Gordon Conference Lecture

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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
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• 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

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Phase-Field Models
Main 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).
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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
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Cahn-Allen Equation
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Ordering in a BCC Binary Alloy
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Parameter Identification

• 1-D solution
• Interface width
• Surface energy
• Curvature-dependence (expand Laplacian)

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Phase-Field Model
J.S. Langer (1978)
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Free Energy Function
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Phase-Field Equations
Penrose Fife (1990), Fried Gurtin (1993),
Wang et al. (1993)
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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

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• 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

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Anisotropic Equilibrium Shapes
W. Miller G. Chadwick (1969)
Hoffman Cahn (1972)
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Cahn-Hoffman -Vector
Taylor (1992)
Phase field
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Cahn-Hoffman -Vector
Cahn Hoffmann (1974)
Phase field
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Diffuse Interface Formulation
Kobayashi(1993), Wheeler McFadden (1996),
Taylor Cahn (1998)
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Corners Edges In Phase-Field

• changes type
  when -plot is concave.

• where
• interpret as a stress tensor

Fried Gurtin (1993), Wheeler McFadden 97
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Corners/Edges

• Jump conditions give
• where
• and

Bronsard Reitich (1993), Wheeler McFadden
(1997)
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Corners and Edges

Eggleston, McFadden, Voorhees (2001)
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• 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

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Cahn-Hilliard Equation
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Phase Field Equations - Alloy
Wheeler, Boettinger, McFadden (1992)
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Alloy Free Energy Function
One possibility
Ideal Entropy
?L and ?S are liquid and solid regular solution
parameters
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W. George J. Warren (2001)

• 3-D FD 500x500x500
• DPARLIB, MPI
• 32 processors, 2-D slices of data

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Surface Adsorption
McFadden and Wheeler (2001)
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Surface Adsorption
1-D equilibrium
Cahn (1979), McFadden and Wheeler (2001)
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Surface Adsorption
Ideal solution model
Surface free energy
Surface adsorption
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• 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

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

• Numerical results (points) reproduce Aziz
  trapping function

• With characteristic trapping speed, VD, given by

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Nonequilibrium Solute Trapping (cont.)
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• 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

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FCC Binary Alloy
Disordered phase
CuAu
G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A.
Wheeler
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Ordering in an FCC Binary Alloy
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Free Energy Functional
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Equilibrium States in FCC
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Wetting in Multiphase Systems
Kikuchi Cahn CVM for fcc APB (Cu-Au)
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Adsorption in FCC Binary Alloy
Interphase Boundaries
G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A.
Wheeler
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• 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

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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
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Incorporation of L2 into the solid phase
Expt Grugel et al.
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Nucleation in L1 and incorporation of L2 into
solid
Expt Grugel et al.
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• 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

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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
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Phase-Field Model of Electrodeposition
J. Guyer, W. Boettinger, J. Warren, G. McFadden
(2002)
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(No Transcript)
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1-D Equilibrium Profiles
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1-D Dynamics
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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

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Experimental Observation of Dendrite Bridging
Process
(b) t 10 sfs 0.70
(a) t 0 sfs 0.00
(c) t 30 sfs 0.82
125 mm
Photo W. Kurz, EPFL
(e) t 210 sfs 0.97
(d) t 75 sfs 0.94
(f) t 1500 sfs 0.98
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Dendrite side arm bridging
X
Y

• Collision of offset arms - Delayed bridging

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Coalescence of two Grains Using Multi-Grain Model
P Disjoining Pressure
ggb 0.3 gsl 0.1 DT 0 K
ggb 0.3 gsl 0.1 DT 50 K
W. Boettinger (NIST) M. Rappaz (EPFL)
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-Tensor Derivation