PhaseField Methods Jeff Mc Fadden NIST Dan Anderson

Outline 1. Background 2. Surface Phenomena in Diffuse-Interface Models • Surface energy and surface

Phase-Field Models Main idea: Solve a single set of PDEs over the entire domain

Phase-Field Model The phase-field model was developed around 1978 by J. Langer at CMU

Cahn-Allen Equation • Anti-phase boundaries in BCC system • Motion by mean curvature: •

Parameter Identification • 1 -D solution: • Interface width: • Surface energy: • Curvature-dependence

Phase-Field Model • Introduce the phasefield variable: • Introduce free-energy functional: • Dynamics J.

Phase-Field Equations Governing equations: Thermodynamic derivation • Energy functionals: • First & second laws

Sharp Interface Asymptotics • Consider limit in which • Different distinguished limits possible. Caginalp

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

Cahn-Hoffman Equilibrium Shape is given by: Force per unit length in interface: Cahn &

Diffuse Interface Formulation Kobayashi(1993), Wheeler & Mc. Fadden (1996), Taylor & Cahn (1998)

Corners & Edges In Phase-Field • changes type when • Steady case: where •

Corners/Edges • Jump conditions give: (force balance) • where • and Bronsard & Reitich

Corners and Edges Eggleston, Mc. Fadden, & Voorhees (2001)

Cahn-Hilliard Equation Cahn & Hilliard (1958)

Phase Field Equations - Alloy { where Coupled Cahn-Hilliard & Cahn-Allen Equations Wheeler, Boettinger,

Alloy Free Energy Function One possibility Ideal Entropy L and S are liquid and

W. George & J. Warren (2001) • 3 -D FD 500 x 500 •

Surface Adsorption Mc. Fadden and Wheeler (2001)

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

Surface Adsorption Ideal solution model Surface free energy Surface adsorption

Solute Trapping Increasing V At high velocities, solute segregation becomes small (“solute trapping”) N.

Nonequilibrium Solute Trapping • Numerical results (points) reproduce Aziz trapping function • With characteristic

FCC Binary Alloy Disordered phase Cu. Au G. Tonaglu, R. Braun, J. Cahn, G.

Wetting in Multiphase Systems M. Marcinkowski (1963) Kikuchi & Cahn CVM for fcc APB

Adsorption in FCC Binary Alloy Interphase Boundaries Antiphase Boundaries G. Tonaglu, R. Braun, J.

Monotectic Binary Alloy A liquid phase can “solidify” into both a solid and a

Incorporation of L 2 into the solid phase Expt: Grugel et al.

Nucleation in L 1 and incorporation of L 2 into solid Expt: Grugel et

Superconformal Electrodeposition • Cross-section views of five trenches with different aspect ratios – filled

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

Conclusions • Phase-field models provide a regularized version of Stefan problems for computational purposes

Experimental Observation of Dendrite Bridging Process (c) t = 30 s fs = 0.

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

Coalescence of two Grains Using Multi-Grain Model P; Disjoining Pressure ggb = 0. 3

Slides: 53

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Phase-Field Methods Jeff Mc. Fadden 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

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

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).

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

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

Ordering in a BCC Binary Alloy

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

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

Free Energy Function

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

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

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

Cahn-Hoffman Taylor (1992) -Vector

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

Diffuse Interface Formulation Kobayashi(1993), Wheeler & Mc. Fadden (1996), Taylor & Cahn (1998)

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

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

Corners and Edges Eggleston, Mc. Fadden, & Voorhees (2001)

Cahn-Hilliard Equation Cahn & Hilliard (1958)

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

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

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

Surface Adsorption Mc. Fadden and Wheeler (2001)

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

Surface Adsorption Ideal solution model Surface free energy Surface adsorption

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

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

Nonequilibrium Solute Trapping (cont. )

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

FCC Binary Alloy Disordered phase Cu. Au G. Tonaglu, R. Braun, J. Cahn, G. Mc. Fadden, A. Wheeler

Ordering in an FCC Binary Alloy

Free Energy Functional

Equilibrium States in FCC

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

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

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

Incorporation of L 2 into the solid phase Expt: Grugel et al.

Nucleation in L 1 and incorporation of L 2 into solid Expt: Grugel et al.

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

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

1 -D Equilibrium Profiles

1 -D Dynamics

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

Experimental Observation of Dendrite Bridging Process (c) t = 30 s fs = 0. 82 (b) t = 10 s fs = 0. 70 125 mm (a) t = 0 s fs = 0. 00 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

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

Coalescence of two Grains Using Multi-Grain Model P; Disjoining Pressure ggb = 0. 3 gsl = 0. 1 DT = 0 K x Large misorientation P>0 grains “repel” ggb = 0. 3 gsl = 0. 1 DT = 50 K W. Boettinger (NIST) & M. Rappaz (EPFL)

-Tensor Derivation