Modelling Seeded Growth of Nanoparticles Far from Equilibrium

1. Introduction

As synthesis is the art of manipulating atoms at the nanoscale, a fundamental understanding of nucleation and growth mechanisms is essential to leverage the highly sensitive, morphology-dependent chemophysical properties of metallic nanoparticles [1,2,3,4,5]. Nanoparticle morphology, which encompasses size, shape, structural motif, and chemical ordering, is crucial for a range of applications, including catalysis, optoelectronics, biomedical applications and sensing. The appearance of a given nanoparticle morphology depends on our ability to engineer synthetic outcomes precisely.[6,7,8,9,10,11] As an example, in catalysis, tuning nanoparticle morphology can significantly influence their catalytic activity; for instance, a CO molecule cannot adsorb at mixed sites on AuRh nanoparticles [12], and oxygen reduction reaction activity can be estimated from geometric considerations of nanoparticle adsorption sites [13,14,15,16].

Proposing a robust simulation tool to model metallic nanoparticle (MNP) formation requires detecting, analysing, and incorporating a variety of individual mechanisms that occur on different time scales and with different activation energies. The interplay of these subprocesses, spanning from quantum-level chemical bonds to macroscale fluid dynamics, shapes the final morphology of the nanomaterial. Because of this complexity, a multiscale modelling approach is needed. Mechanism-Enabled population balance modelling accounts for not only primary phenomena such as nucleation and growth but also for particle agglomeration and breakage, with a focus on reproducing the particle size distribution. For sizes larger than 5 nm, the model presented in Ref.[17], based on classical nucleation theory and heterogeneous nucleation using the general aerosol dynamic equation should be considered.

In the size range between 1 and 10 nm, classical Molecular Dynamics (cMD), which solves Newton’s laws of motion, enables tracking of structural trajectories and elucidates structural transformations during synthesis across different timescales and temperatures. The resulting nanoparticle morphology can be compared with the putative global minimum, which defines the thermodynamic endpoint the cluster inherently drives toward. Comparisons are usually based on geometric descriptors. Bochicchio et al. [18] focus on global optimisation, thermodynamic stability, and chemical ordering layouts -such as core-shell, onion-like, or random alloy nanoalloys. However, it was recently shown that far-from-equilibrium morphology can ensure better performance.[2,19,20,21,22] With recent developments in machine-learned potentials (MLPs) designed specifically for nanoparticles [23,24,25], the quantum description of 1-10 nm objects can be incorporated into the interparticle potential. Nonetheless, cMD applied to MNP growth relies heavily on semi-empirical potentials, such as tight-binding second-moment approximation potentials or the embedded-atom method (EAM), with a few cases described by machine-learning potentials.[26]

Figure 1. TOC: Schematic representation of how to model seeded-mediated growth of metallic nanoparticles. We distinguish between (i) atomic deposition that can happen either from fixed directions or randomly; (ii) sintering of several seeds. Besides temperature and kinetic velocity, key parameters of atom deposition include the deposition rate, $\rho$, the number of sources, and the directionality of the incoming atoms. In the latter, the initial sintering geometry and the initial seed displacement.

Figure 1. TOC: Schematic representation of how to model seeded-mediated growth of metallic nanoparticles. We distinguish between (i) atomic deposition that can happen either from fixed directions or randomly; (ii) sintering of several seeds. Besides temperature and kinetic velocity, key parameters of atom deposition include the deposition rate, $\rho$, the number of sources, and the directionality of the incoming atoms. In the latter, the initial sintering geometry and the initial seed displacement.

Among the several cMD studies reported in the last three decades following the pioneering works of Manninen [27] - inspired by surface growth models - and Mariscal,[28,29] Grammatikopoulos [30] and Baletto [31,32,33] proposed three growth routes onto an existing metallic seed: (i) one-by-one atomic deposition from random positions tuned by different deposition rates and temperatures; (ii) the coalescence of two or three seeds along their centre of mass axis; (iii) the formation of a seed as obtained from the freezing of a nanodroplet. The effects of relative orientation of facets, e.g.(111)/(111), (111)/(100), have been widely investigated[26,34,35,36,37]. There is agreement that when the impact velocity or initial temperature is high enough, the kinetic energy supply allows surface restructuring, followed by reordering of atoms at the core. Zinzani et al.[38,39] recently showed that the neck has a short lifetime of a few ps, at least for Au-based nanoalloys.

These computational approaches highlight the importance of kinetic trapping, out-of-equilibrium pathways, and structural transitions, and show how competition between flux-landing rates and surface-diffusion energy barriers leads to metastable, faceted geometries or amorphous traps. Their work details how distinct, preformed seeds collide in flight, undergoing structural rearrangements, rigid-body reorientation, eventually temporary interfacial melting, neck formation and surface adatom diffusion to fill necks.[30,31,37,38,39,40,41,42,43,44,45,46]

However, the main limitation of existing approaches is their neglect of directionality/oriented attachment in atomic deposition and/or cluster sintering, and thus its failure to model seeded-growth under certain conditions. While random deposition can mimic gas-phase formation, it may fail when the nanoparticles grow in wet environments where molecules can shadow certain regions of the growing seed, or during cluster-beam deposition.

To overcome such limits, we provide an open-source software tool, NPGrowth, to model the growth of metallic nanoparticles based on cMD as embedded in LAMMPS [47]. The LAMMPS-core was chosen being compatible with a wide range of interatomic potentials including MLPs, EAM, and Rosato-Guillope-Legrande (RGL). NPGrowth allows the deposition of individual atoms/clusters from random or oriented deposition, from one or more sources, onto one or more designed targets. NPGrowth enables the formation of elongated, prolate, and even chiral metallic nanoparticles. NPGrowth also provides tools to simulate nanoparticle sintering events with flexibility in the initial configuration. To demonstrate the capabilities of NPGrowth, we present a few toy-model simulations pf oriented atomic deposition on copper seeds. We, further, discuss in detail the dynamics during directional sintering of four Cu seeds. To highlight the morphological differences, we employ a recently developed general characterisation tool, pySNOW [48,49,50]. Following the current development of morphological descriptors [26,50,51], we introduce a nanoparticle ID that reports a small set of geometric descriptors to identify and classify nanoparticles, that we hope to be flexible enough to interest the community.

2. Methodology

NPGrowth is a lightweight, open-source code that enables workflows by coupling a declarative description of the deposition/sintering geometry with a modified version of the fix deposit command in LAMMPS. The code is openly available at https://github.com/nanoMLMS/NPGrowth_V2, together with example input files. The SI Appendix A provides an overview of the code and its use. To provide a consistent framework for describing the different growth scenarios, we summarise the terminology adopted throughout this work in Table 1.

As depicted in Figure 2, NPGrowth automates the modelling of seeded-mediated growth, enabling the modelling of the formation of metallic nanoparticles even under extreme conditions, i.e. chemical supersaturation, where any corner or vertex acts like an exposed peak[8,9,37,52], leading to branched and chiral structures too.

Simulating MNP growth by atomic deposition requires a standard molecular dynamics setup and a deposition protocol that specifies the deposition parameters. NPGrowth keeps the cMD setup and the deposition setup fully decoupled, with LAMMPS handling the cMD engine. The user writes the cMD setup as an ordinary LAMMPS input script. The deposition setup is described declaratively in JSON files and compiled by a Python script, generate_depo_input.py, into a LAMMPS include file containing the corresponding source and target regions, and fix deposit commands. This separation lets the user reuse the same deposition configuration across different potentials, thermostats, or seed geometries without modification.

Related to the deposition protocol, LAMMPS’s native fix deposit inserts atoms with a velocity sampled independently along each Cartesian axis, which does not allow a beam to be aimed at a chosen location on an arbitrarily shaped, off-centre seed cluster. When a highly directed, collimated atomic beams hits the nanoseed, mimicking a Glancing Angle Deposition or any experimental set ups with preferential direction, atoms preferentially land on the “windward” sides of the seed leading to surface bumps, in a highly non-equilibrium kinetic regime governed by ballistic shadowing, forcing an isotropic seed into an elongated nanoparticle, nanoneedle or even a nanorod [53,54]. The key methodological novelty of NPGrowth is a pre-processing modification of LAMMPS fix_deposit.cpp: instead of accepting a fixed target point, the target keyword is redefined to accept the ID of a LAMMPS region. At each insertion attempt, a candidate position is drawn inside the source region (subject to a minimum-distance constraint from existing atoms), a point is drawn uniformly inside the associated target region, and the velocity direction of the inserted particle is set to point from the former to the latter. In contrast, its magnitude is drawn independently from a user-specified speed range. This makes the deposition direction a direct, geometric consequence of the relative placement of both the source and target regions. Atoms are classified by their coordination number and assigned to the growing seed. Only atoms that have become part of the growing seed are coupled to the thermostat. In contrast, newly deposited atoms evolve under plain NVE dynamics until they attach, mimicking the physically distinct regimes of an incoming versus an incorporated atom.

2.1. Nanoparticle Characterisation Using pySNOW

We employ our recently developed pySNOW -a Python Suite for the NanO-World-[49] to characterise the morphology of MNPs. pySNOW is an agile and open access numerical tool that reads nanoparticles’ trajectories/position and calculates various geometrical descriptors, i.e. the maximum pair distance, the maximum radial distance, the radius of gyration, and the relative shape anisotropy, structural descriptors, like the common neighbour analysis (CNA)[5,42,55], and local atomic environment descriptors, such as the atop generalized coordination number (aGCN) [56,57], and the CNA patterns (CNAPs)[49,50,58].

As in Ref. [50,59], the pair distance distribution function (PDDF) is crucial in determining the architecture of a MNP. Indeed, the PDDF provides: (i) its first minimum is the best estimate of the cutoff radius to identify the nearest neighbours; (ii) if the position of its second peak does not correspond to the lattice parameter, the structure is amorphous; (iii) the maximum pair-atomic distance, $D_{max}$, is an estimate of the nanoparticle size. In addition, the shape of the PDDF relates to the shape of the nanoparticle [59,60]. In general, the shape and sharpness of subsequent peaks and minima highlight the degree of ordering of the structure over the range of distances considered.

Another robust global descriptor is the radial atomic density (RAD), which counts atoms in concentric shells around the nanoparticle center of mass (CoM). Twice the maximum distance of RAD, $2R_{max}$, provides an estimate of the dimension of the whole MNP as it defines the sphere needed to include the whole nanoparticle. Moreover, integrating the RAD gives the number of atoms within a sphere of radius r, $N\left(r\right)={\int }_0^{R}RADdr$. We introduce the branching factor f by fitting:

$$N\left(r\right)\propto r^f.$$

(1)

Completely packed spheres have $f\approx 3$, while branched or dendritic structures have $f\lesssim 2.3$. f values close in between show the formation of grooves and reentrances or flattened/oblate shapes.

The radius of gyration of a rigid body is defined as the radial distance from a point having a moment of inertia equal to the effective mass distribution of the body if the total mass of the body is concentrated at that point. One can refer to the radius of gyration of a nanoparticle, i.e. the measure of the extent by which atoms are spread away from the center of mass (${\mathrm{CoM}}_w$) of the nanoparticle.

$$G=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left({\mathbf{r}}_i-{\mathbf{r}}_{\mathrm{CoM}}w\right)}^2}.$$

We also use the gyration radius variation over time

$$\Delta G=G\left(t\right)-G\left(0\right).$$

(2)

to quantify distortion relative to the atom initial distribution. In particular, during sintering simulations we compute the per-seed value of $\Delta G$, to investigate seed-specific, geometry-dependent distortions during the processes. In this case, the sum in Eq. runs over the number of atoms in each seed, $N=N_{seed}$.

The $\alpha =G/R_{max}$ ratio gives information regarding the sphericity of the nanoparticle, with sphere featuring $\alpha \to$ 0.63, while elongated/rod-like shapes showing $\alpha$ below 0.58.

Several global shape descriptors derive from the gyration tensor, $S_{mn}$:

$$S_{mn}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{r}_i^{\left(m\right)}{r}_i^{\left(n\right)}$$

(3)

where $r_i^{\left(\gamma \right)}$ is the $\gamma -$th Cartesian component of the position-vector of the i-th atom, and $m,n\in 1,2,3$. Among them, the relative shape anisotropy ${\kappa }^2$,

$${\kappa }^2={\displaystyle \frac{3}{2}}{\displaystyle \frac{{\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2}{{({\lambda }_1+{\lambda }_2+{\lambda }_3)}^2}}-{\displaystyle \frac{1}{2}},$$

(4)

where the ${\lambda }_i$ are the eigenvalues of S, is a robust shape descriptor. $\alpha$ ranges from 0 for spheres and Platonic solids, such as icosahedra (Ih) or cubes (Cb). In contrast, flattened/pancake geometries yield low but non-zero anisotropy, with a non-zero ${\kappa }^2$ below 0.25. For elongated shapes such as nanorods, ${\kappa }^2$ scales directly with the aspect ratio, approaching 1 for nanochains/nanotubes, with prolate shapes having a non-zero ${\kappa }^2$.

On the basis of f, and $\alpha ,{\kappa }^2$, we can classify the MNP shape into: branched ($f\le 2.3$), compact ($f>2.3$). For the latter, one can distinguish spheroids, oblate and prolate depending on their ${\kappa }^2$ and $\alpha$: ${\kappa }^2<0.01$ for spheroids, with ${\kappa }^2>0.25$ stands for prolate, otherwise the shape is likely oblate or a concave spherical geometry.

We perform a CNA and CNAP to classify a MNP into geometrical families. CNA labels each pair of nearest neighbours from the local connectivity of their common neighbours. Each pair exhibits a signature, a triplet of integers, $(r,s,t)$- where r is the number of nearest-neighbours common to both atoms; s is the number of bonds between shared neighbours and t is the longest chain which can be made from bonding s atoms if they are nearest neighbours-. The CNAP of an atom i is the string $(f_1{[r,s,t]}_1;f_2{[r,s,t]}_2;\dots ;f_k[r,s,t]k)$ containing all the $k-th$ CNA-signatures $[r,s,t]k$i is in. We note that the frequency of each signature correlates with the nominal coordination of the atom i, ${\sum }_k{f}_k\left(i\right)=CN\left(i\right)$.

Here we consider FCC, twin FCC, and multiple-twinned nanoparticles,[3] as decahedral (Dh) and Ih. Counting how many pairs contribute to selected CNA signatures provides information on the overall nanoparticle’s structure: FCC requires a zero (5,5,5) and a zero (4,2,2), while twin FCC has a non-zero (4,2,2). Multiple-twinned NPs have a positive (5,5,5) and a high (4,2,2). The FCC inner core can contain other defects, with a significant (4,1,1) component, mainly during the sintering process. Surface faceting follows: (111) facets features a high (3,1,1), while (2,1,1) labels (100) facets. Distortions contribute as (3,2,2) for multiple-twinned shapes. To further distinguish decahedra from icosahedra, we consider the local environment of each atom through its CNA pattern. For example following Ref.[50], [12(4,2,1)] is a FCC bulk; [6(4,2,1), 6(4,2,2)] labels atoms on a twin plane; [10,(4,2,2), 2(5,5,5)] is those atoms lying on fivefold axis. The [12,(5,5,5)] pattern refers to the Ih-centre. On the other hand, a zero [12,(5,5,5)] with a high [10(4,2,2), 2(5,5,5)] stands for a Dh.

Besides the importance of associating atoms to their CNAPs, and each CNAP with specific atomic environments, we provide a short CNAP-summary as (i) the number of different CNAPs, ${\mathcal{N}}_{CNAP}$ and (ii) the number of singleton CNAPs -CNAPs appearing for only one atom. ${\mathcal{N}}_{CNAP}$ decreases when the symmetry of the shape increases, being 1 in a perfect bulk. to ${\mathcal{N}}_{CNAP}\to N$ for a completely disordered situation. Singleton CNAP, ${\mathcal{U}}_{CNAP}$ provides a further estimate of the number of local defects, such as surface adatoms, islands, vacancy, and interstitial.

3. Results

As the novelty of NPGrowth lies in its ability to mimic directional deposition, we present a few examples, summarised in Table 2, that can lead to asymmetric, grooved or branched shapes. Although we do not pretend to reproduce any specific experimental situation our examples would represent synthesis under far-from-equilibrium conditions. For the sake of completeness, we contrast this with random 1-by-1 and N-by-N atomic deposition, see Figure 3. We further consider the soft-landing of four Cu seeds from three different sintering geometries: (i) along a line, (ii) at the vertices of a square, (iii) at the vertices of a tetrahedron. Figure 4 depicts these sintering geometries.

3.1. Oriented Atomic Deposition on Cu-Seeds

We consider and discuss the four examples reported in Figure 3 and their shape classification, as in Table 2. Oriented atomic deposition simulations are carried out using a timestep of 2 fs and a Langevin thermostat with a damping parameter of 0.1 ps. Interatomic interactions are described using the embedded-atom method (EAM) potential Cu.u3, developed by Foiles et al. [61]. After each deposition event, the system evolves using thermostat-coupled MD, allowing the deposited atoms and the initial seed to relax and evolve according to the imposed deposition conditions. Temperature is kept at 300K.

Figure 3 reports the pySNOW analysis of the final snapshots of four examples listed in Table 2. The final snapshots are (a) a concave cube, (b) a branched, spiky shape, (c) a flattened disk resembling an "onigiri," and (d) a star shape with small reentrances/grooves, but still classified as a spheroid.

The PDDF exhibits a second peak for all structures, indicating a geometrical order. At the same time, we note that the PDDF profiles vary significantly. A branched shape leads to an almost flat region after reaching a peak. Grooves/reentrances of starred shapes are present but not too deep, the MNP retains a PDDF profile and a ${\kappa }^2$ value similar to that of compact shapes, although with less peaks at large distances. Moreover, the atop generalised coordination number (aGCN, [57]) distribution reveals a notably higher concentration of low-coordinated atoms, with aGCN below 4, while flattened and compact shapes have clear peaks in correspondence of facets coordination and a drop of atoms with aGCN around 8.5 and 10. Concavities of all sorts lead to a significant number of atoms with aGCN mainly around 10.

We note that branched MNPs, named spike, exhibit a wide variety ${\mathcal{N}}_{CNAP}$, with 56% being singletons and with a 2% of atoms exhibiting an individual local environment, against the 0.6% from random deposition.

In all the examples, the combination of an initial FCC seed, a fast deposition rate and a low simulation temperature, show that the outcomes have a twinned FCC structure, with a zero (5,5,5), no sign of fivefold symmetries, and the (4,2,2) spanning from 5-10%, with the highest occurrence on the onigiri. The CNAP analysis shows no Ih centres in any snapshot, as ${\mathcal{N}}_{[12,(5,5,5\left)\right]}$ is zero. A common feature is the surface, with (3,1,1) being considerably high in all snapshots, suggesting the formation of (111) facets and indicating that reentrances have a (111) character, as the star has a (3,1,1) occurrence around 14.5%.

3.2. Oriented Sintering of Four Cu-Seeds

We soft-land four Cu-Ih₁₄₇ seeds from an equilibrated NVT MD trajectory. We run simulations with a 2.5 fs timestep and a Langevin thermostat with a time-coupling constant of 100 timesteps. We model the system’s potential energy surface with a custom machine learning potential (MLP) built using the FLARE package [63,64]. Supplementary information (Sec. B) provides more details on the MLP derivation.

First, we carry out a series of short simulations to characterise the occurrence of four-way sintering as a function of the sintering geometry and of the initial displacement between the centre of mass of the seeds, $D_{seed}$, which spans from a distance such that atoms from different seeds are in contact to the one such that they are at a minimum distance close to the cutoff of the potential, where inter-seed interactions are negligible. For each of the three geometries, we perform 10 independent simulations over a range of initial displacement distances, using different random seeds that generate distinct initial cluster orientations. Each simulation runs for 1 ns at 300 K.

Figure 4 reports the probability of forming a certain number of NPs as a function of the initial $D_{seed}$ separation and sintering geometry. The final number of distinct MNPs varies, with simultaneous sintering of all four seeds occurring only under certain conditions. Sintering geometry plays a key role - more compact arrangements (Square or Tetrahedron) generally lead to a single nanoparticle up to relatively large $D_{seed}$. In contrast, for the Line geometry, this happens only when $D_{seed}$ is less than roughly 16 Å; for larger values, the seeds tend to sinter pairwise or in similar, separated ways. This behaviour arises from stochastic, thermostat-induced displacements, which can push seeds beyond the potential’s interaction range. These kicks, together with the different relative orientations of the seeds, are also responsible for the variability of results at fixed geometry and $D_{seed}$.

Then, we perform longer simulations to investigate the dynamics of the sintering processes at 300 K and 600 K. Both temperatures are well below the melting temperature of a 588-atom Cu-NP, as shown from melting and freezing cycles reported in Suppl. Info (Figure A4). For each sintering geometry, we run six independent simulations, ensuring significant variation in the randomised initial relative orientations of the seeds to improve the statistical significance of the results. Energy differences between these isomers are significant, e.g. ≈ 20 meV/atom (≈ 11.76 eV in total) comparing line vs tetrahedron geometry at 300 K (see Suppl. Info. Figure A5). Let us discuss the architectures obtained from the sintering process and their dependence on the initial geometry of the four seeds. Results of individual simulations are in the IDs reported in Suppl. Info. Figures A6-A11. Looking at the time evolution of the PDDF, we note that after the initial sintering stages, a clear peak corresponding to the lattice parameter suggests the formation of ordered architectures. We note that only one simulation ended in a structure that should be identified as amorphous (inline attachment at 600 K, Figure A7, fourth panel). Figure 5 compares the PDDF of the last snapshots, collected at 10 ns, averaged over the six simulations per each case.

The impact of the initial sintering geometry is clear, especially on the nanoparticles’ size. At 300 K, the maximum of the PDDF ranges from 25 Å to more than 35 Å, with sizes longer than 32 Å only in the Line case. At 600 K, surface diffusion is faster and tends to smooth differences among the sintering geometries, but it is still not enough on the 10 ns timescale to eliminate them. Indeed, we see well-defined peaks up to $distance\sim 18$ Å and no size differences between Tetrahedron and Square sintering geometries, with an average MNP size around 25 Å. In contrast, Line sintering yields sizes of about 30 Å. The PDDF profile at short distances is similar to all cases, with the second peak being more pronounced at 600 K than at 300 K. The long-distance region, $distance>6$Å, of the PDDF highlights significant structural differences in the MNP shape. While the PDDF drops sharply for the Tetrahedron and Square sintering geometry, it decays smoothly for the Line case (Figure 5).

Figure 6 shows the time evolution of key shape descriptors of the grown nanoparticles in terms of ${\kappa }^2$ and the per-seed gyration radius variation, $\Delta G$.

A clear effect of the sintering geometry on the overall shape is evident. Sintering from a Tetrahedron geometry yields spherical nanoparticles ( ${\kappa }^2<0.01$) for both temperatures. In contrast, MNPs adopt a prolate shape with ${\kappa }^2>0.01$ when sintered in the Line arrangement. Square-like initial geometries give an intermediate result with a strong temperature dependence. The dynamical mechanisms governing the sintering are also different depending on the initial sintering geometry, as quantified by $\Delta G$ (Figure 6). While sintering from Square and Tetrahedron geometries leads to distortion relative to the initial positions that is uniform across the seeds, this is not the case in the Line geometry, where the two innermost seeds change considerably, with the two outermost significantly less deformed. This analysis highlights how, in this case, the external seeds compress the internal ones, which are deformed by this process, while the former substantially retain their initial shape, as further highlighted by the number of conserved Ih centres - panel c of Figure 7. We also note the change in $\Delta G$ at 5 ns for the Tetrahedron geometry, which reveals a thermally activated process for all MNPs. This change should be related with Figure 7 and indicates an increment in the FCC signature associated to a restructuring of the MNP.

Figure 7 helps in quantifying structural ordering and local atomic environments. The individual CNA and aGCN analysis per each individual simulations is in Suppl. Info., mid and right columns of Figures A6-A11. As expected, structural rearrangements depend on temperature, but they also differ due to the sintering geometry. At 300 K, the contribution of FCC-bulk atoms is lower than other cases. The counts of CNA patterns for FCC, twinning plane, and fivefold-axis environments remain stable after a brief initial variation, and the number of conserved Ih centre environments is not cancelled, indicating that the initial seed structure is largely unchanged. As reported in Suppl. Info. Table A2, if the sintering occurs simultaneously on different facets of the seed(s), it leads to structures with a higher degree of disorder, as quantified by the number of ${\mathcal{N}}_{CNAP}$ and ${\mathcal{U}}_{CNAP}$, with a clear trend as Tetrahedron>Square>Line at 300 K. Furthermore, almost 40% of the atoms in MNP sintered from a Tetrahedron geometry have their own local environment, and among them three quarters are defects. At 600 K, however, there is sufficient kinetic energy to allow restructuring, as several diffusion mechanisms are activated. In this case, the number of different patterns is much lower, but still clearly depends on the initial geometry (Line>Square>Tetrahedron). Prominent Ih-related patterns rapidly drop (right column of Figure 7), with [12(5,5,5)] being zero below 1 ns, showing that the initial atomic environment-level structure is lost. FCC and twinning-plane environments grow significantly throughout the simulations as internal ordering develops. The kinetics differ between the sintering geometries. Line and Square show similar timescales for forming the first FCC-like environments, but are less prone to increasing them. The combination of a high final count of twinning-plane environments with the near-complete absence of Ih centres and the presence of fivefold axes suggests the possible formation of decahedral structures as an outcome of these simulations, as is in fact observed (e.g. in Table 2).

4. Conclusions

We present a new and open source software package, NPGrowth, to model seed-mediated growth on metallic nanoparticles in various situations, including those that can lead to dendritic, spiky, chiral, and non-spherical nanoparticles. The key feature of NPGrowth is a modification of LAMMPS’ fix deposit that allows simulation of a variety of deposition routes compatible with LAMMPS features, including machine-learning potentials.

We aim to introduce the capabilities of the proposed package, not to exhaust or investigate all cases but to show the effect of oriented attachment for the synthesis of copper nanoparticles.

We discuss oriented attachment onto cubic and truncated octahedral seeds, showing how to set up NPGrowth source and target regions to get both branched, star, concave and flattened/oblate motifs. We propose a morphological classification of the obtained nanoparticles employing the pySNOW descriptors. We introduce a nanoparticle ID that includes the PDDF, the CNA and the aGCN distributions plus a few shape and structural quantities. We stress that shape and structure here refer to the envelope enclosing the nanoparticle and to its geometrical shape, respectively. The nanoparticle geometrical character, instead, refers to whether it belongs to FCC, FCC twin, or multiple-twinned (Ih, polyIh, and Dh) geometry. Our examples show the variety of morphologies we can generate with NPGrowth, but they are not exhaustive.

We then analyse in detail the sintering of four Cu-Ih nanoparticles, each with 147 atoms, soft-landed onto one another in three different orientations or sintering geometries. We show an effect on the size distribution due to orientational attachment, with a higher probability of forming a single nanoparticle for 2D or 3D sintering geometries. Inline attachment favours two-by-two aggregation of seeds. In soft-landing sintering at short relative distances among seeds, the initial geometry affects the final shape of the nanoparticle. Analysing the first 10 ns after sintering, we show that inline attachment leads to strongly elongated shapes even at 600 K. A planar attachment, modelled as squarely displaced seeds, generates highly faceted but concave shapes. 3D, isotropic (Tetrahedron) soft-landing favours the formation of an individual, mostly spherical, convex, architecture. Significant internal restructuring occurs depending on the system temperature, and a detailed analysis of the CNA signatures and patterns allows a deep understanding of the underlying physical processes. This analysis shows that the kinetic timescale depends on sintering geometry, with a 5 ns window for major FCC reordering in 3D Tetrahedron sintering while the kinetics is slower for the other initial geometry.

We believe NPGrowth will help the scientific community working on nanosystems to gain atomistic insights into their formation processes and build a sufficiently diverse structural database to screen the best candidates from a large set. Furthermore, the proposed pySNOW characterisation and classification should establish a common nomenclature for identifying nanoparticle morphology (size, shape, motif).

Appendix C.1. Potential Energy of Sintered Configurations

We report here the evolution of the per-atom potential energy in sintering simulations. The plot in Figure A5 shows that a significant energy difference is present between sintered nanoparticles starting with a different geometry.

Figure A5. Potential energy per atom in sintering simulations for different initial geometries and thermostat temperatures.

Figure A5. Potential energy per atom in sintering simulations for different initial geometries and thermostat temperatures.

Appendix C.2. Final structures analysis

Table A2 contains statistical informations about the values of ${\mathcal{N}}_{CNAP}$, $\int CNAP$ and their ratio for the last snapshots of the simulations.

Table A2. Number of unique CNA patterns (${\mathcal{N}}_{CNAP}$), singleton patterns (${\mathcal{U}}_{CNAP}$) and ${\mathcal{U}}_{CNAP}/{\mathcal{N}}_{CNAP}$ ratio for the final snapshot of sintering simulations. For all represented quantities, the main value is the average across the 6 simulations and the value in parentheses is the standard deviation across simulations.

	$\mathit{T}=300$ K			$\mathit{T}=600$ K
	Line	Square	Tetrahedron	Line	Square	Tetrahedron
${\mathcal{N}}_{CNAP}$	203.8 (24.1)	217.5 (23.2)	232.6 (32.2)	115.1 (47.6)	98.5 (40.7)	60.1 (21.2)
$\int CNAP$	144.8 (21.7)	157.3 (23.7)	171.8 (29.9)	68.0 (33.2)	60.1 (31.4)	30.0 (16.1)
ratio	0.71 (0.03)	0.72 (0.03)	0.73 (0.03)	0.57 (0.07)	0.59 (0.07)	0.48 (0.08)

Table A2. Number of unique CNA patterns (${\mathcal{N}}_{CNAP}$), singleton patterns (${\mathcal{U}}_{CNAP}$) and ${\mathcal{U}}_{CNAP}/{\mathcal{N}}_{CNAP}$ ratio for the final snapshot of sintering simulations. For all represented quantities, the main value is the average across the 6 simulations and the value in parentheses is the standard deviation across simulations.

	$\mathit{T}=300$ K			$\mathit{T}=600$ K
	Line	Square	Tetrahedron	Line	Square	Tetrahedron
${\mathcal{N}}_{CNAP}$	203.8 (24.1)	217.5 (23.2)	232.6 (32.2)	115.1 (47.6)	98.5 (40.7)	60.1 (21.2)
$\int CNAP$	144.8 (21.7)	157.3 (23.7)	171.8 (29.9)	68.0 (33.2)	60.1 (31.4)	30.0 (16.1)
ratio	0.71 (0.03)	0.72 (0.03)	0.73 (0.03)	0.57 (0.07)	0.59 (0.07)	0.48 (0.08)

Appendix C.3. IDs of Final Structures

We report here the IDS of the final structures obtained from sintering simulations.

Figure A6. IDs of final structures from simulations with the Line geometry at T=300 K

Figure A6. IDs of final structures from simulations with the Line geometry at T=300 K

Figure A7. IDs of final structures from simulations with the Line geometry at T=600 K

Figure A7. IDs of final structures from simulations with the Line geometry at T=600 K

Figure A8. IDs of final structures from simulations with the Square geometry at T=300 K

Figure A8. IDs of final structures from simulations with the Square geometry at T=300 K

Figure A9. IDs of final structures from simulations with the Square geometry at T=600 K

Figure A9. IDs of final structures from simulations with the Square geometry at T=600 K

Figure A10. IDs of final structures from simulations with the Tetrahedron geometry at T=300 K

Figure A10. IDs of final structures from simulations with the Tetrahedron geometry at T=300 K

Figure A11. IDs of final structures from simulations with the Tetrahedron geometry at T=600 K

Figure A11. IDs of final structures from simulations with the Tetrahedron geometry at T=600 K

Appendix C.4. Time Evolution of Pair Distance Distribution Functions

Figure A12. PDDF time evolution for the sintering simulations at T=300 K. We note that at low T, the PDDF profile are quite similar, with mild deviations at 5 Å.

Figure A12. PDDF time evolution for the sintering simulations at T=300 K. We note that at low T, the PDDF profile are quite similar, with mild deviations at 5 Å.

Figure A13. PDDF time evolution for the sintering simulations at T=600 K. the PDDF differ for all the cases, supporting the presence of various structural motifs, as discussed.

Figure A13. PDDF time evolution for the sintering simulations at T=600 K. the PDDF differ for all the cases, supporting the presence of various structural motifs, as discussed.