Phase-Field Methods
Jeff McFadden NIST
Dan Anderson, GWUBill Boettinger, NISTRich Braun, U DelawareJohn Cahn, NISTSam Coriell, NISTBruce Murray, SUNY BinghamptonBob Sekerka, CMUPeter Voorhees, NWUAdam Wheeler, U Southampton, UK
July 9, 2001
Gravitational Effects in Physico-Chemical Systems: Interfacial Effects
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 ModelsMain idea: Solve a single set of PDEs over the entire domain
Phase-field model incorporates both bulk thermodynamics of multiphase systems and surface thermodynamics (e.g., Gibbs surface excess quantities).
Two main issues for a phase-field model:
Bulk Thermodynamics Surface Properties
Phase-Field ModelThe 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:
•Van der Waals (1893)
•Korteweg (1901)
•Landau-Ginzburg (1950)
•Cahn-Hilliard (1958)
•Halperin, Hohenberg & Ma (1977)
Other diffuse interface theories:
The enthalpy method
(Conserves energy)
The Cahn-Allen equation
(Includes capillarity)
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:
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:
J.S. Langer (1978)
• Introduce free-energy functional:
• Dynamics
Free Energy Function
Phase-Field Equations
Governing equations: • First & second laws
• Require positive entropy production
Penrose & Fife (1990), Fried & Gurtin (1993), Wang et al. (1993)
Thermodynamic derivation• Energy functionals:
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
• Steady case: where
• Noether’s Thm:
• where
• interpret as a “stress tensor”
• changes type when -plot is concave.
Fried & Gurtin (1993), Wheeler & McFadden 97
• Jump conditions give:
• where
• and
Corners/Edges
(force balance)
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
V
C dVcTcfF2
22
2
22),,(
0 22
fF
constant c- 22 Cc f
cF
Coupled Cahn-Hilliard & Cahn-Allen Equations
22
fM
t
-)1( 22 cc f
ccMtc
CC
R'Τ
DpDp M
cMMcM
LSC
BA
())(-1(
)-1(where{
Wheeler, Boettinger, & McFadden (1992)
Alloy Free Energy Function
)())(1()1(
ln)1ln()1(
),(),()1( T)c,,(
ppcc
ccccTR
Tf cTf-cf
LS
BA
Ideal Entropy
L and S are liquid and solid regular solution parameters
One possibility
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 Adsorption1-D equilibrium:
Differentiating, and using equilibrium conditions, gives
where
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
N. Ahmad, A. Wheeler, W. Boettinger, G. McFadden (1998)
At high velocities, solute segregation becomes small (“solute trapping”)
Increasing V
D
DE
VV
VVkVk
/1
/)(
L
E
E
L
SD
D
)k(
)/k(
D
DV
1
1ln1
16
3
Nonequilibrium Solute Trapping
• Numerical results (points) reproduce Aziz trapping function
• With characteristic trapping speed, VD, given by
0 2 4 6 8
ln k E /(k E -1 )
0
20
40
60
VD
(m
/s)
A l-In
A l-C uA l-G e
A l-S n
S i-B i
S i-S n
S i-G eS i-A s
S i-G a
S i-In
S i-S b
V D m easu rem en ts fro m S m ith & A ziz (1 9 9 5 )
Nonequilibrium Solute Trapping (cont.)
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
Disordered
phase
CuAu
G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A. Wheeler
FCC Binary Alloy
Ordering in an FCC Binary Alloy
Free Energy Functional
Equilibrium States in FCC
Wetting in Multiphase SystemsM. 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
Interphase Boundaries
Antiphase Boundaries
G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A. Wheeler
Adsorption in FCC Binary Alloy
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.
Nestler, Wheeler, Ratke & Stocker 00
Expt: Grugel et al.
Incorporation of L2 into the solid phase
2L S 1L
Expt: Grugel et al.
Nucleation in L1 and incorporation of L2 into solid
1L
2L
S
2L2L
Expt: Grugel et al.
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
• Note the bumps over the filled features.
• Cross-section views of five trenches with different aspect ratios
– filled under a variety of conditions.
D. Josell, NIST
Phase-Field Model of Electrodeposition
J. Guyer, W. Boettinger, J. Warren, G. McFadden (2002)
1-D Equilibrium Profiles
1-D Dynamics
• 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
Conclusions
(b) t = 10 sfs = 0.70
(a) t = 0 sfs = 0.00
(e) t = 210 sfs = 0.97
(f) t = 1500 sfs = 0.98
(c) t = 30 sfs = 0.82
(d) t = 75 sfs = 0.94
125 m
Photo: W. Kurz, EPFL
Experimental Observation of Dendrite Bridging Process
Dendrite side arm bridging
Y
X
•Collision of offset arms - Delayed bridging
0
0,2
0,4
0,6
0,8
1
-2,E-08 -1,E-08 0,E+00 1,E-08 2,E-08
Coalescence of two Grains Using Multi-Grain ModelCoalescence of two Grains Using Multi-Grain Model
0
0,2
0,4
0,6
0,8
1
-2,E-08 -1,E-08 0,E+00 1,E-08 2,E-08
gbgb = 0.3 = 0.3 sl sl = 0.1= 0.1
T = 0 KT = 0 K
gbgb = 0.3 = 0.3 sl sl = 0.1= 0.1
T = 50 KT = 50 K
xx
Large misorientationLarge misorientation > 0> 0
grains “repel”grains “repel”
; Disjoining Pressure
W. Boettinger (NIST) & M. Rappaz (EPFL)W. Boettinger (NIST) & M. Rappaz (EPFL)
-Tensor Derivation