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## Phase-Field Methods Jeff McFadden NIST

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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 Gravitational Effects in Physico-Chemical Systems: Interfacial Effects July 9, 2001 NASA Microgravity Research Program**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**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).**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:**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**Parameter Identification**• 1-D solution: • Interface width: • Surface energy: • Curvature-dependence (expand Laplacian):**Introduce the phase-field variable:**• Introduce free-energy functional: • Dynamics Phase-Field Model J.S. Langer (1978)**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)**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**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**Anisotropic Equilibrium Shapes**W. Miller & G. Chadwick (1969) Hoffman & Cahn (1972)**Cahn-Hoffman -Vector**Taylor (1992) Phase field**Equilibrium Shape is given by:**Force per unit length in interface: Cahn-Hoffman -Vector Cahn & Hoffmann (1974) Phase field**Diffuse Interface Formulation**Kobayashi(1993), Wheeler & McFadden (1996), Taylor & Cahn (1998)**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**(force balance)**Corners/Edges • Jump conditions give: • where • and Bronsard & Reitich (1993), Wheeler & McFadden (1997)**Corners and Edges**Eggleston, McFadden, & Voorhees (2001)**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**Cahn & Hilliard (1958)**Cahn-Hilliard Equation**{**where Coupled Cahn-Hilliard & Cahn-Allen Equations Phase Field Equations - Alloy Wheeler, Boettinger, & McFadden (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 500x500x500 • DPARLIB, MPI • 32 processors, 2-D slices of data**Surface Adsorption**McFadden and Wheeler (2001)**where**Differentiating, and using equilibrium conditions, gives Surface Adsorption 1-D equilibrium: Cahn (1979), McFadden and Wheeler (2001)**Surface Adsorption**Ideal solution model Surface free energy Surface adsorption**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**Solute Trapping**IncreasingV At high velocities, solute segregation becomes small (“solute trapping”) N. Ahmad, A. Wheeler, W. Boettinger, G. McFadden (1998)**Nonequilibrium Solute Trapping**• Numerical results (points) reproduce Aziz trapping function • With characteristic trapping speed, VD, given by**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**FCC Binary Alloy**Disordered phase CuAu G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A. Wheeler**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)**Adsorption in FCC Binary Alloy**Antiphase Boundaries Interphase Boundaries G. Tonaglu, R. Braun, J. Cahn, G. McFadden, A. Wheeler**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**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**Incorporationof L2 into the solid phase**Expt: Grugel et al.**Nucleation in L1 and incorporation of L2 into solid**Expt: Grugel et al.**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**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. McFadden (2002)**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 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