for flow in porous media and binary alloy solidification processes", International Journal for Numerical Methods in Engineering, Vol. 60/6, pp. 1103-1138, 2004

10. Materials Process Design and Control Laboratory Lever Rule : (Infinite back-diffusion) Scheil Rule : (Zero back-diffusion) These relationships are based on phase diagrams and certain assumption of the diffusion in solid phase (infinite back-diffusion or zero back-diffusion). These relationships only give the upper bound and the lower bound. According to our experience, numerical results using Scheil rule is closer to experimental results. But neither of them is accurate. We can only turn to meso-scale modeling for better accuracy. COMPUTE VOLUME FRACTION

11. Materials Process Design and Control Laboratory Meso-scale solidification modeling

12. Materials Process Design and Control Laboratory GOVERNING EQUATIONS  A moving solid-liquid interface  Presence of fluid flow  Heat transfer  Solute transport Although equations are even simpler than macro-scale model, numerically it is very hard to handle due to the moving solid-liquid interface.

13. Materials Process Design and Control Laboratory ISSUES RELATED WITH MOVING INTERFACE  Jump in temperature gradient governs interface motion  No slip condition for flow  Gibbs-Thomson relation  Solute rejection flux Requires curvature computation at the moving interface!

14. Materials Process Design and Control Laboratory  Cellular automata  Phase field method  Front tracking  Level set method Techniques for handling moving interface

15. Materials Process Design and Control Laboratory CELLULAR AUTOMATA METHOD Numerically, phase transformation is just change of 0 to 1. The interface phase growth is model as a probability of change 0 to 1. Ref. Kremeyer (1998) Use 0/1 to represent liquid/solid for each pixel.  Advantages CA method might be the most widely used method till now, mainly because it is very easy to implement.  Disadvantage However, the model is so different from the original governing equations. Even for some simple problems, it deviates from the analytical solution significantly.

16. Materials Process Design and Control Laboratory Review paper by Botteringer, Warren, Beckermann, Karma (2002) Ref Langer (1978)  Advantages  Still easy to implement, also widely used.  No essential boundary conditions  (global energy conserving)  Disadvantages  A number of parameters in the phase field equation of little or no physical meaning. To determine their values is not a trivial task.  Require huge grid to be consistent with the sharp interface model. Typical grid sizes: 400 × 400 Karma (1998), 800×800 Goldenfeld (2001), 3000 × 3000 Beckermann (2005), etc. (Due to the large gradient of phase field variable within the diffused interface.) PHASE FIELD METHOD This method diffuses the sharp interface into one with a certain width and uses a variable varying from 0 to 1 to approximately represent interface.

17. Materials Process Design and Control Laboratory FRONT TRACKING METHOD Ref. Tryggvason (1996), J. Heinrich (2001)  Advantages  Solving sharp interface model directly  (physics clear in governing eqs.)  Disadvantages  BC  scheme not strictly energy conserving  Curvature computation (from mark pts) complicated  Difficult to implement Ideas: (1) Uses markers to represent interface (2) Markers are moved using velocity computed from Stefan equation

18. Materials Process Design and Control Laboratory LEVEL SET METHOD Interface motion (level set equation) Ref. S. Osher 1997, Devised by Sethian&Osher  Advantages  Interface geometries can be easily and accurately computed.  Level set equation well studied (FDM with higher order accuracy, FEM)  Disadvantage: Application of boundary conditions still not easy (most applications are restricted to pure materials without melt convection). Signed distance Introduced to this area by J. Dantzig 2000, R. Fedkiw 2003 etc.

19. Materials Process Design and Control Laboratory OUR WORK WITH LEVEL SET METHOD  N. Zabaras, B. Ganapathysubramanian and L. Tan, "Modeling dendritic solidification with melt convection using the extended finite element method (XFEM) and level set methods", Journal of Computational Physics, in press.  L. Tan and N. Zabaras, "A level set simulation of dendritic solidification with combined features of front tracking and fixed domain methods", Journal of Computational Physics, Vol. 211, pp. 36-63, 2006

20. Materials Process Design and Control Laboratory (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. (2) The mean solid-liquid interface temperature in the freezing zone of width 2w is allowed to vary from the equilibrium temperature in a way governed by PRESENT MODEL BASED ON LEVEL SET METHOD  Assumptions Instead of forcing

21. Materials Process Design and Control Laboratory EXTENDED STEFAN CONDITION Without applying essential boundary condition on the moving interface, the numerical scheme satisfies energy conservation. This leads to both accuracy and convenience. Only one parameter k N needs to be specified. How will the selection of k N affect the numerical solution?

22. Materials Process Design and Control Laboratory NUMERICAL STUDY FOR A SIMPLE CASE Pick up a very simple system Initially with left half ice and right half water; whole domain is under-cooled at temperature -0.5. At steady state: Whole domain is ice with temperature 0. Effect of kN k N =0.001 k N =1k N =1000 Temperature distribution in the domain at various time. Conclusion: Large k N converges to classical Stefan problem.

23. Materials Process Design and Control Laboratory STABILITY ANALYSIS 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 most problems of interest.

24. Materials Process Design and Control Laboratory NUMERICAL CONVERGENCE STUDY  Infinite corner problem (2D with analytical solution) After the mesh is refined to 20by20, the error reduces almost quadratically.

25. 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 COVERGENCE BEHAVIOR OF VARIOUS METHODS

26. Materials Process Design and Control Laboratory COVERGENCE BEHAVIOR OF VARIOUS METHODS 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)

27. Materials Process Design and Control Laboratory APPLICATIONS Application: Modeling dendritic solidification (pure materials & binary alloy)  2D crystal growth benchmark problem (pure material)  3D crystal growth benchmark problem (pure material)  Effects of fluid flow on crystal growth (pure material)  Adaptive mesh technique (required for alloys)  2D crystal growth benchmark problem (alloy)  3D crystal growth benchmark problem (alloy)  Effects of fluid flow on crystal growth (alloy)

28. Materials Process Design and Control Laboratory 2D CRYSTAL GROWTH BENCHMARK PROBLEM (1) A small change in under-cooling will lead to a drastic change of tip velocity. (consistent with the solvability theory) (2) An increase of diffusion coefficient in the liquid region tends to make the tip sharper. Effects of solid diffusion coefficient are not obvious.

29. Materials Process Design and Control Laboratory Our diffused interface model with tracking of interface Phase field model without tracking of interface COMPARISON WITH PHASE FIELD METHOD

30. Materials Process Design and Control Laboratory MESH ANISOTROPY STUDY Rotated surface tension Normal surface tension

31. Materials Process Design and Control Laboratory Crystal growth mainly determined by surface tension not initial perturbation. MESH ANISOTROPY STUDY

32. Materials Process Design and Control Laboratory Using a larger domain, perturbations/ second arm dendrites will be developed. FORMATION OF SECONDARY DENDRITE ARMS

33. Materials Process Design and Control Laboratory EXTENSION TO THREE DIMENSION CRYSTAL GROWTH  Applicable to 3d with high under-cooling using a coarse mesh. Temperature and crystal shape at time t=105

34. 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. CRYSTAL GROWTH AT LOW UNDERCOOLING

35. Materials Process Design and Control Laboratory CRYSTAL GROWTH WITH CONVECTION Velocity of inlet flow at top: 0.035 Pr=23.1 Other Conditions are the same as the previous 2d diffusion benchmark problem.

36. Materials Process Design and Control Laboratory CRYSTAL GROWTH WITH CONVECTION  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) For alloys, uniform mesh doesn’t work very well due to the huge difference between thermal boundary layer and solute boundary layer.

37. Materials Process Design and Control Laboratory  Difference between thermal boundary layer and solute boundary layer ADAPTIVE MESHING  Tree type data structure for mesh refinement Coarsen Refine  Implemented for both 2D and 3D.  Coupled with domain decomposition.

38. 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

39. 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

40. 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.

41. Materials Process Design and Control Laboratory 3D CRYSTAL GROWH (Ni-Cu Alloy) 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

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

43. Materials Process Design and Control Laboratory Unfinished Colored by process id DOMAIN DECOMPOSTION

44. 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.

45. Materials Process Design and Control Laboratory CONCLUSION  Successfully applied level set method to single crystal growth (pure material and alloys).  This new method is accurate and efficient. Our method is a very promising method for this purpose. But can we do multi-scale modeling with this method? Multi-scale modeling is my ongoing and planned research work. However, we need to compute thousands of dendrites in practice!

46. Materials Process Design and Control Laboratory  Adaptive meshing is one option for multi-scale modeling of solidification. Estimation of computational effort: 2D promising 3D impossible Necessary for validation purpose! ADAPTIVE MESHING

47. Materials Process Design and Control Laboratory DATA BASE APPROACH A number of runs in the meso-scale volume fraction and macroscopic variables Data base For the interested problem, use the same macro scale model, but with volume fraction from interpolation (not Scheil/Lever rule). Been used in the solid group, Terada and Kikuchi (1995), Lee and Ghosh (1999). Advantages: (1) Very fast. (2) No need to do new developments in coding (non-intrusive). Disadvantage: May only be applicable to certain problems.

48. Materials Process Design and Control Laboratory SUBGRID MODEL Meso-scale grid for each element Macro-scale grid for casting object Simple average based on volume fraction Homogenization (shape, orientation) Boundary condition for temperature, solute, and fluid flow Time step in meso-scale CA Level set Nucleation Binary alloy Multi-component Widely used, but CA method is the underlying micro-scale model due to the huge computational requirement for other methods. Moving of nuclei under convection

49. Materials Process Design and Control Laboratory COMPARISONS StrategyStorageComputational work Accuracy Adaptive meshing One huge matrixVery hugeVery good Sub grid model A number of small matrices ModerateGood Data base approach Just one small matrix Very little(?)

50. Materials Process Design and Control Laboratory COMPUTATIONAL ISSUES Most alloy used in industry is multi-component and usually with multiple solid phases. Limited multi- component capability has been developed using multiphase level set method. But triple points and computational efficiency are still unsolved issues. Nucleation needs to implemented in the meso scale model. Mechanical properties prediction is the final goal. X x x = FX y = FY + w N n

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