phase-field methods jeff mcfadden nist dan anderson, gwu bill boettinger, nist rich braun, u...

Phase-Field Methods

Jeff McFadden NIST

Dan Anderson, GWUBill Boettinger, NISTRich Braun, U DelawareJohn Cahn, NISTSam Coriell, NISTBruce Murray, SUNY BinghamptonBob Sekerka, CMUPeter Voorhees, NWUAdam Wheeler, U Southampton, UK

July 9, 2001

Gravitational Effects in Physico-Chemical Systems: Interfacial Effects

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 ModelsMain 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

Phase-Field ModelThe 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)

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:

Ordering in a BCC Binary Alloy

Parameter Identification

• 1-D solution:

• Interface width:

• Surface energy:

• Curvature-dependence (expand Laplacian):

Phase-Field Model

• Introduce the phase-field variable:

J.S. Langer (1978)

• Introduce free-energy functional:

• Dynamics

Free Energy Function

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:

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

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

Anisotropic Equilibrium Shapes

W. Miller & G. Chadwick (1969)

Hoffman & Cahn (1972)

Cahn-Hoffman -Vector

Taylor (1992)

Cahn-Hoffman -Vector

Equilibrium Shape is given by:

Force per unit length in interface:

Cahn & Hoffmann (1974)

Diffuse Interface Formulation

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

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

• Jump conditions give:

• where

• and

Corners/Edges

(force balance)

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

Corners and Edges

Eggleston, McFadden, & Voorhees (2001)

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

Cahn-Hilliard Equation

Cahn & Hilliard (1958)

Phase Field Equations - Alloy

V

C dVcTcfF2

22

2

22),,(

0 22

fF

constant c- 22 Cc f

cF

Coupled Cahn-Hilliard & Cahn-Allen Equations

22

fM

t

-)1( 22 cc f

ccMtc

CC

R'Τ

DpDp M

cMMcM

LSC

BA

())(-1(

)-1(where{

Wheeler, Boettinger, & McFadden (1992)

Alloy Free Energy Function

)())(1()1(

ln)1ln()1(

),(),()1( T)c,,(

ppcc

ccccTR

Tf cTf-cf

LS

BA

Ideal Entropy

L and S are liquid and solid regular solution parameters

One possibility

W. George & J. Warren (2001)

•3-D FD 500x500x500

•DPARLIB, MPI

•32 processors, 2-D slices of data

Surface Adsorption

McFadden and Wheeler (2001)

Surface Adsorption1-D equilibrium:

Differentiating, and using equilibrium conditions, gives

where

Cahn (1979), McFadden and Wheeler (2001)

Surface Adsorption

Ideal solution model Surface free energy Surface adsorption

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

Solute Trapping

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

At high velocities, solute segregation becomes small (“solute trapping”)

Increasing V

D

DE

VV

VVkVk

/1

/)(

L

E

E

L

SD

D

)k(

)/k(

D

DV

1

1ln1

16

3

Nonequilibrium Solute Trapping

• Numerical results (points) reproduce Aziz trapping function

• With characteristic trapping speed, VD, given by

0 2 4 6 8

ln k E /(k E -1 )

0

20

40

60

VD

(m

/s)

A l-In

A l-C uA l-G e

A l-S n

S i-B i

S i-S n

S i-G eS i-A s

S i-G a

S i-In

S i-S b

V D m easu rem en ts fro m S m ith & A ziz (1 9 9 5 )

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

Disordered

phase

CuAu

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

FCC Binary Alloy

Ordering in an FCC Binary Alloy

Free Energy Functional

Equilibrium States in FCC

Wetting in Multiphase SystemsM. 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

Interphase Boundaries

Antiphase Boundaries

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

Adsorption in FCC Binary Alloy

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

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.

Incorporation of L2 into the solid phase

2L S 1L

Expt: Grugel et al.

Nucleation in L1 and incorporation of L2 into solid

1L

2L

S

2L2L

Expt: Grugel et al.

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

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

Phase-Field Model of Electrodeposition

J. Guyer, W. Boettinger, J. Warren, G. McFadden (2002)

1-D Equilibrium Profiles

1-D Dynamics

• 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

(b) t = 10 sfs = 0.70

(a) t = 0 sfs = 0.00

(e) t = 210 sfs = 0.97

(f) t = 1500 sfs = 0.98

(c) t = 30 sfs = 0.82

(d) t = 75 sfs = 0.94

125 m

Photo: W. Kurz, EPFL

Experimental Observation of Dendrite Bridging Process

Dendrite side arm bridging

Y

X

•Collision of offset arms - Delayed bridging

0

0,2

0,4

0,6

0,8

1

-2,E-08 -1,E-08 0,E+00 1,E-08 2,E-08

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

0

0,2

0,4

0,6

0,8

1

-2,E-08 -1,E-08 0,E+00 1,E-08 2,E-08

gbgb = 0.3 = 0.3 sl sl = 0.1= 0.1

T = 0 KT = 0 K

gbgb = 0.3 = 0.3 sl sl = 0.1= 0.1

T = 50 KT = 50 K

xx

Large misorientationLarge misorientation > 0> 0

grains “repel”grains “repel”

; Disjoining Pressure

W. Boettinger (NIST) & M. Rappaz (EPFL)W. Boettinger (NIST) & M. Rappaz (EPFL)

-Tensor Derivation