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Phase-Field Models of Solidification
Jeff McFadden NIST
Dan Anderson, GWUBill Boettinger, NISTRich Braun, U DelawareSam Coriell, NISTJohn Cahn, NISTBruce Murray, SUNY BinghamptonBob Sekerka, CMUJim Warren, NISTAdam Wheeler, U Southampton, UK
Outline• Background• Phase-Field Models• Numerical Computations
NASA Microgravity Research Program
Atomistic scaleAtomistic scaleÅ
Dendrite scaleDendrite scalem
Grain scaleGrain scalemm
Component scaleComponent scalecm - m
How to connect these various scales ?How to connect these various scales ?
Modeling at various length scalesModeling at various length scales
2 nm
40 m 10 mm
M. Rappaz, EPFL
Liquid decanted during freezing Polished and etched microstructure after freezing
Dendritic Microstructure
Freezing a Pure Liquid
Dendrite
Glicksman
Hele Shaw
Saffman & Taylor
Stefan Problem
Solid
Liquid
• Interface is a surface; • No thickness;• Physical properties:
•Surface energy, kinetics
• Conservation of energy
Surface Energy
• Critical Nucleus and Coarsening
• Grain Boundary Grooves
• Wavelength of instabilities
Critical Nucleus and Coarsening
P. Voorhees & R. Schaefer (1987)
Critical Nucleus:
Coarsening:
Minimize the total surface energy
for a given volume of inclusions
Grain Boundary Grooves
S.C. Hardy (1977)
Wavelength of Instabilities
S. Hardy and S. Coriell (1968)
Ice cylinder growing into
supercooled water, MTT
Instability wavelength
depends on surface energy:
Morphological Instability
Mullins & Sekerka (1963, 1964)
“Point effect” “Constitutional supercooling”
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)
• Description of anti-phase boundaries (APBs)
• 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 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 Thermodynamics
L
’
Phase-Field Model
• Introduce the phase-field variable:
• Introduce free-energy functional:
J.S. Langer (1978)
• 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:
Planar Interface
where
• Particular phase-field equation
• Exact isothermal travelling wave solution:
where
when
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...
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.) useadaptive finite element methods
• Simulation of dendritic growth into an undercooled liquid...
Provatas, Goldenfeld & Dantzig (1999) Dendrite Simulation
Anisotropic Equilibrium Shapes
Cahn & Hoffmann (1972)
W. Miller & G. Chadwick (1969)
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;
Diffuse Interface Formulation
• Recall:
• Suggests:
where:
• Phase-field equation:
where the so-called -vector is defined by:
Corners and Edges
Taylor & Cahn (1998), Wheeler & McFadden (1997)
Eggleston, McFadden, & Voorhees (2001)
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
Inclusion of Surface Properties
•Surface Adsorption
•Wetting in Multiphase Systems
•Solute Trapping
(More than a computational device)
Examples:
Surface Adsorption
McFadden and Wheeler (2001)
Solute Trapping
N. Ahmad, A. Wheeler, W. Boettinger, G. McFadden (1998)
At high velocities, solute segregation becomes small (“solute trapping”)
Results agree well with other trapping models (Aziz 1988)
Wetting in Multiphase SystemsM. 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
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)
Adaptive Meshing
•R. Braun, B. Murray, & J. Soto (1997)
º VLUGR2, vectorized, adaptive finite difference solver
•R. Almgren & A. Almgren (1996)
º 2-D, second-order accurate, semi-implicit
•N. Provatis, N. Goldenfeld, & J. Dantzig (1999)
º 2D, Galerkin FE, dynamically adaptive, quadtree
•M. Plapp & A. Karma (2000)
º Hybrid FD Mesh/diffusion Monte Carlo method
A. Karma & W.-J. Rappel (1997)
•Uniform 300x300x300 mesh
•Grid-corrected anisotropy
W. George & J. Warren (2001)
•3-D FD 500x500x500
•DPARLIB, MPI
•32 processors, 2-D slices of data
J. Jeong, N. Goldenfeld, & J. Dantzig (2001)
Charm++ FEM framework, hexahedral elts, octree, 32 processors, METIS
• 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