BAMC 2001 Reading iffuse Interface Models am A Wheeler iversity of Southampton f McFadden, NIST Anderson, GWU l Boettinger, NIST h Braun, U Delaware n Cahn, NIST tta Nestler, Foundry Inst. Aachen enz Ratke, DLR Sekerka, CMU Outline • Background: History; Microstructure • Phase-field Models • Anisotropy • Solid-solid Phase Transitions • Complex Binary Alloys
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BAMC 2001 Reading Diffuse Interface Models Adam A Wheeler University of Southampton Jeff McFadden, NIST Dan Anderson, GWU Bill Boettinger, NIST Rich Braun,
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BAMC 2001 Reading
Diffuse Interface Models
Adam A Wheeler University of Southampton
Jeff McFadden, NIST
Dan Anderson, GWUBill Boettinger, NISTRich Braun, U DelawareJohn Cahn, NISTBritta Nestler, Foundry Inst. AachenLorenz Ratke, DLRBob Sekerka, CMU
• Microstructure: • evolves on different time an length scales;• involves changes in topology;• physical processes on different scales;• several different phases.
Free Boundary Problems
Solid
Liquid
• Interface is a surface; • No thickness;• Physical properties:
•Surface energy, kinetics
• Conservation of energy
Phase-field Model
• Dynamics
• Introduce free-energy functional:
• Introduce the phase-field variable:
Langer mid 70’s
0 1
Phase-field EquationsGoverning equations: • First & second
laws
Thermodynamic derivation• Energy functionals:
• Require positive entropy production
(Penrose & Fife 90, Wang, Sekerka, AAW et al 93)
Planar Interface
where
• Exact isothermal travelling wave solution:
where
• Particular phase-field equation
when
Sharp Interface Asymptotics
• Consider limit in which
• Different distinguished limits possible.(Caginalp 89…, 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
• First calculations (Kobayashi 91, AAW et al 93)
• State-of-the-art algorithms (Elliot, Provatas et al) useadaptive finite element methods
• Simulation of dendritic growth into an undercooled liquid...