Phase-Field Models of Solidification

Jeff McFadden NIST

Dan Anderson, GWUBill Boettinger, NISTRich Braun, U DelawareSam Coriell, NISTJohn Cahn, NISTBruce Murray, SUNY BinghamptonBob Sekerka, CMUJim Warren, NISTAdam Wheeler, U Southampton, UK

Outline• Background• Phase-Field Models• Numerical Computations

NASA Microgravity Research Program

Atomistic scaleAtomistic scaleÅ

Dendrite scaleDendrite scalem

Grain scaleGrain scalemm

Component scaleComponent scalecm - m

How to connect these various scales ?How to connect these various scales ?

Modeling at various length scalesModeling at various length scales

2 nm

40 m 10 mm

M. Rappaz, EPFL

Liquid decanted during freezing Polished and etched microstructure after freezing

Dendritic Microstructure

Freezing a Pure Liquid

Dendrite

Glicksman

Hele Shaw

Saffman & Taylor

Stefan Problem

Solid

Liquid

• Interface is a surface; • No thickness;• Physical properties:

•Surface energy, kinetics

• Conservation of energy

Surface Energy

• Critical Nucleus and Coarsening

• Grain Boundary Grooves

• Wavelength of instabilities

Critical Nucleus and Coarsening

P. Voorhees & R. Schaefer (1987)

Critical Nucleus:

Coarsening:

Minimize the total surface energy

for a given volume of inclusions

Grain Boundary Grooves

S.C. Hardy (1977)

Wavelength of Instabilities

S. Hardy and S. Coriell (1968)

Ice cylinder growing into

supercooled water, MTT

Instability wavelength

depends on surface energy:

Morphological Instability

Mullins & Sekerka (1963, 1964)

“Point effect” “Constitutional supercooling”

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)

• Description of anti-phase boundaries (APBs)

• 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 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 Thermodynamics

L

’

Phase-Field Model

• Introduce the phase-field variable:

• Introduce free-energy functional:

J.S. Langer (1978)

• 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:

Planar Interface

where

• Particular phase-field equation

• Exact isothermal travelling wave solution:

where

when

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

• Or variation of Hele-Shaw problem...

Numerics

• Advantages - no need to track interface - can compute complex interface shapes

• Disadvantage - have to resolve thin interfacial layers

• State-of-the-art algorithms (C. Elliot, Provatas et al.) useadaptive finite element methods

• Simulation of dendritic growth into an undercooled liquid...

Provatas, Goldenfeld & Dantzig (1999) Dendrite Simulation

Anisotropic Equilibrium Shapes

Cahn & Hoffmann (1972)

W. Miller & G. Chadwick (1969)

Sharp Interface Formulation

• Sharp interface limit:• McFadden & Wheeler (1996)

• is a natural extension of the Cahn-Hoffman of sharp interface theory

• Cahn & Hoffman (1972, 1974)

• is normal to the -plot:

• Isothermal equilibrium shape given by

• Corners form when -plot is concave;

Diffuse Interface Formulation

• Recall:

• Suggests:

where:

• Phase-field equation:

where the so-called -vector is defined by:

Corners and Edges

Taylor & Cahn (1998), Wheeler & McFadden (1997)

Eggleston, McFadden, & Voorhees (2001)

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

Inclusion of Surface Properties

•Surface Adsorption

•Wetting in Multiphase Systems

•Solute Trapping

(More than a computational device)

Examples:

Surface Adsorption

McFadden and Wheeler (2001)

Solute Trapping

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

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

Results agree well with other trapping models (Aziz 1988)

Wetting in Multiphase SystemsM. Marcinkowski (1963)

Kikuchi & Cahn CVM for fcc APB (CuAu)

R. Braun, J. Cahn, G. McFadden, & A. Wheeler (1998)

Phase-field model with 3 order parameters

Early Phase-Field Calculations

•G. Caginalp & E. Socolovsky (1991, 1994)

•R. Kobayashi (1993, 1994)

• A. Wheeler, B. Murray, R. Schaefer (1993)

•2nd order accurate finite differences on 2-D uniform mesh

•Explicit time-stepping for phase-field equation

•Implicit (ADI) for energy equation

•Mesh convergence an issue

•Vector machines (Cray)

•Roldan Pozo (benchmarks on PC cluster at NIST)

Adaptive Meshing

•R. Braun, B. Murray, & J. Soto (1997)

º VLUGR2, vectorized, adaptive finite difference solver

•R. Almgren & A. Almgren (1996)

º 2-D, second-order accurate, semi-implicit

•N. Provatis, N. Goldenfeld, & J. Dantzig (1999)

º 2D, Galerkin FE, dynamically adaptive, quadtree

•M. Plapp & A. Karma (2000)

º Hybrid FD Mesh/diffusion Monte Carlo method

A. Karma & W.-J. Rappel (1997)

•Uniform 300x300x300 mesh

•Grid-corrected anisotropy

W. George & J. Warren (2001)

•3-D FD 500x500x500

•DPARLIB, MPI

•32 processors, 2-D slices of data

J. Jeong, N. Goldenfeld, & J. Dantzig (2001)

Charm++ FEM framework, hexahedral elts, octree, 32 processors, METIS

• 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