Download presentation

Presentation is loading. Please wait.

1
**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

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
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).

4
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:

5
**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:

6
**Ordering in a BCC Binary Alloy**

7
**Parameter Identification**

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

8
**Phase-Field Model Introduce the phase-field variable:**

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

9
Free Energy Function

10
**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)

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 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

13
**Anisotropic Equilibrium Shapes**

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

14
Cahn-Hoffman -Vector Taylor (1992) Phase field

15
**Cahn-Hoffman -Vector Equilibrium Shape is given by:**

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

16
**Diffuse Interface Formulation**

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

17
**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

18
**Corners/Edges Jump conditions give: (force balance) 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 Equation**

22
**Phase Field Equations - Alloy**

Coupled Cahn-Hilliard & Cahn-Allen Equations where { 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
**McFadden and Wheeler (2001)**

Surface Adsorption McFadden and Wheeler (2001)

26
**Cahn (1979), McFadden and Wheeler (2001)**

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

27
**Surface Adsorption Ideal solution model Surface free energy**

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
**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)

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
**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

38
**Adsorption in FCC Binary Alloy**

Interphase Boundaries Antiphase 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
**Incorporation of 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)

47
**1-D Equilibrium Profiles**

48
1-D Dynamics

49
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

50
**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

51
**Dendrite side arm bridging**

X Y Collision of offset arms - Delayed bridging

52
**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)

53
-Tensor Derivation

Similar presentations

OK

Analysis of Hydrodynamic and Interfacial Instabilities During Cooperative Monotectic Growth Cooperative monotectic growth Sources of flow with a fluid-fluid.

Analysis of Hydrodynamic and Interfacial Instabilities During Cooperative Monotectic Growth Cooperative monotectic growth Sources of flow with a fluid-fluid.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on south african culture religion Ppt on supply chain management of dell Ppt on overhead service connection Economics ppt on lpg Ppt on steps in designing hrd system Ppt on adr and drive download Ppt on special types of chromosomes in human Free ppt on food security in india Ppt on child labour pdf Ppt on water conservation methods