Acta Materialia 55 (2007) 2539–2553
www.actamat-journals.com
Modeling of precipitate-free zone formed upon homogenization
in a multi-component alloy
Ch.-A. Gandin
a,b,*
,
A. Jacot
c,d
a
b
d
CEMEF UMR CNRS-ENSMP 7635, Ecole des Mines, BP207, 06904 Sophia Antipolis, France
LSG2M UMR CNRS-INPL-UHP 7584, Ecole des Mines, Parc de Saurupt, 54042 Nancy, France
c
LSMX, MX-G, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
CALCOM ESI SA, PSE, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Received 29 June 2006; received in revised form 20 November 2006; accepted 20 November 2006
Available online 8 February 2007
Abstract
A comprehensive model is presented for the simulation of microstructure evolution during industrial solidification and homogenization processing of aluminum alloys. The model combines on the one hand microsegregation due to long-range diffusion during solidification and subsequent heat treatment with, on the other hand, precipitation in the primary Al phase. The thermodynamic data are
directly obtained from a CALPHAD (CALculation of PHAse Diagrams) approach to thermodynamic equilibrium in multicomponent
systems. The model is applied to the prediction of structure and segregation evolutions in a 3003 aluminum alloy for typical industrial
solidification and homogenization sequences. It is shown that: (i) accounting for the nucleation undercooling of the eutectic/peritectic
structures solidifying from the melt is essential to retrieval of the measured volume fractions of intergranular precipitates; (ii) calculations
of intragranular precipitation are generally not applicable if long-range diffusion is neglected; (iii) the precipitate-free zone can be
quantitatively predicted only based on the coupling between intergranular and intragranular precipitation calculations.
2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Precipitate-free zone; Homogenization; Precipitation; Modeling; Aluminum alloy
1. Introduction
Precipitate-free zones (PFZ) form during the heat treatment of metallic alloys [1,2]. A PFZ consists of a region of
the intragranular primary Al matrix phase with no or very
limited amounts of precipitates. It is delimited by a precipitate-rich zone of the same intragranular primary Al matrix
phase on one side and intergranular precipitates at the
grain boundary on the other side. Fig. 1 shows a typical
PFZ observed in a homogenized 3003 aluminum alloy.
The microstructure is composed of several primary grains
of the Al-fcc phase, also hereafter referred to as the matrix
phase, surrounded by intergranular precipitates and containing intragranular precipitates in the core. The role of
*
Corresponding author.
E-mail address: [email protected] (Ch.-A. Gandin).
PFZ on mechanical properties is clearly attested in the literature. Its effect has been demonstrated for the fracture
toughness of aluminum alloys, more specifically for Al–Li
alloys (2000 series) [3] and Al–Mg–Zn alloys (7000 series)
[4]. Similarly, the yield strength, the ultimate tensile
strength and the plastic strain to fracture of a c 0 -strengthened nickel-base superalloy have been linked to the width
of the PFZ [5,6].
As reviewed by Maldonado and Nembach [5], several
explanations are given in the literature for the origin of
PFZ. The main mechanism that prevails is the competition
between intragranular and intergranular precipitations.
Indeed, both transformations require the diffusion of the
same chemical solute elements of the supersaturated matrix
to take place. The width of the PFZ thus depends on the
diffusion of the solute elements toward the intergranular
precipitates compared to the potency of the intragranular
1359-6454/$30.00 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2006.11.047
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Ch.-A. Gandin, A. Jacot / Acta Materialia 55 (2007) 2539–2553
Compositions
Volume fractions
Size distributions
Mean radius
Densities
r
O
alpha (Al(Fe,Mn)Si)
intragranular precipitates
Precipitate Free Zone (PFZ)
10 µm
Al(Fe,Mn)6 ) and
Al66Mn
Mn ((A
alpha (Al(Fe,Mn)Si)
intergranular precipitates
Fig. 1. Homogenized microstructure of a 3003 aluminum alloy. A PFZ forms as a result of solutal interactions between the intergranular Al6Mn and
alpha precipitates and the intragranular alpha precipitates [28]. Note the difference in size between the intergranular alpha precipitates compared to the
fine intragranular alpha precipitates. The intergranular alpha precipitates form as a result of the transformation of the coarse intergranular Al6Mn
precipitates produced during solidification, while the intragranular alpha precipitates nucleate and grow in the primary Al grains.
precipitates to nucleate and grow in the primary Al matrix
phase.
Several models have been proposed describing the development of the PFZ during ageing [7–9] and homogenization treatments [10,11]. While being able to predict
comprehensively the width of the PFZ, these models are
based on one or several of the following approximations:
binary alloy; stoichiometric composition of the precipitates; simplified phase diagrams; isothermal heat treatment;
and analytical solutions of the mathematical problem of
long-range diffusion in the matrix. The initial conditions
for microstructure calculations during homogenization
heat treatments are also generally approximated by simplified solidification paths.
In the present contribution we develop, a coupled precipitation–homogenization model which does not use any
of these approximations. A particle size distribution
(PSD) method for the description of precipitation [12,13]
has been coupled with a pseudo-front tracking (PFT)
method for the prediction of segregation profiles and
intergranular precipitates during solidification and heat
treatment [14,15]. Both the PFT and PSD methods use
direct coupling with the equilibrium computing program
Thermo-Calc [16] and an appropriate thermodynamic
database [17]. As a result of the interaction between precipitation and long-range diffusion, the PFZ is predicted. The
model thus represents a necessary elementary brick of a
through process modeling approach that aims to predict
the final properties of a metal part by integrating the role
of each of the individual thermo-mechanical forming steps
on microstructure evolution [18]. Comparison of the model
is performed with measured data for a 3003 aluminum
alloy solidified as an extrusion billet using the direct-chill
continuous casting process and then homogenized at
600 C [19,20].
2. Modeling
2.1. Solidification and homogenization modeling: the PFT
method
The evolution of microstructure during solidification is
described with the PFT method [14,15]. The PFT method
permits the calculation of the evolution of solid/liquid
interfaces that are governed by anisotropic interfacial energies and the diffusion of several solute species. Growth of
the primary Al phase (fcc) from the liquid (l) is described
by solving the diffusion equations in both phases for each
solute element:
@xmi
¼ r ½Dmi rxmi with 8i 2 ½1; n and m ¼ fcc;l
@t
ð1Þ
where n is the number of alloying elements in the multicomponent system, xmi the concentration of element i in
phase m, and Dmi is the diffusion coefficient.
The position and velocity of the interface being part of
the problem, a solute balance has to be formulated at the
fcc/l interface:
fcc l
l Dfcc
i ½rxi n þ Di ½rxi n
¼ ðxli xfcc
i Þv n
8i 2 ½1; n
ð2Þ
where v* is the interface velocity and n* is the normal vector
to the interface pointing towards the liquid. The superscript
‘‘*’’ denotes quantities taken at the curved fcc/l interface.
Ch.-A. Gandin, A. Jacot / Acta Materialia 55 (2007) 2539–2553
The concentrations xli and xfcc
in each phase are deduced
i
from the phase diagram taking into account the curvature
of the interface.
The solution of this problem is obtained with an explicit
finite volume method using a fixed grid. The displacement
of the interface is handled through state transitions of the
finite volume cells, from l to (fcc + l) and from (fcc + l)
to fcc. Further details of the method can be found in
Ref. [14]. The calculations are directly coupled with the
phase diagram software Thermo-Calc [16] and the Al-Data
database [17] using the optimized coupling scheme
described in Ref. [14].
As the liquid becomes undercooled for another solid
phase, prediction of phase transformations in the intergranular regions is started out. In this approach, the intergranular regions are considered as a mixture of liquid and
solid phases. The following assumptions are made: (i) the
composition of the intergranular liquid is uniform when
the first intergranular precipitation phase starts to form;
(ii) the intergranular region is locally in thermodynamic
equilibrium and all phases have uniform concentrations;
(iii) back diffusion affects the intergranular region in an
uniform manner.
A solute balance is performed over the intergranular
region, Xm, for all the solute elements:
Z
Z
Z
@xmi
fcc
dX Ji n dC ¼
ðxmi xfcc
i Þv n dC
@t
Cfcc=m
Xm
Cfcc=m
8i 2 ½1; n
ð3Þ
fcc
fcc where Jfcc
i ¼ Di ½rxi n is the back-diffusion flux of
m
element i, xi the average concentration of element i in
the intergranular region, Cfcc/m the contour of the fcc/m
interface which separates the primary Al phase and the
intergranular region, v* the velocity of the fcc/m interface,
and n* is the normal vector
to Cfcc/m pointing toward Xm.
R
fcc
Introducing Ufcc
¼
J
n dC and discretizing (3),
i
Cfcc=m i
we obtain:
m
m
fcc
Ufcc
i dt ¼ V m dxi þ dV m ðxi xi Þ
8i 2 ½1; n
ð4Þ
where Vm is the volume of Xm and the symbol d expresses
small increments.
The mixture composition can be expressed as:
xmi ¼
p
X
gm xmi
8i 2 ½1; n
ð5Þ
m¼1
where p is the number of phases in the mixture, xmi the equilibrium concentration of element i in phase m, and gm is the
volume fraction of phase m in the intergranular region.
Introducing Eq. (5) into (4), one obtains:
!
p
p
X
X
fcc
dgm xmi þ
gm dxmi
Ui dt ¼ V m
m¼1
þ dV m
m¼1
p
X
m¼1
!
gm xmi
!
xfcc
i
8i 2 ½1; n
ð6Þ
2541
Assuming that thermodynamic equilibrium is satisfied in
the intergranular mixture, the temperature, T, can be related to the phase diagram:
fcc
8m 2 ½1; p with 8m 6¼ fcc
ð7Þ
T ¼ T m=fcc xfcc
1 ; . . . ; xn
fcc
where T m=fcc ðxfcc
1 ; . . . ; xn Þ expresses the solidus (or solvus)
temperature as a function of the concentration in the primary Al matrix phase. This function is evaluated with the
phase diagram software Thermo-Calc [16] and the thermodynamic database Al-Data [17].
The concentrations of the other phases are given by the
tie-lines:
m=fcc
xmi ¼ k i
fcc
fcc
ðxfcc
1 ; . . . ; xn Þxi
8m 6¼ fcc and 8i 2 ½1; n
8m 2 ½1; p with
ð8Þ
m=fcc
ki
where the
are partition coefficients defined with respect to the primary Al matrix phase (fcc), which are also
obtained from Thermo-Calc and Al-Data.
The back-diffusion contribution, Ufcc
i , results form the
resolution of Eq. (1) for m = fcc only. The calculation is
made with the same explicit finite volume method as for
the primary Al phase calculation.
If the thermal history is known, the evolution of the system can be described by solving Eqs. (6)–(8) which form a
set of n + (n + 1)(p 1) equations and comprises np + p
+ 1 unknowns: the fdxmi g, the {dgm} and dVm. By appending
the following equations to the system, the problem
becomes closed:
Xp
dgm ¼ 0
ð9Þ
m¼1
k¼
dV m
dV m þ dgfcc V m
ð10Þ
Eq. (10) expresses the proportion of the primary Al matrix
phase formed on the fcc/m interface, which is directly related to the variation of the mixture volume, dVm, with respect to the total amount (fcc formed on fcc/m interface
and within Xm). By selecting an appropriate value for the
eutectic distribution parameter, k, it is possible to distinguish the behaviors of divorced and coupled eutectics. If
k is set to 1 (i.e., dgfcc = 0), eutectic Al will form only on
top of primary Al (divorced eutectics), whereas for k = 0
(dVm = 0) it will be distributed in the intergranular region
and the primary Al phase boundaries will remain stationary (coupled eutectics).
Each secondary phase, m, is attributed a nucleation undercooling, DT mnucl , which is a parameter of the model. The
phase m is introduced in the calculation as the following
condition is satisfied:
T þ DT mnucl 6 T m=fcc ðxm1 ; . . . ; xmn Þ
8m 2 ½1; p with 8m 6¼ fcc
ð11Þ
When the volume fraction of a phase reaches 0, the phase is
withdrawn from the calculation.
After the liquid has disappeared from the intergranular
phase mixture, the calculation is continued so that evolution of the intergranular precipitates during homogenization
2542
Ch.-A. Gandin, A. Jacot / Acta Materialia 55 (2007) 2539–2553
can be predicted. The effect of intragranular diffusion upon
growth or dissolution of intergranular phases is taken into
account with the same formalism as for solidification. This
process is, however, strongly influenced by precipitation
taking place in the primary Al phase, i.e., intragranular
precipitation. A precipitation model is therefore coupled
to the homogenization calculation.
2.2. Modeling of intragranular precipitation: the PSD
method
The PSD method [12,13] is used to track the evolution of
the intragranular precipitates formed in the supersaturated
primary Al matrix phase. It is an extension of models previously proposed in the literature [21,22]. Its formulation is
adapted to handle non-stoichiometric precipitates formed
in multi-component alloys. It is also made compatible for
coupling with the PFT model [14,15] by considering mass
balances for an open system. Indeed, long-range diffusion
simulated by the PFT model does modify the local average
composition with time, thus changing the precipitation
kinetics.
The conservation equation for the density of m-phase
precipitates, Nm, growing in a supersaturated primary Al
matrix phase (fcc) can be written as:
@N m
¼ r ðN m vmþ nmþ Þ þ S m 8m 2 ½1; q
ð12Þ
@t
where vm+ is the growth rate of the precipitates of radius rm
at time t, nm+ is the normal vector to the m/fcc interface
pointing towards the primary Al matrix phase and q is
the number of precipitation phases formed in the primary
Al matrix phase. The superscript ‘‘+’’ denotes quantities
taken at the m/fcc interface. Assuming the precipitates remain spherical and are surrounded by a steady-state diffusion field, the growth rate for the m-phase precipitates, Nm,
is the solution of the second Fick’s law given by [23]:
mm ¼
fccþ
Dfcc
xfcc
i
i xi
mþ
rm xi xfccþ
i
8m 2 ½1; p;
8i 2 ½1; n
ð13Þ
where xfcc
i is the average composition of element i in the matrix. A local equilibrium at the interface between the matrix
and the precipitates is assumed. A thermodynamic calculation is therefore used to determine the compositions of element i at the (m/fcc)-interface in the m-phase precipitates
and in the matrix, respectively xmþ
and xfccþ
. The diffusion
i
i
coefficient of solute element i in the primary Al matrix
phase, Dfcc
i , is computed following an Arrhenius law.
The source term entering Eq. (12) for the m-phase precipitates, Sm, is modeled by an heterogeneous nucleation law
given by [24–27]:
DGmhom f ðhm Þ
S m ¼ ðN mmax N mtot ÞZb exp 8m 2 ½1; q
kBT
ð14Þ
N mmax
is the density of heterogeneous sites available
where
for the nucleation of m-phase precipitates and N mtot is the ac-
tual total density of precipitates. Coefficient b accounts for
the rate at which solute atoms from the matrix can join the
nucleus. For a binary alloy, its expression is given by b ¼
2
4
4pðrm Þ Dfcc xfcc =ðkfcc Þ where kfcc is the diffusion distance
in the primary Al matrix phase. For a multi-component alloy it is chosen to consider the element with a combination
of a slow diffusion rate in the matrix and a low composition
as the limiting factor for increasing the size of the nucleus.
fcc
The minimum product Dfcc
i xi over all elements i in the primary Al matrix phase is thus used and one can write
2
fcc 4
fcc
b ¼ 4pðrm Þ MinðDfcc
i xi =ðk Þ . The critical energy barrier
for the formation of new m-phase precipitates of critical ra
dius rm ¼ 2rm=fcc V m =DGmn is given by DGmhom ¼ ð4=3Þprm=fcc
2
m/fcc
ðrm Þ , where r
is the interfacial energy of the m/fcc interface, Vm is the molar volume of the m-phase and DGmn is the
driving force for nucleation of the m-phase precipitates in
the primary Al matrix phase. The wetting function is given
by the relationship f(hm) = (1/2)(2 + cos hm) (1 cos hm)2,
where hm is the wetting angle of a m-phase nucleus with its
heterogeneous nucleation site [27]. The Zeldovitch’s factor
accounts for the fluctuation of the size of the nucleus due to
the emission of solute atoms from the nucleus back into the
matrix phase. Its estimation
is computed
using the
relationpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ship Z ¼ ðDGmn Þ2 = 8pV m ðrm=fcc Þ3=2 N Av k B Tf ðhm Þ , with T
the temperature, kB Boltzmann’s constant and NAv Avogadro’s number.
A balance equation for the composition of solute element i can be written as:
XX4
m
r
3
3
fcc
pðrm Þ N m ðxmi xfcc
8m 2 ½1; q; 8i 2 ½1; n
i Þ ¼ xi xi
ð15Þ
where xi is the average composition of i. The precipitate
composition xmi is assumed uniform. It is simply given by
the interface composition xmþ
i . It can vary with temperature
since the precipitates are not stoichiometric. Also noticeable
is the possibility for the average composition of element i,
xi, to vary with time. This is useful when considering the
coupling with the long-range diffusion computed by the
PFT model as explained below. Although it is not used in
the present contribution, the PSD method was developed
for the concomitant interaction of several families of precipitates in the same primary Al matrix phase. This is made visible with the summation of m-phase precipitates in Eq. (15).
Several families of precipitates could either correspond to
several m-phases precipitating simultaneously in the same
matrix, or to one phase with several nucleation parameters
(e.g., for application to the nucleation of a single precipitating phase on several families of heterogeneous sites), or also
to a combination of both.
The driving force for nucleation of the m-phase precipitates in the primary Al matrix phase is computed with an
ideal solution thermodynamic approximation:
fcc n
X
x
DGmn ¼ Rg T
xmi 1 ln fcci 1
8m 2 ½1; q
ð16Þ
xi
i¼1
Ch.-A. Gandin, A. Jacot / Acta Materialia 55 (2007) 2539–2553
i¼1
Loop on time-steps
Loop on all cells
Fluxes in the primary Al phase
New local compositions in the primary Al phase
PSD model
Interdendritic phases
PFT model
Equilibrium compositions of element i at the (m/fcc)interface, respectively in the primary Al matrix phase and
1
in the m-phase precipitates, entering Eq. (16), xfcc
and
i
m1
xi , are given by equilibrium calculations for known values of the temperature, T, and the average composition
of elements i, xi. The Gibbs–Thomson effect is accounted
for based on a modification of the solubility product, Km,
with respect to its equilibrium value, Km 1, also computed
using the equilibrium phase diagram compositions:
2 rm=m V m
K m ¼ K m 1 exp m
with
r Rg T
n
n
Y
Y
xmþ
1 xmi 1
i and K m 1 ¼ ½xfcc
8m 2 ½1; q
K m ¼ ½xfccþ
i
i
2543
Loop on the primary
Nucleation driving force
Al phase cells
Iteration loop
i¼1
Precipitate size distribution
ð17Þ
and xmþ
are the composition of elements i at
where xfccþ
i
i
the (m/fcc)-interface in the primary Al matrix phase and
in the m-phase precipitates, respectively. Thermo-Calc
[16] and the Al-Data database [17] are used for the calcu1
lations of the equilibrium compositions xfcc
and xmi 1
i
entering both Eqs. (16) and (17).
2.3. Coupling of the PFT and PSD methods
Fig. 2 shows a schematic flow chart of the coupling
between the PFT and PSD models. During a time-step,
the variations of the matrix composition due to long-range
diffusion are calculated with the PFT model by solving Eq.
(1) for the primary Al maxtrix phase (m = fcc) and sent to
the PSD model. Nucleation, growth and coarsening of
the intragranular precipitates are predicted by the PSD
model by performing separate calculations for each cell
of the PFT finite volume grid that is located in the primary
Al phase region of the calculation domain. The primary Al
matrix compositions in the finite volume cells, xfcc
i , are
modified by the PSD and fed back to the PFT model.
The time-step is determined so as to satisfy the stability criterion of the explicit time integration scheme used by the
PFT model. The PSD model also makes use of a finite volume method to solve Eq. (12), but with an implicit time
integration scheme. In addition, iteration procedures are
required to ensure convergence of the average matrix compositions and the nucleation rate in the PSD model as
shown in Fig. 2.
Simulation results of the coupled PFT–PSD model consist of time evolution of profiles in the primary Al phase for
the compositions, size distributions and volume fractions
of precipitates, as well as volume fractions of intergranular
phases. As a result, the width of a possible PFZ can be calculated. The position of the PFZ is arbitrarily determined
as the location in the primary Al phase where the volume
fraction of the intragranular precipitates falls below 10%
of the same quantity averaged over the entire primary Al
phase. The calculations were performed in one space
dimension using a spherical calculation domain of
8.6 lm, which corresponds to half of the secondary arm
New local compositions in the primary Al phase
Fig. 2. Partial flow chart of the model showing the principle of the
coupling between the PFT and PSD methods integrated in the timestepping algorithm.
spacing reported in Ref. [20]. Axes sketched in Fig. 1
illustrate the coordinate system used as well as the typical
profiles deduced from simulations.
3. Experimental
Li and Arnberg [19,20] and Dehmas [28] provide identification of the phases defining the PFZ formed upon the
homogenization heat treatment of a 3003 aluminum alloy
of composition Al-0.58 wt.% Fe-1.15 wt.% Mn-0.20 wt.%
Si-0.08 wt.% Cu. The solidification grain structure is made
up of globular dendrites of the primary Al matrix phase
which are bounded by Al6(Mn,Fe) and a-Al(Fe,Mn)–Si
intergranular precipitates. For simplicity, hereafter the two
phases will be referred to as alpha and Al6Mn, respectively.
During the heating stage of the homogenization treatment,
alpha precipitates appear in the intragranular regions as a
result of precipitation from the supersaturated primary Al
matrix phase. Concomitantly, a eutectoid reaction partly
transforms the intergranular Al6Mn precipitates into intergranular alpha precipitates and intergranular eutectoid Al.
This reaction requires the diffusion of solute species from
the primary Al matrix phase to the intergranular region. Similar observations were previously reported by Alexander and
Greer in an Al-0.5 wt.% Fe-1.0 wt.% Mn-0.2 wt.% Si aluminum alloy [29]. Li and Arnberg provide measured data that
will be used hereafter [19,20].
4. Results
4.1. Calculation conditions
The model was used to describe the evolution of the
microstructure in a 3003 aluminum alloy having the same
composition as the alloy characterized by Li and Arnberg
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Ch.-A. Gandin, A. Jacot / Acta Materialia 55 (2007) 2539–2553
[19,20] (composition in at.%: 0.283% Fe, 0.570% Mn, and
0.194% Si), yet neglecting the influence of Cu. The influence of interface curvature upon thermodynamic equilibrium at the primary Al/liquid interface was neglected in
the PFT calculations. The cooling rate was assumed to be
0.1 C/s during solidification and between 1.5 and 3 C/s
after the liquid had entirely disappeared. The eutectic distribution parameter, k, was set to 1, i.e., a fully divorced
eutectic structure was assumed.
For the homogenization/precipitation calculations, the
as-cast material is ramped up to 600 C at 50 C/h, maintained for 7 h and cooled down to room temperature at
50 C/h, following the experimental conditions of Li and
Arnberg [19]. Diffusion coefficients are taken from Ref.
[30]. Table 1 summarizes the three sets of simulations presented in this contribution, while Table 2 lists the values of
all other parameters. The parameters changed are essentially the nucleation temperature of the intergranular
Al6Mn and alpha precipitates. Before undertaking coupled
simulations (AwP and CwP), calculations using only the
PFT model (A, B and C) and only the PSD model (P) were
carried out independently. This allows for a better understanding of the individual behavior of each model.
4.2. Simulation of solidification microstructure
The evolution of the phase fractions during solidification is presented in Fig. 3. In calculation A (black curves),
both intragranular Al6Mn (dotted lines) and alpha (dashed
lines) were considered for the computation of thermodynamic equilibrium in the intergranular region, assuming
no nucleation undercooling of these phases. In this case,
the intergranular precipitates obtained after solidification
and cooling down to room temperature are essentially of
the alpha phase, which is more stable than Al6Mn at low
temperature. This is clearly not realistic since observation
(full symbols in Fig. 3) indicates the opposite: an overall
volume fraction of intergranular precipitates of 2.9% in
the as-cast state, with a minor proportion of intergranular
alpha precipitates representing only 5% of the intergranular phases [19]. It is therefore expected that the conditions
for the nucleation of intergranular alpha precipitates are in
reality not met during solidification. To account for this
behavior, calculation B, in which the alpha phase is
excluded from thermodynamic equilibrium, was carried
out. As shown in Fig. 3, a final amount of 3.75% Al6Mn
is predicted with calculation B, which is substantially more
than the measured total amount of intergranular precipitates (2.9%). Calculations performed with cooling rates or
diffusion coefficients modified by one order of magnitude
or more did not allow coming close to 2.9%. It was concluded that nucleation of intergranular Al6Mn precipitates
is likely to occur long after the liquidus of the Al6Mn phase
has been reached. A series of calculations based on
adjusted nucleation temperatures for intergranular Al6Mn
and alpha precipitates was then carried out until the measured percentages were approximately retrieved. The
obtained nucleation temperatures are indicated in Table 1
(calculation C) and the results are displayed as thick grey
lines in Fig. 3.
The solidification paths obtained in calculations A, B
and C are shown in Fig. 4 only considering Fe and Mn,
i.e., slow diffusing elements. When no nucleation undercooling is considered (case A), the Mn content of the liquid
and primary Al matrix phases decrease upon cooling as
soon as the liquidus of the first of the intergranular precipitates is reached, i.e., Al6Mn. This is obviously due to the
formation of the Mn-rich Al6Mn intergranular precipitates. When a nucleation undercooling is introduced (case
C), the equilibrium Mn concentrations in liquid and primary Al keep increasing until Al6Mn intergranular precipitates are formed. The end of solidification temperature is
then considerably different for calculation C (639.5 C,
open circles) as compared with calculation A (646.4 C)
and B (642.8). In calculation C, the primary Al matrix
phase composition reaches 1.3 wt.% Mn, compared to only
0.5 wt.% Mn for calculations A and B. Primary Al is thus
more supersaturated in case C when the nucleation of the
Table 1
List of the simulations with values of the nucleation parameters used for the intergranular Al6Mn and alpha precipitates and the intragranular alpha
precipitates
Simulation
identifier
Nucleation temperature of
intergranular Al6Mn
precipitates (C)
Nucleation temperature of
intergranular alpha
precipitates (C)
Nucleation parameters of
intragranular alpha precipitates for
coupled PFT–PSD simulations
Nucleation parameters of
intragranular alpha precipitates for
uncoupled PFT–PSD simulations
A
AwP
655.25a
655.25a
646.45a
646.45a
–
B
C
CwP
655.25a
634.58
634.58
–
208.75
208.75
P
634.58
208.75
–
ralpha/fcc = 0.15 J m2
halpha = 45
3
21
N alpha
max ¼ 1:5 10 m
–
–
ralpha/fcc = 0.15 J m2
halpha = 45
3
21
N alpha
max ¼ 1:5 10 m
–
a
No nucleation undercooling.
–
–
ralpha/fcc = 0.15 J m2
halpha = 45
3
21
N alpha
max ¼ 1:5 10 m
Ch.-A. Gandin, A. Jacot / Acta Materialia 55 (2007) 2539–2553
2545
Table 2
Values of the parameters and physical properties used for the 3003 aluminum alloy
Description
Symbol
Value
Unit
Nominal composition
Fe
Mn
Si
0.58
1.15
0.2
wt.%
wt.%
wt.%
Diffusion coefficients
Dfcc
Fe
3.62 · 101 exp(214,000/RgT)
Dfcc
Min
Dfcc
Si
Cfcc
Gibbs–Thomson coefficient
exp(211,500/RgT)
m2 s1
1.38 · 10
5
exp(117,600/RgT)
m2 s1
Km
6
m3
9 · 10
V
fcc
Diffusion distance
1.35 · 10
1.5 · 107
alpha
Molar volume
m2 s1
2
2.86 · 10
k
10
m
PFT number of cell
PFT system size
40
8.6
lm
PSD number of classes
PSD system size
500
1
lm
All nucleation parameters are reported in Table 1.
Time [s]
300
200
1.6
100
0
C
1.0
0.80
0.60
Case A: Al6Mn
0.00
0.40
10-3
Case C: alpha
0
100
200
300
400
500
600
10-2
5˚ C
6 53
˚C
8˚ C
42.
at 6
4˚ C
46.
at 6
end
nd
Be
ath
0.01
65
path
Case A: alpha
0.02
Al6Mn
liquidus
is reached
liquid
Case C: Al6Mn
0.03
a
end
1.2
ry Al p
Mn concentration [wt%]
Case B: Al6Mn
Prima
Volume fraction of intergranular
precipitates [1]
0.04
C
9.5˚
t 63
1.4
Measured: Al6Mn ( ), alpha ( )
A
10-1
100
Fe concentration [wt%]
101
700
Temperature [˚C]
Fig. 3. Calculated evolutions of the volume fractions of Al6Mn (dotted
lines) and alpha (dashed lines) intergranular precipitates during the
solidification of a 3003 aluminum alloy. Measurements of the volume
fraction of intergranular phases are shown with solid symbols [20].
Mn-rich intergranular precipitates is delayed. As a result,
the material can experience more intragranular precipitation upon subsequent heat treatments.
4.3. Evolution of intergranular precipitates during
homogenization heat treatment
The influence of the as-cast state on homogenization
kinetics can be assessed by comparing calculations A and
C in Fig. 5a, which shows the time evolution of the temperature and of the volume fractions of intergranular phases
for the two calculations. The comparison shows that the
as-cast state has a considerable influence on the evolution
of the volume fraction during the heating stage of the
homogenization treatment. However, once the temperature
Fig. 4. Calculated solidification path (equilibrium Fe and Mn compositions in liquid and primary Al) for a 3003 aluminum alloy using
assumptions A, B and C for the nucleation of intergranular Al6Mn and
alpha precipitates (see Table 1). The open symbols indicate the compositions when the last liquid disappears.
plateau is reached, the same global equilibrium is obtained
and time evolutions are very similar.
The discussion hereafter will be focused on conditions
C, which are the most pertinent for comparisons with
respect to the experimental data, since they permit us to
start the simulation of the heat treatment from a realistic
as-cast state.
The evolution of intergranular precipitates during the
heat treatment can be decomposed into a series of four
stages which are labeled from I to IV in Fig. 5a. At the
onset of heating (T < 400 C, stage I in Fig. 5a), intergranular Al6Mn precipitates transform progressively into intergranular alpha precipitates by thermal activation of Si
diffusion. In the calculation, the kinetics of this phase
transformation is indeed governed by the diffusion of Si
in the primary Al phase, which is required to form the
2546
Ch.-A. Gandin, A. Jacot / Acta Materialia 55 (2007) 2539–2553
0.04
Al6Mn intergranular precipitates
alpha intergranular precipitates
Case A
Case C
700
I
II
III
IV
500
400
0.02
300
Temperature [˚C]
Volume fraction of intergranular
precipitates [1]
600
0.03
200
0.01
100
0.00
0
0
5
10
15
20
25
30
Time [h]
0.04
Case C
Case CwP
Al6Mn intergranular precipitates
alpha intergranular precipitates
700
I
II
III
IV
500
400
0.02
300
Temperature [˚C]
Volume fraction of intergranular
precipitates [1]
600
0.03
200
0.01
100
0.00
0
0
5
10
15
20
25
30
Time [h]
Fig. 5. Calculated time evolutions of the volume fraction of intergranular Al6Mn precipitates and intergranular alpha precipitates formed upon an
industrial homogenization heat treatment of a 3003 aluminum alloy (a) without precipitation of intragranular alpha precipitates and considering no
nucleation undercooling of the intergranular precipitates (case A, black lines) or adjusted nucleation undercooling of the intergranular precipitates (case C)
and (b) without (case C) or with precipitation of intragranular alpha precipitates (case CwP, black lines). The temperature history is also indicated (thin
plain line).
Si-rich alpha phase. Between 400 and 600 C (stage II in
Fig. 5a), intergranular alpha precipitates transform back
into intergranular Al6Mn precipitates. This evolution is
due to the increase of the Si solubility in the primary Al
matrix. The Fe and Mn resulting from the dissolution of
intergranular alpha precipitates cannot be accommodated
in the matrix and this leads to the formation of intergranular Al6Mn precipitates. Formation of intergranular
Al6Mn precipitates is also enhanced by incoming fluxes
of Fe and Mn from the matrix, which are due to the Mn
and Fe microsegregation profiles inherited from solidification. These interpretations are supported by the left
Ch.-A. Gandin, A. Jacot / Acta Materialia 55 (2007) 2539–2553
columns of Fig. 6 which display the solute profiles at different temperatures during heating.
When the 600 C homogenization plateau is reached, the
equilibrium volume fraction of alpha precipitates is
obtained (0.015) and remains constant since the Si solubility in the primary Al matrix no longer varies (stage III in
Fig. 5). Intergranular Al6Mn precipitates keep growing
due to long-range diffusion of Mn. Equilibrium
CaseCC
case
0.12
Case CwP
case CwP
0.12
20˚C - 400˚C
0.10
2547
0.10
20˚C - 300˚C
(b1)
0.08
450˚C
0.06
500˚C
Composition of Fe [wt%]
Composition of Fe [wt%]
(a1)
0.04
600˚C (start)
0.02
0.08
400˚C
0.06
0.04
600˚C (end)
0.02
600˚C (end)
600˚C (start)
450˚C
500˚C
0.00
0.00
20˚C-450˚C
1.00
1.00
Composition of Mn [wt%]
Composition of Mn [wt%]
(b2)
500˚C
(a2)
0.80
600˚C (start)
0.60
600˚C (end)
0.40
20˚C - 300˚C
400˚C
0.80
450˚C
600˚C (start)
0.60
500˚C
600˚C (end)
0.40
case CwP
case C
0.20
0.20
20˚C
0.20
20˚C
0.20
200˚C
200˚C
(b3)
300˚C
300˚C
Composition of Si [wt%]
Composition of Si [wt%]
(a3)
case C
0.10
600˚C (start, end)
1
0.00
2
3
4
600˚C (start, end)
500˚C
400˚C
450˚C
0
0.10
500˚C
400˚C
0.00
case CwP
5
Position [μm]
6
7
8
450˚C
0
1
2
3
4
5
6
7
8
Position [μm]
Fig. 6. Composition profiles of Fe (row1), Mn (row 2) and Si (row 3) in the primary Al matrix phase (fcc) at selected temperatures during heating and
homogenization, calculated without considering intragranular alpha precipitation (case C, left column) and for a coupled precipitation–homogenization
calculation (case CwP, right column). The profiles corresponding to the beginning and end of the isothermal plateau are labeled as ‘‘600 C (start)’’ and
‘‘600 C (end)’’, respectively.
Ch.-A. Gandin, A. Jacot / Acta Materialia 55 (2007) 2539–2553
Center
Periphery
Case P
700
0.02
600
500
400
0.01
300
200
100
0.00
0
5
10
15
20
Time [h]
25
30
0
Measured intergranular Al6Mn+alpha precipitates ( )
Measured intragranular alpha precipitates ( )
0.05
700
Al6Mn+alpha
precipitates
Case C
0.04
600
Case AwP
500
0.03 Case CwP
400
300
0.02
Case CwP
0.01
alpha
precipitates
Temperature [˚C]
The precipitation model was first used without coupling
with the long-range diffusion model considered in the PFT
model. In such a way the result of the PSD precipitation
model alone could be analyzed. In this approximation,
hereafter referred to as calculation P, the composition profiles of the as-cast state at room temperature (20 C) shown
in Fig. 6 are used as the average compositions, xi, entering
Eq. (15). These average composition profiles remain constant during the heat treatment applied, i.e., upon heating,
holding and cooling to room temperature. As a consequence, the system is closed locally with respect to solute
transfer at all positions along the composition profiles of
the primary Al matrix phase. It is also closed with respect
to the intergranular area. This is clearly an approximation
which is released in calculation CwP presented in the successive section. It must be stated that despite the average
compositions, xi, remaining constant, the compositions of
the precipitates, xalpha
, as well as the composition of the
i
matrix, xfcc
,
evolve
as precipitation/dissolution of the
i
intragranular alpha precipitates phase takes place. Eq.
(15) is the mass balance linking these compositions with
the size and density of the intragranular precipitates, ralpha
and Nalpha, respectively.
The calculation predicts the time evolution of the size
distribution and density of the intragranular alpha precipitates for a series of locations going from the centre
to the periphery of the grain. The volume fractions of
intragranular precipitates for the extreme locations (centre and periphery) are shown in Fig. 7a as a function of
time. Even though the calculated volume fractions at
these two locations are considerably different, they both
indicate substantial intragranular precipitation during
the first part of the heating ramp, followed by dissolution of the intragranular precipitates during the second
part. Once the isothermal plateau is reached, the volume
fraction of intragranular precipitates remains constant,
indicating that equilibrium has been reached. During
the final cooling stage (stage IV), reformation of intragranular alpha precipitates is predicted by the model.
To allow for comparison with the experimental data of
Li and Arnberg [19], (reported in Fig. 7a with symbols),
the results of calculations P were averaged over the
grain, taking into account the spherical morphology
assumption that was also used for the homogenization
Average
Volume fraction of precipitates [1]
4.4. Evolution of intragranular precipitates during
homogenization heat treatment
Measured intragranular alpha precipitates ( )
Temperature [˚C]
composition of Fe is almost reached in the core of the
grains at the beginning of the temperature plateau, while
it requires several hours for the Mn to be uniformly distributed in the primary Al matrix phase (see the two profiles at
600 C in Fig. 6a2). After about 2 h at 600 C, the equilibrium volume fraction of Al6Mn precipitates (0.026) is
reached. During cooling, the alpha phase forms at the
expense of the Al6Mn phase due to the decrease of Si solubility and limited diffusion of Mn (stage IV in Fig. 5).
Volume fraction of precipitates [1]
2548
200
100
Case AwP
0.00
0
5
10
15
20
Time [h]
25
30
0
Fig. 7. Calculated time evolutions of the volume fraction of intergranular
precipitates formed during the homogenization heat treatment of a 3003
aluminum alloy without (a) and with (b) coupling of precipitation and
homogenization calculations. Corresponding experimental data for the
volume fractions of intergranular precipitates [20] and the volume fraction
of intragranular alpha precipitates [19] are plotted with symbols.
calculation. The comparison shows that the volume fraction of the intragranular precipitates is overestimated by
the model. Also, the progressive intragranular precipitate
dissolution observed by Li and Arnberg during holding
is not predicted. This indicates that the uncoupled calculation seems to miss some significant aspects of the phase
transformations.
4.5. Coupled homogenization and precipitation calculations
For a better understanding of the interactions between
short- and long-range diffusions leading to the formation
of intragranular and intergranular alpha precipitates,
respectively, the microstructure evolution resulting from
coupling homogenization and precipitation (calculation
CwP, is detailed hereafter based on the transformation
stages I–IV defined previously.
Ch.-A. Gandin, A. Jacot / Acta Materialia 55 (2007) 2539–2553
4.5.1. Stage I
During heating up to 400 C (stage I in Fig. 5b), thermal
activation of Si diffusion in the primary Al phase allows for
intergranular Al6Mn precipitates to transform into intergranular alpha precipitates and for intragranular alpha
precipitates to nucleate. This is clearly visible in Fig. 8,
where the density and the average radius of the intragranular precipitates have been represented as a function of the
temperature. It can be seen that the intragranular precipitate density increases drastically between 300 and 400 C.
In the vicinity of the intergranular precipitates, the supersaturation is reduced due to the Fe and Mn microsegregation
profiles inherited from solidification and cooling, while the
Si composition is almost uniform due to the high value of
its Fourier number for the system considered (see solute
profiles in Fig. 6 at 20 C). As a consequence, the density
of intragranular precipitates is substantially lower in this
region. This effect is visible in Fig. 9, which shows profiles
of the volume fraction, density and average size of the
intragranular precipitates at different temperatures during
heating. As a result of limited nucleation in the depleted
outer region of the grains, a PFZ is initiated. This can be
seen in Fig. 10, where the size of the PFZ has been represented as a function of time. The intragranular precipitate
radii shown in Figs. 8 and 9, indicate that at 400 C during
heating, intragranular precipitate growth is not yet very
important, although the maximum of the number density
of the intragranular precipitates has already reached. As
a consequence, the matrix is still very rich in Fe and Mn
at 400 C (see Fig. 6b1 and Fig. 5b2). Comparatively, Si
is almost fully depleted (Fig. 6b3), thus limiting further
nucleation of the Si-rich intragranular alpha precipitates.
4.5.2. Stage II
Phase transformations taking place during stage II are
much more complex for calculation CwP than for calculation C. In CwP, between 400 and 500 C, rapid growth of
120
100
1200
80
900
60
600
40
300
0
250
20
350
450
550
Temperature [˚C]
650
Average radius of the precipitates [nm]
Density of the precipitates [μm-3]
1500
0
Fig. 8. Evolution upon heating of the density (dashed lines, case CwP)
and mean radius (plain lines, case CwP) of the intragranular alpha
precipitates. Corresponding experimental data are plotted with thinner
lines using triangles for the density and squares for the radius [19].
2549
intragranular alpha precipitates takes place (Fig. 7b). Near
460 C, simultaneous growth of intragranular alpha precipitates and dissolution of intergranular alpha precipitates
can be observed (see volume fraction of intergranular alpha
precipitates in Fig. 5b and intragranular alpha precipitates
in Fig. 7b at T = 460 C/time @ 9 h). This phenomenon is
due to the dependency of the Si solubility in the primary
Al matrix with respect to Fe and Mn concentrations, combined with the presence of Fe and Mn segregation profiles.
The regions close to the intergranular precipitates are poor
in Fe and Mn and the equilibrium concentration of Si in
the primary Al matrix is higher than in the core
(Fig. 6b3, 450 C). This leads to the dissolution of intergranular alpha precipitates, while, deeper in the primary
Al phase, intragranular precipitates continue to grow and
incorporate the Si released by the intergranular precipitates. The volume fraction of the intragranular precipitates
reaches its maximum about 2 lm from the grain boundary
(Fig. 9a1, 450 C), which also corresponds to the highest
Mn content along the profile (Fig. 6b2, 450 C). One can
note that the phenomenon of simultaneous growth and dissolution of a same phase obtained in this calculation is a
particularity of multi-component systems which could not
be observed in a binary alloy. Between 500 and 550 C,
the solubility of Mn, Fe and Si increases (see concentrations at the fcc/mixture interface at 500 and 600 C in
Fig. 6b1–6b3) and intragranular precipitates are partly dissolved (Figs. 7b and 9a1). Intergranular alpha precipitates
grow slightly at the expense of intergranular Al6Mn precipitates due to the release of Si by the dissolution of intragranular precipitates in the primary Al matrix (see
Fig. 5b for 10 h < time < 11 h). Above 550 C, dissolution
of both intragranular and intergranular alpha precipitates
is observed owing to a general increase of Si solubility in
the primary Al matrix (Figs. 5b and 7b). As in calculation
C, intergranular Al6Mn precipitates grow at the expense of
alpha precipitates.
4.5.3. Stage III
At the beginning of the temperature plateau, dissolution
of intergranular alpha precipitates continues due to longrange diffusion of Fe and Mn (see the four profiles at
600 C in Figs. 6b1 and 5b2). This leads to further growth
of intergranular Al6Mn precipitates (Fig. 5b) and dissolution of intragranular precipitates in their vicinity, thus
extending the PFZ (see Fig. 7).
4.5.4. Stage IV
During cooling, intergranular Al6Mn precipitates transform into intergranular alpha precipitates as in the uncoupled calculation. Between 600 and 500 C, intragranular
alpha precipitates grow slightly due the solubility decrease.
4.6. Comparisons with experimental data
The volume fractions of intergranular phases and precipitates obtained with the coupled calculation (CwP) can
2550
Ch.-A. Gandin, A. Jacot / Acta Materialia 55 (2007) 2539–2553
500˚C
600˚C
0.01
450˚C
400˚C
20˚C 200˚C 300˚C
0.00
1022
(a2)
1021
Density of precipitates [m-3]
(b1)
Volume fraction of precipitates [1]
(a1)
10
450˚C 400˚C
500˚C
20
600˚C
1019
300˚C
1018
0.02
400˚C 200˚C
300˚C
450˚C
500˚C 20˚C
0.01
600˚C
0.00
1020
Density of precipitates [m-3]
Volume fraction of precipitates [1]
0.02
1017
1016
1015
(b2)
20˚C
1019
200˚C 300˚C
400˚C
450˚C 500˚C 600˚C
1018
1017
1016
1015
200˚C
1014
180
(a3)
600˚C
60
Average radius of precipitates [nm]
Average radius of precipitates [nm]
1014
70
50
40
30
500˚C
20
450˚C
400˚C
10
20˚C 200˚C 300˚C
0
0
1
2
3 4 5 6
Position [μm]
(b3)
20˚C 200˚C 300˚C
450˚C
400˚C
500˚C
170
160
600˚C
150
140
130
120
110
7
8
0
1
2
3 4 5 6
Position [μm]
7
8
Fig. 9. Calculated evolutions of profiles of (1) volume fraction, (2) density and (3) mean radius for the intragranular alpha precipitates at selected
temperatures of the heating (a) and cooling (b) sequences of the homogenization treatment (case CwP).
be compared with the experimental results of Li and Arnberg [19,20] in Fig. 7b. One can note that the agreement
with the measured volume fractions of intergranular alpha
and Al6Mn precipitates is considerably better than for calculation P, where intragranular precipitation is not coupled
with long range diffusion (Fig. 7a). In particular, the volume fraction of the intragranular precipitates is better predicted. Furthermore, the progressive dissolution of
intragranular precipitates during holding is correctly captured. It was shown to be due to long-range diffusion of
Fe and Mn combined with the transformation of the intergranular alpha precipitates into intergranular Al6Mn pre-
cipitates. The Mn rejected upon dissolution of the
intragranular precipitates is transported toward the boundary of the grains and used for the growth of intergranular
Al6Mn precipitates. This effect is also visible in the Mn
profiles that are shown in Fig. 6b2 at the beginning and
end of the 600 C temperature plateau.
Calculation CwP can be compared with calculation
AwP where no undercooling of the intergranular precipitates was considered. The volume fraction of precipiates
obtained in calculation CwP (Fig. 7b) shows a considerably better agreement with the experimental data as compared with calculation AwP. This is clearly related to the
0.6
700
Case AwP
Case CwP
0.5
600
0.4
500
400
0.3
300
0.2
Temperature [˚C]
Volume fraction of Precipitate Free Zone [1]
Ch.-A. Gandin, A. Jacot / Acta Materialia 55 (2007) 2539–2553
200
0.1
100
0.0
0
0
5
10
15
20
25
Time [h]
Fig. 10. Calculated time evolution of the volume fraction of the PFZ
without (case AwP, grey line) and with (case CwP, black line) nucleation
undercooling of the intergranular precipitates. Corresponding experimental data are plotted using dots [19]. The temperature history is also
indicated (thin plain line).
better description of the as-cast state, which is fairly well
reproduced in calculation C (Fig. 3). Although the volume
fraction of intragranular precipitates is somewhat overesti-
2551
mated in calculation CwP, the overall precipitation kinetics is well captured and the deviations are fairly
systematic. This indicates that the physical mechanisms
seem to be properly taken into account. The same comment applies when comparing the time evolution of the
total volume fraction of intergranular precipitates in
Fig. 7b. Similarly, the volume fraction of PFZ in the
microstructure shown in Fig. 10, the distribution of intragranular precipitates upon heating and holding shown in
Fig. 11 and the electrical conductivities shown in Fig. 12
compare well with the measurements of Li and Arnberg
[20]. The agreement between experiment and simulation
can be considered as very satisfactory if one bears in mind
the complexity of the system and all the uncertainties associated with the physical properties and measurements. The
measurement of the average volume fraction of intragranular precipitates and its size distribution is a difficult task,
especially in presence of a PFZ, since large variations are
expected between the centre and the periphery of the
grains. In such a case, the analysis of a large number of
samples is required to obtain an acceptable experimental
error. The systematic difference between experimental
and calculated electrical conductivities shown in Fig. 12
could also be due to inaccurate coefficients linking the
compositions to the conductivity and/or to deviations
from the nominal chemistry.
30
350 ˚C
400 ˚C
500 ˚C
580 ˚C
Frequency [%]
60
50
600 ˚C, 0 h
600 ˚C, 1 h
600 ˚C, 4 h
600 ˚C, 7 h
25
Frequency [%]
70
40
30
20
15
10
20
5
10
0
0
0
20
40
60
Sizes [nm]
80
100
0
50
100
150
200
Sizes [nm]
250
300
30
350 ˚C
400 ˚C
500 ˚C
580 ˚C
70
50
Frequency [%]
Frequency [%]
60
600 ˚C, 0 h
600 ˚C, 1 h
600 ˚C, 4 h
600 ˚C, 7 h
25
40
30
20
15
10
20
5
10
0
0
0
20
40
60
Sizes [nm]
80
100
0
50
100
150
200
Sizes [nm]
250
300
Fig. 11. Calculated (top, case CwP) vs. measured (bottom) evolutions of the size distribution of the intragranular alpha precipitates upon (a) heating and
(b) holding [19].
Ch.-A. Gandin, A. Jacot / Acta Materialia 55 (2007) 2539–2553
700
Case C
Case CwP
24
600
22
500
400
20
300
200
18
Temperature [˚ C]
Electrical conductivity [m Ohm-1 mm-2]
2552
100
16
0
5
10
15
Time [h]
20
25
0
Fig. 12. Time evolution of the electrical conductivity without (case C, grey
lines) and with (case CwP, black lines) intragranular alpha precipitation in
the primary Al matrix phase. Corresponding experimental data are plotted
using crosses [19]. The temperature history is also indicated (thin plain
line). The relation for the calculation of electrical conductivity, r [19]: 1/
fcc
fcc
r ¼ 0:0267 þ 0:032wfcc
Fe þ 0:033wMn þ 0:0068wSi .
Besides being evidence of the importance of considering
intragranular precipitation and intergranular phase transformations as a whole, one of the major results of this analysis is the drastic effect of the solidification stage. The
nucleation temperature of the intergranular phases formed
as intergranular precipitates during solidification and cooling to room temperature is found to play a key role in the
determination of the time evolution of the microstructure
and PFZ formation.
5. Conclusions
A methodology has been proposed to couple solidification, homogenization and precipitation calculations within
a comprehensive simulation approach. The model was
applied to the prediction of microstructure formation in a
3003 aluminum alloy. The main findings can be summarized as follows.
Good agreement between calculated and experimental
volume fractions of intergranular precipitates for the
materials in the as-cast state could only be obtained
after having introduced a nucleation undercooling for
Al6(Mn,Fe) (Al6Mn) and assuming that conditions
for a-Al(Fe,Mn)–Si (alpha) intergranular precipitates
to nucleate in the intergranular regions are not met during solidification. This result indicates that the intergranular precipitates in 3xxx alloys are likely to be
formed under conditions that are far from thermodynamic equilibrium.
A good prediction of the amount of precipitates and size
of PFZ obtained after a homogenization heat treatment
is only possible if the as-cast state is correctly described.
This includes the large variations of the matrix concentrations between the centre and the periphery of the
grains which is an important aspect of the as-cast state.
This result, which was already pointed out in Ref. [18], is
not surprising since the as-cast state defines the initial
supersaturation of the primary Al matrix phase prior
homogenization.
The evolutions of intergranular and intragranular precipitates are closely coupled phenomena which cannot
be considered separately in a homogenization model.
Only coupled calculations succeeded in predicting the
measured amount of alpha precipitates. This is due to
the interaction between short- and long-range diffusion
of the same species required for the formation of intergranular and intragranular precipitates.
Sophisticated microstructure parameters such as the distribution of the intragranular alpha precipitates or the
time evolution of the intragranular composition profiles
can be quantitatively predicted by the model. As a result
of coupling between intergranular and intragranular
precipitation the formation of a PFZ can be described.
Currently, the description of intergranular precipitates is
limited to the type of phase and volume fractions. For the
extrusion processes following casting and homogenization,
the size distribution is a key parameter which should thus
be addressed in further investigations. One should also
mention that in its current form the coupled homogenization–precipitation model requires large computational
resources. Efforts are required to make the present model
more efficient and practical for industrial applications.
Acknowledgements
This research has been conducted as a continuation of
the VIRCAST project (Virtual Cast House, Growth Project, OFES Grant 99-00644-2). The authors would like to
thank the European Community and the Office Fédéral
de l’Education et de la Science, Bern, Switzerland, as well
as the companies Alcan (France), Alcan (Switzerland), Calcom-ESI (Switzerland), Corus (The Netherlands), Elkem
(Norway), Hydro (Norway) and Hydro (Germany) for
the financial support provided within the VIRCAST project. The authors also gratefully acknowledge Professor
L. Arnberg, NTNU, Trondheim, Norway for providing
the experimental data.
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