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
Outline• Background: History; Microstructure
• Phase-field Models• Anisotropy• Solid-solid Phase Transitions• Complex Binary Alloys
600 BC
History
1500 BC
Crystallisation of Alum 1556AD
Freezing a Pure Liquid
Dendrite
Glicksman
Hele Shaw
Saffman & Taylor
Simple Binary Alloy
Solidification
Billia et al
Bernard Convection
Cerisier
Microstructure• Solidification of a material yields complex interfacial structure• Important to the physical properties of the casting
Cast agricultural aluminium transmission housing from Stahl Specialty Co.
Nickel Silver (50 microns) http://microstructure.copper.org/
Cu-Cr Alloy (50 microns) http://microstructure.copper.org/
Microstructure
• 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...
Provatas, Goldenfeld & Dantzig (99) Dendrite Simulation
Surface Energy Anisotropy
• Recall:
• Suggests:
where:
• Phase-field equation:
where the so-called -vector is defined by:
Sharp Interface Formulation• Sharp interface limit:• McFadden & AAW 96
• is a natural extension of the Cahn-Hoffman of sharp interface theory
• Cahn & Hoffman (1972,4)
• is normal to the -plot:
• Isothermal equilibrium shape given by
• Corners form when -plot is concave;
Corners & Edges In Phase-Field
• Steady case: where
• Noether’s Thm:
• where
• interpret as a “stress tensor”
• changes type when -plot is concave.
AAW & McFadden 97
• Jump conditions give:
• where
• and
Corners/Edges
• Weak shocks(force balance)
FCC Binary Alloy (CuAu)
• • Order parameters:
• Four sub-lattices with occupation densities:
Braun, Cahn McFadden & AAW 97
• Symmetries of FCC imply
where
• Dynamics:
Dynamics
• Bulk states:
• Disodered:• CuAu:• Cu3Au:• Mixed modes:
Bulk States
CuAu(L10)
Cu3Au(L12)
Interfaces• IPB: Disorder-Cu3Au in (y,z)-plane
• Surface energy dependence on interface orientation
Kikuchi & Cahn (1977)
Summary• FCC models predicts:
• surface energy dependence and hence equilibrium shapes;• internal structure of interface.
• FCC & phase-field fall into a general class of (anisotropic) multiple-order-parameter models;
Two Immiscible Viscous Liquids
where
denotes which liquid; assume
Anderson, McFadden & AAW 2000
Binary Alloys
Can extend these ideas to binary alloys:
Results in pdes involving a composition (a conserved order parameter) temperature and one (or more) non-conserved order parameters
Simple Binary Alloy
The liquid may solidify into a solid with a different composition
AAW, Boettinger & McFadden 93
Eutectic Binary Alloy
In eutectic alloys the liquid can solidify into two different solid phases which can coexist together
Nestler & AAW 99 AAW Boettinger & McFadden 96
Experiments: Mercy & Ginibre
Varicose Instability
Expts: G. Faivre
Simulation of Wavelength Selection
Growth of Eutectic Al-Si Grain
SEM Photograph
Monotectic Binary Alloy
A liquid phase can “solidify” into both a solid and a different liquid phase.
Nestler, AAW, Ratke & Stocker 00
Expt: Grugel et al.
Incorporation of L2 in to the solid phase
2L S 1L
Expt: Grugel et al.
Nucleation in L1 and incorporation of L2 in to solid
1L
2L
S
2L2L
Expt: Grugel et al.
• Phase-field models provide a regularised version of Stefan problems
• Develop a generalised -vector and -tensor theory for anisotropic surface energy; corners & edges
• Can be generalised to
• models of internal structure on interfaces;
• include material deformation (fluid flow);
• models of complex alloys;
• Computations:
• provides a vehicle for computing complex realistic microstructure;
• accuracy/algorithms.
Conclusions
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