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Phase-Field Methods Jeff McFadden NIST

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.

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Phase-Field Methods Jeff McFadden NIST

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  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 Gravitational Effects in Physico-Chemical Systems: Interfacial Effects July 9, 2001 NASA Microgravity Research Program

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

  3. Two main issues for a phase-field model: Bulk Thermodynamics Surface Properties 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).

  4. The Cahn-Allen equation The enthalpy method (Conserves energy) (Includes capillarity) • Van der Waals (1893) • Korteweg (1901) • Landau-Ginzburg (1950) • Cahn-Hilliard (1958) • Halperin, Hohenberg & Ma (1977) Other diffuse interface theories: 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:

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

  6. Ordering in a BCC Binary Alloy

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

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

  9. Free Energy Function

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

  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 • Background • Surface Phenomena in Diffuse-Interface Models • Surface energy andsurface energy anisotropy • Surface adsorption • Solute trapping • Multi-phase wetting in order-disorder transitions • 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 Taylor (1992) Phase field

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

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

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

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

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

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

  21. Cahn & Hilliard (1958) Cahn-Hilliard Equation

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

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

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

  27. Surface Adsorption Ideal solution model Surface free energy Surface adsorption

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

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

  30. Nonequilibrium Solute Trapping • Numerical results (points) reproduce Aziz trapping function • With characteristic trapping speed, VD, given by

  31. Nonequilibrium Solute Trapping (cont.)

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

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

  34. Ordering in an FCC Binary Alloy

  35. Free Energy Functional

  36. Equilibrium States in FCC

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

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

  39. Outline • Background • Surface Phenomena in Diffuse-Interface Models • Surface energy and surface energy anisotropy • Surface adsorption • Solute trapping • Multi-phase wetting in order-disorder transitions • 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. Expt: Grugel et al. Nestler, Wheeler, Ratke & Stocker 00

  41. Incorporationof 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 • Background • Surface Phenomena in Diffuse-Interface Models • Surface energy and surface energy anisotropy • Surface adsorption • Solute trapping • Multi-phase wetting in order-disorder transitions • Recent phase-field applications • Monotectic solidification • Phase-field model of electrodeposition

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

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

  46. 1-D Equilibrium Profiles

  47. 1-D Dynamics

  48. Conclusions • 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

  49. Experimental Observation of Dendrite Bridging Process (c) t = 30 sfs = 0.82 (b) t = 10 sfs = 0.70 (a) t = 0 sfs = 0.00 125 mm Photo: W. Kurz, EPFL (d) t = 75 sfs = 0.94 (e) t = 210 sfs = 0.97 (f) t = 1500 sfs = 0.98

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