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**NASA Microgravity Research Program**

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

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**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

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

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

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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:**

Introduce free-energy functional: Dynamics J.S. Langer (1978)

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Free Energy Function

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**Phase-Field Equations**

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

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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 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

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**Anisotropic Equilibrium Shapes**

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

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Cahn-Hoffman -Vector Taylor (1992) Phase field

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**Cahn-Hoffman -Vector Equilibrium Shape is given by:**

Force per unit length in interface: Cahn & Hoffmann (1974) Phase field

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**Diffuse Interface Formulation**

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

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**Corners & Edges In Phase-Field**

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

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**Corners/Edges Jump conditions give: (force balance) where and**

Bronsard & Reitich (1993), Wheeler & McFadden (1997)

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

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**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

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**Cahn-Hilliard Equation**

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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**

One possibility Ideal Entropy L and S are liquid and solid regular solution parameters

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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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**McFadden and Wheeler (2001)**

Surface Adsorption McFadden and Wheeler (2001)

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**Cahn (1979), McFadden and Wheeler (2001)**

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

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**Surface Adsorption Ideal solution model Surface free energy**

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**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

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**N. Ahmad, A. Wheeler, W. Boettinger, G. McFadden (1998)**

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

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**Nonequilibrium Solute Trapping**

Numerical results (points) reproduce Aziz trapping function With characteristic trapping speed, VD, given by

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**Nonequilibrium Solute Trapping (cont.)**

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**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

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**FCC Binary Alloy Disordered phase CuAu**

G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A. Wheeler

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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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**Adsorption in FCC Binary Alloy**

Interphase Boundaries Antiphase Boundaries G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A. Wheeler

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**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

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**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

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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 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

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**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

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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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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

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**Experimental Observation of Dendrite Bridging Process**

(c) t = 30 s fs = 0.82 (b) t = 10 s fs = 0.70 (a) t = 0 s fs = 0.00 125 mm Photo: W. Kurz, EPFL (d) t = 75 s fs = 0.94 (e) t = 210 s fs = 0.97 (f) t = 1500 s fs = 0.98

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**Dendrite side arm bridging**

X Y Collision of offset arms - Delayed bridging

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**Coalescence of two Grains Using Multi-Grain Model**

P; Disjoining Pressure x Large misorientation P > 0 grains “repel” ggb = gsl = 0.1 DT = 0 K ggb = gsl = 0.1 DT = 50 K W. Boettinger (NIST) & M. Rappaz (EPFL)

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-Tensor Derivation

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