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 July 9, 2001 Gravitational Effects in Physico-Chemical Systems: Interfacial Effects NASA Microgravity Research Program
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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.
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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)