1. NASA Microgravity Research Program
Phase-Field Methods
Jeff McFadden
NIST
Dan Anderson, GWU
Bill Boettinger, NIST
Rich Braun, U Delaware
John Cahn, NIST
Sam Coriell, NIST
Bruce Murray, SUNY Binghampton
Bob Sekerka, CMU
Peter Voorhees, NWU
Adam Wheeler, U Southampton, UK
Gravitational Effects in Physico-Chemical Systems: Interfacial Effects
July 9, 2001
NASA Microgravity Research Program

2. Outline Background Surface Phenomena in Diffuse-Interface Models
Surface energy and surface energy anisotropy
Surface adsorption
Solute trapping
Multi-phase wetting in order-disorder transitions
Recent phase-field applications
Monotectic growth
Phase-field model of electrodeposition

3. Phase-Field Models
Main idea: Solve a single set of PDEs over the entire domain
Two main issues for a phase-field model:
Bulk Thermodynamics Surface Properties
Phase-field model incorporates both bulk thermodynamics of multiphase systems and surface thermodynamics (e.g., Gibbs surface excess quantities).

4. Phase-Field Model
The phase-field model was developed around 1978 by J. Langer at CMU as a computational technique to solve Stefan problems for a pure material. The model combines ideas from:
The enthalpy method
(Conserves energy)
The Cahn-Allen equation
(Includes capillarity)
Van der Waals (1893)
Korteweg (1901)
Landau-Ginzburg (1950)
Cahn-Hilliard (1958)
Halperin, Hohenberg & Ma (1977)
Other diffuse interface theories:

5. Cahn-Allen Equation J. Cahn and S. Allen (1977)
M. Marcinkowski (1963)
Anti-phase boundaries in BCC system
Motion by mean curvature:
Surface energy:
“Non-conserved” order parameter:

6. Ordering in a BCC Binary Alloy

7. Parameter Identification
1-D solution:
Interface width:
Surface energy:
Curvature-dependence (expand Laplacian):

8. Phase-Field Model Introduce the phase-field variable:
Introduce free-energy functional:
Dynamics
J.S. Langer (1978)

9. Free Energy Function

10. Phase-Field Equations
Governing equations:
First & second laws
Require positive entropy
production
Thermodynamic derivation
Energy functionals:
Penrose & Fife (1990), Fried & Gurtin (1993), Wang et al. (1993)

11. Sharp Interface Asymptotics
Consider limit in which
Different distinguished limits possible.
Caginalp (1988), Karma (1998), McFadden et al (2000)
Can retrieve free boundary problem with

12. Outline Background Surface Phenomena in Diffuse-Interface Models
Surface energy and surface energy anisotropy
Surface adsorption
Solute trapping
Multi-phase wetting in order-disorder transitions
Recent phase-field applications
Monotectic solidification
Phase-field model of electrodeposition

13. Anisotropic Equilibrium Shapes
W. Miller & G. Chadwick (1969)
Hoffman & Cahn (1972)

14. Cahn-Hoffman -Vector
Taylor (1992)
Phase field

15. Cahn-Hoffman -Vector Equilibrium Shape is given by:
Force per unit length in interface:
Cahn & Hoffmann (1974)
Phase field

16. Diffuse Interface Formulation
Kobayashi(1993), Wheeler & McFadden (1996), Taylor & Cahn (1998)

17. Corners & Edges In Phase-Field
changes type when plot is concave.
Steady case: where
Noether’s Thm:
where
interpret as a “stress tensor”
Fried & Gurtin (1993), Wheeler & McFadden 97

18. Corners/Edges Jump conditions give: (force balance) where and
Bronsard & Reitich (1993), Wheeler & McFadden (1997)

19. Corners and Edges
Eggleston, McFadden, & Voorhees (2001)

20. Outline Background Surface Phenomena in Diffuse-Interface Models
Surface energy and surface energy anisotropy
Surface adsorption
Solute trapping
Multi-phase wetting in order-disorder transitions
Recent phase-field applications
Monotectic solidification
Phase-field model of electrodeposition

21. Cahn-Hilliard Equation

22. Phase Field Equations - Alloy
Coupled Cahn-Hilliard & Cahn-Allen Equations
where {
Wheeler, Boettinger, & McFadden (1992)

23. Alloy Free Energy Function
One possibility
Ideal Entropy
L and S are liquid and solid regular solution parameters

24. W. George & J. Warren (2001) 3-D FD 500x500x500 DPARLIB, MPI
32 processors, 2-D slices of data

25. McFadden and Wheeler (2001)
Surface Adsorption
McFadden and Wheeler (2001)

26. Cahn (1979), McFadden and Wheeler (2001)
Surface Adsorption
1-D equilibrium:
where
Differentiating, and using equilibrium conditions, gives
Cahn (1979), McFadden and Wheeler (2001)

27. Surface Adsorption Ideal solution model Surface free energy

28. Outline Background Surface Phenomena in Diffuse-Interface Models
Surface energy and surface energy anisotropy
Surface adsorption
Solute trapping
Multi-phase wetting in order-disorder transitions
Recent phase-field applications
Monotectic solidification
Phase-field model of electrodeposition

29. N. Ahmad, A. Wheeler, W. Boettinger, G. McFadden (1998)
Solute Trapping
Increasing V
At high velocities, solute segregation becomes small (“solute trapping”)
N. Ahmad, A. Wheeler, W. Boettinger, G. McFadden (1998)

30. Nonequilibrium Solute Trapping
Numerical results (points) reproduce Aziz trapping function
With characteristic trapping speed, VD, given by

31. Nonequilibrium Solute Trapping (cont.)

32. Outline Background Surface Phenomena in Diffuse-Interface Models
Surface energy and surface energy anisotropy
Surface adsorption
Solute trapping
Interface structure in order-disorder transitions
Recent phase-field applications
Monotectic solidification
Phase-field model of electrodeposition

33. FCC Binary Alloy Disordered phase CuAu
G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A. Wheeler

34. Ordering in an FCC Binary Alloy

35. Free Energy Functional

36. Equilibrium States in FCC

37. Wetting in Multiphase Systems
M. Marcinkowski (1963)
Kikuchi & Cahn CVM for fcc APB (Cu-Au)
R. Braun, J. Cahn, G. McFadden, & A. Wheeler (1998)
Phase-field model with 3 order parameters

38. Adsorption in FCC Binary Alloy
Interphase Boundaries
Antiphase Boundaries
G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A. Wheeler

39. Outline Background Surface Phenomena in Diffuse-Interface Models
Surface energy and surface energy anisotropy
Surface adsorption
Solute trapping
Multi-phase wetting in order-disorder transitions
Recent phase-field applications
Monotectic solidification
Phase-field model of electrodeposition

40. Monotectic Binary Alloy
A liquid phase can “solidify” into both a solid and a different liquid phase.
Expt: Grugel et al.
Nestler, Wheeler, Ratke & Stocker 00

41. Incorporation of L2 into the solid phase
Expt: Grugel et al.

42. Nucleation in L1 and incorporation of L2 into solid
Expt: Grugel et al.

43. Outline Background Surface Phenomena in Diffuse-Interface Models
Surface energy and surface energy anisotropy
Surface adsorption
Solute trapping
Multi-phase wetting in order-disorder transitions
Recent phase-field applications
Monotectic solidification
Phase-field model of electrodeposition

44. Superconformal Electrodeposition
Cross-section views of five trenches with different aspect ratios
filled under a variety of conditions.
Note the bumps over the filled features.
D. Josell, NIST

45. Phase-Field Model of Electrodeposition
J. Guyer, W. Boettinger, J. Warren, G. McFadden (2002)

47. 1-D Equilibrium Profiles

48. 1-D Dynamics

49. Conclusions
Phase-field models provide a regularized version of Stefan problems for computational purposes
Phase-field models are able to incorporate both bulk and surface thermodynamics
Can be generalised to:
include material deformation (fluid flow & elasticity)
models of complex alloys
Computations:
provides a vehicle for computing complex realistic microstructure

50. Experimental Observation of Dendrite Bridging Process
(c) t = 30 s fs = 0.82
(b) t = 10 s fs = 0.70
(a) t = 0 s fs = 0.00
125 mm
Photo:
W. Kurz, EPFL
(d) t = 75 s fs = 0.94
(e) t = 210 s fs = 0.97
(f) t = 1500 s fs = 0.98

51. Dendrite side arm bridgingX Y
Collision of offset arms - Delayed bridging

52. Coalescence of two Grains Using Multi-Grain Model
P; Disjoining Pressure x
Large misorientation
P > 0
grains “repel”
ggb = gsl = 0.1
DT = 0 K
ggb = gsl = 0.1
DT = 50 K
W. Boettinger (NIST) & M. Rappaz (EPFL)

53. -Tensor Derivation