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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 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

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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

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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

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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)

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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:

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Ordering in a BCC Binary Alloy

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

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

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Free Energy Function

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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:

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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

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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

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

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Cahn-Hoffman -Vector Phase field Taylor (1992)

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

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

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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

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

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

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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

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Cahn-Hilliard Equation Cahn & Hilliard (1958)

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

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

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

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Surface Adsorption McFadden and Wheeler (2001)

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

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Surface Adsorption Ideal solution modelSurface free energySurface adsorption

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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

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

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

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Nonequilibrium Solute Trapping (cont.)

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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

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

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Ordering in an FCC Binary Alloy

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Free Energy Functional

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Equilibrium States in FCC

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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

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

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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

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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.

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

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

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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

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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

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

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1-D Equilibrium Profiles

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1-D Dynamics

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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

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(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

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

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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)

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-Tensor Derivation

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