1. NASA Microgravity Research Program
Phase-Field Models of Solidification
Jeff McFadden
NIST
Outline
Background
Phase-Field Models
Numerical Computations
Dan Anderson, GWU
Bill Boettinger, NIST
Rich Braun, U Delaware
Sam Coriell, NIST
John Cahn, NIST
Bruce Murray, SUNY Binghampton
Bob Sekerka, CMU
Jim Warren, NIST
Adam Wheeler, U Southampton, UK
NASA Microgravity Research Program

2. Modeling at various length scales
10 mm
2 nm
Atomistic scale Å
Dendrite scale m
Grain scale mm
How to connect these various scales ?
Component scale
cm - m
M. Rappaz, EPFL

3. Dendritic Microstructure
Polished and etched microstructure after freezing
Liquid decanted during freezing

4. Freezing a Pure Liquid
Hele Shaw
Saffman & Taylor
Dendrite
Glicksman

5. Stefan Problem Interface is a surface; No thickness; Solid
Liquid
Interface is a surface;
No thickness;
Physical properties:
Surface energy, kinetics
Conservation of energy

6. Surface Energy Critical Nucleus and Coarsening Grain Boundary Grooves
Wavelength of instabilities

7. Critical Nucleus and Coarsening
Minimize the total surface energy
for a given volume of inclusions
P. Voorhees & R. Schaefer (1987)

8. Grain Boundary Grooves
S.C. Hardy (1977)

9. Wavelength of Instabilities
Ice cylinder growing into
supercooled water,
Instability wavelength
depends on surface energy:
S. Hardy and S. Coriell (1968)

10. Morphological Instability
“Constitutional supercooling”
“Point effect”
Mullins & Sekerka (1963, 1964)

11. 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:

12. Cahn-Allen Equation J. Cahn and S. Allen (1977)
M. Marcinkowski (1963)
Description of anti-phase boundaries (APBs)
Motion by mean curvature:
Surface energy:
“Non-conserved” order parameter:

13. Ordering in a BCC Binary Alloy

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

15. 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 Thermodynamics L g g’ b
Phase-field model incorporates both bulk thermodynamics of multiphase systems and surface thermodynamics (e.g., Gibbs surface excess quantities).

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

17. Free Energy Function

18. 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)

19. Planar Interface Particular phase-field equation where
Exact isothermal travelling wave solution:
where
when

20. 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
Or variation of Hele-Shaw problem...

21. Numerics Advantages - no need to track interface
- can compute complex interface shapes
Disadvantage - have to resolve thin interfacial layers
State-of-the-art algorithms (C. Elliot, Provatas et al.) use
adaptive finite element methods
Simulation of dendritic growth into an undercooled liquid...

22. Provatas, Goldenfeld & Dantzig (1999) Dendrite Simulation

23. Anisotropic Equilibrium Shapes
W. Miller & G. Chadwick (1969)
Cahn & Hoffmann (1972)

24. Sharp Interface Formulation
Sharp interface limit:
McFadden & Wheeler (1996)
is a natural extension of the Cahn-Hoffman of sharp interface theory
Cahn & Hoffman (1972, 1974)
is normal to the plot:
Isothermal equilibrium shape given by
Corners form when plot is concave;
Phase field

25. Diffuse Interface Formulation
Recall:
Suggests:
where:
Phase-field equation:
where the so-called -vector is defined by:

26. Corners and Edges Taylor & Cahn (1998), Wheeler & McFadden (1997)
Eggleston, McFadden, & Voorhees (2001)

27. Cahn-Hilliard Equation

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

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

31. Inclusion of Surface Properties
Examples:
Surface Adsorption
Wetting in Multiphase Systems
Solute Trapping
(More than a computational device)

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

33. N. Ahmad, A. Wheeler, W. Boettinger, G. McFadden (1998)
Solute Trapping
At high velocities, solute segregation becomes small (“solute trapping”)
Results agree well with other trapping models (Aziz 1988)
N. Ahmad, A. Wheeler, W. Boettinger, G. McFadden (1998)

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

35. Early Phase-Field Calculations
G. Caginalp & E. Socolovsky (1991, 1994)
R. Kobayashi (1993, 1994)
A. Wheeler, B. Murray, R. Schaefer (1993)
2nd order accurate finite differences on 2-D uniform mesh
Explicit time-stepping for phase-field equation
Implicit (ADI) for energy equation
Mesh convergence an issue
Vector machines (Cray)
Roldan Pozo (benchmarks on PC cluster at NIST)

36. Adaptive Meshing R. Braun, B. Murray, & J. Soto (1997)
VLUGR2, vectorized, adaptive finite difference solver
R. Almgren & A. Almgren (1996)
2-D, second-order accurate, semi-implicit
N. Provatis, N. Goldenfeld, & J. Dantzig (1999)
2D, Galerkin FE, dynamically adaptive, quadtree
M. Plapp & A. Karma (2000)
Hybrid FD Mesh/diffusion Monte Carlo method

37. A. Karma & W.-J. Rappel (1997) Uniform 300x300x300 mesh
Grid-corrected anisotropy

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

39. J. Jeong, N. Goldenfeld, & J. Dantzig (2001)
Charm++ FEM framework, hexahedral elts, octree, 32 processors, METIS

40. 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