BAMC 2001 Reading Diffuse Interface Models Adam A Wheeler University of Southampton Jeff McFadden, NIST Dan Anderson, GWU Bill Boettinger, NIST Rich Braun, U Delaware John Cahn, NIST Britta Nestler, Foundry Inst. Aachen Lorenz Ratke, DLR Bob 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)

Cu-Cr Alloy (50 microns)

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 Equations Governing 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) use adaptive 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 Phase field 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 99AAW 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 Expt: Grugel et al.

Nucleation in L1 and incorporation of L2 in to solid 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