# The phase stability of large-size nanoparticle alloy catalysts at ab initio
quality using a nearsighted force-training approach

Cheng Zeng† †School of Engineering, Brown University, Providence, Rhode
Island, 02912, United States.
‡ School of Chemical and Biomolecular Engineering, Georgia Institute of
Technology, Atlanta, Georgia 30318, United States.
*Corresponding author: Email<EMAIL_ADDRESS>Tel: +1 401-863-2153
Sushree Jagriti Sahoo‡ †School of Engineering, Brown University, Providence,
Rhode Island, 02912, United States.
‡ School of Chemical and Biomolecular Engineering, Georgia Institute of
Technology, Atlanta, Georgia 30318, United States.
*Corresponding author: Email<EMAIL_ADDRESS>Tel: +1 401-863-2153
Andrew J. Medford‡ †School of Engineering, Brown University, Providence, Rhode
Island, 02912, United States.
‡ School of Chemical and Biomolecular Engineering, Georgia Institute of
Technology, Atlanta, Georgia 30318, United States.
*Corresponding author: Email<EMAIL_ADDRESS>Tel: +1 401-863-2153
Andrew A. Peterson${}^{{\dagger},\text{*}}$ †School of Engineering, Brown
University, Providence, Rhode Island, 02912, United States.
‡ School of Chemical and Biomolecular Engineering, Georgia Institute of
Technology, Atlanta, Georgia 30318, United States.
*Corresponding author: Email<EMAIL_ADDRESS>Tel: +1 401-863-2153

###### Abstract

Co–Pt alloyed catalyst particles are integral to commercial fuel cells, and
alloyed nanoparticles are important in many applications. Such systems are
prohibitive to fully characterize with electronic structure calculations due
to their relatively large sizes of hundreds to thousands of atoms per
simulation, the huge configurational space, and the added expense of spin-
polarized calculations. Machine-learned potentials offer a scalable solution;
however, such potentials are only reliable if representative training data can
be employed, which typically also requires large electronic structure
calculations. Here, we use the nearsighted-force training approach that allows
us to make high-fidelity machine-learned predictions on large nanoparticles
with $>$5,000 atoms using only small and systematically generated training
structures ranging from 38–168 atoms. The resulting ensemble model shows good
accuracy and transferability in describing the relative energetics for Co–Pt
nanoparticles with various shapes, sizes and Co compositions. It is found that
the fcc(100) surface is more likely to form a L10 ordered structure than the
fcc(111) surface. The energy convex hull of the icosahedron shows the most
stable particles have Pt-rich skins and Co-rich underlayers, and is in
quantitative agreement with one constructed by brute-force first-principles
calculations. Although the truncated octahedron is the most stable shape
across all sizes of Pt nanoparticles, a crossover to icosahedron exists due to
a large downshift of surface energy for CoPt nanoparticle alloys. The
downshift can be attributed to strain release on the icosahedron surface due
to Co alloying. We introduced a simple empirical model to describe the role of
Co alloying in the crossover for Co–Pt nanoparticles. With Metropolis Monte-
Carlo simulations we additionally searched for the most stable atomic
arrangement for a truncated octahedron with equal Pt and Co compositions, and
also we studied its order–disorder phase transition. We validated the most
stable configurations with a new highly scalable density functional theory
code called SPARC. From the outermost shell to the center of a large Co–Pt
truncated octahedron, the atomic arrangement follows a pattern: Pt
$\rightarrow$ Co $\rightarrow$ L12(Pt3Co) $\rightarrow$ L12(PtCo3)
$\rightarrow$ L10(PtCo) $\rightarrow$ $\cdots$ $\rightarrow$ L10(PtCo).
Lastly, the order–disorder phase transition for a Co–Pt nanoparticle exhibits
a lower transition temperature and a smoother transition, compared to the bulk
Co–Pt alloy.

Keywords: Co–Pt nanoparticles, Nearsighted force training, Global
optimization, Morphology crossover, Order–disorder transition

TOC Graphic:

##  1 Introduction

Bimetallic nanostructures have received increasing attention in the past two
decades [1, 2, 3]. Among the family of bimetallic nanostructures, Co–Pt
nanoparticles have a wide range of applications in heterogeneous catalysis and
magnetic storage [4, 5, 6, 7]. Co–Pt nanoparticles have been commercialized as
electrocatalysts in fuel-cell electric vehicles such as Toyoto Mirai [8, 9].

The size, composition, shape, and orderliness of Co–Pt nanoparticles all play
a pivotal role in controlling the structure and hence the chemical and
physical properties, including the catalytic activity. We provide several
examples below, and note they are not all in agreement with one another. For
example, Li et al. have shown that a core-shell Co–Pt nanoparticle with an
ordered core loses fewer Co atoms in electrocatalysis applications than that
with a random core [4]. Others have shown the magnetic and optical properties
are closely associated with the orderliness of the structure [10, 11]. Yang et
al. suggested, based on Monte-Carlo simulations, that disordering is initiated
at the surface due to the reduced coordination, hence lowering the ordering
temperature [12]. Müller and Albe pointed out that surface segregation of one
element can have a large impact on the ordering [13]. Alloyeau et al. showed
in both experiments and simulations that the shape and size affect the
order–disorder transition, indicating that a larger nanoparticle tends to show
a higher transition temperature and the size effect is uniquely determined by
the smallest length of a nanoparticle [14]. Alloyeau’s large-scale simulation
was based on tight-binding potentials fitted to experimental and first-
principles calculations, and the most stable structure for an equal-
composition nanoparticle was identified to be a fully L10 ordered truncated
octahedron [14]. However, first-principles calculations by Gruner et al.
suggested that the most stable shape of a Co–Pt nanoparticle at small sizes is
not the L10 ordered regular truncated octahedron, but a multiply twinned
icosahedron.

Pure Pt nanoparticles are normally stable in the single crystal structure,
whereas Pt alloy nanoparticles can exist in multiply-twinned structures such
as an icosahedron [15, 16, 17, 18], although the shape of Pt nanoparticles can
be controlled by capping materials [19]. It is thus crucial to understand the
characteristics of structure motifs, such as icosahedron and octahedron. An
icosahedron is created by packing twenty tetrahedra in a matter that they
share a common vertex, leading to close-packed surfaces, but distorted
tetrahedra. (See Figure 1.) This distortion leads to a high internal strain
with a relatively low surface energy [20]. Thus, icosahedra are normally more
stable at small sizes, where the surface energy prevails over the volume
contribution [21].

In contrast, an octahedron structure preserves the bulk lattice symmetry—it
can be obtained by directly cutting a single crystal. The normal octahedron
only has fcc(111) facets and has no internal strain. To lower the total
energy, the six “tips” of the octahedra can be removed, creating a truncated
octahedron [22] (Figure 1). Although this raises the energy per surface area
by creating six fcc(100) facets, it lowers the total amount of surface area,
which suggests it may be more stable at larger sizes [23]. The cuboctahedron
can be conceptualized in similar way: starting with a cube with six fcc(100)
facets, the eight corners are cut off, exposing eight new triangular fcc(111)
facets (Figure 1).

The trade-off between surface and volume contributions lead to possible
crossovers in stability among structural motifs as the particle size changes;
for pure nanoparticles, past researchers have used simple empirical
thermodynamic models to describe this phenomenon [24, 25, 22, 23]. In the case
of bimetallic nanoparticles, crossovers among various shapes have rarely been
reported. Zhu et al. used an empirical model fit to density functional theory
(DFT) calculated properties to show that alloying of Pd can extend the
stability of icosahedron Pd–Au nanoparticles beyond that of pure Au
nanoparticles because of the stress release when two different metals are
mixed [26]. Since the empirical potential was fit to a small number of
properties, the prediction accuracy over a large range of different
nanoparticle structures is uncertain.

Figure 1: Atomic configurations and geometrical shapes of nanoparticles in
varioius shapes.

The crossovers in bimetallic nanoparticles are not likely to be fully
understood without reliable descriptions of the atomic interactions. In an
ideal world, such studies would employ electronic structure calculations
directly. DFT offers a good compromise between accuracy and computational
cost. However, model systems of practically-sized nanoparticles, with sizes of
1,000–10,000 atoms, are notoriously expensive for even single calculations,
due to the famous $\mathcal{O}(N^{3})$ scaling, where $N$ indicates the system
scale, such as the number of electrons, number of atoms or number of basis
functions. For nanoparticle structure exploration, the picture is more grim
due to the huge combinatorial space that must be explored to describe the
range of sizes, crystal structures, facets, and alloy (dis-)ordering that may
be encountered. Further, as a magnetic system, Co–Pt particles have an added
expense in electronic structure due to the requirement for spin-polarized
calculations. These combinations of factors make the rigorous exploration of
Co–Pt structure–function relationship out of reach for methods like DFT.

Machine-learned inter-atomic potentials have gained momentum in fitting
potential energy surfaces calculated by ab initio calculations [27, 28, 29,
30, 31, 32, 33]. However, a fundamental problem arises in using atomistic
machine learning to perform large-scale simulations: for high-fidelity
potentials, the training data should closely resemble the ultimate structures
being predicted, and the generation of such training data for nanoparticles
can be very costly, as described above. To circumvent this issue, we recently
released a “nearsighted force-training” (NFT) [34] approach to generate small-
size training data to systematically learn the forces and energies of large
structures. In this approach, a bootstrap ensemble [35] (or any other
reasonable uncertainty metric) is used to identify the most uncertain atoms in
a particular structure. Atomic “chunks” centered on these uncertain atoms are
removed, and calculated at large enough size that the central atom’s force can
be calculated with fidelity by DFT. Only the force on the central atom is used
in the loss function, thus adding targeted data and avoiding noise associated
with boundary atoms. We demonstrated that this approach successfully built and
relaxed nanoparticles containing up to 1,415 atoms in previous work.

Here we apply the NFT approach to build a robust ensemble model for Co–Pt
nanoparticles. Using these well-validated neural network models, we optimize
the structures of Co–Pt alloys of simple bulk, fcc(111) surfaces, fcc(100)
surfaces, icosahedron and octahedron nanoparticles of various sizes and
compositions. Moreover, we aim to address a number of key problems regarding
the thermodynamic stability of Co–Pt nanoparticles, including the crossover
among Pt and Co–Pt structure motifs, the most stable atomic arrangement of a
Co–Pt truncated octahedron, and the order–disorder phase transition of Co–Pt
truncated octahedron.

##  2 Methods

###  2.1 Model creation with nearsighted force training

We used the nearsighted-force training (NFT) approach [34] to generate the
machine-learning model without the need for large-sized training data, which
is expensive to obtain. In this method, we started with a small training set
consisting of 18 Co–Pt bulk structures calculated in density functional theory
(DFT), sampled following the initialization procedure we previously
described.[34] We trained a bootstrap ensemble [35] of Behler–Parinello-type
machine-learning models to this minimal training set; details of the model
parameters are described later. When this ensemble model is applied to a new
structure, it gives a systematic estimate of the uncertainty on each atom in
that structure.

We started the NFT active-learning procedure on a “rattled” Pt192Co68
cuboctahedron. That is, we used the ensemble to identify the most uncertain
atom in the structure, then we pulled out a “chunk” of this atom and its
neighbors within a cutoff distance of 8 Å. This chunk was calculated in DFT,
and solely the force on the central atom was added to the training set. The
ensemble was re-trained, and the procedure was repeated on the next most
uncertain atom. We terminated this process after 9 NFT steps, because the
uncertainty did not improve in two consecutive steps, and we wanted our model
to be optimized for low-force structures.

Next we performed a relaxation on this structure, and used the NFT model to
extract uncertain chunks along the relaxation trajectory, with the procedure
we described earlier [34], leading to a systematic improvement of the
description of this relaxed particle. We generally stopped the active learning
process when any of three criteria were met: the uncertainty was below the
convergence criterion, the number of retraining steps exceeded a pre-defined
number, or the target uncertainty was not lowered for two continuous steps.

After we completed the NFT procedure on the Pt192Co68 particle, we continued
to improve our model by training on a variety of octahedron and icosahedron
nanoparticles, allowing us to find atomic chunks representing diverse local
chemical environments that are unique and informative for potential energy
surfaces of Co–Pt nanoparticles. The icosahedron nanoparticles include Pt1415,
fully disordered Pt736Co679, CoPt/2Pt core-shell with a disordered core
(Pt1150Co265), CoPt/1Pt core-shell with a disordered core (Pt3092Co1991), and
CoPt/2Pt core-shell with a disordered core (Pt3694Co1389). The cuboctahedron
nanoparticles include Pt192Co68 with a disordered core, a Pt1415, a core-shell
CoPt/Pt cluster Pt975Co440 with a disordered core, a fully L10 ordered Co–Pt
cluster Pt736Co679 and a fully disordered Co–Pt cluster Pt736Co679. In total,
2064 atomic “chunks” in sizes from 38 to 168 atoms were extracted from those
nanoparticles. Adding the 18 bulk cells, we had in total 2082 training images.

The statistics of these 2064 generated atomic chunks are discussed in the SI,
including the distributions of forces on the central atom, number of atoms,
force prediction residuals and energy prediction residuals. The nanoparticle
trajectories, training images (atomic chunks), force and energy ensemble
models are included as supporting datasets.

###  2.2 Model structure

We took a bootstrap approach [35] to sample the training images for each
neural network model, our ensemble consisted of 10 neural network models.
Ensemble averages were used as the predictions for both energy and forces.
Atomic uncertainties were proportional to standard deviations of the ensemble
force predictions, as used in Zeng et al.. [34] Each member of neural network
models is a Behler-Parrinello type. [27] Neural network models were
constructed using an open-source atomistic machine learning package (AMP)
developed in our group. [36] Fast force and energy inferences were carried out
with n2p2, which is mainly developed by Singraber et al..[37] Fast
fingerprinting was performed with SIMPLE-NN implemented with AMPTorch. [38,
39] Gaussian symmetry functions with a cutoff radius of 6.5 Å were used to
encode the local chemical environments. For the feature vector of Co elements,
it consists of 12 G2 and 8 G4 symmetry functions, where Pt feature vector
comprises 13 G2 and 7 G4 symmetry functions. Full details of the symmetry
functions are included in the supporting data set in the JSON format. 3000
epochs were used for training. A simple structure of (20, 5, 5, 1) was
employed for the neural network topology to mitigate overfitting. A L2
regularization (L${}_{2}=0.001$) was used to avoid large atomic neural network
weights, hence alleviating the overfitting. We trained both a force ensemble
model and an energy ensemble model on our 2082 images, referred to hereafter
as the force model and energy model, respectively. This was done in order to
allow our model sizes to be smaller and our training procedures to be faster,
with each individual model optimized for its own prediction of interest. The
force model was trained on forces of central atoms of atomic chunks and on
both energy and forces of bulk cells. The energy model was trained on the
total energies of the bulk cells and atomic chunks.

###  2.3 Electronic structure calculations

DFT calculations for bulk, atomic chunks and 201-atom nanoparticles were
carried out with the GPAW code [40]. The Perdew–Burke–Ernzerhof (PBE)
exchange–correlation functional with a plane wave cutoff of 350 eV was used
[41]. To achieve a fast convergence, a Fermi-Dirac smearing of 0.1 eV was
utilized and the energetics were extrapolated to 0 K. Calculations for atomic
chunks were sampled at the $\Gamma$-point of the Brillouin zone, where
calculations for bulk cells used a k-point grid of 12$\times$12$\times$12\.
For atomic structures including cobalt, spin polarization was included. When
only platinum was present, the calculations were spin-paired. The lattice
constant of bulk Pt was found to be 3.936 Å. Atomic chunks were placed in a
non-periodic box where the shortest distance to the box wall is at least 5 Å.
Self-consistent field (SCF) calculations were considered to be converged when
the energy difference between the last three steps is less than 0.0001
eV/electron. Structure optimizations used a MDMin algorithm until the maximum
atomic force was not larger than 0.05 eV/Å.

DFT calculations of the 586-atom particles were done using the highly parallel
“Simulation Package for Ab-initio Real-space Calculations” (SPARC) code.[42,
43, 44] To the best of our knowledge, this calculation represents the largest
size of Co–Pt nanoparticles that has ever been studied directly with DFT. A
mesh spacing of 0.13 Å (0.25 Bohr radii) was used in a $\Gamma$-point
calculation with the PBE functional and the PseudoDojo pseudopotentials [45],
and all calculations were run until the energy converged to within $2.7\times
10^{-4}$ eV/atom ($1\times 10^{-5}$ Ha/atom). Atomic forces were computed and
compared with GPAW results for smaller systems, which showed the mean absolute
force error between the codes is below 0.025 eV/Å for all systems tested (see
Figure S3 in the SI). Particles were surrounded by 3.5 Angstrom of vacuum in
each direction with Dirichlet boundary conditions in all directions.

Note that unlike our prior publication [34], in this work we dealt with a
magnetic system. The underlying assumption for NFT to be applicable for
magnetic systems is that in the electronic ground state there exists a unique
mapping from atomic positions to spin states (or magnetic moments) which also
display a strong locality. In general, forces may depend on the initial guess
of magnetic moments, since a poor initial guess may lead to a different local-
minimum spin configuration. Thus, we have used a consistent initial-guess
strategy for all calculations, with an initial magnetic moment of 0
$\mu_{\mathrm{B}}$ for Pt atoms and 2.1 $\mu_{\mathrm{B}}$ for Co atoms.

###  2.4 Global optimization techniques

We aimed to explore a wide range of the potential energy surface with an
emphasis on structures of nanoparticles in various shapes. We were
particularly interested in the global minima for a given shape, size and alloy
composition. However, searching global minima using a brute-force approach is
computationally prohibitive. For example, if we consider a small fixed-shape
147-atom nanoparticle with 73 Co and 74 Pt atoms, the number of possible
atomic arrangement is already as large as $147!/(73!\times 74!)\approx
10^{44}$. Although symmetry can reduce the complexity, attempting to exhaust
the search space is inaccessible even with machine-learned potentials,
especially for large nanoparticles in the size of thousands of atoms. Instead,
we used global optimization techniques; specifically genetic algorithms and
Metropolis Monte-Carlo simulations.

####  2.4.1 Genetic algorithms

Genetic algorithms, inspired from evolutionary theory, have become popular in
the past two decades for optimizing structures [46, 47, 48, 49, 50]. The
genetic algorithm was performed on both Co–Pt surfaces and a type of Co–Pt
icosahedron to construct energy convex hulls. The genetic algorithm was set up
with the Atomic Simulation Environment (ASE) [51] based on the procedure
implemented by Lysgaard et al. [49] and Bossche et al. [50]. We used a rigid
structure; that is, all derived structures are not allowed to relax.
Structural relaxation was performed afterwards if needed. To make comparisons
between different compositions, we define the negation of the mixing energy
$E_{\mathrm{f}}(\mathrm{Pt}_{x}\mathrm{Co}_{y})$ of a structure
$\mathrm{Pt}_{x}\mathrm{Co}_{y}$ as the fitness score to propagate the
algorithm:

$E_{\mathrm{f}}(\mathrm{Pt}_{x}\mathrm{Co}_{y})={E}(\mathrm{Pt}_{x}\mathrm{Co}_{y})-\frac{x}{x+y}{E}(\mathrm{Pt}_{x+y})-\frac{y}{x+y}{E}(\mathrm{Co}_{x+y})$
(1)

where $E(\mathrm{Pt}_{x}\mathrm{Co}_{y})$, $E(\mathrm{Pt}_{x+y})$, and
$E(\mathrm{Co}_{x+y})$ denote the ML model calculated per-atom energies of the
corresponding structures. We first studied fcc(111) and fcc(100) surfaces. For
the fcc(111) and fcc(100) surfaces, we used a $4\times 4\times 5$ supercell in
size of 80 atoms with 13 Å separation between slabs in the direction
orthogonal to the surface. The initial generation was populated with 120
surfaces using randomly chosen compositions. The lattice constants of pure Co
and Pt slabs were determined by the force model, and for the mixed slabs, the
lattice constant is linearly interpolated based on Vegard’s law. We used three
ASE operators to create the next generation. “CutSpliceCrossover”, as
introduced by Deaven and Ho [46], takes two parent structures, then cuts them
in a random plane and combine the halves from two parent slabs together to
form an offspring. The second operator “RandomSlabPermutation” was used to
randomly permute two atoms of different types. The last
“RandomCompositionMutation” changes the composition of the slab by mutating
one element to the other. The probability of the above three operators are
respectively 0.6, 0.2, and 0.2. In addition, we used a variable function named
“RankFitnessPopulation” to uphold the composition diversity at each generation
so that optimization is on a full range of compositions. We ran for 100
generations.

This approach was also used to build the convex hull of a 147-atom Co–Pt
icosahedron nanoparticle. The initial generation was populated with 100
members, the composition was randomly chosen, and the lattice constant was
obtained by a linear interpolation. For the icosahedron nanoparticle, we were
interested in the fittest Co/Pt composition, hence we did not restrict the
algorithm to keep a wide range of compositions in each generation. For the
nanoparticles, four types of operations were utilized to create the offspring,
including “CutSpliceCrossover”, “RandomSlabPermutation”, “MirrorMutation” (to
mirror half of the cluster in a randomly oriented cutting plane while
discarding the other half) and “SymmetricSubstitue” (to permute all atoms
within a shell of the symmetric particle), and the corresponding operation
probabilities are 3/6, 1/6, 1/6, 1/6, respectively. This was also run for 100
generations. After the runs were completed, we chose to study structures after
the 80th generation. We pinpointed the Co–Pt icosahedron with the most
negative formation energy, which is named as the fittest Co–Pt icosahedron. We
calculated the $c_{\ce{Pt}}$ of the fittest structure, and we selected
structures whose mole fractions of Pt are close to the fittest one based on a
Gaussian function. In total, 232 Co–Pt icosahedron nanoparticles were
selected, and energetics of their relaxed structures were used to construct
the energy convex hull for the 147-atom Co–Pt icosahedra.

###  2.5 Metropolis Monte Carlo simulations

Previous works suggested that while a genetic algorithm was more efficient to
search a wide range of compositions, Metropolis Monte Carlo simulations were
found to be more effective for structures with fixed compositions [52, 13,
12]. We thus employed such Metropolis calculations, in the canonical ensemble.
At each elementary Monte Carlo step, two neighboring atoms were exchanged and
the energy change was calculated for the exchange. The new structure was
accepted if the energy change was negative, or it is accepted based on the
Boltzmann probability if the energy change was positive. The number of Monte
Carlo steps was determined in a way that on average at least 40 swaps were
performed for each atom in the structure. We used these simulations at 300 K
to find the putative global minima of truncated octahedra Pt96Co105 and
Pt300Co286, and we compared them to the fully L10 ordered counterpart at full
DFT levels of theory. We also used these simulations at temperatures ranging
from 300 K to 1800 K to study the order–disorder phase transitions for Co–Pt
bulk and nanoparticles with nearly equal compositions of Co and Pt. For each
Monte Carlo trajectory at a given temperature, the order parameter was
calculated as the average over configurations after a burn-in period.

##  3 Results & Discussion

The key objective of this study is to train robust machine learning models
that can predict stable structures of nanoparticles, which we will use to
distinguish the phase stability of bimetallic nanoparticles with various
shapes and atomic arrangements.

###  3.1 Comparison to literature structures and ab initio calculations

We first validated our ML models by a comparison with published ab initio
calculations; additional validation calculations are reported in Sections 3.2
and 3.4 and comparisons to literature-reported structures and trends are
contained in Sections 3.2, 3.3, and 3.4.

Gruner et al. used first-principles calculations to compare the energetics of
several structural motifs of Co–Pt alloy nanoparticles in reference to an L10
ordered cuboctahedron [53]. We created a number of 561-atom nanoparticles for
Pt296Co265 that are either identical or close in atomic arrangements to those
used in Gruner’s ab initio calculations, since the exact structure was not
always reported. Different randomness should play a negligible role in the
energetics because only a small fraction of atoms are randomly positioned. The
nanoparticles included an L10 ordered cuboctahedron, a disordered icosahedron,
an icosahedron with alternating Co and Pt shells, and a core–shell icosahedron
with a Co-rich second shell. We relaxed these structures with the force model
and then we calculated the energetics with the energy model. The comparison
between ML predictions and ab initio calculations by Gruner et al. is shown in
Figure 2. We note that the work of Gruner et al. used a smaller cutoff (268
eV) than the one (350 eV) we used for DFT calculations on atomic chunks, which
may account for some discrepancy to the literature. One can see a very good
agreement for both cuboctahedron and icosahedron nanoparticles, and the
overall order for all structures presented is exactly captured by the ML
models, with the mean ensemble prediction agreeing very well and the parity
line within the error bars.

In addition, we created a 147-atom Pt icosahedron and cuboctahedron, relaxed
it with the ML model, and compared the absolute energy difference to that
obtained by DFT calculations we performed in the GPAW calculator. The DFT and
ML-predicted energies for both structures are presented in Figure S4 of SI.
Although the exact energetics for each shape can differ by 13.6–27.2 meV/atom
between ML predictions and DFT calculations, the relative energy difference
between those two shapes is much closer; Pt icosahedron is more stable than Pt
cuboctahedron by 7 meV/atom using ML models versus 8.6 meV/atom using DFT
calculations. This suggests that the ML models are able to distinguish the
thermodynamic stability across various shapes of nanoparticles, and different
atomic arrangements for a given shape.

Figure 2: Energetics of Pt296Co265 nanoparticles in various shapes in
reference to the L10 ordered cubocahedron: ML predictions versus DFT
calculations by Gruner et al. [53]. The DFT calculations used a smaller cutoff
(268 eV) compared to 350 eV used to obtain the training data for ML models.
The error bar represents the ensemble halfspread as defined by Peterson et al
[35].

###  3.2 Energy convex hull of Co–Pt surfaces and nanoparticles

We next turn to studying the relative stability of alloyed structures, by
using these potentials to construct “convex hulls” that plot the alloy-
formation energy versus composition. Negative energies indicate the alloy is
stable relative to the pure components.

We began by examining the formation energy of two ordered bulk alloys, PtCo
and Pt3Co. The formation energy for PtCo and Pt3Co are -0.24 and -0.14
eV/atom, respectively, close to values by experiments and empirical potentials
[54, 11]. As a comparison, the DFT-calculated formation energies for PtCo and
Pt3Co are -0.10 and -0.06 eV/atom, respectively. This indicates that Co and Pt
atoms have a strong tendency of being mixed.

Figure 3: Energy convex hulls of fcc(100) (a) and fcc(111) (b) surface PtCo
alloys as a function of Co compositions. Solid squares are the stable
structures and circles refer to unstable structures. Top and side views of the
fittest configurations are shown on the right.

To gain insights into the atomic arrangement near a Co–Pt surface, we built
energy convex hulls for 5-layer Co–Pt fcc(100) and fcc(111) surfaces using the
neural-network-enhanced genetic algorithm. Figure 3(a) shows the energy convex
hull for fcc(100), where $x_{\rm{Co}}$ represents the mole fraction of Co
atoms. We extracted the fittest fcc(100) surface (that with the lowest
formation energy), and its composition is around $x_{\rm{Co}}=0.4$. A side
view of the global minima, shown in the figure, implies that it is an L10
ordered structure, forming alternating Pt and Co layers with the outermost
layer being Pt. Although the convex hull of fcc(100) is not symmetric, we can
infer that this lack of symmetry is constrained by the number of layers (5),
and if it were increased it would approach a more symmetric form. The
structure at $x_{\rm{Co}}=0.2$ is close to L12 ordered, which suggests the
atomic arrangement in a fcc(100) surface is inclined to form an ordered
structure. We anticipate that compositions of stable structures may change if
we increase the thickness of the surface where bulk contributions become more
dominant (e.g., global minima closer to 0.5 for an infinitely thick surface).
Yet atomic arrangement patterns, in particular near surfaces, should hold for
thicker surfaces. We will see this to be the case when we examine large
nanoparticles.

In comparison, the convex hull of fcc(111) surface alloys is much smoother,
representing more flexibility in atomic arrangements in this more closely-
packed facet. In terms of the fittest configuration Pt41Co39, the general
trend still holds that Pt tends to segregate at the surface while being
depleted at the subsurface. A small amount of Co shows up at the surface,
although Co–Co direct connections are not present. In brief, the main features
of atomic arrangements for both fcc(100) and fcc(111) surfaces are a surface
Pt layer and a subsurface Co layer. Besides, the fcc(100) surface is more
likely to form an ordered structure than the fcc(111) surface.

Figure 4: Energy convex hull of a 147-atom Co–Pt Ih as a function of Co
compositions. Solid squares are the stable structures found by ML models, and
circles refer to unstable structures of ML predictions. Up-triangles refer to
the stable structures excerpted from Noh et al. [55].

Next we turned to build the convex hull of a 147-atom Co–Pt icosahedron.
Figure 4 shows the formation energy for this system as a function of mole
fractions of Co atoms, centered on the Pt-rich region where the global minimum
lies. The convex hull from a DFT-based study conducted by Noh et al. [55] is
also included for comparison. The trend of formation energy versus composition
demonstrates that the ML-predicted convex hull agrees very well with the ab
initio results. The prediction discrepancy of formation energies is less than
0.03 eV/atom across the composition range shown in Figure 4. The discrepancy
could likely be reduced by adding additional atomic chunks extracted from
uncertain 147-atom Co–Pt icosahedra—here, we did not seek such an improvement
since we considered the prediction accuracy to be satisfactory. On the Pt-rich
side (increasing from $x_{\ce{Co}}=0$), the formation energy rapidly drops
with the addition of Co. In contrast, there exists a wide flat region
($0.3\leq x_{\rm{Co}}\leq 0.5$) where either addition or removal of Co atoms
barely changes the formation energy.

The most significant feature (also reported by Noh et al.) along the convex
hull is the formation of a Pt skin on the surface with a Co-rich layer
directly below the Pt skin. Among the eight Co–Pt icosahedra along the hull in
Figure 4, seven structures are covered by a full Pt skin, with the exception
being the one with the lowest overall platinum composition
($x_{\ce{Co}}=0.44$), where the skin contains 78% Pt. The average Co
composition of the second shell is 74%.

It is well-known in fuel-cell catalysis that Pt–Co alloyed catalysts exhibit a
platinum skin, which is generally considered to be formed by dealloying of the
cobalt under electrochemical conditions, due to the difference in
electrodeposition potential between Co and Pt. The Pt skin is contracted
relative to what would be found on a pure-Pt particle, which has been
concluded to increase the catalyst’s activity. These results show that Pt is
also thermodynamically most stable on the surface (for a fixed Pt:Co
composition). This may suggest a greater long-term stability to these
catalysts than if the skin layer were present due to cobalt dissolution alone.

On the left side of the convex hull, the ML model identifies a stable
structure with composition Pt83Co64, where Co atoms on the surface occupy the
center of fcc(111) surfaces. In this same region, Noh et al.’s calculations
show a structure with composition Pt80Co67, where Co atoms on the surface sit
at the corners. To validate whether the center occupancy represents a stable
atomic arrangement, we constructed icosahedron structures with stoichiometry
Pt80Co67 where surface Co atoms occupy both types of sites, and compared the
energies with both DFT and ML calculators. In both cases, the corner occupancy
was predicted to be more favorable, with DFT energies showing a difference of
$\sim$14 meV/atom, which indicates that the terrace center occupancy is a low-
energy state as well. However, this configuration was not captured in the DFT
calculations by Noh et al. [55]. The configurations of two types of Pt80Co67,
together with their ML and DFT energies, are provided in Figure S5 of SI.
Since both calculators correctly show the corner site to have lower energy,
this indicates that the two procedures captured different minima structures
purely by the stochasticity of the genetic algorithm itself, and not due to an
issue with the ML fidelity.

###  3.3 Crossovers among morphology in Pt and Co–Pt nanoparticles

##### Platinum particles.

In this section, we aim to provide physical insights into the distinct
crossover behavior of Co–Pt nanoparticles. We first focus on pure Pt
nanoparticles, where the crossover between different shapes has been
extensively investigated based on well-parameterized empirical potentials [22,
23, 56]. The energies of each structure in such studies were fit to an
empirical thermodynamic model, dividing the total potential energy ($U$) for a
nanoparticle of a specific shape into contributions from volume, surface, and
edges:

$\frac{U}{N}=A+BN^{-1/3}+CN^{-2/3}$ (2)

where $N$ is the total number of atoms, and $A$, $B$, and $C$ are parameters
corresponding to the volume, surface, and edge contributions, respectively.
These parameters are unique to each nanoparticle shape. As $N$ increases, the
edge contribution becomes less important, and we will show that this term can
be dropped in our size range of interest.

Discrepancy exists in the literature, even for the crossover of pure Pt
nanoparticles. For example, Uppenbrink et al.concluded that the crossover
between icosahedron and cuboctahedron occurs at around 550 atoms for both pure
Pt and pure Pt nanoparticles, and a decahedron is found in a narrow range of
sizes [22]. In contrast, Baletto et al., using a different empirical
potential, concluded that the crossover between icosahedron and regular
truncated octahedron should occur in a small size ($<$100 atoms) and decahedra
can exist in a wider range of sizes [23]. For simulations on small-size Pt
nanoparticles, either cuboctahedron or regular truncated octahedron has been
used in previous works [57, 18, 14]. Although controlling experimental
conditions can open up possibilities for a variety of shapes of pure Pt
nanoparticles, it is well acknowledged that multiply-twinned structures rarely
form [16].

We used the well-validated ML models to predict energetics of typical
structure motifs of Pt nanoparticles across a size range of 201 to 6266 atoms,
including seven cuboctahedron, five truncated octahedron, and seven
icosahedron nanoparticles; decahedron is not considered because it is usually
only an intermediate state and it has been rarely reported in experiments [16,
58]. We then fit the predicted energetics of each structure type as a function
of $N$ to a simplified version of equation (2) in which we dropped the edge
term ($CN^{-2/3}$).

The results are shown in Figure 5, which shows per-atom energy versus
$N^{-1/3}$. (Results including edge terms are included in Figure S6 of SI.)
First, we note that the data points show little significant curvature about
the straight lines, which implies that the neglect of edge terms is justified
for this range of particle size. From the fit parameters displayed on the
plot, one can see that volume contributions of single-crystal cuboctahedron
and truncated octahedron are almost identical, while that of icosahedron is
larger. We attribute this to the distorted internal structure of the
icosahedron. As we expect, the surface contribution is always positive, with
the order of cuboctahedron $>$ icosahedron $>$ truncated octahedron. It is
reasonable that the surface contribution of cuboctahedron is larger than that
of icosahedron because more fcc(100) facets are exposed on the cuboctahedron
surface compared to all fcc(111) facets on the icosahedron surface. It was
also found that the surface contribution of truncated octahedron is lower than
icosahedron, probably because the distorted internal structure of icosahedron
also has a profound impact on its surface energy (i.e., the icosahedron
surface may be distorted as well).

This analysis shows the truncated octahedron to be the most stable Pt shape
across this size range (roughly 200-7000 atoms). To the best of our knowledge,
it is the first time that the unique high stability of truncated octahedron is
identified, and it explains the observation that truncated octahedron is the
structure of pure Pt nanoparticles most frequently found in experiments. [18,
16] A crossover exists between cuboctahedron and icosahedron, which is
estimated to be at $N=538$, in agreement with the result of Uppenbrink et al..
[22] However, these lines are nearly overlapping which presumably makes the
precise location of the crossover very sensitive to the fidelity of the
interatomic potential used, perhaps explaining the large variation seen in the
literature for the location of this crossover.

Figure 5:  Energies of relaxed structure motifs of Pt nanoparticles, plotted
as per-atom energy ($U/N$) versus $N^{-1/3}$. COh, TOh and Ih represent
cuboctahedron, truncated octahedron and icosahedron, respectively.

##### Bimetallic particles.

Using a similar logic, we extended this analysis to the crossover in Co–Pt
nanoparticles. Section 3.2 concluded that the most outstanding feature for
Co–Pt nanoparticles is an outermost Pt skin with a Co-rich second shell. Thus
for truncated octahedron and icosahedron, we created Co–Pt structure motifs
enforcing the above feature, with the remaining Co atoms randomly placed in
the core. As a comparison, we also included a cuboctahedron with an
L10-ordered core surrounded by a pure Pt skin. To account for the Co
composition effect on the energetics, we introduced a revised empirical model,
as shown in the following equations. In the interest of having fewer
parameters, we assume edge sites can be neglected (as we justified earlier for
pure Pt particles in this size range).

$\frac{U}{N}=A(x_{\ce{Co}})+B(x_{\ce{Co}})\cdot N^{-1/3}$ (3)

where

$A(x_{\ce{Co}})=x_{\ce{Co}}A_{\mathrm{Co}}+(1-x_{\ce{Co}})A_{\rm{Pt}}+\alpha
x_{\ce{Co}}(1-x_{\ce{Co}})$ (4)

and

$B(x_{\ce{Co}})=B_{0}(1-\kappa x_{\ce{Co}})$ (5)

Here, $A_{\rm{Pt}}$ and $B_{0}$ are respective volume and surface contribution
found from pure Pt nanoparticle results. $A_{\mathrm{Co}}$ is the volume
contribution in a pure Co nanoparticle; this was found from a bulk
calculation. $\alpha$ describes the enthalpy of mixing and $\kappa$ describes
the reduction of the Pt surface energy due to alloying the bulk with Co; both
of these were taken as free variables fit to the Co–Pt particles in question.
The ML-calculated energetics for cuboctahedron, icosahedron, and truncated
octahedron are shown in Figure 6(a–c), and the fitted results for each shape
at each Co composition are represented by a family of dashed lines. The fitted
results are summarized in Figure 6(d) using a Co composition of 35%, which
falls within the range of investigated compositions. Co alloying lowers both
volume and surface contributions, suggesting a strong tendency of mixing Co
and Pt atoms; however, the dependencies differ with Co–Pt nanoparticle shapes.
The fitted enthalpy of mixing and reduction in Pt surface contribution due to
Co alloying for each shape of Co–Pt nanoparticle, together with fitted volume
and surface contributions in a pure Pt nanoparticle and volume contribution of
Co from a bulk calculation, are listed in Table 1.

Table 1: Fitted enthalpy of mixing ($\alpha$) and reduction of Pt surface energy due to Co alloying ($\kappa$) for each shape of Co–Pt nanoparticles. Fitted volume ($A_{\rm{Pt}}$) and surface contributions ($B_{0}$) in a pure Pt nanoparticle are also listed. The volume contribution of Co ($A_{\mathrm{Co}}$) obtained from a bulk calculation is shown as well.  | $\alpha$ | $\kappa$ | $A_{\ce{Pt}}$ | $B_{0}$ | $A_{\mathrm{Co}}$
---|---|---|---|---|---
| [eV/atom] | [eV/atom4/3] | [eV/atom] | [eV/atom4/3] | [eV/atom]
Cuboctahedron | -0.913 | 0.485 | -6.251 | 3.480 | -7.528
Icosahedron | -0.850 | 0.924 | -6.235 | 3.348 | -7.528
Truncated octahedron | -0.966 | 0.624 | -6.249 | 3.213 | -7.528
Figure 6: Energies of relaxed structure motifs of Co–Pt nanoparticles, plotted
as per-atom energy ($U/N$) versus $N^{-1/3}$. COh, TOh and Ih represent
cuboctahedron, truncated octahedron and icosahedron, respectively. (a)
Cubocahedron, (b) Icosahedron, (c) Truncated octahedron, (d) Fitted results
using Eq. 3, at a Co composition of 35%. The Co compositions are indicated by
texts next to each data point. The fitted family of lines are indicated by
dash lines; each line represents the fit for the fixed Co composition
indicated by the nearest point. Cross-sectional view of the structure motif
for Co–Pt nanoparticles are included as an inlet. The number of atoms are
presented in the twin axis at the top.

Over most of the size range, the truncated octahedron is the most stable
shape, as it is for pure Pt particles. We can deduce, by extrapolation, that
at small particle sizes ($<$333 atoms) icosahedron becomes more stable. We
infer that this is because the addition of Co in the core switches the order
of the surface contribution coefficients for alloyed truncated octahedron
(2.512) and icosahedron (2.266) compared to that for pure Pt truncated
octahedron (3.213) and icosahedron (3.348), adding that the order of volume
contributions remain unchanged for alloyed truncated octahedron (-6.916) and
icosahedron (-6.881) versus that in pure Pt truncated octahedron (-6.249) and
icosahedron (-6.235), as shown in Figure 5 and Figure 6(d). This can be
explained by the strain/stress release on the distorted surface of icosahedron
when a smaller element such as Co is introduced into the subsurface and core.
Specifically, we define the average local strain for a surface atom $i$ as:

$\varepsilon_{i}=\frac{1}{M}\sum_{j\in\textrm{NN}_{i}}\frac{d_{ij}-d_{\rm{Pt}}}{d_{\rm{Pt}}}$
(6)

where $d_{ij}$ is the interatomic distance between atoms $i$ and $j$,
$d_{\rm{Pt}}$ is the DFT-calculated lattice constant for an optimized bulk Pt
(3.936 Å), $\textrm{NN}_{i}$ represents the summation over the nearest surface
neighbors of atom $i$, and $M$ is the number of surface neighbors. We
calculated the average local strains on the terrace fcc(111) sites for
cuboctahedron and icosahedron particles in sizes from 561 to 5083 atoms with
and without Co alloying, and report the result in Table 2. As we can see, the
surface atoms on the cuboctahedron start in compressive strain for pure Pt,
and alloying with Co only further compresses the surface atoms. In contrast,
the icosahedron starts in tensile strain in the pure system, so alloying with
Co allows strain relief and crosses into the compressive regime. As a result,
the absolute strain increases with alloying for cuboctahedron and decreases
for icosahedron, confirming stress release in icosahedron nanoparticles upon
Co alloying.

If we increase the Co composition to 40%, we found that the crossover between
icosahedron and truncated octahedron shifts to a larger size of 570 atoms,
further extending the range of stability for icosahedron. Of course, the
crossover may also depend on the surrounding environment and surface
reactions, which are not considered in this study. Here we main aim to
providing the physical insights for differences in crossover for Pt and Co–Pt
nanoparticles, and the structures used to analyze the crossover for Co–Pt
alloy systems can probably be further optimized. The method presented here can
readily be extended.

Table 2: Strain levels (engineering strain, percent) for particles in sizes of 561, 923, 1415, 2057, 2869, 3871 and 5083 atoms with and without Co alloying. The $\pm$ indicates the standard deviation across particle sizes.  | Cuboctahedron | Icosahedron
---|---|---
Pure Pt | (-1.321 $\pm$ 0.176)% | (+2.333$\pm$0.215)%
Alloyed | (-3.116 $\pm$ 0.068)% | (-1.244 $\pm$ 0.190)%

###  3.4 Order–disorder phase transition in Co–Pt truncated octahedrons

Before discussing the order–disorder phase transition, we need to first
investigate the stable structure of a Co–Pt nanoparticle. Two questions should
be answered in this regard—first, is the stable structure ordered? Second, if
it is ordered, how? The truncated octahedron structure was chosen for this
analysis based on the crossover analysis, as it is the most thermodynamically
stable shape for large sizes and is also the structure most commonly reported
in experiments [16, 59, 4]. First, we performed Metropolis simulations at a
temperature of 300 K on particles with composition Pt300Co286. We picked a
structure after more than 58,600 steps, equivalent to 100 swaps per atom on
average, and we treated it as the putative global minima. We relaxed the
structure using the force model.

We compared the energy of this structure with its fully-ordered L10
counterpart, using both our ML energy model and with DFT, using the highly-
scalable SPARC code. To our knowledge, this DFT validation calculation on a
spin-polarized 586-atom structure represents the largest Co–Pt nanoparticle
that has been directly validated by a full ab initio method. The shell-by-
shell atomic arrangement of both structures are shown in Figure 7. For the ML-
found minimum, an alternating preference for Pt and Co atoms can be seen
starting with a Pt-rich surface with the subsurface layer fully occupied by
Co. Co atoms on the surface of this structure are more likely to occupy
terrace fcc(111) sites and to connect with Pt atoms on the surface, consistent
with the previous findings in the genetic algorithm study. This observation is
validated by aforementioned SPARC DFT calculations to prove that L10 ordered
Co–Pt truncated octahedron alloy is not the most stable structure, but ordered
in a different pattern as shown in the configurations in Figure 7(a). Both the
SPARC DFT calculations and the ML-calculations conclude the Metropolis-derived
structure found by ML models is lower in energy than the L10 structure, with
ML predicting 0.097 eV/atom and DFT calculating 0.057 eV/atom. The DFT-maximum
atomic forces for the ML-found minimum and L10 ordered structures are 0.42 and
0.27 eV/Å, respectively, which are within ML-predicted maximum atomic
uncertainty of forces, 0.43 and 0.42 eV/Å, respectively. Since the Metropolis-
found structure has a larger maximum force, we expect its energy may decrease
more if it were re-optimized at the DFT level; this would likely have the
effect of reducing the energy differences between the ML and DFT estimates.
(Performing full relaxations at the DFT level, even with a highly scalable
code like SPARC, would be extremely costly due to the scaling of DFT, the spin
polarization, and the large number of degrees of freedom in the system.) We
performed a similar study on a smaller Co–Pt truncated octahedron Pt96Co105,
and the comparison between ML predicted putative minima against the L10 one is
included in Figure S7 of SI.

Figure 7: Composition depth profile of a truncated octahedron Pt300Co286: the
putative global minima found by ML models (a), and the fully L10 ordered Co–Pt
nanoparticle alloy (b). Atomic arrangement at each shell and the total number
of atoms are provided at the top.

We also performed Metropolis simulations at a temperature of 500 K on a much
larger nanoparticle with 6266 atoms ($\sim$6 nm). We started with a fully L10
ordered structure, and we would like to see where the thermodynamic
fluctuations lead the structure. We took out a structure after 330,000 steps,
showing the configuration for each shell in Figure 8. It is clear that that
the first four outermost shells change to a distinct orderliness, whereas we
find an almost unchanged L10-like structure going from the 5th shell to the
center of the truncated octahedron. Interestingly, the optimized structure for
a truncated octahedron with nearly equal compositions of Pt and Co follows a
pattern of atomic arrangement as:

$\ce{Pt\to
Co\to\mathrm{L}1_{2}(Pt3Co)\to\mathrm{L}1_{2}(PtCo3)\to\mathrm{L}1_{0}(PtCo)\to\cdots\to\mathrm{L}1_{0}(PtCo)}$

So far, we can conclude that the optimal particle for a truncated octahedron
with equal compositions displays concentric Pt and Co shells at the outermost
two shells, then respective Pt-rich and Co-rich L12 ordered at the third and
fourth shell, and fully L10 ordered close to the center.

Figure 8: Depth profile of compositions and configurations at each shell for a
6 nm Co3102Pt3164 optimized by Metropolis Monte Carlo simulations.

To find the temperature for order–disorder phase transition, we carried out a
series of Metropolis simulations at various temperatures We studied the
order–disorder phase transition for two structures; one is a 500-atom bulk
cell Pt250Co250 and the other is a 1289-atom truncated octahedron Pt632Co657.
We employed a long-range order (LRO) parameter ($\Phi$) introduced by Cowley
[60] to describe the order–disorder transition, and it takes the form of:

$\Phi=\max_{i\in\\{x,y,z\\}}(\\{\Phi_{i}\\})\qquad\mbox{where}~{}\Phi_{i}=|p_{\mathrm{A},i}-1/2|+|p_{\mathrm{B},i}-1/2|$
(7)

where $p_{\mathrm{A},i}$ and $p_{\mathrm{B},i}$ are the occupation
probabilities on each sublattice of the L10 phase evaluated in an ordering
direction $i$. As the stable structure of truncated octahedron found at 300 K
is similar to that of the aforementioned 6266-atom structure, only exhibiting
L10 ordering from the 5th shell to the center, we only consider those L10
ordered shells for the order–disorder transition in the 3.3-nm truncated
octahedron nanoparticle Pt632Co657. Figure 9 shows the order parameters
calculated at various temperatures by Metropolis simulations on a bulk cell
and a nanoparticle. Although there isn’t a sharp phase transition, we can see
that the loss of order occurs in the vicinity of the experimentally observed
transition temperature of $\sim$850∘C [61, 62], although our calculations
appear to predict it to be slightly higher at a temperature of $\sim$1050∘C.
This deviation is similar to what has been found by well-validated empirical
inter-atomic potentials. [14] For the 1289-atom truncated octahedron, there
exists a much smoother transition region where the transition temperature is
found to be around 900∘C, which is 150∘C lower than that of a bulk. The size
effect agrees well with experimental observations and simulations in the work
of Alloyeau et al.[14], in which order–disorder phase transition temperature
is lowered by at least 175 ∘C. This phenomenon can be understood by the
surface induced disordering due to the reduced coordination hence an overall
lowered order–disorder transition temperature [12].

Figure 9: Long-range order parameter for a 500-atom bulk cell and a 1289-atom
3.3 nm truncated octahedron calculated by Metropolis simulations at various
temperatures. TOh is short for truncated octahedron.

##  4 Conclusions

Based on an NFT approach, robust neural network models have been developed for
Co–Pt nanoparticles of up to several thousand atoms in size, using training
sets containing images with no more than 168 atoms/image. This work also
demonstrates that the NFT approach is applicable to multi-element magnetic
nanoparticles. The resulting models can readily be improved by addressing
uncertain local chemical environments when necessary. By pairing these models
with genetic algorithms and Metropolis Monte Carlo simulations, we have
presented a thorough study of the stable structures of Co–Pt nanoparticles. We
summarize the key findings below, which not only refine existing
understandings of the thermodynamic stability of Co–Pt nanoparticles, but also
offer guidelines for the synthesis of nanoparticle catalysts in experiments.
The experimental guidelines include, but are not limited to, using temperature
to control the orderliness of the nanoparticle and tuning surface energy with
capping agent targeting desired nanoparticle shapes.

  1. 1.

Co–Pt nanoparticles exhibit a strong tendency to form alternating layers near
the surface, with a platinum-rich skin and a cobalt-rich underlayer. This was
seen in many systems throughout this study, and the concentric nature
continued through the 4th shell in the case of a 6-nm (6266-atom) structure.

  2. 2.

Co–Pt fcc(100) surfaces also exhibit a strong tendency to form an L10 ordered
structure featuring alternating Pt and Co layers. Co–Pt fcc(111) surfaces show
more flexibility of the atomic arrangement while the major feature is also the
alternating layers.

  3. 3.

The truncated octahedron is the most stable shape for Pt nanoparticles of
moderate to large sizes (200–7000 atoms), due to its low surface and volume
energies This explains its frequent appearance in experiments. The stability
of icosahedron and cuboctahedron particles is always less, but these two
shapes exhibit a crossover in stability at a size of $\sim$500 atoms.

  4. 4.

The truncated octahedron is the most stable shape for large Co–Pt
nanoparticles, while the icosahedron is more stable for smaller nanoparticles.
An composition-dependent empirical model was introduced to study the crossover
among structural motifs in Co–Pt nanoparticles. The addition of Co improves
the stability of icosahedron, leading to a crossover between icosahedron and
truncated octahedron at the size of $\sim$333 atoms for a given Co composition
of 35%. It can be rationalized by the significant stress release on the
distorted fcc(111) surfaces of icosahedron when a smaller element is
introduced in the core. The crossover moves to a larger size when more Co
atoms are added.

  5. 5.

Metropolis simulations reveal that the most stable atomic arrangement of a
Co–Pt truncated octahedron with nearly equal Co and Pt compositions is not
fully L10 ordered, as often found by well-parameterized empirical potentials.
Instead, it displays a more complex pattern going from the outermost shell to
the center of truncated octahedron, which is confirmed by large-scale DFT
calculations on SPARC. The outermost shell is Pt segregated, followed by a Pt
depleted second shell. The third and fourth shells are L12-like structures
rich in Pt and Co, respectively. Deeper shells all exhibit L10-like atomic
arrangement.

  6. 6.

The order–disorder phase transition for a bulk and nanoparticle has been
studied based on a long-range order parameter. Nanoparticles show a lower
transition temperature and a much smoother transition compared to a bulk Co–Pt
alloy.

  7. 7.

The energy convex hull for a 147-atom Co–Pt icosahedron constructed by neural
network models is quantitatively accurate compared to brute-force ab initio
calculations, and a new low-energy atomic arrangement for Pt80Co67 is
identified.

#### SUPPORTING INFORMATION

Representative cubocahedron and icosahedron structures used to generate the
training data (atomic chunks) and the corresponding average atomic
uncertainties for the relaxed structures of those full-size nanoparticles
(S-1), statistics for the atomic chunks (S-2), validation of force consistency
between GPAW and SPARC (S-3), energies and configurations for a 147-atom
icosahedron and cuboctahedron (S-4), energies and configurations for two
Pt80Co67 icosahedra (S-5), fitted size dependency of pure Pt nanoparticle
energies including edge terms (S-6), and shell-by-shell compositions and
configurations of a Pt96Co105 truncated octahedron (S-7) are included in the
supporting information.

#### SUPPORTING DATASET

The supporting dataset is saved in a private repository and will be made
available to public after this work is accepted for publication. Scripts to
reproduce some simulations will also be provided along with the supporting
dataset.

#### AUTHOR INFORMATION

##### Corresponding author.

*Email<EMAIL_ADDRESS>Telephone +1 401-863-2153.

##### Acknowledgments.

The authors thank Phanish Suryanarayana and Qimen Xu for their support in
running the SPARC code and providing feedback on convergence issues. This work
was supported by the U.S. Department of Energy’s Basic Energy Science,
Computational Chemical Sciences Program Office, under Award No. DE-SC0019441.
Cheng Zeng acknowledges support from the Presidential Fellowship at Brown
University. This work was undertaken using the computational resources at the
Brown University Center for Computation and Visualization (CCV).

## References

  * [1] Ristau, R.A.; Barmak, K.; Lewis, L.H.; Coffey, K.R.; Howard, J.K.  On the relationship of high coercivity and L10 ordered phase in CoPt and FePt thin films.  _Journal of Applied Physics_ 1999; 86, 4527–4533.  Publisher: American Institute of Physics.
  * [2] Sun, S.; Murray, C.B.; Weller, D.; Folks, L.; Moser, A.  Monodisperse FePt Nanoparticles and Ferromagnetic FePt Nanocrystal Superlattices.  _Science_ 2000; 287, 1989–1992.  Publisher: American Association for the Advancement of Science.
  * [3] Ferrando, R.; Jellinek, J.; Johnston, R.L.  Nanoalloys: From Theory to Applications of Alloy Clusters and Nanoparticles.  _Chemical Reviews_ 2008; 108, 845–910.  Publisher: American Chemical Society.
  * [4] Li, J.; Sharma, S.; Wei, K.; Chen, Z.; Morris, D.; Lin, H.; Zeng, C.; Chi, M.; Yin, Z.; Muzzio, M.; Shen, M.; Zhang, P.; Peterson, A.A.; Sun, S.  Anisotropic Strain Tuning of L10 Ternary Nanoparticles for Oxygen Reduction.  _Journal of the American Chemical Society_ 2020; 142, 19209–19216.
  * [5] Sun, X.; Huang, Y.; Nikles, D.E.  FePt and CoPt magnetic nanoparticles film for future high density data storage media.  _International Journal of Nanotechnology_ 2004; 1, 328–346.
  * [6] Jun, Y.w.; Choi, J.s.; Cheon, J.  Heterostructured magnetic nanoparticles: their versatility and high performance capabilities.  _Chemical Communications_ 2007; , 1203–1214.
  * [7] Christodoulides, J.A.; Huang, Y.; Zhang, Y.; Hadjipanayis, G.C.; Panagiotopoulos, I.; Niarchos, D.  CoPt and FePt thin films for high density recording media.  _Journal of Applied Physics_ 2000; 87, 6938–6940.
  * [8] Yoshida, T.; Kojima, K.  Toyota MIRAI Fuel Cell Vehicle and Progress Toward a Future Hydrogen Society.  _The Electrochemical Society Interface_ 2015; 24, 45.
  * [9] Wang, C.; Spendelow, J.S.  Recent developments in Pt–Co catalysts for proton-exchange membrane fuel cells.  _Current Opinion in Electrochemistry_ 2021; 28, 100715\.
  * [10] Mehaddene, T.; Kentzinger, E.; Hennion, B.; Tanaka, K.; Numakura, H.; Marty, A.; Parasote, V.; Cadeville, M.C.; Zemirli, M.; Pierron-Bohnes, V.  Lattice dynamics and migration enthalpies in CoPt3 and FePd.  _Physical Review B_ 2004; 69, 024304.  Publisher: American Physical Society.
  * [11] Andreazza, P.; Pierron-Bohnes, V.; Tournus, F.; Andreazza-Vignolle, C.; Dupuis, V.  Structure and order in cobalt/platinum-type nanoalloys: from thin films to supported clusters.  _Surface Science Reports_ 2015; 70, 188–258.
  * [12] Yang, B.; Asta, M.; Mryasov, O.N.; Klemmer, T.J.; Chantrell, R.W.  The nature of A1–L10 ordering transitions in alloy nanoparticles: A Monte Carlo study.  _Acta Materialia_ 2006; 54, 4201–4211.
  * [13] Müller, M.; Albe, K.  Lattice Monte Carlo simulations of FePt nanoparticles: Influence of size, composition, and surface segregation on order-disorder phenomena.  _Physical Review B_ 2005; 72, 094203.  Publisher: American Physical Society.
  * [14] Alloyeau, D.; Ricolleau, C.; Mottet, C.; Oikawa, T.; Langlois, C.; Le Bouar, Y.; Braidy, N.; Loiseau, A.  Size and shape effects on the order–disorder phase transition in CoPt nanoparticles.  _Nature Materials_ 2009; 8, 940–946.  Number: 12 Publisher: Nature Publishing Group.
  * [15] He, D.S.; He, D.; Wang, J.; Lin, Y.; Yin, P.; Hong, X.; Wu, Y.; Li, Y.  Ultrathin Icosahedral Pt-Enriched Nanocage with Excellent Oxygen Reduction Reaction Activity.  _Journal of the American Chemical Society_ 2016; 138, 1494–1497.  Publisher: American Chemical Society.
  * [16] Xia, Y.; Xiong, Y.; Lim, B.; Skrabalak, S.E.  Shape-Controlled Synthesis of Metal Nanocrystals: Simple Chemistry Meets Complex Physics?  _Angewandte Chemie International Edition_ 2009; 48, 60–103.  _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/anie.200802248.
  * [17] Wu, J.; Qi, L.; You, H.; Gross, A.; Li, J.; Yang, H.  Icosahedral Platinum Alloy Nanocrystals with Enhanced Electrocatalytic Activities.  _Journal of the American Chemical Society_ 2012; 134, 11880–11883.  Publisher: American Chemical Society.
  * [18] Shao, M.; Peles, A.; Shoemaker, K.  Electrocatalysis on Platinum Nanoparticles: Particle Size Effect on Oxygen Reduction Reaction Activity.  _Nano Letters_ 2011; 11, 3714–3719.  Publisher: American Chemical Society.
  * [19] Ahmadi, T.S.; Wang, Z.L.; Green, T.C.; Henglein, A.; El-Sayed, M.A.  Shape-Controlled Synthesis of Colloidal Platinum Nanoparticles.  _Science_ 1996; 272, 1924–1925.  Publisher: American Association for the Advancement of Science.
  * [20] Mackay, A.L.  A dense non-crystallographic packing of equal spheres.  _Acta Crystallographica_ 1962; 15, 916–918.  _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1107/S0365110X6200239X.
  * [21] Cleveland, C.L.; Landman, U.  The energetics and structure of nickel clusters: Size dependence.  _The Journal of Chemical Physics_ 1991; 94, 7376–7396.  Publisher: American Institute of Physics.
  * [22] Uppenbrink, J.; Wales, D.J.  Structure and energetics of model metal clusters.  _The Journal of Chemical Physics_ 1992; 96, 8520–8534.  Publisher: American Institute of Physics.
  * [23] Baletto, F.; Ferrando, R.; Fortunelli, A.; Montalenti, F.; Mottet, C.  Crossover among structural motifs in transition and noble-metal clusters.  _The Journal of Chemical Physics_ 2002; 116, 3856–3863.  Publisher: American Institute of Physics.
  * [24] Nishioka, K.  Thermodynamics of a liquid microcluster.  _Physical Review A_ 1977; 16, 2143–2152.  Publisher: American Physical Society.
  * [25] Northby, J.A.; Xie, J.; Freeman, D.L.; Doll, J.D.  Binding energy of large icosahedral and cuboctahedral Lennard-Jones clusters.  _Zeitschrift für Physik D Atoms, Molecules and Clusters_ 1989; 12, 69–71.
  * [26] Zhu, B.; Guesmi, H.; Creuze, J.; Legrand, B.; Mottet, C.  Crossover among structural motifs in Pd–Au nanoalloys.  _Physical Chemistry Chemical Physics_ 2015; 17, 28129–28136.  Publisher: The Royal Society of Chemistry.
  * [27] Behler, J.; Parrinello, M.  Generalized Neural-Network Representation of High-Dimensional Potential-Energy Surfaces.  _Physical Review Letters_ 2007; 98, 146401.
  * [28] Bartók, A.P.; Payne, M.C.; Kondor, R.; Csányi, G.  Gaussian Approximation Potentials: the accuracy of quantum mechanics, without the electrons.  _Physical Review Letters_ 2010; 104, 136403.
  * [29] Rupp, M.; Tkatchenko, A.; Müller, K.R.; von Lilienfeld, O.A.  Fast and Accurate Modeling of Molecular Atomization Energies with Machine Learning.  _Physical Review Letters_ 2012; 108, 058301.
  * [30] Behler, J.  Four Generations of High-Dimensional Neural Network Potentials.  _Chemical Reviews_ 2021; 121, 10037–10072.  Publisher: American Chemical Society.
  * [31] Deringer, V.L.; Bartók, A.P.; Bernstein, N.; Wilkins, D.M.; Ceriotti, M.; Csányi, G.  Gaussian Process Regression for Materials and Molecules.  _Chemical Reviews_ 2021; 121, 10073–10141.  Publisher: American Chemical Society.
  * [32] Unke, O.T.; Chmiela, S.; Sauceda, H.E.; Gastegger, M.; Poltavsky, I.; Schütt, K.T.; Tkatchenko, A.; Müller, K.R.  Machine Learning Force Fields.  _Chemical Reviews_ 2021; 121, 10142–10186.  Publisher: American Chemical Society.
  * [33] Botu, V.; Batra, R.; Chapman, J.; Ramprasad, R.  Machine Learning Force Fields: Construction, Validation, and Outlook.  _The Journal of Physical Chemistry C_ 2017; 121, 511–522.  Publisher: American Chemical Society.
  * [34] Zeng, C.; Chen, X.; Peterson, A.A.  A nearsighted force-training approach to systematically generate training data for the machine learning of large atomic structures.  _The Journal of Chemical Physics_ 2022; 156, 064104.  Publisher: American Institute of Physics.
  * [35] Peterson, A.A.; Christensen, R.; Khorshidi, A.  Addressing uncertainty in atomistic machine learning.  _Physical Chemistry Chemical Physics_ 2017; 19, 10978–10985.
  * [36] Khorshidi, A.; Peterson, A.A.  Amp: A modular approach to machine learning in atomistic simulations.  _Computer Physics Communications_ 2016; 207, 310–324.
  * [37] Singraber, A.; Behler, J.; Dellago, C.  Library-Based LAMMPS Implementation of High-Dimensional Neural Network Potentials.  _Journal of Chemical Theory and Computation_ 2019; 15, 1827–1840.
  * [38] Lee, K.; Yoo, D.; Jeong, W.; Han, S.  SIMPLE-NN: An efficient package for training and executing neural-network interatomic potentials.  _Computer Physics Communications_ 2019; 242, 95–103.
  * [39] Shuaibi, M.; Lei, X.; Comer, B.; Chen, R.Q.; Musa, E.; Adams, M.  AMPtorch: Atomistic Machine Learning Package (AMP) - PyTorch.  https://github.com/ulissigroup/amptorch 2020.
  * [40] Enkovaara, J.; Rostgaard, C.; Mortensen, J.J.; Chen, J.; Dułak, M.; Ferrighi, L.; Gavnholt, J.; Glinsvad, C.; Haikola, V.; Hansen, H.A.; Kristoffersen, H.H.; Kuisma, M.; Larsen, A.H.; Lehtovaara, L.; Ljungberg, M.; Lopez-Acevedo, O.; Moses, P.G.; J Ojanen; Olsen, T.; Petzold, V.; Romero, N.A.; Stausholm-Møller, J.; Strange, M.; Tritsaris, G.A.; Vanin, M.; Walter, M.; B Hammer; Häkkinen, H.; Madsen, G.K.H.; Nieminen, R.M.; Nørskov, J.K.; Puska, M.; Rantala, T.T.; Schiøtz, J.; Thygesen, K.S.; Jacobsen, K.W.  Electronic structure calculations with GPAW: a real-space implementation of the projector augmented-wave method.  _Journal of Physics: Condensed Matter_ 2010; 22, 253202\.
  * [41] Perdew, J.P.; Burke, K.; Ernzerhof, M.  Generalized Gradient Approximation Made Simple.  _Physical Review Letters_ 1996; 77, 3865–3868.
  * [42] Xu, Q.; Sharma, A.; Comer, B.; Huang, H.; Chow, E.; Medford, A.J.; Pask, J.E.; Suryanarayana, P.  SPARC: Simulation Package for Ab-initio Real-space Calculations.  _SoftwareX_ 2021; 15, 100709.
  * [43] Ghosh, S.; Suryanarayana, P.  SPARC: Accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory: Isolated clusters.  _Computer Physics Communications_ 2017; 212, 189–204.
  * [44] Ghosh, S.; Suryanarayana, P.  SPARC: Accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory: Extended systems.  _Computer Physics Communications_ 2017; 216, 109–125.
  * [45] van Setten, M.J.; Giantomassi, M.; Bousquet, E.; Verstraete, M.J.; Hamann, D.R.; Gonze, X.; Rignanese, G.M.  The PseudoDojo: Training and grading a 85 element optimized norm-conserving pseudopotential table.  _Computer Physics Communications_ 2018; 226, 39–54.
  * [46] Deaven, D.M.; Ho, K.M.  Molecular Geometry Optimization with a Genetic Algorithm.  _Physical Review Letters_ 1995; 75, 288–291.  Publisher: American Physical Society.
  * [47] Johnston, R.L.  Evolving better nanoparticles: Genetic algorithms for optimising cluster geometries.  _Dalton Transactions_ 2003; , 4193–4207Publisher: The Royal Society of Chemistry.
  * [48] Vilhelmsen, L.B.; Hammer, B.  Systematic Study of ${\mathrm{Au}}_{6}$ to ${\mathrm{Au}}_{12}$ Gold Clusters on MgO(100) $F$ Centers Using Density-Functional Theory.  _Physical Review Letters_ 2012; 108, 126101.
  * [49] Lysgaard, S.; Landis, D.D.; Bligaard, T.; Vegge, T.  Genetic Algorithm Procreation Operators for Alloy Nanoparticle Catalysts.  _Topics in Catalysis_ 2014; 57, 33–39.
  * [50] Van den Bossche, M.; Grönbeck, H.; Hammer, B.  Tight-Binding Approximation-Enhanced Global Optimization.  _Journal of Chemical Theory and Computation_ 2018; 14, 2797–2807.  Publisher: American Chemical Society.
  * [51] Hjorth Larsen, A.; Jørgen Mortensen, J.; Blomqvist, J.; Castelli, I.E.; Christensen, R.; Dułak, M.; Friis, J.; Groves, M.N.; Hammer, B.; Hargus, C.; Hermes, E.D.; Jennings, P.C.; Bjerre Jensen, P.; Kermode, J.; Kitchin, J.R.; Leonhard Kolsbjerg, E.; Kubal, J.; Kaasbjerg, K.; Lysgaard, S.; Bergmann Maronsson, J.; Maxson, T.; Olsen, T.; Pastewka, L.; Peterson, A.; Rostgaard, C.; Schiøtz, J.; Schütt, O.; Strange, M.; Thygesen, K.S.; Vegge, T.; Vilhelmsen, L.; Walter, M.; Zeng, Z.; Jacobsen, K.W.  The atomic simulation environment—a Python library for working with atoms.  _Journal of Physics: Condensed Matter_ 2017; 29, 273002\.
  * [52] Boes, J.R.; Kitchin, J.R.  Modeling Segregation on AuPd(111) Surfaces with Density Functional Theory and Monte Carlo Simulations.  _The Journal of Physical Chemistry C_ 2017; 121, 3479–3487.  Publisher: American Chemical Society.
  * [53] Gruner, M.E.; Rollmann, G.; Entel, P.; Farle, M.  Multiply Twinned Morphologies of FePt and CoPt Nanoparticles.  _Physical Review Letters_ 2008; 100, 087203.  Publisher: American Physical Society.
  * [54] Oriani, R.; Murphy, W.K.  Thermodynamics of ordering alloys—IV heats of formation of some alloys of transition metals.  _Acta Metallurgica_ 1962; 10, 879–885.
  * [55] Noh, S.H.; Seo, M.H.; Seo, J.K.; Fischer, P.; Han, B.  First principles computational study on the electrochemical stability of Pt–Co nanocatalysts.  _Nanoscale_ 2013; 5, 8625–8633.  Publisher: The Royal Society of Chemistry.
  * [56] Baletto, F.; Ferrando, R.  Structural properties of nanoclusters: Energetic, thermodynamic, and kinetic effects.  _Reviews of Modern Physics_ 2005; 77, 371–423.  Publisher: American Physical Society.
  * [57] Tang, L.; Han, B.; Persson, K.; Friesen, C.; He, T.; Sieradzki, K.; Ceder, G.  Electrochemical Stability of Nanometer-Scale Pt Particles in Acidic Environments.  _Journal of the American Chemical Society_ 2010; 132, 596–600.  Publisher: American Chemical Society.
  * [58] Wang, X.; Vara, M.; Luo, M.; Huang, H.; Ruditskiy, A.; Park, J.; Bao, S.; Liu, J.; Howe, J.; Chi, M.; Xie, Z.; Xia, Y.  Pd@Pt Core–Shell Concave Decahedra: A Class of Catalysts for the Oxygen Reduction Reaction with Enhanced Activity and Durability.  _Journal of the American Chemical Society_ 2015; 137, 15036–15042.  Publisher: American Chemical Society.
  * [59] Li, J.; Sun, S.  Intermetallic Nanoparticles: Synthetic Control and Their Enhanced Electrocatalysis.  _Accounts of Chemical Research_ 2019; 52, 2015–2025.  Publisher: American Chemical Society.
  * [60] Cowley, J.M.  Short- and Long-Range Order Parameters in Disordered Solid Solutions.  _Physical Review_ 1960; 120, 1648–1657.  Publisher: American Physical Society.
  * [61] Leroux, C.; Cadeville, M.C.; Pierron-Bohnes, V.; Inden, G.; Hinz, F.  Comparative investigation of structural and transport properties of L10-NiPt and CoPt phases: the role of magnetism.  _Journal of Physics F: Metal Physics_ 1988; 18, 2033–2051.  Publisher: IOP Publishing.
  * [62] Le Bouar, Y.; Loiseau, A.; Finel, A.  Origin of the complex wetting behavior in Co-Pt alloys.  _Physical Review B_ 2003; 68, 224203.  Publisher: American Physical Society.