# Harvesting nucleating structures in nanoparticle crystallization:
The example of gold, silver and iron

Arthur France-Lanord  Sorbonne Université, Muséum National d’Histoire
Naturelle, UMR CNRS 7590, Institut de Minéralogie, de Physique des Matériaux
et de Cosmochimie, IMPMC, F-75005 Paris, France  Sarath Menon  ICAMS, Ruhr-
Universität Bochum, Bochum, Germany  Julien Lam<EMAIL_ADDRESS>Univ.
Lille, CNRS, INRA, ENSCL, UMR 8207, UMET, Unité Matériaux et Transformations,
F 59000 Lille, France  Centre d’élaboration des Matériaux et d’Etudes
Structurales, CNRS (UPR 8011), 29 rue Jeanne Marvig, 31055 Toulouse Cedex 4,
France

###### Abstract

The thermodynamics and kinetics of crystallization of nanoparticles, as
opposed to bulk phases, may be influenced by surface and size effects. We
investigate the importance of such factors in the crystallization process of
gold, silver, and iron nanodroplets using numerical simulations in the form of
molecular dynamics combined with path sampling. This modeling strategy is
targeted at obtaining representative ensembles of structures located at the
transition state of the crystallization process. A structural analysis of the
transition state ensembles reveals that both the average size and location of
the critical nucleation cluster are influenced by surface and nanoscale size
effets. Furthermore, we also show that transition state structures in smaller
nanodroplets exhibit a more ordered liquid phase, and differentiating between
a well-ordered critical cluster and its surrounding disordered liquid phase
becomes less evident. All in all, these findings demonstrate that
crystallization mechanisms in nanoparticles go beyond the assumptions of
classical nucleation theory.

##  I Introduction

The formation of nanocrystals plays a crucial role in numerous research fields
including material scienceRathinavel2021Jun ; Farzin2021Mar ; Baig2021Mar ,
geologyHochella2008Mar ; Schafer2012Feb , environmental scienceRonkko2017Jul ;
Kulmala2014Apr ; Bundschuh2018Dec  and biologyMathivanan2021Jun ;
Prasad2014Dec . Beyond these applied interests, crystal formation transposed
to nanoscale systems exhibits many additional challenges when compared to its
bulk analogue. Indeed, the preponderance of surface and the natural occurrence
of finite-size effects drastically increase the complexity of nucleation
mechanismsLee2016Jun ; Thanh2014Aug ; Jun2022Apr ; Zhang2012Mar . As such,
classical nucleation theory – which was widely employed to understand crystal
formation in bulk phases –, was proven inadequate to describe many
experimental observations of the nanocrystal nucleation, including the
competition between several polymorphic phasesLam2018Mar ; Amodeo2020Oct ;
Wells2015Apr ; Tarrat2023Apr , and the emergence of pre-nucleation
clustersRamamoorthy2020Aug ; Ramamoorthy2020Aug ; Schiener2015Jun .

Along with experimental measurements, numerical simulations have been pivotal
in providing further insights on the inherent mechanismsSchoonen2022Dec ;
Lutsko2019Apr , including atomistic observations based on molecular
dynamicsQuigley2011Jan ; Pavan2015Nov ; Lam2018Mar ; Amodeo2020Oct . In this
research field, which consists in observing in silico the birth of
crystalsSosso2016Jun , the necessity to overcome free energy barriers is the
major challenge, be it for bulk phases or nanoscale systems. In this context,
while rare-event sampling methods have been extensively developed to help
characterizing nucleation processesGiberti2015Mar ; Pietrucci2017Nov ;
Bolhuis2002Oct ; Bolhuis2021Apr , their application to the formation of
nanocrystals have so far been limited to very few examplesQuigley2011Jan ;
Pavan2015Nov ; Amodeo2020Oct . Although highly informative, these studies
focused on enhanced sampling (in the form of metadynamicsLaio2002escaping ) to
overcome free energy barriers, which requires biasing the dynamics along a
specified collective variable. This bias influences the dynamics as well as
the sampled configurations, in a possibly unrealistic fashion, depending on
many factors, including the quality of the collective variablebussi2020using .

In this work, we employ an alternative simulation strategy which consists in
combining metadynamics along with committor analysispeters2006using  and
transition path samplingdellago2002transition  in the form of aimless
shootingPeters2006aimless ; Peters2007extensions . Metadynamics is only used
to formally identify, through committor analyses, structures close to or
belonging to the transition state ensemble. These are then used as starting
points to sample unbiased reactive trajectories and transition state
configurations, using aimless shooting. We therefore obtain ensembles of
critical structures located on top of the free energy barrier, without the
need for biasing along any collective variable. We apply this methodology to
investigate crystal formation in nanodroplets composed of three different
metals, namely gold, silver, and iron, which are currently considered in many
technological applicationsZhang2020 ; Yaqoob2020May ; Ebrahiminezhad2018Feb .
Upon decreasing the droplet size, we demonstrate that the critical nucleus
gets smaller, and tends to move towards the droplet’s surface. In addition, as
observed in bulk crystallization and denoted as pre-nucleation mechanisms, the
surrounding liquid becomes more ordered when compared to a normal liquid.
Finally, the interface between the critical nucleus and its surroundings gets
more diffuse. Altogether, our results strongly suggest the existence of
crystallization processes beyond classical nucleation theory.

##  II Methodology

We study three different materials, namely gold, silver, and iron. Their
interactions are modeled using three different EAM interatomic
potentialsMendelev2003 ; Williams2006May ; Chamati2004Nov  chosen for their
good match with experimental data regarding both bulk and surface
propertiesCombettes2020Sep . Throughout this work, molecular dynamics (MD)
simulations are carried out using the LAMMPS package (v. 23 Jun
2022)thompson2022lammps , supplemented by PLUMED (v. 2.7.5)bonomi2019promoting
; tribello2014plumed  for enhanced sampling as well as for the definition of
the collective variables. The time step is set to $1$ fs and a Nosé-Hoover
chain of three thermostats is applied with a damping factor of $100$ fs. To
measure crystal ordering, the averaged Steinhardt parameter ($\bar{q_{6}}$) is
computed with a cutoff equal to 3.5 Å which roughly corresponds to the end of
the first peak of the radial distribution in the liquid regime. We then use
the $\bar{q_{6}}$ value of 0.35 as a threshold discriminating liquid and
crystal structures lechner2008accurate . During a simulation, more than one
crystalline cluster can appear simultaneously. Therefore, we define
$N_{\text{crys}}$ as the size of the largest cluster with a cutoff parameter
equal to 4 Åusing the Cluster analysis as implemented in Ovito (v.
3.7.7)Stukowski2009Dec . Finally, we consider four different nanoparticles
made of 500, 1000, 1500 and 2000 atoms.

With our current simulation protocol, we purposely assume that the atomic
precursors concerned by the nanoparticle synthesis coalesce at some point to
form our initial liquid nanodroplets. Yet, depending on the thermodynamic
conditions, it is also possible that nanocrystals form through atom-by-atom
growth directly in the ordered phase without going through the liquid phase.
Few works attempted at modeling the whole gas to nanocrystal process which
required the use of unrealistically fast cooling ratesForster2019Oct ;
Zhao2015Jan ; Kesala2007May ; Rossi2023Jun . Meanwhile, authors also chose to
deposit atoms on an already formed solid seed, while varying the deposition
rateGoniakowski2010Apr ; Wells2015Apr ; Xia2021May . This therefore involves
investigating a different process than in the present work, namely a
deposition (or desublimation) transition. The methodology we propose here,
consisting in combining metadynamics with committor analysis and aimless
shooting simulations, can of course still be employed in this alternative
situation.

##  III Results

###  III.1 Measuring the melting temperature

Our goal is to sample many configurations at the transition state ensemble of
the crystallization process of nanoparticles. In order to compare different
sizes as well as different materials, we position ourselves at the same degree
of supercooling, which requires measuring the melting temperature. To this
end, we start by generating crystalline configurations by carving spheres from
a bulk crystal that are then structurally optimized at 0 K for each materials
and each nanoparticle sizes. We start simulations in the canonical ($nVT$)
ensemble at 700 K, well bellow the melting temperature, and progressively ramp
up the temperature up to 1700 K. Five different simulation lengths are
considered: 10, 9, 8, 7, 6 and 5 ns which correspond to a range in heating
ramp going from 100 K/ns to 200 K/ns. From the recorded trajectories, we
measure the temperature evolution of $N_{\text{crys}}$ [See Fig. 1.a] and
observe that the heating ramp does not significantly influence the results. We
then measure a melting temperature by fitting the obtained curves with a
hyperbolic tangent function and extracting a critical temperature, for which
corresponding snapshots are displayed in Fig. 1.c. Fig. 1.b presents the
melting temperatures averaged over the five considered heating ramp, and
confirms that the melting temperature is negatively correlated with the
nanoparticle size, as observed in experimentsLai1996Jul . We note that the
obtained melting temperatures is highly dependent on some arbitrary choices
including $N_{\text{crys}}$ as the collective variable, the hyperbolic tangent
fit, and the employed empirical interatomic potential. As such, the obtained
values for $T_{\text{melt}}$ should not really be compared with experimental
results although our approach remains very similar to recent attempts at
measuring melting temperatures in nanoparticlesZeni2021Oct ; Delgado-
Callico2021 ; Rossi2018Jun . What is important here is to use a consistent
method for all materials and droplet sizes, in order to control the degree of
supercooling employed in the remaining part of the study.

Figure 1: Measurement of the melting temperature as a function of the droplet
size. (a) Crystal proportion as a function of the temperature during five
different heating rates shown in different dash types. (b) Extracted melting
temperature averaged over the five different heating rates. (c) Snapshot
corresponding to the onset of melting shown for Au nano-crystal.

###  III.2 Obtaining a first transition state configuration

In the following, we perform all simulations at $T=0.85\,T_{\text{melt}}$, for
all nanoparticle sizes and materials. In order to find a first transition
state connecting the liquid and crystalline states, we perform metadynamics
simulations with fixed Gaussian heights Laio2002escaping ; bussi2020using ;
Laio2008Nov . Simulations are initialized in the liquid regime obtained from
the previously melted simulations and thermalized at the chosen temperature
during $100$ ps. Then, the metadynamics bias is chosen along the effective
entropy $S$, derived from liquid state theory:

$S=-2\pi\rho\int_{0}^{+\infty}\left[g(r)\ln
g(r)-g(r)+1\right]r^{2}\mathrm{d}r,$ (1)

where $\rho$ is the atomic density measured in the whole simulation box and
$g(r)$ is the radial distribution function. Every 1 ps, we add bias in the
form of Gaussian functions of width and height respectively equal to 0.5
$k_{B}$ and 1 kJ/mol for all sizes and materials. In all cases, we perform
three independent simulations using different initial configurations and
atomic velocities. Results reported in Fig. 2 show that $S$ allows for
overcoming the free energy barrier of nucleation, which is consistent with
previous works employing the same collective variablePiaggi2017Jul ;
Piaggi2017Sep ; Amodeo2020Oct ; Lam2023Jan . More importantly, it appears that
larger droplets seem to require longer simulations in order to reach a first
transition. However, it does not necessarily mean that the associated free
energy barrier is larger, since a proper measurement of the free energy
barrier requires overcoming numerous technical challenges especially related
to the definition of a representative collective variable.Lam2023Jan  Since
this is not the scope of this work, we will not use metadynamics calculations
to estimate a free energy barrier, but instead only to obtain a first
transition state configuration. In particular, the committor probability
$P_{\text{crys}}$, i.e. defined as the probability that a configuration
$\mathbf{X}$ will reach the crystalline state before reaching the liquid
state, is measured for configurations extracted from the metaydnamics
simulations. For that purpose, we initialize unbiased molecular dynamics
simulations with atomic velocity drawn from the Maxwell-Boltzmann
distribution, and monitor the committment of the system towards the liquid or
the crystalline metastable states. Both basins are defined using
$\overline{q_{6}}$ averaged over all of the atoms. For all the materials, the
crystalline basin corresponds to $\overline{q_{6}}>0.35$. Since iron liquid is
inherently less ordered than the two noble metals, two different values are
chosen for the liquid basin of gold and silver ($\overline{q_{6}}<0.225$) and
iron ($\overline{q_{6}}<0.175$). At this stage, we are solely interested in
coarsely characterizing transition states: we therefore only used 10
trajectories to measure $P_{\text{crys}}$, and defined a first transition
state as matching $P_{\text{crys}}\in[0.3,0.7]$. From each of the three
metadynamics simulations performed for all of the considered materials and
droplet sizes, we managed to extract one transition state configuration and
several unbiased trajectories reaching either the crystal or the liquid
regime.

Figure 2: Metadynamics results for different droplet sizes and materials. In
each case, the three different curves correspond to three different initial
positions/velocities, with the remaining simulation parameters are kept
constants.

###  III.3 Sampling many transition state configurations and reactive paths

Aimless shootingPeters2006aimless ; Peters2007extensions  is a transition path
sampling method which does not require biasing following a aparticular
collective variable. The principle is the following: a configuration
$\mathbf{X}(t)$, which is part of an initial path, is propagated backward and
forward in time (initial velocities are drawn from the Maxwell-Boltzmann
distribution). If the new path connects both basins, it is accepted and used
for the next iteration. If it connects the same basins, it is rejected; in
this case, the former path is used for the next iteration. Then, configuration
$\mathbf{X}(t\pm\Delta t)$ – i.e. the configuration separated by a time step
$\Delta t$ before or after configuration $\mathbf{X}(t)$ in the current path –
is selected as a new shooting point. Iterating allows to sample transition
pathways, which, since they obviously pass through the separatrix, leads also
to sampling configurations from the transition state ensemble. The acceptance
criterion acts as an effective restoring force towards the separatrix:
biaising is performed in trajectory space instead of configuration space.
Selecting the right value for $\Delta t$ is not trivial. A too small time step
would lead to sampling highly correlated configurations, while a too large
value would lead to a low path acceptance ratio. We set $\Delta t$ equal to 2
ps, which leads to a global average acceptance ratio of 28.6 %. For each of
the three initial paths, we perform three independent aimless shooting
simulations with randomized initial velocities. In total, we sample 47163
configurations from the accepted transition paths, with detailed statistics
for each system reported in Table 1.

Table 1: Statistics of the aimless shooting simulations for each system: number of iterations and number of accepted configurations, with bold numbers being sums over all nine independent runs (three initial paths, three velocity initializations).  System |  $N_{\text{iterations}(\Delta t)}$ |  $N_{\text{accepted}}$
---|---|---
|  |  avg |  min |  max |  avg |  min |  max
Au |  500 |  453 |  258 |  822 |  127 |  87 |  172
|  | 4073 | 1141
|  1000 |  350 |  236 |  479 |  118 |  61 |  182
|  | 3146 | 1063
|  1500 |  298 |  221 |  384 |  79 |  58 |  102
|  | 2678 | 711
|  2000 |  231 |  198 |  286 |  82 |  62 |  118
|  | 2077 | 738
Ag |  500 |  668 |  447 |  1062 |  187 |  21 |  336
|  | 6015 | 1683
|  1000 |  534 |  424 |  875 |  153 |  57 |  221
|  | 4807 | 1374
|  1500 |  479 |  371 |  683 |  134 |  61 |  204
|  | 4312 | 1204
|  2000 |  391 |  296 |  586 |  117 |  49 |  169
|  | 3518 | 1054
Fe |  500 |  561 |  404 |  837 |  153 |  103 |  215
|  | 5053 | 1379
|  1000 |  582 |  525 |  653 |  148 |  101 |  182
|  | 5240 | 1333
|  1500 |  394 |  330 |  431 |  99 |  29 |  141
|  | 3546 | 890
|  2000 |  300 |  215 |  364 |  85 |  64 |  107
|  | 2698 | 766

Finally, for each material and droplet size, among the accepted
configurations, we randomly select between 110 and 210 structures for which we
compute $P_{\text{crys}}$, this time using 50 independent unbiased dynamics
instead of 10. Altogether with this procedure, we sample configurations from
the transition state ensemble, which we define as $0.3\leq P_{\text{crys}}\leq
0.7$. In average, for each droplet size and materials, we obtain 90 structures
matching this criterion and which are used for the following structural
analysis.

###  III.4 Structural features of the transition states

Fig. 3 shows several measurements averaged over all of the obtained transition
state configurations. For $N_{\text{crys}}$, we use the definition already
introduced in section II. $x_{\text{crys}}$ corresponds to the ratio between
the radial position of the mass barycenter of the largest crystalline cluster
and the radius of the droplet. Finally, we defined $q_{6}^{\text{crys}}$ and
$q_{6}^{\text{liq}}$ as the averaged values of $\overline{q_{6}}$ when
selecting respectively only the atoms in the largest crystalline cluster, and
the remaining ones. These are descriptors of the structuration of the critical
nucleus and its surroundings. First, one can observe that in panels (b)-(e),
gold and silver seem to produce quantitatively similar results, which can be
attributed to both systems having the same crystal structure (face centered
cubic) and very similar lattice parameters (4.08 Å and 4.09 Å respectively for
gold and silver). Meanwhile, for iron, values of $N_{\text{crys}}$ and
$\overline{q_{6}}$ both in the liquid and the crystalline regime seem to be
smaller than what is obtained for gold and silver. This can be explained by
iron being intrinsically less ordered both in the crystalline and the liquid
regime. In particular, since iron crystallizes in a body-centered cubic
lattice, its overall $\overline{q_{6}}$ is deemed to be
smallerlechner2008accurate  and if one defines a threshold of 0.30 instead of
0.35, we already see that $N_{\text{crys}}$ now gets much closer to values
obtained with both gold and silver [See open points in Fig. 3]. After
examining differences and similarities between the different materials, we
will have a closer look at the influence of the nanoparticle size. First,
$N_{\text{crys}}$ strongly depends on the nanoparticle size as it
monotonically increases with larger nanoparticles [See Fig. 3.a]. Since we
intentionally performed the simulations at the same degree of supercooling –
which would have led to similar critical nucleus sizes in bulk –, this results
is far from being trivial and advocates for the crucial role of surfaces and
finite size effects. Secondly, we observe that the value of $x_{\text{crys}}$
slightly yet consistently increases when going towards smaller nanoparticles.
This means that the nuclei seem to get closer to the surface, favoring a
somewhat heterogeneous nucleation mechanism, as opposed to homogeneous
nucleation within the core of the nanoparticle as obserbed in the largest
droplet sizes. In Fig. 3(d,e), we focus on the ordering quality of the
critical nucleus ($q_{6}^{\text{crys}}$) and of the rest of the nanoparticle
($q_{6}^{\text{liq}}$). When decreasing the nanoparticle size,
$q_{6}^{\text{crys}}$ and $q_{6}^{\text{liq}}$ have opposite trends, with the
critical nucleus showing reduced crystallinity, and the surrounding liquid
domain getting more structured. Compared to the evolution of
$N_{\text{crys}}$, it seems that the decrease in the size of the crystalline
cluster is compensated by a pre-ordered surrounding liquid phase. Observation
of such a pre-ordered phase goes against classical nucleation theory and has
already been observed in the bulk solidification of nickel and of molybdenum,
using transition path sampling techniquesdiaz2018maximum ; Menon2020Sep .

Figure 3: (a) For each droplet sizes and material types, one typical image of
the critical structures is shown. Coloring from blue to red and atomic radius
from small to large correspond to an increase in the value of
$\overline{q_{6}}$. (b-e) Influence of the droplet size on the structure of
the critical structures in terms of the size of the largest crystalline
cluster (b), the radial position of the largest crystalline cluster (c), the
value of $\overline{q_{6}}$ averaged over atoms of the largest crystalline
cluster (d) and of the remaining atoms (e). We note that in (b) open circle
correspond to $N_{\text{crys}}$ defined with a smaller threshold values for
$\overline{q_{6}}>0.30$.

In order to further investigate this striking result, we display in Fig. 4 two
kinds of $\overline{q_{6}}$ distributions. Fig. 4.a compares the
$\overline{q_{6}}$ distribution in the whole nanoparticle depending if the
system is in the liquid, crystalline or critical state. It appears that while
the crystalline and liquid regimes are not affected by changing $N$, the
critical nucleus exhibits a shift in the distribution when decreasing the
nanoparticle size, which is consistent with our previous findings. More
importantly, we also observe a broadening of the peak, which suggests that
nucleation for the smallest droplet requires a more diverse structural
landscape. In addition, in Fig. 4.b, one can see the spatial distribution of
$\overline{q_{6}}$ measured with respect to the center of the critical
nucleus. First, the plateau at distances far from the critical nucleus, where
atoms are mostly in the liquid state, is again consistent with our previous
findings since we observe a higher value for the smallest nanoparticles,
particularly clear for $N=500$. More importantly, while classical nucleation
theory predicts a step-like behavior with the system being either in the
liquid or the crystalline regime, a more complex scenario seems to emerge
here, with a decreasing gradient in ordering from the center of the
crystalline nucleus. Increasing the nanoparticle size leads to a steeper
gradient, with a less diffuse interface between the crystalline nucleus and
the rest of the droplet. It is therefore clear that spatial confinement
affects the crystallization mechanism, with greater departure from classical
nucleation theory with stronger nanoscale effects.

Altogether, our results advocated for a departure from classical nucleation
theory at nanoscale, including diffuse interfaces between the critical cluster
and the rest of the structure, as well as the existence of a pre-ordered
liquid phase favoring the emergence of crystals, which is in-line with
previous results obtained so far in bulk systemsdiaz2018maximum . Here, we
show that such behavior is enhanced by surface and finite-size effects
inherently present at the nanoscale.

Figure 4: (a) Distribution of $\overline{q_{6}}$ measured for each droplet
sizes and each materials when the whole nanoparticle is in the crystalline,
critical and liquid regimes respectively in green, red and blue. (b) Spatial
distribution of $\overline{q_{6}}$ measured with respect to the center of the
critical nucleus for each droplet sizes and each materials and averaged over
all of the obtained transition states. In dashed line, we plot one example of
the same distribution but for a single transition state to show that the
gradient is found in all cases.

##  IV Conclusions

Using numerical simulations in the form of molecular dynamics, we have
investigated the crystallization mechanism of gold, silver, and iron
nanoparticles of various sizes, with a focus on structurally analyzing
configurations located at the transition state between the crystalline and
liquid states. For each composition, after determining the melting temperature
as a function of the nanoparticle size, we perform simulations in the
supercooled regime. Initial reactive trajectories are obtained using biased
simulations in the form of metadynamics, from which we coarsely identify
transition states and pathways, using unbiased committor analysis. We then
obtain many brute-force reactive trajectories using transition path sampling
in the form of aimless shooting, from which we extract representative
transition state structures. An analysis of these structures reveals several
systematic trends that are directly related to the reduction in size down to
the nanoscale, i.e. surface and finite-size effects: with smaller nanoparticle
sizes, the critical nucleus gets smaller and less ordered, and is shifted
towards the surface of the nanoparticle. In the meantime, the surrounding
atoms show increased ordering, with the emergence of a pre-ordered liquid
phase for the smallest nanoparticles. Finally, with smaller nanoparticle
sizes, it is less evident to differentiate between the critical nucleus and
the rest of the system, as the interface between both phases gets more
diffuse, and both phases are less contrasted. Overall, this indicates a
departure from classical nucleation theory induced by nanoscale, which hints
at the importance of characterizing explicitly crystallization kinetics and
thermodynamics for nanoparticles without resorting to classical models relying
on assumptions on transformation mechanisms. Future directions would include
analyzing the entire transformation pathway from the liquid to the crystalline
phase, a complementary analysis to the present one focusing on the transition
state ensemble. In closing, we expect that our original approach can easily be
transferred towards different nanoscale systems including oxydes and
nanoalloys to unravel complex mechanisns associated with polymorphs stability
and chemical ordering.

## Acknowledgment

This study was supported by the French National Research Agency (ANR) in the
framework of its “Jeunes chercheuses et jeunes chercheurs” program,
ANR-21-CE09-0006 and by the EUR grant NanoX n° ANR-17-EURE-0009 in the
framework of the ”Programme des Investissements d’Avenir”. Computational
resources have been provided by CALMIP, by Jean Zay and by TGCC.

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