Modelling of the motion of phase interfaces; coupling of thermodynamics and kinetics

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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering

Modelling of the motion of phase interfaces; coupling of thermodynamics and kinetics

John ÅgrenDept of Materials Science and Engineering

Royal Institute of TechnologyS-100 44 Stockholm

Ackowledgement:Joakim Odqvist, Henrik Larsson, Klara Asp

Annika Borgenstam, Lars Höglund, Mats Hillert

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Content• Sharp interface• Finite thickness interface

- solute drag theories- Larsson-Hillert

• Diffuse interface- phase field

• No interface• Conclusions

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Modeling of the local state of phase interface (Hillert 1999)

Finite interfacethickness – continuousvariation in properties

Sharp interface - no thickness

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Diffuse interface: no sharp boundary betweeninterface and bulk (phase-field method)

No interface: Only bulk propertiesare used, i.e. not even an operating interfacialtieline calculated.

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The rate of isobarothermal diffusional phase transformations in an N component system may be predicted by solving a set of

N-1 diffusion equations in each phase : 2(N-1)

Boundary conditions at interface: - Composition on each side 2(N-1)- Velocity of interface +1

2N-1These boundary conditions must obey:Flux balance equations N-1

Net number of extra conditions neededat phase interface: N

Sharp interface(Stefan Problem)

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How to formulate the N extra conditions?

Baker and Cahn (1971): N response functions. For example in a binary system :

Simplest case: Local equilibrium.- The interfacial properties do not enter into the problem except for the effect of interfacial energy of a curved interface (Gibbs-Thomson) and the interface velocity does not enter.

0),(

0),(

,,2

,,1

Tvxxf

Tvxxf

BB

BB

0),(),(

0),(),(

TxTx

TxTx

BBBBB

BABAA

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Dissipation of driving force at phase interface

The driving force across the interface is consumed by two independent processes:- Transformation of crystalline lattice- Change in composition by trans-

interface diffusion

The processes are assumed independent and thus each needs a positive driving force.

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FeCFeCC xxNNz //

Carbon diffuses across interface from to .

Driving force for trans-interfacediffusion:

Driving force for change ofcrystalline lattice:

All quantities expressed per mole of Fe atoms.

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CCCC

Cs

sFe

CCCCs

C

CCCtC

zzL

z

M

V

V

v

zzLV

v

LJ

//2

/

C and Fe are functions of the compositionon each side of the interface and may be describedby suitable thermodynamic models of the and phase, respectively.

Trans-interface diffusion(Hillert 1960)

Combination with finite interface mobility yields:

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Combination with finite interface mobility yields:

0. 0 Since

BA

BBBB

Bm

mB

BBBB

Bm

mA

xxL

x

M

V

V

v

xxL

x

M

V

V

v

//2

//2

)1(

0

A and B are functions of the compositionon each side of the interface and may be describedby suitable thermodynamic models of the and phase, respectively.

Trans-interface diffusion substitutional system

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x

x

x

For a given interfacevelocity the equationsmay be solved toyield the compositionon each side of interface.

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”Sharp” interface with with representative composition (Ågren 1989)

BiBA

iA

mm

ABBiB

tm

BiB

mABBB

tB

tA

xxG

xxG

xxV

vLJJ

)(

)(

BiB

BB

iBm

mB

BiB

BB

iBm

mA

xxL

x

M

V

V

v

xxL

x

M

V

V

v

)1(2

2

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Aziz model (1982) Similar as the previous sharp interface

models but:

/1

:

:,:

/)(

//

/

/

iB

eq

B

i

BB

BBm

itB

tA

D

vkk

Df

aa

aafV

DJJ

:finds one constant tscoefficien activity Assuming

interface of Thickness

interface in ydiffusivit in tcoefficien activity

interface of side and on activity B : and

where

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Finite interface thickness

Diffusion inside the interface solute dragProperty

Distance y

Solute drag theory (Cahn, Hillert and

Sundman etc)

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• Solution of steady state equation inside interface.

• Integration of dissipation over interface:

• Total driving force

dy

dy

dJ

v

VG ABt

Bmsd

m

)(

).(

,()(

)(

yx

xv

xyf

xxV

v

yLJJ

B

B

BAB

BBm

ABBB

tB

tA

calculate thus may we and givenFor

). yields model specific A

n

kkkk

totm xG

1

int )(

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Available driving force across phase-interface (Hillert et al. 2003)

0

)(

int

1

int

kkkk

kkkkk

n

kkkk

totm

JxJJ

JxJxx

xG

when

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dy

dy

dJ

v

VG ABt

Bmsd

m

)( M

VvG mi

m

For a given composition of growing

n

kkkk

totm xG

1

int )(

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Driving force = Dissipation:

The compositions on each sideof the phase interface dependon interface velocity and they app-roach each other.

Above a critical velocity trans-formation turns partitionless. ),(

),(

vTgx

vTfx

k

k

0),(

0),(

vuvTgu

vuvTfueq

BB

eqBB

as

as

eqBu

eqBu

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Alloy 1

Driving diffusion in

Maximum possible growth rate for alloy 1

Non-parabolic growth in earlystages.

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Alloy 2

Maximum possible (partitionless) growth rate for alloy 2

partitionless growth possible

partitionless growth with spikein possible

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Calculated limit of the massive transformation in Fe-Ni alloys(dashed lines) calculated for two different assumptions onproperties of interface.

(Odqvist et al. 2002)

Exp: Borgenstam and Hillert 2000

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Fe-Ni-CThe interfacial tieline depends on the growth rate.(Odqvist et al. 2002)

Decreasing growth rate

- At high growth rates the state is close to paraequilibrium.

- At slower rates there is a gradual change towards NPLE.

- For each alloy composition there is a maximum size which can be reached under non-partitioning conditions. This size may be reached before there is carbon impingement. See also Srolovitz 2002.

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Thickness of M spike in : DM / v.

Local equilibrium, i.e. quasi paraequilibrium impossible if DM / v < atomic dimensions.

v = 4.8 10-7 ms-1

DM / v = 3.5 10-14 m

Ferrite formation under ”practical” conditionsin Fe-Ni-C (Oi et al. 2000)

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Simulations Fe-Ni-C Odqvist et al. 2002

700 oC, uNi=0.0228

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Larsson-Hillert (2005)

kk

kk

kk

m

kk RT

expxRT

expxz~VRTM

J

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less ideal entropy of mixing

kk xx Absolute reaction

rate theory of vacancydiffusion:

Lattice-fixed frameof reference

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km JVv

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Simulation 1 (close to local equilibrium)

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/B

eq x

/B

eq x

distance

nucleates here

Simulation 2a vacancy diffusionwith composition gradient

0TBx

Initial composition

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RT

xexp

RT

xexpxx

z~VRTM

RTexpx

RTexpx

z~VRTM

J

jjjjkk

m

k

kk

kk

m

kk

2121

21

22

Solute trapping impossible!

Add term for cooperative mechanism

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distance

Simulation 2bvacancy + cooperative mechanism

nucleates here

/B

eq x

/B

eq x

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distance

Simulation 2c - other directionvacancy + cooperative mechanism

nucleates here

/B

eq x

/B

eq x

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Simulation 3Ternary system A-B-C, C is interstitial

C

kk x

xu

1

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B

NPLEu

0Bu

0Cu

nucleates here

Simulation 3

B,AC MM 1000

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221 m

m

G

VM

GM

:equation Allen-Cahn

conserved not

jjj

m

mj

jkjk

kk

cc

G

Vc

G

c

GLJ

Jc

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''

1

equations) 1-(n

:conserved ionConcentrat

Diffuse interfaces – phase-field method

Thermodynamic and kinetic properties of diffuse interfaces are needed, e.g.

.mkm VcG /),(

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ABBBB

Bmm

WxWxxW

xWGG

)1()(

)(1 22

Phase-field modelling of solute drag in grain-boundary migration(Asp and Ågren 2006)

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Example 1 – grain boundary segregation

Equilibrium solute segregation

RT

WWxx

x

G

ABB

gbB

B

m

)()1(exp

0

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Equilibrium solute segregation by Cahn

kT

xECC

)(exp0

/

2

ln16

18

gbAB

AmA

kRTWW

VW

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Two grains devided by a planar grain boundary• Concentration distribution at t1=30 sec and

t2=1 h

t1 t2

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• Concentration profile at t1=30 sec and t2=1 h

t1 t2

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Example 2 – dynamic solute drag

• Initial state: A circular grain with =1 surrounded by another grain where =0. Homogenous concentration of xB=0.01.

Phase-field at t=0

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• Concentration profile as a function of time

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• Velocity as a function of curvature

• Concentration xB in the grain boundary

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• Smaller initial grain• Concentration profile as a function of time


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• Smaller initial grain

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Phase-field method contains similar steps as in sharp and finite interface modelling:

• Some properties of interface modelled.• Solution of a diffusion equation to obtain

concentreation profile.• Cahn-Allen equation plays a similar role as

the equation for interfacial friction.

But: Thickness of interface must be treated not only as a numerical parameter but as a physical quatity.

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Loginova et al. 2004

Example: Formation of WS-ferrite in steels

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Loginova et al. 2004

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No Interface at allSvoboda et al. 2004 (by Onsager extremum principle)Ågren et al. 1997 (from other principles)

C

i ii

ii

eff

mm

m

C

iiieff

Mxxx

M

VM

VGxM

dt

d

1

2

1

1

21

where

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• Very efficient method – quite simple calculations.

• No details about the phase interface.• Remains to investigate its accuracy.

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ConclusionsConclusionsConclusions

• Sharp interface methods are computationally simple but may show problems with convergency.

• Solute drag models may be better than sharp interface models but have similar convergency problems.

• Larsson-Hillert method very promising• Phase-field approach very powerful – no

convergency problems – heavy computations.

• No-interface methods – quick calculations, accuracy remains to be investigated.

Modelling of the motion of phase interfaces; coupling of thermodynamics and kinetics

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