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