On the formation of intermetallics in rapidly solidifying Al–Fe–Si alloys

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On the formation of intermetallics in rapidly solidifying Al–Fe–Si alloys
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 159–166
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CALPHAD: Computer Coupling of Phase Diagrams and
Thermochemistry
journal homepage: www.elsevier.com/locate/calphad
On the formation of intermetallics in rapidly solidifying Al–Fe–Si alloys
Dmitri V. Malakhov a,∗ , Damon Panahi a , Mark Gallerneault b
a
Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4L7
b
Novelis Global Technology Centre, 945 Princess Street, Kingston, Ontario, Canada, K7L 5L9
article
info
Article history:
Received 24 November 2009
Received in revised form
19 January 2010
Accepted 19 January 2010
Available online 7 February 2010
Keywords:
Al–Fe–Si alloy
Rapid solidification
Deep supercooling
Driving force
Compositional similarity
abstract
It is known from previous experimental observations that a rapid solidification of Al–Fe and Al–Fe–Si
alloys results in the formation of various metastable phases. Despite attempts to explain why particular
intermetallics form and to accurately predict a sequence of their precipitation, a fully satisfactory and
prognostic explanation remains to be found. In this communication, it is conjectured that after the concept
of the driving forces for the onset of precipitation is adapted to take into account important differences
between the Al-rich FCC solution and all other solid phases, it can be used to identify intermetallics whose
formation is thermodynamically permissible for a given supercooling. If the composition of some of the
‘‘possible phases’’ is close to the composition of the remaining liquid, then the precipitation of these
particular phases is facilitated, because corresponding nucleation events do not require a long-range
diffusion, which might be slow in supercooled melts.
© 2010 Elsevier Ltd. All rights reserved.
1. Introduction
Most metals (Mg and Cu are noteworthy exceptions) exhibit a
low solubility in solid Al. All commercial wrought aluminum alloys contain at least some iron, whose growing concentration is
caused by an increasing use of recycled aluminum in manufacturing. It should be noted that recycling also increases the silicon concentration, because it is virtually impossible to prevent pieces of
Si-rich cast alloys from entering the wrought alloy production
route. Consequently, it is not surprising that recycled Al alloys will
inevitably show an increase in the amount of type of intermetallic
phases containing Fe and Si. Such intermetallics frequently differ
from those predicted via the Scheil mode of solidification and from
the thermodynamic properties of phases [1–5]; these ‘‘unexpected
phases’’ were considered metastable in the literature.
Although the nature of phases in an as-cast Al alloy undoubtedly
depends on its composition, a cooling rate can be a more influential
factor in defining the overall phase portrait. If slow freezing (e.g.
in a thick-wall graphite crucible) is employed, then one usually
observes intermetallics conforming to the Scheil formalism [6,7]
and the thermodynamic properties of phases. Let us consider, for
the sake of determinacy and simplicity, an alloy with 0.3 wt.% Fe
(for brevity, an alloy containing x wt.% Fe is designated as Al–xFe
everywhere below) and assume that it solidifies according to the
Scheil scheme. According to Fig. 1, Al13 M4 forming via a eutectic
∗
Corresponding author.
E-mail address: [email protected] (D.V. Malakhov).
0364-5916/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.calphad.2010.01.006
reaction (refer to sublattice models of phases in Tables 1 and 2;
notice that M means Fe and Mn, which usually reside on the same
sublattice) is the only intermetallic phase expected in the solidified
alloy. 1
In Fig. 2, which was constructed for the alloy with 0.3 wt.% Fe
and 0.2 wt.% Si (named, indeed, as Al–0.3Fe–0.2Si), the calculations
suggest that the presence of silicon does not drastically change the
prediction made for the binary Al–Fe case in the sense that Al13 M4
still predominates in the solidified alloy, which also contains small
fractions of α -AlFeSi and β -AlFeSi.
In both cases, the Scheil-based predictions unequivocally point
to Al13 M4 as the major intermetallic, but in reality it is seen only
in the interior (last-to-freeze) portions of an ingot produced via
direct-chill casting, i.e., within the region where the cooling rate
is lower. Near the ingot surface, Al13 M4 is not the major phase.
In a portion of the ingot where the cooling rate is high, other
intermetallics predominate. Among them, one often observes the
orthorhombic Al6 M, orthorhombic Alx Fe (5.6 ≤ x ≤ 5.8),
monoclinic Alx Fe (4.5 ≤ x ≤ 5.0) and body-centered tetragonal
Alm Fe (4.0 ≤ m ≤ 4.4). They all are referred to in the literature
as metastable, because they are not seen in the equilibrium
Al–Fe phase diagram. If solidification occurs with a very high
heat extraction rate (strip casting, impulse atomization and melt
sputtering, for example), then mainly metastable phases will be
observed.
1 All calculations were carried out with Thermo–Calc version R and COST 507
database [8].
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D.V. Malakhov et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 159–166
Fig. 1. Eutectic formation of Al13 M4 in the solidifying Al–0.3Fe alloy.
Table 1
Sublattice models and compositional limits of some intermetallic phases in the
Al–Fe system.
Phase
Sublattice model
xFe
Al11 M4
Al12 M
Al13 M4
Al4 M
Al6 M
Al11 (Fe, Mn)4
Al12 (Fe, Mn)1
Al0.6275 Fe0.235 (Al, Va)0.1375
Al4 (Fe, Mn)1
Al6 (Fe, Mn)1
Min
Max
0.2667
0.0769
0.235
0.2725
0.2
0.1429
Table 2
Sublattice models and compositional limits of some intermetallic phases in the
Al–Fe–Si system.
Phase
Sublattice model
xFe
Min
Al13 M4
α -AlFeSi
β -AlFeSi
δ -AlFeSi
γ -AlFeSi
τ1
τ3
Max
Min
on it here; it would be sufficient to delineate its main features.
For the sake of clarity and determinacy, let us consider a liquid
solution Al1−xLFe −xLSi FexLFe SixLSi , where xLFe and xLSi are mole fractions of
iron and silicon in it. Further, we assume that the thermodynamic
properties of the solution are known, i.e., that the molar Gibbs
energy GL (T , P , xFe , xSi ) and its partial derivatives are calculable.
Let us fix temperature and pressure and builda plane, g, tangent
to the molar Gibbs energy at the point xLFe , xLSi :
g (xFe , xSi ) = GL xLFe , xLSi +
xSi
Al0.6275 Fe0.2350 (Al, Si, Va)0.1375
0.235 0.2725
Al0.6612 Fe0.1900 Si0.0496 (Al, Si)0.0992
0.19
Al13 Fe3 Si3
0.15
Al0.55 Fe0.15 Si0.3
0.15
Al3 Fe1 Si1
0.2
Al2 Fe2 Si1
0.4
Al2 Fe1 Si1
0.25
Fig. 2. Accumulation of intermetallics in the solidifying Al–0.3 wt.%Fe–0.2 wt.%Si
alloy.
+
Max
0
0.1375
0.0496 0.1488
0.15
0.3
0.2
0.2
0.25
It is commonly found that by increasing the cooling rate, a
greater level of supercooling (i.e., temperature below the equilibrium solidification point) of the melt can be achieved. Consequently, instead of repetitively referring to rates of cooling or heat
extraction, one can summarize the essence of the experimental observations by stating that the greater the supercooling of the melt,
the easier it gives birth to metastable intermetallics. Within the
context of the problem in hand, the terms ‘‘supercooled liquid alloy’’ and ‘‘supersaturated liquid alloy’’ can be used interchangeably,
because they mean that the formation of an infinitesimal amount
of at least one solid phase is thermodynamically favorable, i.e., that
there exists a positive driving force for its formation from the melt.
If the problem is viewed from such an angle, then it can be approached by invoking the concept of the driving forces for the onset of precipitation originated by Miroshnichenko [9] and further
elaborated by Cahn [10] and Hillert [11].
∂ GL
∂ xSi
∂ GL
∂ xFe
xFe =xL
Fe
xSi =xL
Si
xFe − xLFe
xSi − xLSi .
xFe =xL
Fe
xSi =xL
Si
It can be shown that if at least a portion of the molar Gibbs energy of a solid phase is situated below this tangent plane, then
there is a positive driving force for its precipitation from the melt.
If the phase is stoichiometric, then the driving force is uniquely defined. If the phase is a solution, then there are an infinite number
of its compositions for which the driving force is positive. A particular composition corresponding to the maximal driving force can
be found by employing the well-known parallel tangent construction [11]. If temperature exceeds the liquidus temperature, then all
driving forces for the onset of precipitation are, of course, negative.
If the liquid phase is cooled below liquidus, then depending upon
supercooling attained, one or several solid phases would have positive driving forces. In the latter case, a phase with the maximal
driving force can be identified. Then it can be postulated that this
particular phase possesses the highest nucleation potential2 and
thus has the greatest likelihood to nucleate for the given supersaturation. An example of how this scheme can be applied to predict
the first-forming intermetallic phase via a diffusion-controlled reaction at the interface between Cu and binary eutectic solders is
given in [13].
What does this approach predict for the aluminum alloy with
0.3 wt.% Fe and 0.2 wt.% Si? According to the construct in Fig. 3,
2. A direct application of the driving forces concept
Since this conception and its applications were discussed in
great details by Purdy et al. [12], there is no need to elaborate
2 In order to avoid a too frequent usage of ‘‘driving forces’’ and ‘‘driving forces
for the onset of nucleation’’, ‘‘nucleation potential’’ is also used. In this work, these
terms are considered as equivalent.
D.V. Malakhov et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 159–166
Fig. 3. Driving forces for the onset of precipitation of solid phases from the
Al–0.3Fe–0.2Si alloy.
FCC is the only solid phase, which may precipitate in a wide temperature region stretching from 659 °C (the liquidus temperature)
to 508 °C. At 508 °C, the driving force for the formation of Al13 M4
becomes positive, but its nucleation potential will exceed that of
FCC only if the melt is supercooled below 367 °C, i.e., 292 degrees
below liquidus! If the liquid is supercooled below 146 °C, which is
513 degrees below liquidus, then α –AlFeSi will have the maximal
driving force.
Such calculations, while providing a consistent rationale are
largely irrelevant, because such enormous supercoolings can
never be achieved in current commercial processing practices. In
practice, it has been firmly established that if the composition of
the melt falls into the primary FCC region (this condition is satisfied
for an overwhelming majority of wrought Al alloys), then only
small, 2–4 °C, supercooling is needed to trigger the formation of
this phase. This happens, because an abundant presence of suitable
sites (small oxide particles, grain refiners, etc.) inherited from
previous processing chains facilitate heterogeneous nucleation of
the primary phase. It is worth emphasizing that the formation
of the Al-rich solid solution, whose composition is very close
to the initial composition of the melt, is not limited by a longrange diffusion especially in view of the fact that growing FCC
dendrites can dynamically trap solutes. A non-faceted interface
between α -Al and liquid favors both nucleation and a subsequent
growth of the FCC phase. It should also be noted that the growth
is aided by the high thermal conductivity of the Al-rich solid
solution, which allows the system to easily dissipate the latent
heat of solidification. These characteristics distinguish the primary
aluminum from intermetallic phases. Such differences suggest that
the melt is never significantly supersaturated with respect to FCC,
i.e., that the driving force for its formation is always close to zero
during solidification provided that the rate of heat extraction is not
extraordinary high.
3. Coupling the driving forces concept and the Scheil solidification mode
Based on the arguments presented in the end of the previous
section, it is reasonable to assume that the accumulation of the
FCC phase occurs according to the Scheil formalism. A rejection
161
Fig. 4. Driving forces for the onset of precipitation of intermetallics from the
Al–0.3Fe melt when only liquid and FCC phases are in equilibrium along Scheil’s
path.
of solute(s) from growing α -Al dendrites into the remaining liquid is accompanied by a decrease of melt temperature, which will
eventually lead to a positive driving force for the start of precipitation of an intermetallic phase. Since all (or almost all) sites for the
heterogeneous nucleation have been already consumed by FCC nuclei, and since surfaces of rapidly growing FCC dendrites can hardly
be deemed as a source of new sites, the appearance of this phase
originates from homogeneous nucleation. It is known that pure liquid metals can easily be supercooled by tens or even hundreds of
degrees below their melting points if heterogeneous nucleation is
avoided [14]. It can be argued that a degree of supercooling triggering a noticeable homogeneous nucleation of ordered intermetallic
phases cannot be less, because, firstly, their composition is usually quite different from that of the parent liquid phase. Secondly,
an atomically smooth interface between a faceted sub-critical intermetallic nucleus and a surrounding melt has a low accommodation factor for the liquid atoms [15]. A wide range of attainable
supercoolings implies that various phases may possess the maximal driving force, i.e. that a phase with the greatest nucleation potential at a certain temperature would not necessarily predominate
if temperature changes.
Merging the driving forces concept and the Scheil solidification
path leads to the following recipe regarding the formation of
intermetallic phases in solidifying aluminum alloys:
1. assume that a very low supersaturation is needed for the
nucleation of the FCC solution and that while growing it is
always in a local equilibrium with the liquid phase;
2. do not allow intermetallic phases to precipitate from the melt;
3. follow the Scheil solidification path and attain supercooling,
which is deemed reasonable for the given casting conditions;
4. for the chosen temperature, use concentrations of solutes in
the remaining liquid (which will undoubtedly be quite different
from the initial concentrations) and then compute the molar
driving forces for the formation of all intermetallics (if none is
positive, then supersaturation must be increased);
5. identify a phase with the maximal driving force and declare it as
the most likely to form from the melt for the given supercooling
and, hence, to be seen in the solidified substance.
How does this procedure work for the binary Al–0.3Fe alloy? The
answer to this question is given in Fig. 4.
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D.V. Malakhov et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 159–166
Fig. 5. Driving forces for the onset of precipitation of intermetallics from the
Al–0.3Fe–0.2Si melt which solidifies according to Scheil’s scheme, and in which only
liquid and FCC phases are in equilibrium.
Fig. 6. Accumulation of Fe in the remaining liquid during solidification of the
Al–0.3Fe melt.
Apparently, the algorithm does not yield a trustworthy prediction, because neither Al13 M4 nor Al5 M2 have been detected in
rapidly solidified Al–Fe alloys in a wide range of iron concentrations. Al6 M, which is commonly observed in such alloys, has a
much lower driving force for its formation than these two erroneously predicted phases. It is also worth mentioning that in the
presence of Mn, Al12 M becomes an important part of the phase portrait, although its nucleation potential is by far lower than that of
Al13 M4 and Al5 M2 .
As Fig. 5 evidences, the above procedure also leads to a similar
(and equally inaccurate) prediction for the ternary case, because
Al13 M4 is again identified as the intermetallic, which should dominate, but which, in fact, is not observed in the alloy’s microstructure.
4. A positive driving force: a necessary, but not sufficient
condition
Let us revisit the binary Al–Fe case and focus on two phases:
Al13 M4 and Al6 M. What if a possible reason due to which the former intermetallic is never seen in rapidly solidified alloys, while
the latter one, which is usually observed, originates from a significant compositional difference between the two? In Al6 M, the
mole fraction of iron xFe = 1/7 ≈ 0.1429. In the case of
Al13 M4 (sublattice model in Table 2), the mole fraction varies from
(xFe )min = 0.2350 to (xFe )max = 0.2350/(0.2350 + 0.6275) ≈
0.2725. Let us compare these quantities with a temperaturedependent mole fraction of iron in the remaining solidifying
liquid. The comparison, given in Fig. 6, indicates that there is a supercooling at which the mole fractions of iron in the liquid phase
and Al6 M are identical, and that the magnitude of supercooling
is not unreasonably huge. However, the results warrant further
clarifications. It should be noted that Scheil’s assumption that the
liquid is compositionally uniform is an idealization; there are various sources (density differences, convection, etc.) of concentration
gradients between growing dendrite branches. In a practical sense,
such sources can be operational exactly where intermetallics eventually form. However, there is little doubt that within a given temperature range, the compositions of the liquid phase and Al6 M are
close to each other. In contrast, Al13 M4 is so enriched with iron that
its composition never closely matches that of the remaining liquid.
It is interesting to note that among many phases specified in
Fig. 4, only Al12 M and Al4 M have compositions comparable to
Fig. 7. Accumulation of iron and silicon in the remaining liquid during solidification
of the Al–0.3Fe–0.2Si melt.
that of the solidifying liquid within a wide range of supercoolings
stretching from T liquidus to 300 ◦ C below it. The result for Al12 M is
especially remarkable, because, despite its low (though positive!)
driving force, a similarity of two mole fractions is attained at
low supercoolings. This might be a reason why this phase is so
frequently observed in rapidly solidified alloys containing both Fe
and Mn.
Now let us switch attention to the ternary case and compare the
mole fractions of iron and silicon along two curves shown in Fig. 7
with the mole fractions of iron and silicon in intermetallics given
in Table 2.
The compositions of Al13 M4 , δ -AlFeSi and τ3 are different from
the composition of the remaining liquid over the entire temperature region. Hence, it is not surprising that these phases are not
observed in rapidly solidified Al alloys, even though they all have
large positive driving forces within a wide range of supercooling.
In contrast, α -AlFeSi, β -AlFeSi and γ -AlFeSi, i.e., phases with compositions not too dissimilar to that of a supercooled liquid, are detected in Al–Fe–Si alloys.
D.V. Malakhov et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 159–166
Fig. 8. Driving forces for the onset of precipitation from the solidifying
Al–0.4Fe–0.1Si melt.
This is consistent with experimental observations that the
presence of silicon suppresses the formation of Al6 M (typical in the
Al–Fe system, though it is especially pronounced in the Al–Fe–Mn
system) and results in the formation of ternary Si-containing
Al M
phases. Although a good partial compositional match xFe6 ≈ xLFe
can be attained, Al6 M does not dissolve silicon, which means that
compositionally it will never be close to a supercooled liquid. This
rationalization is consistent with a similar observation related to
the solid-state transformations in certain steels. It is found that a
high silicon concentration can block the formation of cementite
during austenite → ferrite transformations. The reason is that
Si, whose solubility in carbides, in general, and in cementite, in
particular, is very low, must leave a region where nucleation may
occur, i.e., the site within which it cannot be accommodated. A
slow solid-state diffusion hampers such an escape of silicon atoms,
which, in turn, impedes the cementite nucleation.
A persuasive argument that a compositional similarity matters
in the solid-state processes was presented by Byeong-Joo Lee [16],
who analyzed the formation of silicides during Co/Si, Ti/Si and
Ni/Si interfacial reactions. It was demonstrated that if more than
one silicide had positive driving forces for the start of their
precipitation, then the one whose composition was the closest to
the compositions of the phase existed at the interface would form a
continuous layer and suppress the growth of other compounds. Lee
also mentioned that such a blockage of the growth could happen
even if the compositionally favorable silicide nucleated later than
others.
The analogy is also consistent with the observation that although at deep supercoolings the driving forces for the precipitation of such iron silicides as FeSi, Fe5 Si3 , Fe3 Si7 and FeSi2 are large
and positive, they are never seen in rapidly solidified aluminum
alloys whose composition belongs to the region of primary solidification of the FCC phase. All these binary compounds are so rich in
iron and silicon that their composition will always be very different
from the composition of the liquid phase despite steadily growing
xFe and xSi in the remaining melt.
Figs. 5, 8 and 9 provide evidence that a change of melt composition has a perceptible influence on the driving forces for the formation of intermetallics. Despite an increased relative difference
between the driving force of Al13 M4 and those of other phases
caused by a higher Fe and lower Si concentrations, Al13 M4 is still
not observed in rapidly solidified alloys whose compositions are
similar to that for which the curves in Fig. 8 were calculated. If
163
Fig. 9. Driving forces for the start of precipitation from the solidifying
Al–0.2Fe–0.3Si melt.
Fig. 10. Accumulation of solutes during solidification of the Al–0.4Fe–0.1Si melt.
the Si concentration is increased and the Fe concentration is lowered, then the driving forces of several phases are situated within
a narrow corridor, which suggests that though a phase possesses
the maximum driving force for a given supercooling, this does not
automatically indicate a propensity for it to form (see Fig. 9).
Slight changes in the initial melt composition lead to large
changes of the composition of a solidifying liquid, which, in turn,
results in different intermetallics having a compositional similarity
with the remaining liquid phase. By comparing Figs. 7, 10 and 11,
one can appreciate a magnitude of such changes.
There are many ways to quantitatively characterize a difference
between the compositions of the remaining liquid and a particular
intermetallic. Let us, for the sake of determinacy, associate a
temperature-dependent compositional dissimilarity  with the
Euclidean distance:
=
q
xintermetallic
− xLFe
Fe
2
2
+ xintermetallic
− xLSi .
Si
Figs. 12–14, which were constructed for three selected intermetallics, are in accord with previously mentioned experimental
findings, i.e., with the observation that Al13 M4 is found only if low
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D.V. Malakhov et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 159–166
Fig. 11. Accumulation of solutes during solidification of the Al–0.2Fe–0.3Si melt.
Fig. 13. Compositional dissimilarities for the Al–0.4Fe–0.1Si case.
Fig. 12. Compositional dissimilarities for the Al–0.3Fe–0.2Si case.
Fig. 14. Compositional dissimilarities for the Al–0.2Fe–0.3Si case.
rates of heat extraction are employed while α -AlFeSi and β -AlFeSi
are the phases commonly seen in Al–Fe–Si alloys. Although Al13 M4
possesses a homogeneity range, the least dissimilarity value is attained when the mole fraction of iron is equal to (xFe )min = 0.2350.
Hence, only one curve is given for this phase in Figs. 12–14. In the
case of α -AlFeSi, each single curve in Figs. 12 and 13 corresponds
to the mole fraction of silicon providing the best compositional
match. Two intersecting curves in Fig. 14 are a reminder that a nonstoichiometry of an intermetallic phase should not be overlooked
in such calculations.
Figs. 12–14 demonstrate that by altering the Fe/Si ratio, one can
facilitate the formation of different intermetallics. This prediction
does not contradict what was observed in the solidification of Al
alloys, for which the Fe/Si ratio was recognized as one of important
(though not necessarily the most important) factors responsible for
phase portraits of as-cast alloys.
To reiterate, the main assumption in this work is that a rapid
solidification occurs so quickly that there is insufficient time for
diffusional fluxes in the liquid phase to deliver atoms of Fe and Si
to homogeneous nucleation sites if the compositions of a corresponding intermetallic and the liquid are far apart. It should also be
noted that the fluxes must be precisely balanced, because usually
ordered intermetallic phases are either stoichiometric compounds
or compounds with a narrow homogeneity regions. In other words,
it is hypothesized that a compositionally invariant transformation
rather than a diffusion-controlled process is responsible for nucleation events.
Although reliable temperature dependencies of the diffusion
coefficients of Fe and Si in liquid aluminum are available [17], it
is not yet possible to accurately formulate and then rigorously
solve a corresponding mass transfer problem. Primarily, this is
due to limited information about solutes’ concentration fields in
pools/films of the rejected liquid between arms of fast growing FCC
dendrites. Even less is known about the diffusivities of components
in intermetallics, and there is a paucity of information related to
the details of the nucleation mechanism(s). For example, perhaps
there is an almost instantaneous co-formation of several unit cells
resulting in a nucleus, but it is also possible that the nucleus is
assembled when several pre-existing cells meet. In spite of these
complexities, it is useful to roughly assess the reasonability of the
main assumption.
D.V. Malakhov et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 159–166
Let us consider a ‘‘generic intermetallic’’, for which the unit
cell is a 10 Å × 10 Å × 10 Å cube containing 100 atoms. Now
assume that xFe = 0.25 in this phase. Let us further hypothesize
that a cube-shaped nucleus is made of 512 unit cells, with 8
cells in each dimension. Thus, the embryo contains 51 200 atoms
of which 12 800 atoms are Fe atoms. The surface area of the
nucleus is given by A = 6 × (8 × 10 × 10−10 )2 = 3.84 ×
10−16 m2 . Let us presume that T = 400 ◦ C = 673.15 K (which
corresponds to a significantly supercooled remaining liquid), and
that the solidification time t = 10−3 s (which corresponds to
the lower range of rapid solidification). According to [17], the
diffusion coefficient of iron in liquid Al is described by D = 2.34 ×
10−7 exp(−35 × 103 /(RT )) m2 /s. Hence, for the temperature
chosen we obtain D ≈ 4.5 × 10−10 m2 /s. The question is: ‘‘Is
the flux of Fe atoms sufficient to deliver 12 800 to the nucleus?’’
Knowing that J = −D1C /1z and recalling that 1C = 1x/Vm , one
can write J = −D1x/ (Vm 1z ). We accept that the molar volume
is constant and equal to 10 cm3 /mole, i.e., to 10−5 m3 /mole. If
1z is defined as the distance between secondary dendrites at the
end of solidification, a value of 1z = 20 µm = 2 × 10−5 m
would not be too unreasonable. Choosing 1x = 0.01, we can
calculate the flux: J = 4.5 × 10−10 × 10−2 / 10−5 × 2 × 10−5 =
0.0225 mole/ m2 × s . Multiplication of the flux by A = 3.84 ×
10−16 m2 and t = 10−3 s gives 8.64 × 10−21 as the total number
of moles of atoms of iron delivered to the nucleus via liquid-state
diffusion, and we can multiply this by Avogadro’s number NA =
6.022 × 1023 atom/mole to end up with 5203 atoms of Fe. The
authors clearly understand that such a calculation is speculative,
but its purpose is to give at least an indication that the relative
sluggishness of liquid-state diffusion should not be overlooked at
low temperatures.
5. Conclusions
The approach suggested in this work makes use of Scheil’s formalism, the driving forces concept, and the idea of a compositional
similarity. It allows one to make a reasonable prediction about
the type(s) of intermetallic phase(s) in a rapidly solidified Al alloy
whose composition falls into the region of primary crystallization
of the FCC phase by following the next steps:
1. assume that a low supercooling (2–4 °C) below liquidus triggers
the nucleation of the Al-rich FCC phase and that the local
equilibrium is always sustained at the FCC/liquid interface;
2. presume that intermetallic phases do not precipitate from the
melt, follow the Scheil solidification path, and reach supercooling congruous with a particular mode of rapid solidification;
3. for the state attained, use concentrations of solutes in the remaining liquid and compute the molar driving forces for the
formation of all intermetallics;
4. among all intermetallics with positive driving forces, identify
these with compositions close to the composition of the remaining liquid, i.e., phases whose homogeneous nucleation and
growth does not necessitate a long-range diffusion.
An ability of the approach to explain experimental observations
and its apparent self-consistency should not disguise the fact that
it rests on two precarious assumptions.
The first is that the problem can be tackled without an explicit
consideration of the interfacial energy. This statement should not
be interpreted as if the interfacial energy is immaterial. Certainly,
its contribution to the height of the nucleation energy barrier is
crucial and therefore cannot be ignored. The existence of the interfacial energy also means that an infinitesimal positive driving
force is not sufficient to trigger the nucleation, but that a finite
165
positive value is required. Let us revisit, for instance, Fig. 4, and
realize that such a requirement causes each curve to shift downward and that for each intermetallic phase, a magnitude of the
change, i.e., a new location of the curve, is governed by a shape
of the nucleus and a corresponding interfacial energy. Although
nothing specific is known about these characteristics, it is useful
to recall that almost all intermetallics are ordered compounds and
that their interfaces with the liquid phase are atomically smooth.
This connotes similar values of the interfacial energy and therefore
suggests that all curves shift down in unison, i.e., their mutual positions do not change dramatically. A very similar viewpoint was
expressed in [16]: ‘‘Generally, the information on the interface energy and the misfit strain energy is lacking. Further, if the interfacial energy and the misfit strain energy terms for the nucleation of
various silicides in a metal-Si system can be assumed to have similar values so that the differences of each term between the competing silicides are smaller than the differences of driving forces
between the silicides, the driving force can be regarded as a decisive factor. In this case, the driving force of formation can be a good
approximate criterion for prediction of the first-forming silicide.’’
Although the qualitative arguments are not helpful in quantifying
the decrease of the driving forces due to the interfacial energy, it is
worth emphasizing that this work is primarily focused on deeply
supercooled (tens and hundreds of degrees) melts, for which the
thermodynamic driving forces are so large (approaching and even
exceeding 10 kJ/mole) that the surface energy term is not as important as it would be for a moderately supercooled liquid phase.
The second assumption is that the intermetallics nucleate homogeneously. It has to be admitted that a heterogeneous nucleation is justifiably considered as the only active mechanism in real
systems. A hypothesis that the homogeneous nucleation is in effect
requires either indisputable experimental evidence or a convincing
argument. In this work, it was conjectured that sites existing in the
melt and suitable for heterogeneous nucleation were consumed in
the very beginning of solidification by the FCC phase, and no new
sites were subsequently generated. The latter supposition is debatable, because there is a likelihood of heterogeneous nucleation at
dendrite arms broken due to impingement and detached from a
growing dendrite as well as at arms not in the immediate vicinity
of a rapidly propagating tip. Since the FCC phase grows faster than
an ordered intermetallic compound, such a mechanism will likely
result in intragranular precipitates. However, such a prediction
contradicts a general observation that intermetallics in rapidly solidified Al alloys are usually situated along grain boundaries, which
may mean that the heterogeneous nucleation on dendrite arms is
rather limited. Let us also recall that a high concentration of sites
at which intermetallic phases could nucleate would likely result in
many small particles. Microstructural findings, however, are not in
a good agreement with this. Finally, an intense generation of new
sites in the course of solidification would significantly facilitate the
formation of intermetallic phases with high thermodynamic driving forces including, indeed, Al13 M4 . An absence of this phase in
rapidly solidified Al–Fe and Al–Fe–Si alloys is hardly compatible
with an abundance of sites suitable for heterogeneous nucleation.
A workability of the method proposed here was illustrated by
applying it to relatively simple cases of the Al–Fe and Al–Fe–Si.
Whether it becomes a useful tool for predicting phase portraits of
multicomponent aluminum alloys is yet to be seen.
Acknowledgement
The authors are grateful to the Ontario Centres of Excellence
(OCE) for a financial support.
166
D.V. Malakhov et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 159–166
Appendix. Supplementary data
Supplementary data associated with this article can be found,
in the online version, at doi:10.1016/j.calphad.2010.01.006.
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