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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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Content• Sharp interface• Finite thickness interface
- solute drag theories- Larsson-Hillert
• Diffuse interface- phase field
• No interface• Conclusions
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Modeling of the local state of phase interface (Hillert 1999)
Finite interfacethickness – continuousvariation in properties
Sharp interface - no thickness
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
x
x
x
For a given interfacevelocity the equationsmay be solved toyield the compositionon each side of interface.
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
”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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Finite interface thickness
Diffusion inside the interface solute dragProperty
Distance y
Solute drag theory (Cahn, Hillert and
Sundman etc)
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
• 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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Alloy 1
Driving diffusion in
Maximum possible growth rate for alloy 1
Non-parabolic growth in earlystages.
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Alloy 2
Maximum possible (partitionless) growth rate for alloy 2
partitionless growth possible
partitionless growth with spikein possible
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Simulations Fe-Ni-C Odqvist et al. 2002
700 oC, uNi=0.0228
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Larsson-Hillert (2005)
kk
kk
kk
m
kk RT
expxRT
expxz~VRTM
J
22
less ideal entropy of mixing
kk xx Absolute reaction
rate theory of vacancydiffusion:
Lattice-fixed frameof reference
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
km JVv
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Simulation 1 (close to local equilibrium)
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
/B
eq x
/B
eq x
distance
nucleates here
Simulation 2a vacancy diffusionwith composition gradient
0TBx
Initial composition
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
distance
Simulation 2bvacancy + cooperative mechanism
nucleates here
/B
eq x
/B
eq x
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
distance
Simulation 2c - other directionvacancy + cooperative mechanism
nucleates here
/B
eq x
/B
eq x
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Simulation 3Ternary system A-B-C, C is interstitial
C
kk x
xu
1
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
B
NPLEu
0Bu
0Cu
nucleates here
Simulation 3
B,AC MM 1000
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
221 m
m
G
VM
GM
:equation Allen-Cahn
conserved not
jjj
m
mj
jkjk
kk
cc
G
Vc
G
c
GLJ
Jc
22
''
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Example 1 – grain boundary segregation
Equilibrium solute segregation
RT
WWxx
x
G
ABB
gbB
B
m
)()1(exp
0
22
Equilibrium solute segregation by Cahn
kT
xECC
)(exp0
/
2
ln16
18
gbAB
AmA
kRTWW
VW
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Example 1 – grain boundary segregation
Two grains devided by a planar grain boundary• Concentration distribution at t1=30 sec and
t2=1 h
t1 t2
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Example 1 – grain boundary segregation
• Concentration profile at t1=30 sec and t2=1 h
t1 t2
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Example 1 – grain boundary segregation
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Example 2 – dynamic solute drag
• Concentration profile as a function of time
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Example 2 – dynamic solute drag
• Velocity as a function of curvature
• Concentration xB in the grain boundary
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
• Smaller initial grain• Concentration profile as a function of time
Example 2 – dynamic solute drag
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Example 2 – dynamic solute drag
• Smaller initial grain
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Loginova et al. 2004
Example: Formation of WS-ferrite in steels
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
Loginova et al. 2004
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
• Very efficient method – quite simple calculations.
• No details about the phase interface.• Remains to investigate its accuracy.
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ALEMIGrenoble May 8-9, 2006Materials Science and Engineering
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.
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