Prediction of precipitation kinetics in Al alloys · New project ClaNG+ Collaboration ... • Alloy...
Transcript of Prediction of precipitation kinetics in Al alloys · New project ClaNG+ Collaboration ... • Alloy...
Institute ofPhysical Metallurgy and Metal Physics
Prediction of precipitation kinetics
in Al alloys
GTT workshop / Herzogenrath (Germany) / June 20-22, 2007
E. Jannot, V. Mohles, G. Gottstein
2 GTT Workshop / June 20-22, 2007
New project ClaNG+
Collaboration between
• Hydro (Aluminum industry)
• GTT technologies
• IMM: Institut für Metallkunde und Metallphysik
Extension of the use of ClaNG (tool for the
prediction of precipitation kinetics)
• To other alloys extend the thermodynamic databank
(input of the model)
• To other process steps couple precipitation with
other phenomena (Deformation, RX)
Started 01/2007…
3 GTT Workshop / June 20-22, 2007
Aluminum sheet production
Evolution: start with a higher content of recycled products
Goal: control the end- properties of the material
Requirement: predict the precipitation state at each step
Casting - Homogenization - Hot Rolling - Cold Rolling - Annealing
break down tandem-mills
5 GTT Workshop / June 20-22, 2007
ClaNG model overview
INPUT
OUTPUTStarting state:
• Alloy concentrations
• Primary phase volume fraction (density & size)
• Dendrite arm spacing
Physical parameter:
• Phase diagram
• Diffusivity
• Interfacial energies
Matrix:
• Alloy concentrations
Primary & secondary phases:
• Volume fraction(density and size distribution)
• Composition
Material properties:
• Conductivity…
Industrial process:
• Time-Temperature curve
• (dislocation density)
Nucleation Growth &
Coarsening
Thermodynamic *Free Energy curves
Equilibrium calculation
Kinetic Diffusion
Phase
Composition
Update
State at time tTemperature, Alloy
conc…
State at time t + tNew conc., phase vol. frac…
* : performed using ChemApp from GTT technologies
6 GTT Workshop / June 20-22, 2007
Thermodynamic calculations (I)
At every interface, one assumes that a local equilibrium is achieved after a short time
The equilibrium conditions are calculated using the ChemApp application developed by GTT technologies
For these calculations, a thermodynamic databank is required
• The AlTT databank (Thermotech) is used for the moment
• Light version containing 8 elements (Al, Cr, Cu, Fe, Mg, Mn, Si, Ti)
For each possible phase (Mg2Si, Apha…),
perform the chemical reaction corresponding
the matrix decomposition occurring at the
interface between the matrix and the phase
focus
here
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Thermodynamic calculations (II)
Use ChemApp:
• Set conditions (temperature, concentrations in the matrix)
• Enter 2 phases: matrix + another phase
• Perform equilibrium
• Extract information
Equilibrium concentrations ci in the Al matrix
Equilibrium concentrations ci in the phase
Chemical potentials of the elements
Derive the Chemical Driving Force gu
x Mole
Matrix 0
Gm0
Matrix 1
Gm1
1 Mole
Phase
Gm
(1-x) Mole
i
1
i
0
iiu )(cg
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Nucleation
When gu is known, the critical radius rc can be derived
• User input = interfacial energy
The nucleation rate is then given by the classical theory of Becker & Döring
The nucleation rate is often underestimated
• Reason: the thermodynamic quantities (gu, ) are not precisely defined for nanoparticles
• corrected by a “tuning” factor
texp
Tk
rGfexpZNN
B
Ccorr0
u
mc
g
V2r
Usual value 1.0 0.1
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Growth & Coarsening
Assumption: precipitate growth always diffusion controlled in Al alloys
ClaNG treats growth and coarsening in a single equation
• A particle above the critical radius grows
• A particle below the critical radius dissolves
The growth law used in ClaNG derives from Zener’s formulation
• All particles are considered spherical
r
D
)r(cc
)r(cc
dt
drv
/
/0
rTR
V2expc)r(c
g
m/
eq
/
c0
c
c/(r)
xr
Matrix
Gibbs-Thomson concentration
at the interface
/
.eqc
Phase
(sphere)
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Evolution with time
particle distribution at t
radius of the particle
de
ns
ity
particle distribution at t+dt
radius of the particle
de
ns
ity+ t
g(ri)
g(ri+1)
ri
ri+1
For every radius ri at t+t, one determines its image g(ri) at t using a Runge-Kutta method
tNdr)t,r(fdr)t,r(f)t(N)tt(N nucl
i
r
)r(g
r
)r(g
ii
1i
1i
i
i
Ni(t+
t)
(Robson, Acta Mater. 51-2003)
Discretization in radius classes (1 nm width)
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Phase composition update
During the heat treatment, the equilibrium phase composition changes
• The concentration in the precipitate should be consequently updated
Diffusion inside the precipitate does the work, but unfortunately the diffusion coefficient in the phase is generally unknown
• Hence this phenomenon can’t be accurately modeled
• Pragmatic modeling: use of a “tuning” factor to level diffusion strength
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Conclusions
Each step is dependent on the results of an
thermodynamic equilibrium calculation
The classical nucleation and growth theory
provides the frame for the precipitation kinetics
calculation
However still hard to provide a 100% physical
based model
• “Tuning” parameters still required
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Microstructure after homogenization
AA 3104 investigated
Still inhomogeneous after homogenization
• Complex structure with 3 different regions
Primary phases or
constituents (3%)
Phase Beta = Al6(Fe,Mn)
Precipitation free zone
(30%)
Dendrites with secondary
phases or dispersoides
(67%)
Phase Alpha = Al12Mn3Si1.8
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Formation of dispersoides
0
100
200
300
400
500
600
700
0 5 10 15 20 25 30
Zeit / h
Te
mp
era
tur
/ °C
2 m
2 m
2 m
2 m
2 m
2 m
2 m
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Prediction quality
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
3.1
3.3
3.5
3.7
3.9
0 3 6 9 12 15 18 21 24
Homogenization time [h]
Re
sis
tiv
ity
[1
0^
-8 O
hm
.m]
-
100
200
300
400
500
600
Ho
mo
ge
niz
ati
on
te
mp
. [°
C]
Rsimu
Rexp_Center
Rexp_Edge
Temp.
-
20
40
60
80
100
120
0 3 6 9 12 15 18 21 24Homogenization time [h]
Av
era
ge
Ra
diu
s [
nm
]
-
100
200
300
400
500
600
Ho
mo
ge
niz
ati
on
te
mp
. [°
C]
Alpha
Mg2Si
Exp_Center
Exp_Edge
1.0E+18
1.0E+19
1.0E+20
1.0E+21
1.0E+22
0 3 6 9 12 15 18 21 24Homogenization time [h]
De
ns
ity
[m
^-3
]
-
100
200
300
400
500
600
Ho
mo
ge
niz
ati
on
te
mp
. [°
C]
Alpha
Mg2Si
Exp_Center
Exp_Edge
AA3104 Fe Mg Mn Si
Nom. [c] 0.45 1.02 0.85 0.18
Arm [c] - Start 0.02 1.03 0.75 0.20
Arm [c] - End 0.03 1.03 0.69 0.13
Alpha = 0.2 Jm-2 and fcorr(Alpha) = 0.12
Mg2Si = 0.2 Jm-2 and fcorr(Mg2Si) = 1.00
Simulation takes less than 30 min,
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Conclusions
The model is able to deliver good predictions of the precipitation kinetics
• Used by Hydro since 2004
Acceptable sensitivity to empirical factors:
• Nucleation rates are very sensitive to their “tuning” parameters but you can’t do without
• Other factors allow simulations to 100% fit the experimental results
Results are however very sensitive to the starting conditions
• Model casting step…
19 GTT Workshop / June 20-22, 2007
Problematic
The goal of the project is to extend the use of
ClaNG to different alloy systems
Hence, you need to have a robust growth law
which is correct for the whole range of alloys
ClaNG uses today a growth law based on the
one proposed by Zener
Question: Is this law the right one?
r
D
)r(cc
)r(cc
dt
drv
/
/0
20 GTT Workshop / June 20-22, 2007
Survey of possible growth laws
Many groups have proposed modified versions of this
growth law these last years
Robson et al. (good for Al3Sc vs. Al3Zr)
Kozeschnik et al. (derived from the extremum principle)
Hoyt, Morral, Purdy, Olson… (valid in case of coarsening)
r
D
)r(cc
)r(cc'r i
/
ii
/
i
0
i
'r)C[r
)C[]D[
ieq
i
/
ii
m
c
)r(cln.RTc
r
V2
+Differential equation
(r, r’) to solve
1
i i
0
i
20
ii
g
mu
Dc
cc
rTRr
V2g
'r
Originally developed for steels
21 GTT Workshop / June 20-22, 2007
GT effect in multicomponent alloy (I)
Problem: GT effect not correctly accounted when
multicomponent alloy or not pure solute phase
Solve it in the “static” case for a stoichiometric phase
Classical expression
i
/
i
0
i c)r()r(c)r(1c
Matrix0 Matrix1
1 Mole
Phase
1-ε(r)
ci0 ci
/(r) ciβ
GM0 GM1 G +
ε(r)
r
V2 m
ieq
i
/
ii
m
C
)r(cln.RTc
r
V2
“Master” equation for the GT effect
i0
i
iiu
m
c
c)r(1ln.RTcg
r
V2
Equation with only one unknown ε(r)
Equilibrium influenced by the radius r
rTR
V2expc)r(c
g
m/
eq
/
22 GTT Workshop / June 20-22, 2007
GT effect in multicomponent alloy (II)
Plotting as function of r is difficult but plotting r as
function of is straightforward
r() nearly always has the same shape and its
inverse (r) can then be extrapolated
a
ceq
r
r1)r(
The parameter a is easily
found by interpolation
When coarsening, a
reaches the limit 1.0
eq
ii
eq
i
0
ieq
cc
cc
Same value for all
components
23 GTT Workshop / June 20-22, 2007
Growth law N°1
Assumption: the slowest diffuser drives the dance
(among the elements contained in the phase)
a
ceq
SlowEltSlowElt1
r
r1
r
D)r(
r
Dvr
Problem: is it satisfactory?
24 GTT Workshop / June 20-22, 2007
Growth law N°2 (I)
Principle: the GT effect is a degree of freedom and the
system will use it to maximize the growth rate
oR
r
i
/
i
0
i dx)x()r(
)x(TR
)x(cD)x(j i
g
iii Generalized
1st Fick’s law
oo R
r i
2
i
ig
2R
r ii
ig
2
2/
i
0
i dx)x(cx
1
D
)r(jTRrdx
)x(cD
)x(jTR
x4
x4)r(
for steady-state diffusion, 4 x2 ji(x) = cst. for a spherical geom.
oR
r ii
ig/
i
0
i dx)x(cD
)x(jTR)r(
Do some
calculus
Assumption: coars. & GT effect small ci(x) cst. goes out of the integral
)r(cD
)r(jTRr
r
1
R
1
)r(cD
)r(jTRr)r(
/
ii
ig
o
/
ii
ig
2
/
i
0
i
If R0 >> r
25 GTT Workshop / June 20-22, 2007
Growth law N°2 (II)
)r(cD
)r(jTRr)r(
/
ii
ig/
i
0
i
)c)r(c(r)r(j i
/
ii
i
/
i
0
iim
u )r(cr
V2g
Stephan’s interface
mass balanceGT effect
i/
ii
/
iig
im
u)r(cD
))r(cc(rTRrc
r
V2g
1
0
i
i
i
i
g
mu
2 )1c
c(
D
c
rTRr
V2g
vr
The growth rate is only dependent on the intrinsic diffusion
coefficients (interdiffusion coefficients not necessary)
It is valid for coarsening & small GT effect ( r > 30 nm)
Very close to Kozeschnik
ci/(r) ci
0 near the
critical radius
26 GTT Workshop / June 20-22, 2007
Prediction quality
The 1st law provides the best results: kinetics given by the
diffusion of the slowest element (Mn)…
0.0E+00
5.0E-08
1.0E-07
1.5E-07
2.0E-07
2.5E-07
0.0 2.4 4.8 7.2 9.6 12.0 14.4 16.8 19.2 21.8
time [h]
pa
rtic
le r
ad
ius
[m
]
-
100
200
300
400
500
600
700
tem
pe
ratu
re [
°C]Law1
Law2
Exp
Temp
27 GTT Workshop / June 20-22, 2007
The end
Thank you for your attention,
A version of the model is available on SimWeb
• Contact: [email protected]
If you have questions…
29 GTT Workshop / June 20-22, 2007
Calculation of the nucleation rate
Clear separation between the contributions
• Diffusivities are type dependant: hom. or het.
The effect on the energy of the critical radius is
accounted by a separate correction factor
texp
Tk
rGfexpZNN hom
B
C.corr
homhomhom
0
homhom
texp
Tk
rGffexpZNN het
B
Ccorr
hom
corr
hethethet
0
hethet
hethomtot NNN
HOMOGENEOUS NUCL. HETEROGENEOUS NUCL.
)m10(atomsofnumberN 3290
hom
)(.funct.geomN0
het
TR
QexpD)T(D
g
bulk0
bulkhom
TR
QfexpD)T(D
g
bulkpipe0
bulkhet
Usual value 0.6
Usual value ?
30 GTT Workshop / June 20-22, 2007
Dendrite flux
Dendrite arm focused modeling
• Start with the concentrations in the arm
The interactions between the center of the
dendrite (dispersoides) and the edge
(constituents) can be summarized by an
exchange of solutes ( ji )
Assumption: the concentrations near the
constituents are linked to the concentrations
in the dendrite arm by a simple proportional
law
Arm
iii
Arm
i
Edge
iiiii c
DccDcDj
Dispersoides Constituents
= dentride armspacing
ji
Arm
i
32
i c)1(3
4t)1(4j
Arm
ii
Edge
i c)1(c
Solute Flux:
Mass Conservation:tcDac Arm
iii
Arm
i
ai: empirical value, kept constant
during the simulation
Change of concentration in the
dendrite arm:
CiEdge
CiArm
.Nom
i
Arm
i cc
31 GTT Workshop / June 20-22, 2007
Efficiency (I)
temp [C]
0
100
200
300
400
500
600
700
- 3 6 9 12 15 18 21 24 27 30 33
time [h]
Rho [m^-2]
1.00E+09
1.00E+10
1.00E+11
1.00E+12
1.00E+13
1.00E+14
1.00E+15
1.00E+16
1.00E+17
- 3 6 9 12 15 18 21 24 27 30 33
time [h]
Elt. c [%w.]
Al 90.10
Cr 0.80
Cu 4.00
Fe 0.30
Mg 1.50
Mn 1.00
Si 0.80
Ti 1.50