Key Issues in Solidification Modeling—

1. Scales: Process Micro-strcture ~10 mm ~0.5 m Key Issues in Solidification Modeling— Vaughan Voller, University of Minnesota, Aditya Birla Chair Validation and Verification: Do governing equations model the correct physics? Is approximate solution a solution of governing Equations ? Ferreira et al How can we deal with this problem Micro-macro models How feasible is Direct Modeling of Microstructure? What can it tell us about the process Scale? Prediction Of microstructure

2. In mushy region solute is partitioned at solid-liquid interface Cliquid Csolid 1m Scales: An Example Problem: Macrosegregation—Ingot alloy solidification Result after full solidification is macro-scale areas with concentration above or Below the nominal concentration see Flemings (Solidification Processing) and Beckermann (Ency. Mat) solid crystals + liquid “mushy region” This solute is redistributed at process scale by fluid and solid motions shrinkage solid grain motion liquid alloy convection

3. REV representative ½ arm space Process solid g ~ 50 mm ~5 mm sub-grid model ~0.5 m Key Scales in Macro-segregation Computational grid size Solute value in liquid phase controlled by local diffusion in solid “microsegregation” Solute transport controlled by advection

4. ~ 0.1 m chill A Casting The REV casting ~10 mm 103 101 10-1 10-3 10-5 10-7 10-9 Nucleation Sites heat and mass tran. equi-axed columnar grain formation The Grain Envelope growth ~ mm Time Scale (s) solute diffusion The Secondary Arm Space nucleation ~100 mm interface kinetics 10-9 10 10-3 10-1 Length Scale (m) The Tip Radius ~10 mm The Diffusive Interface f = 1 f = -1 ~1 nm Scales in General Solidification Processes (after Dantzig) Question for later: Can we build a direct-simulation of a Casting Process that resolves to all scales?

5. A solidification model has three components: The Domain: The Grid: The sub-Grid: Examples The Grid Problem Domains Sub-grid --Constitutive -- Controlled by averaged Properties in REV Realizations--Of multi-phase regions Element in numerical Calculation ---REV State described by averaged mixture values Macro Process Effect of morphology on flow REV METER Meso Microstructure The Grain Envelope Solid-liquid interface A representative Arm spacing— Form of Constitutive model T=F(g) f = 1 f = -1 The Diffusive Interface, e.g. NANO-Meter

6. REV representative ½ arm space Process solid g ~ 50 mm ~5 mm sub-grid model ~0.5 m Key Scales in Macro-segregation Computational grid size Solute value in liquid phase controlled by local diffusion in solid “microsegregation” Solute transport controlled by advection Develop a “Macro-Micro Model” (Rappaz) Solve transport equations at macroscopic scale (MACRO) Use sub-grid model to account for microsegregation (MICRO) a “constitutive model”

7. Macro (Process) Scale Equations Equations of Motion (Flows) mm REV Heat: Solute Concentrations: Assumptions for shown Eq.s: -- No solid motion --U is inter-dendritic volume flow If a time explicit scheme is used to advance to the next time step we need find REV values for • T temperature • Cl liquid concentration • gs solid fraction • Cs(x) distribution of solid concentration

8. Process REV representative ½ arm space MICRO: solid g sub-grid model ~ 50 mm microsegregation and solute diffusion in arm space ~5 mm ~0.5 m Computational grid size need to extract from computation Of these Mixture values -- -- -- The Micro-Macro Model MACRO:

9. A C Primary Solidification Solver g Transient mass balance g model of micro-segregation Iterative loop Cl T (will need under-relaxation) Gives Liquid Concentrations equilibrium

10. Micro-segregation Model transient mass balance gives liquid concentration Solute Fourier No. Solute mass density after solidification Solute mass density before solidification Q -– back-diffusion Solute mass density of new solid (lever) liquid concentration due to macro-segregation alone ½ Arm space of length l takes tf seconds to solidify In a small time step new solid forms with lever rule of concentration Need an easy to use approximation For back-diffusion

11. For special case Of Parabolic Solid Growth In Most other cases The Ohnaka approximation and And ad-hoc fit sets the factor Works very well The parameter Model --- Clyne and Kurz, Ohnaka

12. Need to lag calculation one time step and ensure Q >0 m is sometimes take as a constant ~ 2 BUT In the time step model a variable value can be use Due to steeper profile at low liquid fraction ----- Propose The Profile Model Wang and Beckermann

13. An Important wrinkle ---Coarsening Due to dissolution processes some arms will melt and arm-space will coarsen Time 1 Time 2 > Time 1

14. A model by Voller and Beckermann suggests If we assume that solid growth is close to parabolic m =2.33 in Parameter model In profile model Arm-space will increase in dimension with time Coarsening This will dilute the concentration in the liquid fraction—can model be enhancing the back diffusion 

15. Verification: of Micro Models: Constant Cooling of Binary-Eutectic Alloy With Initial Concentration C0 = 1 and Eutectic Concentration Ceut = 5, No Macro segregation , k= 0.1 T Cl As solidification proceeds the concentration in liquid increases. When the eutectic composition is reached remaining liquid solidifies isothermally, Eutectic Fraction In model calculate the transient value of g from Use 200 time steps and equally increment 1 < Cl < 5 Parameter or Profile

16. Verification of Micro Models: Verify approximate model for back-diffusion by comparing solution with FD solution of Fick’s equation in arm space. Parameter or Profile Remaining Liquid when C =5 is Eutectic Fraction

17. Validation of Micro Model: Predictions of Eutectic Fraction With constant cooling Co = 4.9 Ceut = 33.2 k = 0.16 Al-4.9% Cu Comparison with Experiments Sarreal-Abbaschian Met Trans 1986 X-ray analysis determines average eutectic fraction

18. Macro-Micro Model of Solidification Calculate Transient solute balance in arm space predict T Predict g predict Cl Two MICRO Models For Back Diffusion Profile Parameter A A little more difficult to use Robust Easy to Use Poor Performance at very low liquid fraction— can be corrected With this Ad-hoc correction Excellent performance at all ranges Account for coarsening C My Method of Choice

19. Extra Terms Magically vanish Buoyancy Friction between solid and liquid Accounts for mushy region morphology Can requires a solid-momentum equation Modeling the fluid flow could require a Two Phase model, that may need to account for: Both Solid and Liquid Velocities at low solid fractions A switch-off of the solid velocity in a columnar region A switch-off of velocity as solid fraction g  o. An EXAMPLE 2-D form of the momentum equations in terms of the interdentrtic fluid flow U, are:

20. riser liquid Parameter Current estimate mushy empirical y solid chill Verification: of Macro-Micro Model—Inverse Segregation in a Binary Alloy Shrinkage sucks solute rich fluid toward chill – results in a region of +ve segregation at chill 100 mm Flow by simple app. of continuity Fixed temp chill results in a similarity solution

21. Validation: Comparison with Experiments riser liquid mushy y solid chill 100 mm Ferreira et al Met Trans 2004

22. Direct Microstructure Modeling ~ 0.1 m chill A Casting domain The REV ~10 mm grid Nucleation Sites equi-axed columnar domain The Grain Envelope ~ mm sub-grid The Secondary Arm Space ~100 mm Macro-Micro models at process scale The Tip Radius grid ~10 mm The Diffusive Interface f = 1 f = -1 Micro-Nano model for micro-structure ~1 nm constitutive

23. Example: Growth of dendritic crystal in an under-cooled melt (seminar on July 14) Solved in ¼ Domain with A 200x200 grid Growth of solid seed in a liquid melt Initial dimensionless undercooling T = -0.8 Resulting crystal has an 8 fold symmetry

36. Grid independent results with correct dynamics can be readily obtained Tip Velocity Interacting grains grid anisotropy prediction of concentration field Scale of calculations shown 1 mm

37. Process REV ~ 1 mm ~5 mm ~0.5 m Computational grid size So can we use DMS to predict microstructure at the process level? Sub grid scale For 2-D calc at this scale Will need 1018 grids

38. For 2-D calc at this scale Will need 1018 grids Voller and Porte-Agel, JCP 179, 698-703 (2002 1000 20.6667 Year “Moore’s Law” 2055

39. CONCLUSIONS ~ 0.1 m chill A Casting The REV ~10 mm Nucleation Sites equi-axed columnar The Grain Envelope ~ mm The Secondary Arm Space ~100 mm The Tip Radius ~10 mm The Diffusive Interface f = 1 f = -1 ~1 nm Conclusion: Can Currently Build Validated and Verified Models that Can successfully model across ~ 4 decades Of length scales Able to use Macro-Micro Approach To model all scales of Heat and Mass Transport Able to build Local Microstructure models But a long way from DMS Direct microstructure simulation at the process scale In the meantime what Value Added can we get from Local microstructure models Use as generator for constitutive models Use in volume averaging approaches