# Effect of the Composition on the Free Energy of Crystal Nucleation for CuPd
Nanoalloys.

Caroline Desgranges  Jerome Delhommelle<EMAIL_ADDRESS>[

###### Abstract

Using molecular simulation, we study the nucleation process from supercooled
liquid alloys of Cu and Pd. The simulations reveal a complex interplay between
the size of the crystal nucleus and its composition that greatly impacts the
crystallization process on the nanoscale. In particular, we find that the free
energy of nucleation strongly depends on the composition of the alloy, with a
free energy barrier for the equimolar alloy that is more than two times larger
than for the pure metals. We attribute this increase in free energy to the
variations in composition occurring both at the surface and in the core of the
nucleus. The local changes in composition are then analyzed by comparing the
surface energies of the two metals and by taking into account the competition
between crystallization and demixing that takes place at the interface between
the nucleus and the surrounding liquid.

University of North Dakota] Department of Chemistry, University of North
Dakota, Grand Forks, ND 58202, USA
(Phone: 701-777-2495)

##  1 Introduction

Crystal nucleation is an incredibly complex 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, yet ubiquitous, phenomenon, common
to many applications in materials sciences. Its complexity stems from the
constant competition between thermodynamics and kinetics 23 that arises during
the formation of a nanoparticle and dictates the size, the crystal structure,
the shape, and, ultimately, the properties of the nanomaterial. As a result,
the mechanisms, at the atomic scale, accounting for the formation of metallic
and bimetallic nanoparticles have remained elusive. This, in turn, limits our
ability to control the properties of nanoparticles 24, which are increasingly
used in applications as e.g. drug carriers 25, DNA/protein markers 26 and for
high-density magnetic recording 27. In recent years, bimetallic nanoalloys 28
and nanoparticles 29 have become the focus of intense research, as synergistic
effects between the two metals often lead to enhanced catalytic properties for
bimetallic nanoparticles. These systems present the additional challenge that
the composition of the crystal nucleus becomes also a key parameter, in
addition to e.g. the size and structure of the crystal nucleus, for the
rationalization of the nucleation process.

The aim of this work is to use molecular simulation to shed light on the
mechanisms underlying the formation of nanosized crystal nuclei from
supercooled liquids of CuPd alloys. We use a many-body force field, known as
the quantum corrected Sutton Chen - Embedded Atoms Model (qSC-EAM) 30, in our
simulations of CuPd alloys. This force field has been previously shown to
model accurately the properties of crystal phases of the Cu and Pd metals and
of several CuPd alloys, as well as the melting temperatures for these systems
30, 31. To analyze the impact of the composition of the alloy on the
nucleation process, we determine the free energy surface associated with the
formation of the crystal nucleus over the entire range of compositions for the
alloy. For this purpose, for each composition, we perform a series of umbrella
sampling simulations 3, 4, 8 to promote the onset of crystalline order and the
formation of a crystal nucleus. We then carry out a detailed analysis of the
structure, the size as well as the composition of the crystal nucleus for each
composition, and focus on elucidating the processes that occur at the surface
of the nucleus, i.e at the interface of the crystal nucleus with the
surrounding liquid, and in the core of the nucleus. This leads us to
rationalize the nucleation process in these binary systems in terms of the
competition that arise between ordering and demixing as observed during the
crystallization of polymer mixtures 32, 33 and, more recently, in metal alloys
and nanoalloys 8, 34.

The paper is organized as follows. In the next section, we present the qSC-EAM
force field used in the simulations to model the CuPd alloys. We also detail
the method with which we perform the simulations for the crystal nucleation
process, the structural analysis carried out to determine the structure and
composition of the crystal nucleus, both at the surface and in the core of the
nucleus. We then discuss the results for the free energy of nucleation and its
dependence upon the composition of the CuPd alloy. We also present the results
obtained for the evolution in size, structure and composition of the crystal
nucleus as nucleation proceeds, and analyze the impact of the properties for
the two components of the alloy, i.e. their relative sizes as well as the
relative values for their cohesive and surface energies, on the
characteristics of the nucleus. We finally draw the main conclusions from this
work in the last section.

##  2 Simulation methods

###  2.1 Force field for CuPd alloys

Metal alloys are complex to model because of the many-body interactions that
take place in such systems. A strategy that is often used is to employ a force
field based on an embedded atoms approach (EAM) 35, 36, 37, 38. This allows to
simulate large systems of several thousands of atoms over the long simulation
times necessary to give rise to the formation of a crystal nucleus. EAM force
fields include density dependent terms and were initially parametrize to model
the properties of the solid phases of metals. A very successful approach known
as the quantum-corrected Sutton Chen embedded atoms models (qSC-EAM) 36, 30
was shown to accurately model the properties of many transition metals. In
subsequent work, the qSC-EAM potential was also used in simulations of the
solid-liquid transition and of the liquid phase. The results from these
simulations established that the qSC-EAM potential performed very well in
predicting the melting temperatures 30, 39, 31 of single component systems and
binary allows as well as a large number of liquid properties. In this work, we
use the qSC-EAM potential to study the crystallization behavior in CuPd
nanoalloys for different compositions of the mixture. Interactions between
atoms according to the qSC-EAM force field are calculated as the sum of a two-
body contribution and of a many-body term

$U={\sum_{i=1}^{N}\left(\sum_{j\neq i}{1\over
2}\varepsilon_{ij}V(r_{ij})-c_{i}\varepsilon_{ii}(\rho_{i})^{1/2}\right)}$ (1)

where $\rho_{i}$ is the local density associated with an atom $i$ and $r_{ij}$
the distance between two atoms $i$ and $j$. The pair potential is given by

$V(r_{ij})={\left({a_{ij}\over r_{ij}}\right)^{n_{ij}}}$ (2)

and the many-body part by

$\rho_{i}=\sum_{j\neq i}{\left({a_{ij}\over r_{ij}}\right)^{m_{ij}}}$ (3)

To determine the parameters for the interactions between unlike atoms, we use
a set of combining rules which has been shown to perform very well for metal
nanoalloys 39, 31

$\varepsilon_{ij}=\sqrt{\varepsilon_{i}\varepsilon_{j}}$ (4)
$m_{ij}={{m_{i}+m_{j}}\over 2}$ (5) $n_{ij}={{n_{i}+n_{j}}\over 2}$ (6)
$a_{ij}={{a_{i}+a_{j}}\over 2}$ (7)

The parameters used in this work are given in Table 1, with the cutoff
distance set to twice the parameter $a$. The qSC-EAM potential, together with
the combining rules, accurately models the melting temperature ($T_{m}$) of
CuPd alloys across the entire composition range (see Table 2).

Table 1:  Parameters for the qSC-EAM potential. |  $a$ (Å) | $\varepsilon(10^{-2}eV)$ | $c$ | $m$ | $n$
---|---|---|---|---|---
$Cu$ | $3.6030$ | $0.57921$ | $84.843$ | $5$ | $10$
$Pd$ | $3.8813$ | $0.32864$ | $148.205$ | $6$ | $12$

###  2.2 Simulating crystal nucleation in CuPd nanoalloys

We simulate the crystallization process from supercooled liquid phases of the
CuPd nanoalloy. Crystal nucleation is an activated process, as the system has
to overcome a large free energy barrier of nucleation to form a crystal
nucleus of a critical size. According to classical nucleation theory 40, the
critical size corresponds to the size necessary for two contributions, of
opposite signs, to the free energy of the system to cancel out. The first,
energetically favorable, contribution is the ’conversion’ of the metastable
phase (here, the supercooled liquid) into the stable phase (the crystal
phase), while the second, unfavorable, term is associated with the cost of
creating (and increasing the area of) the interface between the crystal
nucleus and the surrounding liquid. As shown by Turnbull 41 , metals often
exhibit a large free energy barrier of nucleation. As a result, metals and
alloys remain liquid, even if they are cooled far below their melting point
without crystallizing. Previous work has connected this behavior with the
icosahedral order that can often be found in supercooled liquid metals and
that prevents crystallization 42, 43. The large free energy barriers of
nucleation for metals and alloys implies that, even for large supercoolings,
the spontaneous formation of a critical nucleus cannot take place during the
time scales spanned by conventional Monte Carlo (MC) or a Molecular Dynamics
(MD) simulations. These methods rely on a Boltzmann sampling scheme, and, as a
result, configurations at the top of the free energy barrier, and thus, of
high free energy are not sampled. Methods have been developed in recent years
to study the nucleation process 44, 3, 45, 46, 47. The non-Boltzmann sampling
technique we use in this work is known as the umbrella sampling method 44.
This method consists of adding a bias potential energy to the total energy of
the system, in order to promote the formation of the crystal nucleus. The bias
potential energy is often chosen to be a harmonic function of a reaction
coordinate, which measures the progress of the system towards the formation of
a crystal nucleus. In this work, we choose as the reaction coordinate the
order parameter 48 $Q_{6}$ as the reaction coordinate. Previous work on
$Q_{6}$ has shown that it is extremely well-suited to study the onset of
crystalline order as discussed in experiments on colloidal suspensions 49 and
in simulations of the crystallization process for many systems 3, 50, 4, 51,
including metals 55, 56, 52, 53, 57, 54, 58. $Q_{6}$ has also been
successfully applied to the study of the crystallization in binary mixtures of
simple fluids 59, 60, as well as of metal alloys 64, 61, 62, 63. We carry out
simulations of the nucleation process on systems of a total of $4,000$ atoms,
for pure Cu and pure Pd, and for three different mole fractions in Pd
($x_{Pd}=0.25$, $x_{Pd}=0.5$ and $x_{Pd}=0.75$). We present in Table 2 the
conditions of crystallization for each of the systems studied in this work at
$P=1$ bar.

Table 2:  Melting temperatures ($T_{m}$) and crystallization conditions ($T_{cryst}$ corresponding to a supercooling of $35$ %) for the different CuPd systems. $x~{}(Pd)$ | $0$ | $0.25$ | $0.5$ | $0.75$ | $1$
---|---|---|---|---|---
$T_{m}~{}(exp)$ 65 | $1356$ | $1405$ |  | $1650$ | $1825$
$T_{m}~{}(sim)$ 31 | $1370$ | $1450$ | $1575$ | $1700$ | $1820$
$T_{cryst}$ | $890.5$ | $942.5$ | $1024$ | $1105$ | $1183$

Throughout the nucleation process, we carry out a detailed structural analysis
of the system to identify which atoms belong to the incipient crystal nucleus
and what type of crystal structure forms as nucleation proceeds. For this
purpose, we first determine the type of environment (liquid-like or crystal-
like) around each atom by keeping track of the correlation between the bond
vectors associated with each atom (in this work, we use a cutoff radius of
$3.3$ Å to determine that two atoms are nearest neighbors). Once an atom has
been identified as having a crystal-like environment, we assign a specific
crystal structure on the basis of another local parameters 48, $w_{6}(i)$ for
each atom $i$. As in previous work, a crystal-like atom $i$ is labelled as
body-centered cubic ($BCC$) if $w_{6}(i)$ is greater than $0$ and is labelled
as closed-packed structure ($CP$) otherwise. We also looked for other types of
crystal structures including the ordered phases $L_{12}$ and $B_{2}$ which
have been reported at low temperature for the $Cu_{3}Pd$ (or $CuPd_{3}$) and
for the CuPd crystals, respectively. However, our analysis only revealed
disordered crystal phases, which is why we only discuss the crystal structure
of the nuclei in terms of the $CP$ and $BCC$ structures. Finally, we also
analyze the evolution of the nucleus in terms of its core and surface atoms.
Core atoms are identified as atoms for which all of the nearest neighbors have
a crystal-like environment. The remaining atoms of the nucleus are then
defined as surface atoms.

##  3 Results and discussion

We start by discussing the results obtained for the free energy of nucleation
of the CuPd nanoalloys. Fig. 1 shows a plot of the free energy barrier of
nucleation as a function of the reaction coordinate $Q_{6}$ for $x_{Pd}=0$
(pure Cu), $x_{Pd}=0.25$, $x_{Pd}=0.5$, $x_{Pd}=0.75$ and $x_{Pd}=1$ (pure
Pd). We observe the following main trends. First, for any value of $x_{Pd}$,
the free energy of the system always starts to increase with $Q_{6}$ before a
maximum is reached. This increase corresponds to the formation of a small
crystal nucleus, which is an unfavorable process due to the predominant cost
of creating the solid-liquid interface. Then, the free energy profile of
nucleation reaches a maximum associated with the formation of the critical
nucleus. The value of the order parameter $Q_{6}^{c}$ for which the critical
nucleus has formed depends on the Pd mole fraction with e.g. $Q_{6}^{c}=0.105$
for $x_{Pd}=0.5$ while it is of about $Q_{6}^{c}=0.08$ for $x_{Pd}=0.25$ or
$x_{Pd}=0.75$ and of about $Q_{6}^{c}=0.06$ for the pure metals. The second
feature of the free energy plot is the increase of free energy as the alloy
starts to approach an equimolar composition. This can best be seen in the case
of $x_{Pd}=0.5$, for which the top of the free energy barrier is reached for a
free energy of $75\pm 5~{}k_{B}T$. This is about $40$% more than the free
energy of nucleation for $x_{Pd}=0.25$ and $x_{Pd}=0.75$ and more than twice
the free energy of nucleation for the pure metals. This result is consistent
with our findings for the $Q_{6}^{c}$ values which are markedly greater for
the alloys than for the pure metals. A greater value for a $Q_{6}^{c}$ means
that the system has to reach a greater level of organization, by forming a
larger crystal nucleus and, as a result, the system will have to overcome a
larger free energy barrier of nucleation to achieve this. This phenomenon is
even more pronounced as we reach the equimolar composition for the alloy,
resulting in the highest free energy barrier obtained in the middle of Fig. 1.
We add that this result is consistent with previous work on the AgPd mixture,
which also revealed an increase in the free energy barriers of nucleation for
AgPd alloys 8.

Figure 1: Free energy of nucleation (/$k_{B}T$) of CuPd nanoalloys against the
mole fraction in Pd, $x_{Pd}$, and the order parameter $Q_{6}$. The plot shows
that the free energy increases as we move towards a mole fraction of $0.5$.

Several reasons can potentially account for this steep increase of the height
of the free energy barrier for the alloy. A first possible explanation lies in
the difference between the sizes for the two atoms, since we have $a=3.603$ Å
for Cu and $a=3.8813$ Å for Pd. The difference between the sizes of the two
types of atom composing the alloy may result in a mismatch during the
nucleation process and hinder the formation of the thermodynamically preferred
$CP$ structure, potentially leading to higher free energy barrier of
nucleation for the alloy. The second factor leading to the greater barrier
observed for the alloys, may result from the energetics for the formation of
the crystal phase. To examine this, we consider the following two quantities,
the cohesive energy and the surface energies. First, comparing the cohesive
energies for Cu and Pd, we find that the qSC-EAM potential predicts a larger
cohesive energy for Pd ($3.89~{}eV$) than for Cu ($3.49~{}eV$). This means
that the ’conversion’ of liquid-like Pd atoms into crystal-like Pd atoms will
result in a greater energetic gain. This can lead to the preferred
incorporation of Pd atoms into the crystal nucleus during the nucleation
process. Such a selection of the composition of the nucleus during its
formation will also lead to a larger free energy barrier of nucleation.
Finally, considering now the relative values for the surface energies, we find
that for Cu, depending on the crystallographic plane considered 30, the qSC-
EAM potential predicts surface energies which are between $1495$ and $1657$
$mJ/m^{2}$, while for Pd exhibits lower surface energies (between $1229$ and
$1416$ $mJ/m^{2}$). This implies that the crystal nucleus will have a lower
surface energy if more Pd atoms are located at the surface, a factor that will
impact the interplay between composition and size of the crystal nucleus.

Figure 2: Fraction of the closed-packed (CP) structures (diamonds) and the
body-centered cubic (BCC) structure (circles) in the nucleus as a function of
its overall size. We show the results for the nanoalloy with $x_{Pd}=0$ in
black, for $x_{Pd}=0.25$ in blue, for $x_{Pd}=0.50$ in green, for
$x_{Pd}=0.75$ in red and for $x_{Pd}=1$ in brown.

To identify the reasons underlying the steep increase in the free energy
barrier for the alloy, we carry out a detailed analysis of the structure and
composition of the nucleus throughout the nucleation process. We show in Fig.
2 the evolution of the fraction for each crystal structure, and in Fig. 3, the
evolution of the composition of the nucleus as nucleation advances. We add
that these two plots display the overall behavior observed for the entire
(surface+core) nucleus. Nucleation starts with the formation of a crystal
nucleus which is predominantly $CP$ but also contains a significant amount of
atoms with a $BCC$ signature (of the order of $20$ % for nuclei containing
less than $50$ atoms). The $BCC$ fraction, however, decreases very quickly
during nucleation and stabilizes around $10$ % for all systems, when the
nucleus reaches a size in excess of $200$ atoms. These results indicate that
regardless of the mole fraction in Pd, the structure of the nucleus is
predominantly $CP$, which tends to show that the structural mismatch due to
the size difference between Cu and Pd atoms does not impact the overall
structure of the nucleus and, as such, that the main cause for the increase in
the free energy of nucleation for the alloys is not due to a structural
difference with the crystal nuclei obtained for pure metals.

Figure 3: Composition of the crystal nucleus as a function of its size (Cu:
solid line and Pd: dashed line). The colors for each composition are assigned
as in Fig. 2.

We now turn to the variations in the composition of the crystal nucleus during
the nucleation process. Fig. 3 shows that, during the early stages of the
nucleation process, the fraction in Pd in the nucleus is always greater than
in the alloy. For instance, for the alloy with $x_{Pd}=0.25$, we find a
fraction in Pd that remains above $30$ % for all sizes of the nuclei below
$100$ atoms. Similarly, for the alloy $x_{Pd}=0.5$, nuclei of less than $50$
atoms are richer in Pd. Even when the value for $x_{Pd}$ is very large
($x_{Pd}=0.75$), we obtain a nucleus that is predominantly composed of Pd
atoms for small sizes ($N<50$). At this stage of the nucleation process, the
nucleus is mainly composed of atoms located at the surface and therefore,
since Pd atoms have a lower surface energy, the nucleus has a high Pd
fraction. Then, we observe for all alloys that the composition changes very
quickly once its size exceeds $50-100$ atoms with a drop in the fraction in Pd
in the nucleus. This is likely due to the nucleus growing and incorporating
the neighboring Cu atoms from the liquid in the immediate vicinity of the
crystal nucleus. This shows that a competition between ordering and demixing
occurs during the nucleation process. Then, we observe an increase in the mole
fraction in Pd in the nucleus (see e.g. the top of Fig. 3 for $x_{Pd}=0.75$).
At the end of the nucleation process, we find that the mole fraction in Pd for
the nucleus is either very close to that of the bulk ($x_{Pd}=0.25$) or
higher, with a Pd fraction in the nucleus of $0.55$ for the system
$x_{Pd}=0.5$ and of $0.8$ in the nucleus for $x_{Pd}=0.75$. This result can be
attributed to the larger cohesive energy of Pd atoms which accounts for the
larger fraction in Pd for larger nuclei. We add, however that the excess of Pd
in the crystal nucleus does not significantly impact the Pd mole fraction in
the remaining liquid. This is due to the small size of the critical nucleus
when compared to the overall system size. For instance, for $x_{Pd}=0.5$, the
Pd fraction in the remaining liquid is of $49.6$ % while it is of $74.3$ %
when $x_{Pd}=0.75$. To ascertain these conclusions, we now move on to the
analysis of the core and of the surface of the crystal nuclei.

First, we examine the structure of the atoms at the surface of the nuclei for
the different composition of the alloys. We observe some subtle deviations
from the behavior discussed in the case of the entire nuclei. As for Fig. 2,
we find that the atoms at the surface are more $BCC$-like at the beginning of
the nucleation process than at the end. However, the evolution of the
structure of the surface towards the $CP$ structure is much slower and much
more gradual. This can be attributed to the greater amount of disorder and the
greater presence of defects at the interface between the nucleus and the
surrounding liquid. This behavior is qualitatively the same for all
compositions of the alloys.

Figure 4: Mole fractions of Cu (solid line) and of Pd (dashed line) as a
function of number of atoms located at the surface of the crystal nucleus.

The composition at the surface of the nucleus closely mirrors the overall
composition of the nucleus during the early stages of the nucleation process.
As a result, the trend observed on the left of Fig. 4 is in line with that
observed for small nuclei in Fig. 3, with nuclei forming first with an excess
of Pd (when compared to the composition of the bulk). This is then compensated
by the inclusion of the Cu atoms from the surrounding liquid. We then see that
the composition of the surface generally tends to be richer in Pd than its
counterpart for the entire nucleus, which confirms the predominant role played
by the lower surface energy of Pd in the nucleation process.

The structure associated with the atoms located at the core of the crystal
nucleus shows some notable differences with that observed for the surface. In
particular, even early in the nucleation process, the structure of the core is
much more of the $CP$ type. Very rapidly, the fraction of the $CP$ structures
exceeds $90$ % (for a number of atoms greater than $100$ in the core) and
becomes greater than $95$ % for all compositions as soon as the size of the
core is beyond $150$ atoms. This means that the surface atoms that were
present in small nuclei (which contain a significant fraction of the $BCC$
structure) quickly convert into the $CP$ structures as they become part of the
core of the nucleus. As for Fig. 2, we observe a similar behavior for all
alloy compositions which point to the fact that the mismatch in size between
the two atoms does not impact strongly the structure of the core of the
nuclei.

Figure 5: Mole fractions of Cu (solid line) and of Pd (dashed line) against
the number of atoms located inside the crystal nucleus.

Fig. 5 shows the variation of the mole fractions in the core as a function of
the the number of atoms inside the nucleus. Since the surface of the nuclei
was initially rich in Pd, we observe a core that starts predominantly with Pd
atoms. This conclusion is found to hold for all alloy compositions. Then, we
observe behaviors that depend on the composition of the alloy. For
$x_{Pd}=0.25$, we find that the drop in the mole fraction in Pd that results
from the incorporation of the Cu atoms from the liquid in the immediate
vicinity, leads to a core that shows a lower concentration in Pd than for the
bulk. This shows the great significance of the processes occurring at the
surface of the nucleus. The system ’chooses’ to keep a high fraction of Pd
atoms, of lower surface energy than Cu atoms, at the surface. This process is
done at the expense of the energy of the core of the nucleus, as there are
fewer Pd atoms of greater cohesive energy than Cu atoms in the core. For
$x_{Pd}=0.5$ and $x_{Pd}=0.75$, we observe that the drop in the Pd mole
fraction observed for core sizes close to $50$ atoms does not durably impact
the composition of the inside of the nucleus since its composition remains
rich in Pd.

(a) (b) (c) (d) (e)

Figure 6: Snapshots of the critical nuclei obtained for each nanoalloy. (a)
$x_{Pd}=0$, (b) $x_{Pd}=0.25$, (c) $x_{Pd}=0.5$, (d) $x_{Pd}=0.75$ and (e)
$x_{Pd}=1$. The atoms belonging to the crystal nucleus are shown as spheres
(Cu is shown in tan and Pd in silver).

We present in Fig. 6 snapshots of the critical nuclei obtained for each alloy
composition. In each case, the entire simulation cell (i. e. truncated
octahedron) is shown with the atoms belonging to the crystal nucleus shown as
spheres. The critical nuclei are the smallest for the two pure metals, with a
size of $223\pm 39$ total atoms for Cu ($152\pm 25$ atoms at the surface and
$71\pm 14$ atoms in the core) and $243\pm 50$ total atoms for Pd ($154\pm 31$
atoms at the surface and $89\pm 20$ atoms in the core). Then for intermediate
mole fractions, we obtain slightly larger critical nuclei. For $x_{Pd}=0.25$,
we have a critical nucleus containing a total of $285\pm 38$ atoms ($177\pm
19$ atoms at the surface and $108\pm 19$ atoms in the core). For
$x_{Pd}=0.75$, we have a critical nucleus containing a total of $265\pm 41$
atoms ($167\pm 21$ atoms at the surface and $98\pm 20$ atoms in the core). The
largest critical nucleus is observed for the equimolar alloy with a total of
$448\pm 45$ atoms ($250\pm 25$ atoms at the surface and $192\pm 20$ atoms in
the core). The increase in size for the critical nucleus as we approach the
equimolar composition, is correlated with the increase in the free energy of
nucleation. It is also consistent with the greater values for $Q_{6}^{c}$
obtained at the top of the free energy barrier for CuPd alloys when compared
to the pure metals.

##  4 Conclusions

In this work, we have calculated the free energy of crystal nucleation for
CuPd alloys and determined its dependence on the composition of the alloy. Our
simulations reveal that the free energy of nucleation increases significantly
as the mole fraction in Pd approaches $0.5$ with e. g. the alloy with $x_{Pd}$
being associated with a free energy barrier $40$ % higher than the alloys with
$x_{Pd}=0.25$ or $x_{Pd}=0.75$, and more than twice that for the pure metals.
We interpret this increase in the free energy of nucleation in terms of their
relative properties, i.e. the relative size of the atoms composing the alloy
and their relative surface and cohesive energies. The simulations shed light
on the predominant role played by the processes occurring at the surface of
the developing crystal nucleus. In particular, the results show that the
composition at the surface of the nucleus varies significantly during the
early stages of the nucleation process, starting with nuclei rich in the
element of lower surface energy (Pd). Then, as a result of the depletion in Pd
that occurs for the liquid in the immediate vicinity of the crystal nucleus,
the composition at the surface of the nucleus undergoes a sharp change with
the rapid incorporation of Cu atoms in the nucleus. These variations in
composition continue throughout the nucleation process, illustrating the
competition between crystallization and demixing in the liquid alloy during
the nucleation process. This has important consequences on the composition of
the nanoparticle, both at the surface and in its core, and will likely impact
its properties for applications e.g. in catalysis.

Partial funding for this research was provided by NSF through CAREER award
DMR-1052808.

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