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Phase-Field Methods Jeff McFadden NIST Dan Anderson, GWU Bill Boettinger, NIST Rich Braun, U Delaware John Cahn, NIST Sam Coriell, NIST Bruce Murray, SUNY.

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Presentation on theme: "Phase-Field Methods Jeff McFadden NIST Dan Anderson, GWU Bill Boettinger, NIST Rich Braun, U Delaware John Cahn, NIST Sam Coriell, NIST Bruce Murray, SUNY."— Presentation transcript:

1 Phase-Field Methods Jeff McFadden NIST Dan Anderson, GWU Bill Boettinger, NIST Rich Braun, U Delaware John Cahn, NIST Sam Coriell, NIST Bruce Murray, SUNY Binghampton Bob Sekerka, CMU Peter Voorhees, NWU Adam Wheeler, U Southampton, UK July 9, 2001 Gravitational Effects in Physico-Chemical Systems: Interfacial Effects NASA Microgravity Research Program

2 Outline 1. Background 2. Surface Phenomena in Diffuse-Interface Models Surface energy and surface energy anisotropy Surface adsorption Solute trapping Multi-phase wetting in order-disorder transitions 3.Recent phase-field applications Monotectic growth Phase-field model of electrodeposition

3 Phase-Field Models Main idea: Solve a single set of PDEs over the entire domain Phase-field model incorporates both bulk thermodynamics of multiphase systems and surface thermodynamics (e.g., Gibbs surface excess quantities). Two main issues for a phase-field model: Bulk Thermodynamics Surface Properties

4 Phase-Field Model The phase-field model was developed around 1978 by J. Langer at CMU as a computational technique to solve Stefan problems for a pure material. The model combines ideas from: Van der Waals (1893) Korteweg (1901) Landau-Ginzburg (1950) Cahn-Hilliard (1958) Halperin, Hohenberg & Ma (1977) Other diffuse interface theories: The enthalpy method (Conserves energy) The Cahn-Allen equation (Includes capillarity)

5 Cahn-Allen Equation J. Cahn and S. Allen (1977) M. Marcinkowski (1963) Anti-phase boundaries in BCC system Motion by mean curvature: Surface energy: “Non-conserved” order parameter:

6 Ordering in a BCC Binary Alloy

7 Parameter Identification 1-D solution: Interface width: Surface energy: Curvature-dependence (expand Laplacian):

8 Phase-Field Model Introduce the phase- field variable: J.S. Langer (1978) Introduce free-energy functional: Dynamics

9 Free Energy Function

10 Phase-Field Equations Governing equations:First & second laws Require positive entropy production Penrose & Fife (1990), Fried & Gurtin (1993), Wang et al. (1993) Thermodynamic derivation Energy functionals:

11 Sharp Interface Asymptotics Consider limit in which Different distinguished limits possible. Caginalp (1988), Karma (1998), McFadden et al (2000) Can retrieve free boundary problem with

12 Outline 1. Background 2. Surface Phenomena in Diffuse-Interface Models Surface energy and surface energy anisotropy Surface adsorption Solute trapping Multi-phase wetting in order-disorder transitions 3.Recent phase-field applications Monotectic solidification Phase-field model of electrodeposition

13 Anisotropic Equilibrium Shapes W. Miller & G. Chadwick (1969) Hoffman & Cahn (1972)

14 Cahn-Hoffman -Vector Phase field Taylor (1992)

15 Cahn-Hoffman -Vector Phase field Equilibrium Shape is given by: Force per unit length in interface: Cahn & Hoffmann (1974)

16 Diffuse Interface Formulation Kobayashi(1993), Wheeler & McFadden (1996), Taylor & Cahn (1998)

17 Corners & Edges In Phase-Field Steady case: where Noether’s Thm: where interpret as a “stress tensor” changes type when -plot is concave. Fried & Gurtin (1993), Wheeler & McFadden 97

18 Jump conditions give: where and Corners/Edges (force balance) Bronsard & Reitich (1993), Wheeler & McFadden (1997)

19 Corners and Edges Eggleston, McFadden, & Voorhees (2001)

20 Outline 1. Background 2. Surface Phenomena in Diffuse-Interface Models Surface energy and surface energy anisotropy Surface adsorption Solute trapping Multi-phase wetting in order-disorder transitions 3.Recent phase-field applications Monotectic solidification Phase-field model of electrodeposition

21 Cahn-Hilliard Equation Cahn & Hilliard (1958)

22 Phase Field Equations - Alloy Coupled Cahn-Hilliard & Cahn-Allen Equations where { Wheeler, Boettinger, & McFadden (1992)

23 Alloy Free Energy Function Ideal Entropy  L and  S are liquid and solid regular solution parameters One possibility

24 W. George & J. Warren (2001) 3-D FD 500x500x500 DPARLIB, MPI 32 processors, 2-D slices of data

25 Surface Adsorption McFadden and Wheeler (2001)

26 Surface Adsorption 1-D equilibrium: Differentiating, and using equilibrium conditions, gives where Cahn (1979), McFadden and Wheeler (2001)

27 Surface Adsorption Ideal solution modelSurface free energySurface adsorption

28 Outline 1. Background 2. Surface Phenomena in Diffuse-Interface Models Surface energy and surface energy anisotropy Surface adsorption Solute trapping Multi-phase wetting in order-disorder transitions 3.Recent phase-field applications Monotectic solidification Phase-field model of electrodeposition

29 Solute Trapping N. Ahmad, A. Wheeler, W. Boettinger, G. McFadden (1998) At high velocities, solute segregation becomes small (“solute trapping”) Increasing V

30 Nonequilibrium Solute Trapping Numerical results (points) reproduce Aziz trapping function With characteristic trapping speed, V D, given by

31 Nonequilibrium Solute Trapping (cont.)

32 Outline 1. Background 2. Surface Phenomena in Diffuse-Interface Models Surface energy and surface energy anisotropy Surface adsorption Solute trapping Interface structure in order-disorder transitions 3.Recent phase-field applications Monotectic solidification Phase-field model of electrodeposition

33 Disordered phase CuAu G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A. Wheeler FCC Binary Alloy

34 Ordering in an FCC Binary Alloy

35 Free Energy Functional

36 Equilibrium States in FCC

37 Wetting in Multiphase Systems M. Marcinkowski (1963) Kikuchi & Cahn CVM for fcc APB (Cu-Au) R. Braun, J. Cahn, G. McFadden, & A. Wheeler (1998) Phase-field model with 3 order parameters

38 Interphase Boundaries Antiphase Boundaries G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A. Wheeler Adsorption in FCC Binary Alloy

39 Outline 1. Background 2. Surface Phenomena in Diffuse-Interface Models Surface energy and surface energy anisotropy Surface adsorption Solute trapping Multi-phase wetting in order-disorder transitions 3.Recent phase-field applications Monotectic solidification Phase-field model of electrodeposition

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

41 Incorporation of L2 into the solid phase Expt: Grugel et al.

42 Nucleation in L1 and incorporation of L2 into solid Expt: Grugel et al.

43 Outline 1. Background 2. Surface Phenomena in Diffuse-Interface Models Surface energy and surface energy anisotropy Surface adsorption Solute trapping Multi-phase wetting in order-disorder transitions 3.Recent phase-field applications Monotectic solidification Phase-field model of electrodeposition

44 Superconformal Electrodeposition Note the bumps over the filled features. Cross-section views of five trenches with different aspect ratios –filled under a variety of conditions. D. Josell, NIST

45 Phase-Field Model of Electrodeposition J. Guyer, W. Boettinger, J. Warren, G. McFadden (2002)

46

47 1-D Equilibrium Profiles

48 1-D Dynamics

49 Phase-field models provide a regularized version of Stefan problems for computational purposes Phase-field models are able to incorporate both bulk and surface thermodynamics Can be generalised to: include material deformation (fluid flow & elasticity) models of complex alloys Computations: provides a vehicle for computing complex realistic microstructure Conclusions

50 (b) t = 10 s f s = 0.70 (a) t = 0 s f s = 0.00 (e) t = 210 s f s = 0.97 (f) t = 1500 s f s = 0.98 (c) t = 30 s f s = 0.82 (d) t = 75 s f s =  m Photo: W. Kurz, EPFL Experimental Observation of Dendrite Bridging Process

51 Dendrite side arm bridging Y X Collision of offset arms - Delayed bridging

52 Coalescence of two Grains Using Multi-Grain Model  gb = 0.3  sl = 0.1  T = 0 K  T = 0 K  gb = 0.3  sl = 0.1  T = 50 K  T = 50 K x Large misorientation  > 0 grains “repel”  ; Disjoining Pressure W. Boettinger (NIST) & M. Rappaz (EPFL)

53 -Tensor Derivation


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