Field Method of Simulation of Phase Transformations in Materials

Alex Umantsev

Fayetteville State University, Fayetteville, NC

What do we need to account for?

Multi-phase states: thermodynamic systems may have multiple stable phase ‘coexisting’ at the same conditions.

Heterogeneous states: phase transformations may go along very complicated paths.

Dynamic structures: rate of transformation can make a difference for the final structure and properties

Nucleation: phase transformations are initiated through the process of overcoming of some kind of a potential barrier.

Phase Field Theory 101: Six Pillars of the Field Theory of Phase Transformations

Order para-meter

Free Energy

Hetero-geneity

Dyna-mics

Fluctua-tions

‘Hydrodynamics’

Field Theory

Landau theory

Ginzburg- Landau

Coarse-graining procedure

X

Y 1 23

ZnCu

blueZnredCu

ZnCu

Cu

NNNN

NNN

C

parameterorder

ionconcentrat

//1

C

R1 0.500 1.000

R2 0.486 0.971

R3 0.487 0.920

Pillar 1: Order Parameter: Symmetries of the System

--Zn atom --Cu atom

Order-disorder transformation in -brass (Cu-Zn alloy)

Pillar 2: Free Energy 432

0 41)(

31

21)(),( +T=Tg Wg

stable phase: ‘crystal’ metastable phase: ‘liquid’ W—potential barrier—energy scale -chemical potential difference /W-driving force

Landau potential

reaction coordinate

Pillar 3: Heterogeneous Systems: Gradient Energy

dv+W,g=G 2])()([21

Length scale:

WenergylInterfacia

WlengthnCorrelatio

:

1: – interface thickness

Order- Disorder Fe-Al Allen, Cahn Circa’82

Pillar 4: Kinetics

G=

dtd

WscaleTime

1:

The Gradient Flow Equation:

M.I. Mendelev, J. Schmalian, C.Z. Wang, J.R. Morris, and K.M. Ho, Phys. Rev. B, 74, 104206 (2006).

W

intV

Pillar 5: Internal fluctuations: Langevin force

TγkΓ

t)δ(t'x)Γδ(x')t',ζ(x't)ζ(x,

0t)ζ(x,

t)ζ(x,)δηδGγ(=

dtdη

B2

T,P

(x, t)—noise: Gaussian, white, additive

One more energy scale: thermal fluctuations-kBT

Pillar 6: ‘Hydrodynamic Modes’: Momentum Flow (Pressure or Stress) Diffusion (Concentration of species) Electromagnetic Field Variation (Waves) Heat Flow (Temperature Variation):

)()( t,Q+T=dtdTC x

dtdetQ 2

ETV

])[(),(

, x

General Heat-Equation

Heat Source

Latent Heat L

‘Thermodynamically consistent heat equation’ A. Umantsev and A. Roytburd. Sov. Phys. Solid State 30(4), 651-655, (1988)

Length Scales

– interfacial thickness

– capillary length

– kinetic length

– thermal length

– radius of curvature 1

2

K

CVl

Ll

L

CTl

Wl

n

E

T

C

I

Cl

l

RL

WCT

Il

lU

C

EC 2

dynamic

timekinetic

L

C2

timerelaxation

W 1

scalesTime

C TE

L W

Truly Multi-Scale Method

0 100 200 300 400

distance into the sample

liquid phase

--------solid phase

--------transition state

-----melting temp-----

temperature

order parameter

D. Danilov

lC l lT X

Å nm m mm Length scales

Tim

e sc

ale

Real Material Modeling Problem A: Convert thermodynamic functions of two or more phases into a continuous Landau-Gibbs Free Energy

Solution 1: LTPT: Symmetry expansion Solution 2: Speculate

G{i; T, P, C}

CALPHAD {T, P, C}

{i}

Problem B: Find the PFM parameters: {Wi, i, i}

Solution 1: Calculate from First Principles Solution 2: Calibrate from experiments or MD/MC simulations

Landau-Gibbs Free Energy

of Au-Sn Alloy

Au-Sn Binary (Bulk) Phase Diagram

Landau-Gibbs Free Energy of Carbon

-0.25 0.25 0.75 1.25-.025

0.250.75

1.25-820

-810

-800

-790

-780

-770

-760

-750

-740

crystallization OPcoordination OP

mola

r G

ibbs f

ree e

nerg

y [

kJ/m

ol]

liquid

graphite

diamond

Measurable Quantities

Interface energy:

Kinetic coefficient:

Wareaunitenergyfree ~

ETL

WgupercoolinsofKvelocitynterfacei

1

Interface thickness: W

lI

~

Equilibrium Fluctuations of OP ∆𝜂 𝑉 𝐤2=

𝑘𝐵𝑇

𝑉𝜕2𝑔𝜕𝜂2

𝜂 + 𝜅 𝐤 2

Non-Equilibrium Fluctuations of OP

time

Dynamic structure factor

Materials Genome Database of Interfacial Properties

Soldering 101

Cross-section

Solder

Cu

IMP

100m

5m

Sn-Solder

Cu-Plate

IMP

Soldering: InterMetallic Phase Growth

Sn-Solder

Cu-Plate

Where was the original interface between the tin and copper?

Liquid-state solder Solid-state solder

Onishi and Fujibuchi’75

Gagliano and Fine’00

baseline

PLC

IMP

Experimental results

Lord & Umantsev, J. Appl. Phys. 98, 063525 (2005) (Centennial Campus, 2004)

Total Free Energy

])()(

),,([22

cfdF r

))()()(

))()()(

)(),,(

(][

(][

s

s

cfcfBw

cfcfAw

cfcf

solidintermet

liquidsolid

liquid

Homogeneous part

normal

liquid

disordered

solid

inter

metallic molecular

liquid

crystallization

ord

eri

ng

Umantsev, JAP ‘07

0.0 0.2 0.4 0.6 0.8 1.0

concentration Cu

-35000

-30000

-25000

-20000

-15000

-10000

fre

e e

ne

rgy J

/mo

l

molecular liquid

atomic liquidsolution

fcc solidsolution

intermetallic compound

Thermocalc® G. Ghosh

1. fori=f()—not invariant against rotation of the reference

frame

2. fori=f(1,2,3,…)—orientation order parameter

3. fori=f(||)= s||+ q||2 —Kobayashi, Warren, Carter

f=+[1-s()]fliquid(c)+[s() -s()]fsolid(c)+ s()fintermet(c)

+(,) fori

0.0 0.2 0.4 0.6 0.8 1.0

order parameter

0

40

80

120

rela

xa

tio

n c

oe

ffic

ien

t

liquid crystall

KWC

G; this work

Ft

),(

Dynamics

Grain Boundary Contribution to Free Energy

Grain-boundary diffusion: Mobility: M=M(,,||)

Evolution Equations

F

t

F

t

)) ((

]),,([

soltinliqsolliq MMMMMM

cFcM

tc

Crystallization

Ordering

Diffusion

Parameters: 3 Diffusion

coefficients: solder

intermetallic substrate

+ 9 interfacial parameters

(interface energies thicknesses, and

mobilities)

Initial Conditions: slab, no nucleation

Scales: Length=0.25nm Time=0.25s

Ft

),(GB orientation

Solid State Soldering

Copper

Tin

Liquid state soldering

Copper

Tin

2D Modeling

solder

disordered

solid

inter

metallic

molecular

liquid

crystallization

ord

erin

g C-concentration

||-grad angle orient

Problem: rotation of grain orientation

Phase Field Simulation of Nucleation at Large Driving Forces

Numerical simulations x

x

t t

x, t are not just grid parameters. They are physical quantities–the noise correlation length and time.

Lifetime: time for a supercritical nucleus to appear in the system. Advantages:1. more reliable theory 2. free-energy landscape

3D as opposed to 2D Correlation properties of the fluctuations are very different.

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

Supercritical Nucleus

20 40 60 80 100

10

20

30

40

50

60

70

80

90

100 -0.2

0

0.2

0.4

0.6

0.8

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

X-Y plane

Y-Z plane

plane Z-X

Time Evolution of the System

escape

equilibrium fluctuations

________________ perturbation theory

xx dtV

tV ,1 10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

-phase

transition state

rectification of fluctuations

Free Energy Barrier

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

3D Structure of Supercritical Nucleus

Shape characterization

1. Volumetric content

2. Eccentricity 3. Roughness 4. Probability

distribution