3 Dimensional Phase Field Modeling of Eutectic Solidification

1. 3 Dimensional Phase Field Modeling of Eutectic Solidification D. J. Lewis, W. J. Boettinger, J. A. Warren National Institute of Standards and TechnologyMetallurgy Division100 Bureau DriveGaithersburg, MD 20899

2. Prior Work on 2d Phase Field Modeling of Eutectics • A. Karma, Phys. Rev. E 49(3), 2245 (1994) (binary eutectics) • K. Elder, J. Gunton, and M. Grant, Phys. Rev. E 54(6), 6476 (1996) • I. Steinbach, et al., Physica D 94, 135 (1996) (multi-phase field) • M. Seeßelberg and J. Tiaden, Modeling of Casting Welding and Advanced Solidification Processes VIII, Eds. B. Thomas and C. Beckermann, TMS, 1998. • A. Wheeler, G. McFadden, W. Boettinger, Proc. R. Soc. A 452, 495 (1996) • B. Nestler, A. Wheeler, Physica D 138, 114 (2000) • F. Drolet, Phys. Rev. E 61(6), 6705 (2000)

3. 2-dimensional Simulations of Eutectic Growth • 2d simulations to validate approach • lamellae pinch-off, nucleation of new lamellae, and interface undercooling minimum with variation in lamellar wavelength.

4. What can 3d simulations provide? (Provides ability to study behavior of fault motion.) Transverse section of Sn-Cd alloy, directionally solidified. • Hogan, Kraft, and Lemkey, Advances in Materials Research, 5, 83 John Wiley and Sons, NY (1971)

5. What can 3d simulations provide? (Provides ability to study ternary eutectics.) Transverse section of Ag-Cu-Mg alloy, directionally solidified. • Cooksey and Hellawell, J. Inst. Metals, 95, 183 (1967)

6. Overall project goal is to model ternary eutectics. • Work completed to date • lamellar nucleation, fault motion, and lamellar coalescence • 3d oscillatory modes • Work in progress and future work • lamellar-rod transition (in progress) • ternary eutectic microstructure selection (future work)

7. TE = 1041K Vb= 0.44 Phase Diagram used in Phase Field Calculations L b a T (K) wt. fraction B

8. Multi-Phase Field Equations for a Binary Alloy • Phase fields indicate if a phase is present ( 1 ) or not present ( 0 ). • Free energy of the system: • Bulk free energy is composed of ideal solutions and standard double well potentials: • The interpolating functions are given by:

9. Surface Energy Model • Model of Spapen and Meyer is based on configurational entropy and scales with temperature. • Their result is equivalent to expression: • a is related to the atomic packing at the solid-liquid interface. • a is used, in this case, to allow independent control of each surface energy when computing Wij and eij.

10. Multi-Phase Field Equations for a Binary Alloy • The Lagrangian is given by: • Equations of motion in frame moving at ‘v’:

11. Compromises in the model: • Interface is too thick (> 1E-5 m) • Kinetics are too slow (Mf ~ DL) • Thermal fluctuations simulated by random variations in the liquid free energy.

12. Simulation of Directional Solidification c = CE , fsolid,i = 0 , fliquid = 1 • Constant incoming composition of liquid. • T gradient in the x direction. • Temperature profile moves with frame. (Fixed in box!) • Solid liquid interface will move in box as S/L interface temperature varies. • No diffusion at mass exit boundary. • Beneficial in that one can simulate larger solidification distance with fewer grid points. v x y

13. List of Simulations • Lamellar coalescence • Lamellar nucleation • Lamellar fault motion • Oscillating mode

14. 3d Simulations of Lamellar Coalescence

16. 3d Lamellae Nucleation(catastrophic spacing selection) Concentration Liquid field (red = 1) Concentration mapped to an isosurface of f=0.1in the liquid phase field or 1/3 in the solid phase field. Solid 1 field (red = 1)

18. 3d Simulations of Lamellar Fault Motion(gradual spacing selection)

20. 3d Simulations of Lamellar Fault Motion(orthogonal slice through volume)

22. 3d Instability

24. Conclusions • We have seen: • Lamellar fault motion to reduce lamellar wavelength • Bulbous lamellar terminations • Non coalescence of lamellar faults • Organization of lamellae from random distribution of particles • Nucleation of new lamellae in lamellar trough • 3d oscillations • First three features have been observed by others in previous experimental studies. • Fourth is consistent with classic theories of morphology selection. • Nucleation across lamellar fault is rapid and occurs on the order of 10-5 seconds.

25. l l Simple Morphology Selection Computation(Regular Eutectics) Orientation relationships can change the position of the lamellar curve. Rods Surface Energy Lamellae 0.28 Volume Fraction Frank and Puttick, Acta. Met. 4, 206 (1956) – Cementite morphology in Fe-C. Cooksey and Hellawell, J. Inst. Met. 95, 63 (1967) – Additional metallic systems. Croker, Fidler, and Smith, Proc. R. Soc. A 335, 15 (1973) – Multiple ceramic systems.

26. Lamellae Surface Energy Rods Volume Fraction Lamellar-Rod Transition in Sn-Cd • Hunt and Chilton, J. Inst. Met., 91, 338-432 (1963)

27. Future Work • Systematic study of parameters that effect lamellar coalescence. • Add anisotropy to simulate orientation relationship between phases. • Implement adaptive grid technique to improve simulation time.

28. l l Brief Overview of Morphology Selection(Regular Eutectics) DT/m = vlK1+K2/l DT Vf and v constant Transition between rod and lamellar occurs at 1/p for an isotropic system l Jackson and Hunt, Trans. AIME 236, 1129 (1966)

29. Surface Energy and Interface Thickness in a Binary Alloy • A first integral may be computed to show that the surface energy in the multi phase field model is given by: • As the alloy concentration becomes more dilute in a two-phase multi-phase-field model the computed quantities approach the expected values for the pure components.

30. Computed Alloy Surface Energy versus Temperature (2 phase fields and 3 interface thickness parameters) gA = gB = 0.1

31. Computed Alloy Interface Thickness versus Temperature(2 phase fields and three interface thickness parameters)

32. Computed Alloy Surface Energy versus Temperature(2 vs. 3 Phase fields)

33. ga,b = 0.5 g = 0.086 g a,b = 2.5 g = 0.093

34. 2d Spacing Selection(Nearly Equal Volume Fraction) • Initial conditions: • 2d ‘lamellae’ • Equilibrium concentration of solute in each phase (flat interface concentration) • Random size distribution less than stable lamellar spacing • Evolution • Smallest lamellae will melt back from the solid-liquid interface and increase the average lamellar width.

35. 2d Spacing Selection(Small Second Phase Volume Fraction) • Initial conditions: • 2d ‘lamellae’ • Equilibrium concentration of solute in each phase (flat interface concentration) • Initial lamellae larger than stable lamellar spacing. • Evolution: • Solute builds up ahead of each solid/liquid interface. • Depressions form as local undercooling due to solute buildup increases. • Nuclei form in troughs to reduce the average lamellar width.

36. Method for computing approximate DT • Average temperature for all grid points where 0.33 < fliquid < 0.9 • Results in a temperature measurement ahead and behind the solid-liquid interface. • Good for determining steady state. • Better measure for quantitative calculations is needed.

37. DT versus Wavelength for a Multi-Phase Field Binary Alloy (Lamellar Morphology) TE-TI (K) l (m) DT is defined as the difference between the equilibrium eutectic temperature and the average temperature of the solid liquid interface. ‘Removing’ the kinetic effects in the PF model would improve agreement between PF and Jackson-Hunt.

38. Phase Diagrams used in Phase Field Calculations L L a b T (K) a b wt. fraction B wt. fraction B TE = 690K TE = 1041K Va= 0.28 (<1/p) Vb= 0.44

39. What can 3d simulations provide?