se-Field Models of Solidification McFadden 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 Outline • Background • Phase-Field Models • Numerical Computations NASA Microgravity Research Program
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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.
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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
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...