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
Outline
1. Background
2. Surface Phenomena in Diffuse-Interface Models
• Surface energy and surface energy anisotropy
• Surface adsorption
• Solute trapping
• Multi-phase wetting in order-disorder transitions
3. Recent phase-field applications
• Monotectic growth
• Phase-field model of electrodeposition
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).
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 The Cahn-Allen equation
(Conserves energy) (Includes capillarity)
•Van der Waals (1893)
•Korteweg (1901)
Other diffuse interface theories: •Landau-Ginzburg (1950)
•Cahn-Hilliard (1958)
•Halperin, Hohenberg & Ma (1977)
Cahn-Allen Equation
• Anti-phase boundaries in BCC system
• Motion by mean curvature:
• Surface energy:
• “Non-conserved” order parameter:
M. Marcinkowski (1963)
J. Cahn and S. Allen (1977)
Ordering in a BCC Binary Alloy
Parameter Identification
• 1-D solution:
• Interface width:
• Surface energy:
• Curvature-dependence (expand Laplacian):
Phase-Field Model
• Introduce the phase-
field variable:
• Introduce free-energy
functional:
• Dynamics
J.S. Langer (1978)
Free Energy Function
Phase-Field Equations
Governing equations: • First & second laws
• Require positive entropy
Thermodynamic derivation production
• Energy functionals:
Penrose & Fife (1990), Fried & Gurtin (1993), Wang et al. (1993)
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
Outline
1. Background
2. Surface Phenomena in Diffuse-Interface Models
• Surface energy and surface energy anisotropy
• Surface adsorption
• Solute trapping
• Multi-phase wetting in order-disorder transitions
3. Recent phase-field applications
• Monotectic solidification
• Phase-field model of electrodeposition
Anisotropic Equilibrium Shapes
W. Miller & G. Chadwick (1969)
Hoffman & Cahn (1972)
Cahn-Hoffman -Vector
Taylor (1992)
Cahn-Hoffman -Vector
Equilibrium Shape is given by:
Force per unit length in interface:
Cahn & Hoffmann (1974)
Diffuse Interface Formulation
Kobayashi(1993), Wheeler & McFadden (1996), Taylor & Cahn (1998)
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
Corners/Edges
• Jump conditions give:
(force balance)
• where
• and
Bronsard & Reitich (1993), Wheeler & McFadden (1997)
Corners and Edges
Eggleston, McFadden, & Voorhees (2001)
Outline
1. Background
2. Surface Phenomena in Diffuse-Interface Models
• Surface energy and surface energy anisotropy
• Surface adsorption
• Solute trapping
• Multi-phase wetting in order-disorder transitions
3. Recent phase-field applications
• Monotectic solidification
• Phase-field model of electrodeposition
Cahn-Hilliard Equation
Cahn & Hilliard (1958)
Phase Field Equations - Alloy
2 C2
2
F f ( c, , T ) c dV
2
V
2 2
F f
2 2 0
F f 2 2
- C c constant
c c
f 2 2
M M (1 - c) M A cM B
{ t
c
t
M C c(1 c)
f - 2 2c
c
C
where
MC
(1 - p( )) D S p ( D L
R'Τ
Coupled Cahn-Hilliard & Cahn-Allen Equations
Wheeler, Boettinger, & McFadden (1992)
Alloy Free Energy Function
One possibility
f ( , c, T) (1-c) f A (T , ) c f B (T , )
RT (1 c) ln(1 c) c ln c
Ideal Entropy c(1 c) S (1 p ( )) L p ( )
L and S are liquid and solid regular
solution parameters
W. George & J. Warren (2001)
•3-D FD 500x500x500
•DPARLIB, MPI
•32 processors, 2-D slices of data
Surface Adsorption
McFadden and Wheeler (2001)
Surface Adsorption
1-D equilibrium:
where
Differentiating, and using equilibrium conditions, gives
Cahn (1979), McFadden and Wheeler (2001)
Surface Adsorption
Ideal solution model Surface free energy Surface adsorption
Outline
1. Background
2. Surface Phenomena in Diffuse-Interface Models
• Surface energy and surface energy anisotropy
• Surface adsorption
• Solute trapping
• Multi-phase wetting in order-disorder transitions
3. Recent phase-field applications
• Monotectic solidification
• Phase-field model of electrodeposition
Solute Trapping
Increasing V
At high velocities, solute segregation
becomes small (“solute trapping”)
N. Ahmad, A. Wheeler, W. Boettinger, G. McFadden (1998)
Nonequilibrium Solute Trapping
• Numerical results (points) reproduce Aziz trapping function
k E V / VD
k (V )
1 V / VD
• With characteristic trapping speed, VD, given by
3 DS ln ( 1/k E ) DL
VD 1
16 DL ( 1 k E )
Nonequilibrium Solute Trapping (cont.)
60
Si-In
VD measurements from Smith & Aziz (1995)
40 Al-In
Al-Sn
VD (m/s)
Si-Bi
Si-Ga
20
Si-Sn
Al-Cu
Al-Ge
Si-Ge
Si-As Si-Sb
0
0 2 4 6 8
ln kE/(kE-1)
Outline
1. Background
2. Surface Phenomena in Diffuse-Interface Models
• Surface energy and surface energy anisotropy
• Surface adsorption
• Solute trapping
• Interface structure in order-disorder transitions
3. Recent phase-field applications
• Monotectic solidification
• Phase-field model of electrodeposition
FCC Binary Alloy
Disordered
phase
CuAu
G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A. Wheeler
Ordering in an FCC Binary Alloy
Free Energy Functional
Equilibrium States in FCC
Wetting in Multiphase Systems
M. Marcinkowski (1963)
Kikuchi & Cahn CVM
for fcc APB (Cu-Au)
Phase-field model with
3 order parameters
R. Braun, J. Cahn, G. McFadden, & A. Wheeler (1998)
Adsorption in FCC Binary Alloy
Interphase Boundaries
Antiphase Boundaries
G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A. Wheeler
Outline
1. Background
2. Surface Phenomena in Diffuse-Interface Models
• Surface energy and surface energy anisotropy
• Surface adsorption
• Solute trapping
• Multi-phase wetting in order-disorder transitions
3. Recent phase-field applications
• Monotectic solidification
• Phase-field model of electrodeposition
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
Incorporation of L2
into the solid phase
Expt: Grugel et al.
L1 S L2
Nucleation in L1 and
incorporation of L2 into solid
L1
L2
Expt: Grugel et al.
L2
S
Outline
1. Background
2. Surface Phenomena in Diffuse-Interface Models
• Surface energy and surface energy anisotropy
• Surface adsorption
• Solute trapping
• Multi-phase wetting in order-disorder transitions
3. Recent phase-field applications
• Monotectic solidification
• Phase-field model of electrodeposition
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
Phase-Field Model of Electrodeposition
J. Guyer, W. Boettinger, J. Warren, G. McFadden (2002)
1-D Equilibrium Profiles
1-D Dynamics
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
Experimental Observation of Dendrite Bridging Process
(c) t = 30 s (b) t = 10 s (a) t = 0 s
125 mm
fs = 0.82 fs = 0.70 fs = 0.00
Photo:
W. Kurz, EPFL
(d) t = 75 s (e) t = 210 s (f) t = 1500 s
fs = 0.94 fs = 0.97 fs = 0.98
Dendrite side arm bridging
X
Y
•Collision of offset arms - Delayed bridging
Coalescence of two Grains Using Multi-Grain Model
1
P; Disjoining Pressure 0,8
0,6
ggb = 0.3 gsl = 0.1
ggb
o
DT = 0 K 0,4
o 0,2
2gsl
0
-2,E-08 -1,E-08 0,E+00 1,E-08 2,E-08
1
0,8
x
0,6
Large misorientation ggb = 0.3 gsl = 0.1
P>0 DT = 50 K 0,4
grains “repel” 0,2
0
-2,E-08 -1,E-08 0,E+00 1,E-08 2,E-08
W. Boettinger (NIST) & M. Rappaz (EPFL)
-Tensor Derivation