Barret Memorial Lecture

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Phase-Field Models of Solidification Jeff
McFadden NIST
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
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Modeling at various length scales
40 ?m
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
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Dendritic Microstructure
Polished and etched microstructure after freezing
Liquid decanted during freezing
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Freezing a Pure Liquid
Glicksman
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Stefan Problem

• Interface is a surface
• No thickness
• Physical properties
• Surface energy, kinetics
• Conservation of energy

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Surface Energy

• Critical Nucleus and Coarsening
• Grain Boundary Grooves
• Wavelength of instabilities

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Critical Nucleus and Coarsening
Critical Nucleus
Coarsening Minimize the total surface energy for
a given volume of inclusions
P. Voorhees R. Schaefer (1987)
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Grain Boundary Grooves
S.C. Hardy (1977)
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Wavelength of Instabilities
Ice cylinder growing into supercooled water,
Instability wavelength depends on surface energy
S. Hardy and S. Coriell (1968)
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Morphological Instability
Point effect
Mullins Sekerka (1963, 1964)
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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 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
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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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Planar Interface

• Particular phase-field equation

• where

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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

• Or variation of Hele-Shaw problem...

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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...

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Provatas, Goldenfeld Dantzig (1999) Dendrite
Simulation
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Anisotropic Equilibrium Shapes
W. Miller G. Chadwick (1969)
Cahn Hoffmann (1972)
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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
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Diffuse Interface Formulation
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Corners and Edges

Taylor Cahn (1998), Wheeler McFadden
(1997) Eggleston, McFadden, Voorhees (2001)
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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
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(No Transcript)
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Inclusion of Surface Properties
Examples

• Surface Adsorption
• Wetting in Multiphase Systems
• Solute Trapping

(More than a computational device)
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Surface Adsorption
McFadden and Wheeler (2001)
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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)
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Wetting in Multiphase Systems
Kikuchi Cahn CVM for fcc APB (CuAu)
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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)

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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

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A. Karma W.-J. Rappel (1997)

• Uniform 300x300x300 mesh
• Grid-corrected anisotropy

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