BAMC 2001 Reading

1. Outline • Background: History; Microstructure • Phase-field Models • Anisotropy • Solid-solid Phase Transitions • Complex Binary Alloys 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 BAMC 2001 Reading

2. 1500 BC Crystallisation of Alum 1556AD 600 BC History

3. Hele Shaw Dendrite Saffman & Taylor Freezing a Pure Liquid Glicksman

4. Bernard Convection Solidification Cerisier Simple Binary Alloy Billia et al

5. 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.

6. Nickel Silver (50 microns) http://microstructure.copper.org/

7. Cu-Cr Alloy (50 microns) http://microstructure.copper.org/

8. Microstructure • Microstructure: • evolves on different time an length scales; • involves changes in topology; • physical processes on different scales; • several different phases.

9. Interface is a surface; • No thickness; • Physical properties: • Surface energy, kinetics • Conservation of energy Solid Liquid Free Boundary Problems

10. Introduce the phase-field variable: • Introduce free-energy functional: 1 0 • Dynamics Phase-field Model Langer mid 70’s

11. Governing equations: • First & second laws • Require positive entropy • production • Thermodynamic derivation • Energy functionals: Phase-field Equations (Penrose & Fife 90, Wang, Sekerka, AAW et al 93)

12. when Planar Interface • Particular phase-field equation • where • Exact isothermal travelling wave solution: • where

13. 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...

14. 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...

15. Provatas, Goldenfeld & Dantzig (99) Dendrite Simulation

16. Recall: • Suggests: where: Surface Energy Anisotropy • Phase-field equation: where the so-called -vector is defined by:

17. Sharp InterfaceFormulation • 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; Phase field

19. Steady case: where • Noether’s Thm: Corners & Edges In Phase-Field • changes type when -plot is concave. • where • interpret as a “stress tensor” AAW & McFadden 97

20. (force balance) Corners/Edges • Weak shocks • Jump conditions give: • where • and

21. Order parameters: FCC Binary Alloy (CuAu) • Four sub-lattices with • occupation densities: Braun, Cahn McFadden & AAW 97

22. Dynamics • Symmetries of FCC imply • where • Dynamics:

23. Bulk States • Bulk states: • Disodered: • CuAu: • Cu3Au: • Mixed modes: CuAu (L10) Cu3Au (L12)

24. Interfaces • IPB: Disorder-Cu3Au in • (y,z)-plane • Surface energy dependence • on interface orientation Kikuchi & Cahn (1977)

25. 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;

26. Two Immiscible Viscous Liquids denotes which liquid; assume where Anderson, McFadden & AAW 2000

27. 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

28. Simple Binary Alloy The liquid may solidify into a solid with a different composition AAW, Boettinger & McFadden 93

30. Eutectic Binary Alloy In eutectic alloys the liquid can solidify into two different solid phases which can coexist together Experiments: Mercy& Ginibre Nestler & AAW 99 AAW Boettinger & McFadden 96

31. Varicose Instability Expts: G. Faivre

32. Simulation of Wavelength Selection

33. Growth of Eutectic Al-Si Grain SEM Photograph

34. Monotectic Binary Alloy A liquid phase can “solidify” into both a solid and a different liquid phase. Expt: Grugel et al. Nestler, AAW, Ratke & Stocker 00

35. Incorporationof L2 in to the solid phase Expt: Grugel et al.

36. Nucleation in L1 and incorporation of L2 in to solid Expt: Grugel et al.

37. Conclusions • 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.