Classical nucleation theory in ordering alloys precipitating with L12 structure.
Emmanuel Clouet∗ and Maylise Nastar
Service de Recherches de Métallurgie Physique, CEA/Saclay, 91191 Gif-sur-Yvette, France
(Dated: October 28, 2018)
By means of low-temperature expansions (LTEs), the nucleation free energy and the precipi-
tate interface free energy are expressed as functions of the solubility limit for alloys which lead to
the precipitation of a stoichiometric L12 compound such as Al-Sc or Al-Zr alloys. Classical nu-
cleation theory is then used to obtain a simple expression of the nucleation rate whose validity is
demonstrated by a comparison with atomic simulations. LTEs also explain why simple mean-field
approximation like the Bragg-Williams approximation fails to predict correct nucleation rates in
such an ordering alloy.
Since its initial formulation in 1927 by Volmer, We-
ber and Farkas and its modification in 1935 by Becker
and Döring the classical nucleation theory (CNT)1,2,3
has been a suitable tool to model the nucleation stage
in phase transformations. The success of this theory re-
lies on its simplicity and on the few parameters required
to predict the nucleation rate. Recently, the use of com-
puter simulations have allowed to assess the applicability
of the theory for solid-solid phase transformations4,5,6,7.
Thanks to a precise control of simulation conditions, it is
possible to get accurate estimations of CNT parameters
and thus to make a direct comparison between theory
predictions and quantities observed during simulations.
One thus gains a deeper understanding of the validity of
the different assumptions used by the CNT.
Previous studies have shown that the capillary approx-
imation, which CNT relies on, gives a precise description
of cluster thermodynamics. Within this approximation,
the free energy of a nucleus is written as the sum of a
volume contribution, the nucleation free energy, and a
surface contribution corresponding to the energy cost to
create an interface between the nucleus and the solvent.
For CNT to agree with atomic simulations, care has to
be taken in the way these two energetic contributions
are obtained. In particular, we have shown that one has
to take into account short range order when calculating
the nucleation free energy in an ordering alloy4. Usual
thermodynamic approximations, like the ideal solid so-
lution or the Bragg-Williams approximation, cannot de-
scribe short range order and thus can predict values of
the cluster size distribution and of the nucleation rate
wrong by several orders of magnitude. This is to contrast
with more sophisticated mean-field approximations like
the cluster variation method (CVM) which provides good
predictions of the nucleation rate4. However, an easy use
of CNT and a clear determination of the missing ingre-
dients in simple mean-field approximations requires an
analytical approach which CVM cannot provide. Such
an approach has to lead to accurate expressions of the
CNT input parameters so as to make the theory predic-
tive without any fitting of its parameters.
In this Letter, we use low-temperature expansions
(LTE)8,9 to derive an analytical formulation of the nucle-
ation free energy and the interface free energy in a binary
system like Al-Sc or Al-Zr, i.e. a supersaturated Al-X
solid solution leading to the nucleation of a stoichiomet-
ric Al3X compound with the L12 structure. This struc-
ture corresponds to an ordering of the fcc lattice with so-
lute X atoms lying on one of the four cubic sublattices8.
LTE are well suited to describe short range order in
dilute solid solution and nearly stoichiometric ordered
compounds10,11,12,13,14 like Al3X compound. The use of
this method in CNT framework allows to obtain a fully
analytical modelling whose only material parameters are
the solubility limit and the solute diffusion coefficient.
To do so, we start from the same atomic diffusion
model previously developed for Al-Sc-Zr system4,15. This
model relies on a rigid lattice with interactions between
first- and second nearest neighbors and uses a thermally
activated atom-vacancy exchange mechanism to describe
diffusion. Despite its simplicity, it has been shown to
lead to predictions in good agreement with experimental
data15,16,17. Within this atomic model, atoms are con-
strained to lie on a fcc lattice and the configurations of
a binary Al-X alloy is fully described by the solute atom
occupation number pn with pn = 1 if the site n is occu-
pied by a solute atom and pn = 0 otherwise. The energy
per atom of a given configuration of the Al-X alloys is
then given by
E = UAl +
(UX − UAl)
(1− pn) pm
(1− pn) pm (1)
where the first and second sums, respectively, run on all
first and second nearest-neighbor pairs of sites, Ns is the
number of lattice sites, UAl (respectively UX) is the en-
ergy per atom when only Al (respectively X) atoms lie
on the fcc lattice and ω
AlX and ω
AlX are the first and
second nearest neighbor order energies. Al-Sc and Al-
Zr thermodynamics are characterized by the order ten-
dency between first nearest neighbors (ω
AlX < 0) and
the demixing tendency between second nearest neighbors
AlX > 0). Eq. 1 is a rewriting for binary alloys of the
atomic model developed in Refs.4,15 when one neglects
vacancy contributions.
http://arxiv.org/abs/0704.1988v1
The nucleation free energy entering CNT is defined by
∆Gnuc(x0X) =
µAl(x
X )− µAl(x
X )− µX(x
, (2)
where µAl(xX) and µX(xX) are the Al and X component
chemical potentials in a solid solution of concentration
xX, and x
X and x
X the concentrations of the equilibrium
and supersaturated solid solution.
LTE are more easy to handle in semi-grand-canonical
ensemble where all quantities are written as functions of
the effective potential µ = (µAl − µX) /2. Definition of
the nucleation free energy then becomes
∆Gnuc(µ) = A(µeq)−A(µ) +
(µeq − µ) , (3)
where µeq is the effective potential corresponding to equi-
librium between the Al solid solution and the Al3X L12
compound. We have defined in Eq. 3 the solid solution
semi-grand-canonical free energy A = (µAl + µX) /2 =
F (x) + (1− 2x)µ, F (x) being the usual canonical free
energy.
A LTE consists in developing the partition function
of the system around a reference state, keeping in the
series only the excited states of lowest energies. Use of
the linked cluster theorem8,9 allows then to express the
corresponding semi-grand canonical free energy as
A(µ) = A0(µ)− kT
gi,n exp (−∆Ei,n(µ)/kT ), (4)
where the energy of the ground state is A0(µ) = UAl +µ
for the Al solid solution and A0(µ) = 3/4 UAl+1/4 UX+
AlX+µ/2 for the Al3X L12 compound. In the sum ap-
pearing in Eq. 4, the excited states have been gathered
according to their energy state i and the number n of
lattice sites involved. LTE parameters corresponding to
the excited states with the lowest energies are given in
Tab. I. All excitation energies only involve a set of iso-
lated atoms or in second nearest neighbor position since
flipping two atoms at nearest neighbor position produces
an excited state with a much higher energy.
The solute concentration in a given phase is obtained
by considering the derivative of the corresponding semi-
grand-canonical free energy. For the solid solution, one
xX(µ) =
∂A(µ)
ngi,n exp (−∆Ei,n(µ)/kT ). (5)
The solid solution and the L12 compound are in equi-
librium when both phases have the same semi-grand
canonical free energy. Considering third order LTE
TABLE I: Coefficients entering in the low temperature ex-
pansion (Eq. 4). The first seven excited states are considered
for the solid solution and the first three excited states for the
Al3X L12 compound. The effective potential is written as
µ = (UX − UAl)/2 + 6ω
AlX + δµ.
Solid solution L12 compound
i n ∆Ei,n(µ) gi,n ∆Ei,n(µ) gi,n
1 1 6ω
AlX − 2δµ 1 6ω
AlX + 2δµ 1/4
2 2 10ω
AlX − 4δµ 3 10ω
AlX + 4δµ 3/4
3 2 12ω
AlX − 4δµ −19/2 12ω
AlX + 4δµ −7/8
4 3 14ω
AlX − 6δµ 15
5 4 16ω
AlX − 8δµ 3
5 3 16ω
AlX − 6δµ −96
6 4 18ω
AlX − 8δµ 83
6 3 18ω
AlX − 6δµ −774
7 5 20ω
AlX − 10δµ 48
7 4 + 20ω
AlX − 8δµ −1569/2
(i = 3), this happens for the effective potential
µeq = (UX − UAl)/2 + 6ω
exp (−6ω(2)AlX/kT ) + 3 exp (−10ω
AlX/kT )
exp (−12ω(2)AlX/kT )
, (6)
corresponding to the solubility
X = exp
−6ω(2)AlX/kT
+ 6 exp
−10ω(2)AlX/kT
− 16 exp
−12ω(2)AlX/kT
. (7)
As these expressions have to be consistent with the ex-
pansion of A, terms with larger exponential arguments
than −12ω(2)AlX are discarded. For equilibrium phases, one
does not need to go further in the expansion than the
third order. Indeed, thermodynamic properties are al-
ready well converged as the solid solution and the L12
compound in equilibrium only slightly deviate from their
respective ground states. On the other hand, Fig. 1
show that an expansion beyond the third order of the
semi-grand canonical free energy A(µ) of the supersat-
urated solid solution and of the corresponding concen-
tration xX(µ) improves the convergence of the nucle-
ation free energy. When only the first excited state
is included in the expansion, LTE leads to the same
value of the nucleation free energy as the ideal solid
solution model. As more excited state are included in
the expansion, the value deduced from LTE converges
to the one calculated with CVM in the tetrahedron-
octahedron approximation4. This is to contrast to the
Bragg-Williams approximation which leads to a worse
prediction of the nucleation free energy than the ideal
solid solution model.
 0  0.5  1  1.5
Ideal solid solution
Bragg−Williams
LTE : 1st order
3rd order
7th order
FIG. 1: Variation with the nominal concentration x0Zr of
the nucleation free energy ∆Gnuc at T = 723 K obtained
with different thermodynamic approximations: CVM, ideal
solid solution, Bragg-Williams and low-temperature expan-
sions (LTE) to different orders.
So as to understand why Bragg-Williams approxima-
tion does so bad, it is worth going back to the canonical
ensemble. When considering only the third order LTE,
thermodynamic quantities can be expressed as functions
of the solid solution nominal concentration x0X. In par-
ticular, the nucleation free energy is given by
∆GnucLTE(x
X) = kT
q (x0X)− q (x
+ 3kT exp
AlX/kT
q (x0X)
2 − q (xeqX )
q (x0X)
− ln [q (xeqX )]
, (8)
where we have defined the function
q (x) =
1 + 4x
6 exp
AlX/kT
. (9)
This expression developed to first order in the concentra-
tions x0X and x
X leads to
∆GnucLTE(x
kT ln
1− xeqX
1− x0X
kT ln
1 + 6e2ω
x0X − x
. (10)
Doing the same development for the nucleation
free energy calculated within the Bragg-Williams
approximation4, we obtain
∆GnucBW(x
kT ln
1− xeqX
1− x0X
kT ln
AlX + 3ω
x0X − x
. (11)
Comparing Eq. 10 with Eq. 11, we see that these two
thermodynamic approximations deviate from the ideal
solid solution model by a distinct linear term. In the
LTE (Eq. 10), the nucleation free energy is only de-
pending on the second nearest neighbor interaction and
the coefficient in front of the concentration difference is
positive. On the other hand, the Bragg-Williams ap-
proximation (Eq. 11) incorporates both first and sec-
ond nearest neighbor interactions into a global parameter
ωAlX = 6ω
AlX + 3ω
AlX. This leads to a linear correction
with a coefficient which can be negative due to the os-
cillating nature of the interactions. In particular, this is
the case for both binary Al-Zr and Al-Sc alloys4. Bragg-
Williams approximation thus leads to a wrong correction
of the ideal model because it does not consider properly
short range order. In the case of a L12 ordered com-
pound precipitating from a solid solution lying on a fcc
lattice, one cannot use such an approximation to calcu-
late the nucleation free energy. On the other hand, Eq. 8
is a good approximation and can be used to calculate
the nucleation free energy even when the second nearest
neighbor interaction ω
AlX is not known. Indeed, this pa-
rameter can be deduced from the solubility limit x
inverting Eq. 7, leading to the relation
AlX = −
kT ln (x
X ) + kT
2/3 −
. (12)
This relation combined with Eq. 8 provides a powerful
way for calculating the nucleation free energy from the
solid solubility.
LTE can be used too to calculate the plane interface
free energy σ100 corresponding to a [100] direction. Due
to the inhomogeneity perpendicular to the interface, the
main contribution arises from broken bonds and excited
states, whose energies are lower than in bulk phases, only
bring a small correction. One thus does not need to go
further than the second order in the expansion. At 0 K,
the isotropic interface free energy σ̄ is obtained by mul-
tiplying σ100 with the geometric factor (6/π)
corre-
sponding to a perfect [100] facetting of the precipitates.
For low temperatures, this is a good approximation to
assume that the same linear relation holds between both
quantities4. The isotropic interface free energy given by
LTE is then
a2σ̄ = (6/π)
AlX − 2kT exp (−4ω
AlX/kT )
−kT exp (−6ω(2)AlX/kT )
, (13)
where a is the fcc lattice parameter.
LTE thus allow to calculate all CNT input parameters
from the knowledge of the solubility limit. The nucle-
ation rate is then obtained from the equation
J st(x0X) = −16Ns
∆Gnuc(x0X)√
kTa2σ̄
(a2σ̄)3
kT [∆Gnuc(x0X)]
, (14)
21.510.5
T = 723 K
T = 773 K
T = 823 K
T = 873 K
T = 723 K
T = 773 K
T = 823 K
T = 873 K
T = 723 K
T = 773 K
T = 823 K
T = 723 K
T = 773 K
T = 823 K
FIG. 2: Variation with nominal concentration and tempera-
ture of the steady-state nucleation rate Jst for Al3Zr (top)
and Al3Sc (bottom) precipitations. Symbols correspond to
kinetic Monte Carlo simulations and lines to classical nucle-
ation theory
whereDX is the X impurity diffusion coefficient in Al. We
thus obtain a fully analytical expression of the nucleation
rate. Using the same experimental data, i.e. solubility
limits and diffusion coefficients, as the ones used to fit
the atomic diffusion model of kinetic Monte Carlo simu-
lations, we can compare CNT predictions with nucleation
rate observed in simulations4. A good agreement is ob-
tained both for Al-Zr and Al-Sc binary alloys (Fig. 2).
The combination of LTE with CNT thus allows to build
a quantitative modeling of nucleation relying on a very
limited number of material parameters. Such a model
can be directly applied to aluminum alloys where a L12
compound precipitates from a supersaturated solid so-
lution as this is the case with Zr, Sc or other rare earth
elements like Er, Tm, Yb and Lu18. Li too precipitates in
aluminum with a L12 structure, but this system requires
another statistical approximation than LTE. Indeed, this
approach based on LTE, requires that the precipitating
phase only slightly deviates from its perfect stoichiome-
try and that the solute solubility remains low. Provided
these conditions are fulfilled, it could be applied to al-
loys other than aluminum alloys. More generally, LTE
demonstrate that the oscillating nature of the interac-
tions in an alloy with an ordering tendency has to be
taken into account by the CNT and requires a better
statistical description than the Bragg-Williams approxi-
mation which treats all interactions on the same footing.
The authors would like to thank Y. Le Bouar and A.
Finel for helpful discussions on LTE, and B. Legrand, F.
Soisson and G. Martin for their invaluable help.
∗ Electronic address: emmanuel.clouet@cea.fr
1 G. Martin, in Solid State Phase Transformation in Met-
als and Alloys (Les Éditions de Physique, Orsay, France,
1978), pp. 337–406.
2 K. F. Kelton, in Solid State Physics, edited by H. Ehren-
reich and D. Turnbull (Academic Press, 1991), vol. 45, pp.
75–177.
3 D. Kashchiev, Nucleation : basic theory with applications
(Butterworth Heinemann, Oxford, 2000).
4 E. Clouet, M. Nastar, and C. Sigli, Phys. Rev. B 69,
064109 (2004).
5 V. A. Shneidman, K. A. Jackson, and K. M. Beatty, Phys.
Rev. B 59, 3579 (1999).
6 F. Soisson and G. Martin, Phys. Rev. B 62, 203 (2000).
7 F. Berthier, B. Legrand, J. Creuze, and R. Tétot, J. Elec-
troanal. Chem. 561, 37 (2004); 562, 127 (2004).
8 F. Ducastelle, Order and Phase Stability in Alloys (North-
Holland, Amsterdam, 1991).
9 C. Domb and M. S. Green, eds., Phase Transitions and
Critical Phenomena, vol. 3 (Academic Press, London,
1974).
10 C. Woodward, M. Asta, G. Kresse, and J. Hafner, Phys.
Rev. B 63, 094103 (2001).
11 A. F. Kohan, P. D. Tepesch, G. Ceder, and C. Wolverton,
Comput. Mater. Sci. 9, 389 (1998).
12 M. Asta, S. M. Foiles, and A. A. Quong, Phys. Rev. B 57,
11265 (1998).
13 R. W. Hyland, M. Asta, S. M. Foiles, and C. L. Rohrer,
Acta Mater. 46, 3667 (1998).
14 Y. Le Bouar, A. Loiseau, and A. Finel, Phys. Rev. B 68,
224203 (2003).
15 E. Clouet, L. Laé, T. Épicier, W. Lefebvre, M. Nastar, and
A. Deschamps, Nat. Mater. 5, 482 (2006).
16 E. Clouet, A. Barbu, L. Laé, and G. Martin, Acta Mater.
53, 2313 (2005).
17 E. Clouet and A. Barbu, Acta Mater. 55, 391 (2007).
18 K. E. Knipling, D. C. Dunand, and D. N. Seidman, Z.
Metallkd. 2006, 246 (2006).
mailto:emmanuel.clouet@cea.fr

  By means of low-temperature expansions (LTEs), the nucleation free energy and
the precipitate interface free energy are expressed as functions of the
solubility limit for alloys which lead to the precipitation of a stoichiometric
L12 compound such as Al-Sc or Al-Zr alloys. Classical nucleation theory is then
used to obtain a simple expression of the nucleation rate whose validity is
demonstrated by a comparison with atomic simulations. LTEs also explain why
simple mean-field approximation like the Bragg-Williams approximation fails to
predict correct nucleation rates in such an ordering alloy.

Classical nucleation theory in ordering alloys precipitating with L12 structure.
Emmanuel Clouet∗ and Maylise Nastar
Service de Recherches de Métallurgie Physique, CEA/Saclay, 91191 Gif-sur-Yvette, France
(Dated: October 28, 2018)
By means of low-temperature expansions (LTEs), the nucleation free energy and the precipi-
tate interface free energy are expressed as functions of the solubility limit for alloys which lead to
the precipitation of a stoichiometric L12 compound such as Al-Sc or Al-Zr alloys. Classical nu-
cleation theory is then used to obtain a simple expression of the nucleation rate whose validity is
demonstrated by a comparison with atomic simulations. LTEs also explain why simple mean-field
approximation like the Bragg-Williams approximation fails to predict correct nucleation rates in
such an ordering alloy.
Since its initial formulation in 1927 by Volmer, We-
ber and Farkas and its modification in 1935 by Becker
and Döring the classical nucleation theory (CNT)1,2,3
has been a suitable tool to model the nucleation stage
in phase transformations. The success of this theory re-
lies on its simplicity and on the few parameters required
to predict the nucleation rate. Recently, the use of com-
puter simulations have allowed to assess the applicability
of the theory for solid-solid phase transformations4,5,6,7.
Thanks to a precise control of simulation conditions, it is
possible to get accurate estimations of CNT parameters
and thus to make a direct comparison between theory
predictions and quantities observed during simulations.
One thus gains a deeper understanding of the validity of
the different assumptions used by the CNT.
Previous studies have shown that the capillary approx-
imation, which CNT relies on, gives a precise description
of cluster thermodynamics. Within this approximation,
the free energy of a nucleus is written as the sum of a
volume contribution, the nucleation free energy, and a
surface contribution corresponding to the energy cost to
create an interface between the nucleus and the solvent.
For CNT to agree with atomic simulations, care has to
be taken in the way these two energetic contributions
are obtained. In particular, we have shown that one has
to take into account short range order when calculating
the nucleation free energy in an ordering alloy4. Usual
thermodynamic approximations, like the ideal solid so-
lution or the Bragg-Williams approximation, cannot de-
scribe short range order and thus can predict values of
the cluster size distribution and of the nucleation rate
wrong by several orders of magnitude. This is to contrast
with more sophisticated mean-field approximations like
the cluster variation method (CVM) which provides good
predictions of the nucleation rate4. However, an easy use
of CNT and a clear determination of the missing ingre-
dients in simple mean-field approximations requires an
analytical approach which CVM cannot provide. Such
an approach has to lead to accurate expressions of the
CNT input parameters so as to make the theory predic-
tive without any fitting of its parameters.
In this Letter, we use low-temperature expansions
(LTE)8,9 to derive an analytical formulation of the nucle-
ation free energy and the interface free energy in a binary
system like Al-Sc or Al-Zr, i.e. a supersaturated Al-X
solid solution leading to the nucleation of a stoichiomet-
ric Al3X compound with the L12 structure. This struc-
ture corresponds to an ordering of the fcc lattice with so-
lute X atoms lying on one of the four cubic sublattices8.
LTE are well suited to describe short range order in
dilute solid solution and nearly stoichiometric ordered
compounds10,11,12,13,14 like Al3X compound. The use of
this method in CNT framework allows to obtain a fully
analytical modelling whose only material parameters are
the solubility limit and the solute diffusion coefficient.
To do so, we start from the same atomic diffusion
model previously developed for Al-Sc-Zr system4,15. This
model relies on a rigid lattice with interactions between
first- and second nearest neighbors and uses a thermally
activated atom-vacancy exchange mechanism to describe
diffusion. Despite its simplicity, it has been shown to
lead to predictions in good agreement with experimental
data15,16,17. Within this atomic model, atoms are con-
strained to lie on a fcc lattice and the configurations of
a binary Al-X alloy is fully described by the solute atom
occupation number pn with pn = 1 if the site n is occu-
pied by a solute atom and pn = 0 otherwise. The energy
per atom of a given configuration of the Al-X alloys is
then given by
E = UAl +
(UX − UAl)
(1− pn) pm
(1− pn) pm (1)
where the first and second sums, respectively, run on all
first and second nearest-neighbor pairs of sites, Ns is the
number of lattice sites, UAl (respectively UX) is the en-
ergy per atom when only Al (respectively X) atoms lie
on the fcc lattice and ω
AlX and ω
AlX are the first and
second nearest neighbor order energies. Al-Sc and Al-
Zr thermodynamics are characterized by the order ten-
dency between first nearest neighbors (ω
AlX < 0) and
the demixing tendency between second nearest neighbors
AlX > 0). Eq. 1 is a rewriting for binary alloys of the
atomic model developed in Refs.4,15 when one neglects
vacancy contributions.
http://arxiv.org/abs/0704.1988v1
The nucleation free energy entering CNT is defined by
∆Gnuc(x0X) =
µAl(x
X )− µAl(x
X )− µX(x
, (2)
where µAl(xX) and µX(xX) are the Al and X component
chemical potentials in a solid solution of concentration
xX, and x
X and x
X the concentrations of the equilibrium
and supersaturated solid solution.
LTE are more easy to handle in semi-grand-canonical
ensemble where all quantities are written as functions of
the effective potential µ = (µAl − µX) /2. Definition of
the nucleation free energy then becomes
∆Gnuc(µ) = A(µeq)−A(µ) +
(µeq − µ) , (3)
where µeq is the effective potential corresponding to equi-
librium between the Al solid solution and the Al3X L12
compound. We have defined in Eq. 3 the solid solution
semi-grand-canonical free energy A = (µAl + µX) /2 =
F (x) + (1− 2x)µ, F (x) being the usual canonical free
energy.
A LTE consists in developing the partition function
of the system around a reference state, keeping in the
series only the excited states of lowest energies. Use of
the linked cluster theorem8,9 allows then to express the
corresponding semi-grand canonical free energy as
A(µ) = A0(µ)− kT
gi,n exp (−∆Ei,n(µ)/kT ), (4)
where the energy of the ground state is A0(µ) = UAl +µ
for the Al solid solution and A0(µ) = 3/4 UAl+1/4 UX+
AlX+µ/2 for the Al3X L12 compound. In the sum ap-
pearing in Eq. 4, the excited states have been gathered
according to their energy state i and the number n of
lattice sites involved. LTE parameters corresponding to
the excited states with the lowest energies are given in
Tab. I. All excitation energies only involve a set of iso-
lated atoms or in second nearest neighbor position since
flipping two atoms at nearest neighbor position produces
an excited state with a much higher energy.
The solute concentration in a given phase is obtained
by considering the derivative of the corresponding semi-
grand-canonical free energy. For the solid solution, one
xX(µ) =
∂A(µ)
ngi,n exp (−∆Ei,n(µ)/kT ). (5)
The solid solution and the L12 compound are in equi-
librium when both phases have the same semi-grand
canonical free energy. Considering third order LTE
TABLE I: Coefficients entering in the low temperature ex-
pansion (Eq. 4). The first seven excited states are considered
for the solid solution and the first three excited states for the
Al3X L12 compound. The effective potential is written as
µ = (UX − UAl)/2 + 6ω
AlX + δµ.
Solid solution L12 compound
i n ∆Ei,n(µ) gi,n ∆Ei,n(µ) gi,n
1 1 6ω
AlX − 2δµ 1 6ω
AlX + 2δµ 1/4
2 2 10ω
AlX − 4δµ 3 10ω
AlX + 4δµ 3/4
3 2 12ω
AlX − 4δµ −19/2 12ω
AlX + 4δµ −7/8
4 3 14ω
AlX − 6δµ 15
5 4 16ω
AlX − 8δµ 3
5 3 16ω
AlX − 6δµ −96
6 4 18ω
AlX − 8δµ 83
6 3 18ω
AlX − 6δµ −774
7 5 20ω
AlX − 10δµ 48
7 4 + 20ω
AlX − 8δµ −1569/2
(i = 3), this happens for the effective potential
µeq = (UX − UAl)/2 + 6ω
exp (−6ω(2)AlX/kT ) + 3 exp (−10ω
AlX/kT )
exp (−12ω(2)AlX/kT )
, (6)
corresponding to the solubility
X = exp
−6ω(2)AlX/kT
+ 6 exp
−10ω(2)AlX/kT
− 16 exp
−12ω(2)AlX/kT
. (7)
As these expressions have to be consistent with the ex-
pansion of A, terms with larger exponential arguments
than −12ω(2)AlX are discarded. For equilibrium phases, one
does not need to go further in the expansion than the
third order. Indeed, thermodynamic properties are al-
ready well converged as the solid solution and the L12
compound in equilibrium only slightly deviate from their
respective ground states. On the other hand, Fig. 1
show that an expansion beyond the third order of the
semi-grand canonical free energy A(µ) of the supersat-
urated solid solution and of the corresponding concen-
tration xX(µ) improves the convergence of the nucle-
ation free energy. When only the first excited state
is included in the expansion, LTE leads to the same
value of the nucleation free energy as the ideal solid
solution model. As more excited state are included in
the expansion, the value deduced from LTE converges
to the one calculated with CVM in the tetrahedron-
octahedron approximation4. This is to contrast to the
Bragg-Williams approximation which leads to a worse
prediction of the nucleation free energy than the ideal
solid solution model.
 0  0.5  1  1.5
Ideal solid solution
Bragg−Williams
LTE : 1st order
3rd order
7th order
FIG. 1: Variation with the nominal concentration x0Zr of
the nucleation free energy ∆Gnuc at T = 723 K obtained
with different thermodynamic approximations: CVM, ideal
solid solution, Bragg-Williams and low-temperature expan-
sions (LTE) to different orders.
So as to understand why Bragg-Williams approxima-
tion does so bad, it is worth going back to the canonical
ensemble. When considering only the third order LTE,
thermodynamic quantities can be expressed as functions
of the solid solution nominal concentration x0X. In par-
ticular, the nucleation free energy is given by
∆GnucLTE(x
X) = kT
q (x0X)− q (x
+ 3kT exp
AlX/kT
q (x0X)
2 − q (xeqX )
q (x0X)
− ln [q (xeqX )]
, (8)
where we have defined the function
q (x) =
1 + 4x
6 exp
AlX/kT
. (9)
This expression developed to first order in the concentra-
tions x0X and x
X leads to
∆GnucLTE(x
kT ln
1− xeqX
1− x0X
kT ln
1 + 6e2ω
x0X − x
. (10)
Doing the same development for the nucleation
free energy calculated within the Bragg-Williams
approximation4, we obtain
∆GnucBW(x
kT ln
1− xeqX
1− x0X
kT ln
AlX + 3ω
x0X − x
. (11)
Comparing Eq. 10 with Eq. 11, we see that these two
thermodynamic approximations deviate from the ideal
solid solution model by a distinct linear term. In the
LTE (Eq. 10), the nucleation free energy is only de-
pending on the second nearest neighbor interaction and
the coefficient in front of the concentration difference is
positive. On the other hand, the Bragg-Williams ap-
proximation (Eq. 11) incorporates both first and sec-
ond nearest neighbor interactions into a global parameter
ωAlX = 6ω
AlX + 3ω
AlX. This leads to a linear correction
with a coefficient which can be negative due to the os-
cillating nature of the interactions. In particular, this is
the case for both binary Al-Zr and Al-Sc alloys4. Bragg-
Williams approximation thus leads to a wrong correction
of the ideal model because it does not consider properly
short range order. In the case of a L12 ordered com-
pound precipitating from a solid solution lying on a fcc
lattice, one cannot use such an approximation to calcu-
late the nucleation free energy. On the other hand, Eq. 8
is a good approximation and can be used to calculate
the nucleation free energy even when the second nearest
neighbor interaction ω
AlX is not known. Indeed, this pa-
rameter can be deduced from the solubility limit x
inverting Eq. 7, leading to the relation
AlX = −
kT ln (x
X ) + kT
2/3 −
. (12)
This relation combined with Eq. 8 provides a powerful
way for calculating the nucleation free energy from the
solid solubility.
LTE can be used too to calculate the plane interface
free energy σ100 corresponding to a [100] direction. Due
to the inhomogeneity perpendicular to the interface, the
main contribution arises from broken bonds and excited
states, whose energies are lower than in bulk phases, only
bring a small correction. One thus does not need to go
further than the second order in the expansion. At 0 K,
the isotropic interface free energy σ̄ is obtained by mul-
tiplying σ100 with the geometric factor (6/π)
corre-
sponding to a perfect [100] facetting of the precipitates.
For low temperatures, this is a good approximation to
assume that the same linear relation holds between both
quantities4. The isotropic interface free energy given by
LTE is then
a2σ̄ = (6/π)
AlX − 2kT exp (−4ω
AlX/kT )
−kT exp (−6ω(2)AlX/kT )
, (13)
where a is the fcc lattice parameter.
LTE thus allow to calculate all CNT input parameters
from the knowledge of the solubility limit. The nucle-
ation rate is then obtained from the equation
J st(x0X) = −16Ns
∆Gnuc(x0X)√
kTa2σ̄
(a2σ̄)3
kT [∆Gnuc(x0X)]
, (14)
21.510.5
T = 723 K
T = 773 K
T = 823 K
T = 873 K
T = 723 K
T = 773 K
T = 823 K
T = 873 K
T = 723 K
T = 773 K
T = 823 K
T = 723 K
T = 773 K
T = 823 K
FIG. 2: Variation with nominal concentration and tempera-
ture of the steady-state nucleation rate Jst for Al3Zr (top)
and Al3Sc (bottom) precipitations. Symbols correspond to
kinetic Monte Carlo simulations and lines to classical nucle-
ation theory
whereDX is the X impurity diffusion coefficient in Al. We
thus obtain a fully analytical expression of the nucleation
rate. Using the same experimental data, i.e. solubility
limits and diffusion coefficients, as the ones used to fit
the atomic diffusion model of kinetic Monte Carlo simu-
lations, we can compare CNT predictions with nucleation
rate observed in simulations4. A good agreement is ob-
tained both for Al-Zr and Al-Sc binary alloys (Fig. 2).
The combination of LTE with CNT thus allows to build
a quantitative modeling of nucleation relying on a very
limited number of material parameters. Such a model
can be directly applied to aluminum alloys where a L12
compound precipitates from a supersaturated solid so-
lution as this is the case with Zr, Sc or other rare earth
elements like Er, Tm, Yb and Lu18. Li too precipitates in
aluminum with a L12 structure, but this system requires
another statistical approximation than LTE. Indeed, this
approach based on LTE, requires that the precipitating
phase only slightly deviates from its perfect stoichiome-
try and that the solute solubility remains low. Provided
these conditions are fulfilled, it could be applied to al-
loys other than aluminum alloys. More generally, LTE
demonstrate that the oscillating nature of the interac-
tions in an alloy with an ordering tendency has to be
taken into account by the CNT and requires a better
statistical description than the Bragg-Williams approxi-
mation which treats all interactions on the same footing.
The authors would like to thank Y. Le Bouar and A.
Finel for helpful discussions on LTE, and B. Legrand, F.
Soisson and G. Martin for their invaluable help.
∗ Electronic address: emmanuel.clouet@cea.fr
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mailto:emmanuel.clouet@cea.fr
