Types of Size-Dependent Melting in Fe Nanoclusters: a Molecular Dynamics Study
Louis E. S. Hoffenberg Alexander Khrabry Yuri Barsukov Igor D. Kaganovich David B. Graves





    Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08540

    Andlinger Center for Energy and the Environment, Princeton University, Princeton, New Jersey 08540

    Princeton Plasma Physics Laboratory, Princeton, New Jersey 08540

    ikaganov@pppl.gov
    Princeton Plasma Physics Laboratory, Princeton, New Jersey 08540

    dgraves@princeton.edu
Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08540



§ ABSTRACT 
Metallic nanoclusters are of interest in many fields because of their size-dependent catalytic activity. This activity can, in part, be influenced by their melting properties. In this work, the melting phase transitions of Fe_n nanoclusters with n ≤ 100 atoms were investigated using classical many-body molecular dynamics simulations.

Adding a single atom to many cluster sizes induced strong variations in melting point (T_melt), latent heat of melting (Δ H_melt), and onset temperature of isomerization (T_iso). Clusters with size-dependent melting behavior were classified into 3 distinct cluster types: closed-shell, near-closed-shell, and far-from-closed-shell clusters. First-order-like phase transitions were observed only for cluster sizes with particularly symmetric closed shells and near-closed shells with up to a few missing or extra atoms. Near-closed-shell clusters had very low T_iso relative to their T_melt. Far-from-closed-shell clusters exhibited second-order-like phase transitions. Variations in the melting and isomerization behavior of neighboring cluster sizes may have implications for catalytic systems such as the growth of single-wall carbon nanotubes.


Types of Size-Dependent Melting in Fe Nanoclusters: 
a Molecular Dynamics Study
    David B. Graves
    September 9, 2024
===============================================================================





§ INTRODUCTION



Nanoparticles (NPs) – i.e., particulate material with characteristic dimensions under 100 nm – have interesting properties that make them desirable for catalytic<cit.>, optoelectronic<cit.>, and biomedical<cit.> applications, among many others. These properties can depend on many factors like nanoparticle size, composition, degree of crystallinity, and other structural elements. The phase of the NP can have a significant effect on properties since atoms in liquid particles have considerably more mobility than in solid form. For example, it has been proposed that carbon nanotube (CNT) growth on catalytic iron NPs depends in part on carbon precursor adsorption, surface diffusion, and dissolution into the NP<cit.> – all of which can be influenced by the NP phase<cit.>. The transition between solid and liquid NP phases is therefore of potential importance in multiple applications.

The melting points for NPs are known to differ from those of their corresponding bulk materials<cit.>. NP melting temperatures are lower than the bulk melting point due to larger surface-atom-to-volume-atom ratios<cit.>. Surface atoms are bonded to fewer atoms than the inner atoms, so smaller NPs require less energy to melt than larger ones. Their melting points scale according to the Gibbs-Thomson equation:

    T_m, NP = T_m, bulk(1 - 2 σ_sl/Δ H_mρ_s r)

where σ_sl is the solid-liquid interfacial energy, Δ H_m is the bulk latent heat of the melting, ρ is the bulk solid density, and r is the radius of the NP<cit.>.

It is also known that when the NP radius is below some threshold size range (i.e., approaching the nanocluster regime), melting points no longer follow the Gibbs-Thompson equation. Instead, melting temperatures can fluctuate strongly with cluster size, with the addition or subtraction of a single atom (cf. Fig.<ref>). Sometimes, the nanocluster melting points can exceed those of the bulk solid<cit.>.


This variation in melting points is attributed to interrelated geometric and electronic quantum size effects<cit.>, collectively called magic number effects. Magic number clusters are nanoclusters with particularly stable structures due to either configurational symmetry that maximizes bonding between atoms (corresponding to geometric magic numbers) or electronic effects that stabilize certain cluster geometries (i.e., electronic magic numbers)<cit.>. Magic numbers have been documented to affect melting behavior and the Gibbs free energies of formation for small metal clusters<cit.>.

In larger systems and NPs, atoms are frequently approximated as surface or bulk. The transition regime between Gibbs-Thomson NP scaling and the nanocluster fluctuation regime illustrated in Fig.<ref> is thought to occur when the cluster's atoms can no longer fit the surface-bulk binary<cit.>. When clusters fall in this regime, there are fewer bulk atoms and distinct types of surface atoms with different binding energies, which determine cluster energetics. Because certain structures have perfectly closed atomic shells (geometric magic numbers)<cit.>, nanoclusters with more or fewer atoms have notably different binding energies per atom and therefore different melting temperatures<cit.>. Ion calorimetry measurements of Al nanoclusters have revealed that the transition between Gibbs-Thompson NP scaling and nanocluster variation of melting temperatures occurs between clusters of 150 and 342 atoms<cit.>. Simulations of Ni nanoclusters observed the Gibbs-Thompson NP scaling in clusters as small as 90 atoms<cit.>.

The nanocluster size range, which is the focus of this work, is particularly interesting for some catalysis applications such as the catalytic growth of carbon nanotubes with Fe or Fe-containing alloys in floating catalyst chemical vapor deposition (FCCVD)<cit.>. Fe nanoclusters of up to ∼100 atoms (∼1.2 nm in diameter) are most relevant to the growth of single-wall CNTs (SWCNTs)<cit.>. This work focuses on Fe nanoclusters. Despite interest in iron nanoparticles for the catalytic growth of CNTs, among other applications, the melting behavior of Fe nanoclusters has garnered few dedicated studies<cit.>. Furthermore, no study has analyzed the majority of the Fe nanocluster size range. The caloric curves describing Fe cluster melting behavior in this work are used in the accompanying paper<cit.> to determine the free energies of cluster formation in the kinetic modeling of nucleation and growth from condensing vapor.

The process of melting in nanoclusters differs from bulk material melting and NP melting. Bulk melting is described by a sharp increase in atomic mobility of all atoms<cit.> and a steep rise in a caloric curve (a graph of cluster energy vs. temperature) at the melting point, indicating a first-order phase transition. NP melting, on the other hand, is often characterized by surface melting followed by melting of the NP core<cit.>. For nanoclusters, the process of ”melting“ involves a dynamic coexistence between ordered and disordered phases, a phenomenon generally not seen in nanoparticles or bulk materials<cit.>.

Because NP and nanocluster melting generally occurs on length scales and time scales that are difficult to resolve experimentally, molecular simulation is often employed. Monte Carlo (MC) methods can efficiently sample configurational potential energy surfaces and construct caloric curves to describe cluster phase transitions<cit.>. Molecular dynamics (MD) simulations construct time trajectories of atoms by directly integrating Newton’s equations of motion. Forces between atoms are calculated with an interatomic potential (F = -∇ E). Classical molecular dynamics uses models of the interatomic potential with parameters fit to some combination of experimental data and quantum mechanical calculations, such as density functional theory (DFT). 

It is possible to use DFT to compute interatomic potentials at each time step in an MD simulation – sometimes referred to as Born-Oppenheimer MD (BOMD) or ab initio MD<cit.>. This method is more accurate but is considerably more computationally expensive (prohibitively expensive for clusters of tens of atoms). BOMD has been used to simulate the melting of palladium clusters<cit.> and gallium clusters with changing electronic properties or competing stable solid phases<cit.>.

This study uses classical MD simulation because of its accessibility and computational feasibility. Moreover, classical MD lends itself more readily to subsequent analyses involving more complex processes relevant to CNT growth (e.g., surface adsorption/desorption, diffusion, carbon dissolution, and formation of graphitic carbon). Magic numbers in Fe have been studied in small nanoclusters with both experiments<cit.> and spin polarization DFT simulations to capture magnetic properties <cit.>. Although classical MD simulations do not capture detailed electronic magic number effects (e.g., Fe_7 and Fe_15), geometric magic numbers (e.g., Fe_13) and their properties can be extracted and may be relevant to other transition metal atoms apart from Fe.

The paper is organized as follows. Section <ref> details the MD simulation method and the associated nanocluster structural and thermodynamic analysis. Section <ref> summarizes the results of the calculations of Fe nanocluster melting and phase transition characteristics. Section <ref> discusses the relationship between nanocluster structure and melting behavior. Finally, concluding remarks are summarized in Section <ref>.



§ METHODS



Molecular dynamics (MD) simulations were used to investigate cluster melting. In MD, discrete atoms are simulated in a periodic box and their motion is integrated forward in time, abiding by Newton's equations of motion (F = m a).

Classical MD, which uses simple and cheap interatomic potentials, can access larger lengthscales and timescales than quantum mechanical methods. However, some long-time- and length-scale phenomena (e.g., vapor condensation, leading to nucleation and growth of large numbers of NPs) are still prohibitively expensive because MD must account for every atom's movement. Despite these limitations, MD is still useful for gaining insights into atomic-scale phenomena that cannot be experimentally observed, such as the individual nanocluster phase transitions analyzed in this work.

Cluster melting data was obtained with classical MD simulations using the open-source LAMMPS (Large Atomic/Molecular Massively Parallel Simulator) software<cit.> with an embedded atom method Finnis-Sinclair (EAM-FS)<cit.> many-body interatomic potential for Fe. The potential energy of a given atom i is given by:

    E_i = F_α(∑_j ≠ iρ_αβ (r_ij) ) + 1/2∑_j ≠ iϕ_αβ (r_ij)

where F_α is the embedding energy, a function of the (modeled) electron density, ρ_αβ, contributed by neighboring atom j of element β at the site of atom i of element α. ϕ_αβ is a simple pair potential between atoms i and j. The potential was parameterized to reproduce solid and liquid characteristics of Fe<cit.>. Solid state binding energies agreed with those calculated using density functional theory (DFT) (supplementary material Fig.S1-3). Furthermore, the boiling point and latent heat of vaporization were validated using MD simulations of direct vapor-liquid co-existence (supplementary material Fig.S4).

Global minimum energy configurations for Fe_n clusters for up to 100 atoms were collected from Elliot et al.<cit.> (obtained using basin-hopping energy minimization with the EAM-FS potential). The EAM-FS potential used in this work was slightly different from the one employed in Ref.<cit.>, with a smaller cutoff distance of 5.6 Å. For this reason, the cluster configurations were further minimized in LAMMPS with the  command. MD simulations of individual Fe_n clusters were run in the microcanonical ensemble (NVE – constant number of atoms, N volume, V; and energy, E) for 10 ns with a 1 fs timestep. Thermodynamic data was collected from 75 NVE runs for each cluster size at its minimum energy configuration, initiated at temperatures from 100 - 7500 K (without whole-cluster translation and rotation), spanning the solid-liquid phase transition. Time-averaged cluster temperatures were ∼50 K to 3500 K for most cluster sizes because around half of the energy is transformed into potential energy. Canonical ensemble (NVT – constant number of atoms, volume, and temperature) simulations could have been used instead for thermodynamic sampling; however, NVE simulations were sufficient. Sampling with the NVE MD simulations was benchmarked with data from Frantz et al.<cit.> on Lennard-Jones Ar clusters, obtained with a parallel-tempering Monte Carlo approach (supplementary material, Fig.S5).




§ RESULTS


Molecular dynamics simulations were performed for individual clusters with up to 100 atoms at temperatures spanning the melting transition. Upon analysis of cluster melting, four types of clusters were delineated (Fig.<ref>) and will be used throughout this work: 1) Closed-shell clusters are clusters with highly symmetric, closed-shell global minimum energy structures. 2) Near-closed-shell clusters are 1 to a few atoms away from closed-shell structures. Both closed-shell and near-closed-shell clusters are considered magic number clusters. 3) Far-from-closed-shell clusters are distant in size to highly symmetric minimum-energy structures. These are considered non-magic number clusters.



Snapshots of two closed-shell clusters Fe_13 and Fe_78 before, in the middle of, and after melting are shown in Fig.<ref>a. Closed-shell (magic number) sizes – such as Fe_13 and Fe_78 – can have deeper global minima energy isomers in configuration space than far-from-closed-shell nanoclusters like Fe_8 (Fig.<ref>b). For this reason, the range of temperatures over which a cluster melts (escapes the global minimum potential energy well) can vary strongly. Moreover, due to the presence of many configurations that are close in energy and the dynamic-coexistence (isomerization) nature of nanocluster melting<cit.>, a single unambiguous melting point temperature can be difficult to identify.


To quantify the melting behavior of iron clusters, caloric curves of specific [total = potential and kinetic] energy (eV/atom) vs. temperature (K) were obtained for each cluster size; each data point corresponding to an NVE MD simulation at one cluster size and temperature (Fig.<ref>). Temperature was determined from time-averaged cluster kinetic energies (T_MD = 2 E_kin/k_B), corrected for whole-cluster rotational and translational degrees of freedom absent in the single-cluster MD simulations: T = T_MD/3 n - 6 for n ≥ 3 and T = T_MD/3 n - 5 for n = 2.

Caloric curves qualitatively differ by cluster size. Closed-shell magic number clusters (Fe_13, Fe_19, Fe_23, and more not shown – all have perfect or interpenetrating icosahedral structures) exhibit phase transitions with large increases in specific energy upon melting relative to neighboring clusters. This phenomenon can be seen in the large vertical spaces between the curves in the solid region (left) of Fig.<ref>. Many other cluster sizes exhibit no extra increase in specific energy upon melting.



In general, specific energy decreases with cluster size, n: E_n-1 > E_n > E_n+1. A cluster with a higher n has more bonds per atom due to a smaller surface-to-volume ratio, leading to a lower specific energy. The difference in specific energy between adjacent sizes (spacing between curves) decreases with increasing n, as the difference in surface-to-volume ratios decreases with greater n. The trend is violated in the solid phase regime (left) for closed-shell magic number clusters, whose energy curves then cross their n -1 neighbor cluster (Fe_12, Fe_18, and Fe_22 in Fig.<ref>) upon phase transition into the liquid phase regime (right), restoring the trend.

The melting point (T_melt) and the latent heat of melting (Δ H_melt) were calculated from individual caloric curves. Figs.<ref>a-c shows caloric curves for Fe_13,  Fe_15, and  Fe_96. The curve for Fe_13 (<ref>a) is representative of magic number cluster sizes, consisting of both closed-shell clusters and near-closed-shell clusters. Magic number clusters have first-order-like phase transitions with a relatively large T_melt (1415 K for Fe_13) and a large Δ H_melt. The curve for Fe_96 (<ref>b) represents far-from-closed-shell cluster sizes, which have a second-order-like melting transition featuring an abrupt change in slope. Far-from-closed-shell clusters have a small T_melt (555 K for Fe_96) and a negligible Δ H_melt. The caloric curve for Fe_15 represents those of 3 anomalous clusters constituting a subset of near-closed-shell cluster sizes (Fe_15, Fe_16, and Fe_17) with qualitatively different caloric curves. Anomalous caloric curves feature an elongated second-order phase transition with a gradual shift from the solid slope (C_v,solid) to the liquid slope (C_v,liquid). Anomalous clusters have ambiguous but high melting points (932 K for Fe_15) and small latent heat.


Because cluster caloric curves appear different from bulk ones (which usually feature a discontinuity at a first-order phase transition temperature), there is some freedom in determining cluster melting points and latent heat of melting. Linear fits were used on the solid and liquid ends of the caloric curves (red and orange lines in Figs.<ref>a-c) to help define T_melt and Δ H_melt. The fit lines were constructed to incorporate the number of data points (from each end) that minimized the uncertainty on the fit parameters. T_melt was defined as the point on the curve with the maximum distance from the solid and liquid fit lines. This definition worked well for caloric curves with first-order-like transitions. The Δ H_melt was defined as the difference in energy between the solid and liquid fit lines at T_melt. If the maximum distance between the data and the fit lines was below the noise threshold in the data (i.e., outliers determined the melting point of the curve), then the melting point was taken to be the data point associated with the intersection of the fit lines. This definition worked well for second-order-like phase transitions. Uncertainties for T_melt and Δ H_melt were calculated via propagation of the standard deviations of the best-fit line parameters interpolated at T_melt.

These two definitions for T_melt, while sensitive to the number of points taken in the liquid fit line (red), are a reasonable proxy for finding the maximum in C_v for a first-order-like transition and a maximum in dC_v/dT (which is d^2E/dT^2) for a second-order-like transition. C_v calculated directly from the MD data was very noisy when differentiated, especially at higher temperatures for smaller clusters. Clusters below 10 atoms showed more ambiguous melting behavior due to noise in high-temperature MD simulations, so the melting data for Fe_2-Fe_9 can be found in the supplementary material (caloric-curves folder).

Melting points and latent heat had strong variability between cluster sizes (T_melt ranged from 500 K to 1500 K) and generally correlated with each other.  Scatter-plotted data of T_melt and Δ H_melt (Fig.<ref>a) show distinct separation between non-magic number clusters, clustered near the origin, and magic number clusters, which have positively correlated T_melt with Δ H_melt. The anomalous clusters are separate,  spread just above the T_melt axis with a high T_melt, but a small Δ H_melt. Fig.<ref>b exhibits mountain-like “shelves” of magic number cluster sizes with high melting points and latent heats (Fe_19 - Fe_30, Fe_55 - Fe_69, and Fe_76 - Fe_83).



To understand the transitions in melting behavior in the T_melt and Δ H_melt shelves, the structures of the initial cluster configurations were inspected at the start and ends of the Fe_55 - Fe_69 and Fe_76 - Fe_83 shelves (Fig.<ref>). The increase of latent heat of clusters in the Fe_55 - Fe_69 shelf may be explained by a change away from a 180° symmetric structure in Fe_54 (right) to a configuration with 120° symmetry in Fe_55 (left), which is conserved until Fe_64. Images of these clusters can be found in the supplementary material (cluster-snapshots folder).

The increase in latent heat of the Fe_76 - Fe_83 shelf may be due to a strong shift from Fe_75's staggered 4-ring structure (left) and a 120° symmetric structure (right) to Fe_76's stacked 4-ring structure, and polyicosahedral-like structure (right – this polyicosahedral structure is complete in Fe_78). The decrease in melting point and latent heat at the end of the shelves are not easily apparent in the structural motifs of the minimum energy clusters.

Further investigating the cluster phase transitions, the Lindemann index<cit.> (δ) was calculated from each MD trajectory. δ is often used to quantify the melting of the clusters because it measures changes in atomic mobility in a solid, increasing rapidly with an increase in atom movement. In the case of nanoclusters, δ effectively measures the onset of cluster isomerization:

    δ = 2/N(N-1)∑_i < j√(⟨ r_ij^2⟩_t - ⟨ r_ij⟩_t^2)/⟨ r_ij⟩_t,

where N is the number of atoms in the cluster, r_ij is the pairwise distance between two atoms i and j, and ⟨⟩_t indicates an average over the entire MD trajectory. The temperature at which a cluster's δ increases sharply is defined here as the cluster's isomerization temperature, T_iso.

The plots of the Lindemann index vs. temperature for Fe_13, Fe_96, and Fe_15 are given in Figs.<ref>a-c. The onset of atomic mobility described by a stark increase in δ indicates the beginning of cluster isomerization, where atoms move around while maintaining a structure similar to the solid cluster. For these 3 (and most other) clusters, this Lindemann isomerization temperature, T_iso, is much lower than the previously defined T_melt.



The different melting temperatures, T_melt, T_iso, and the solid fit limit of the caloric curve, T_m,start (the temperature at which the energy begins to rise faster due to melting) are discussed below. T_iso is much lower than T_melt for most sizes, especially anomalous and near-closed-shell cluster sizes (Fig.<ref>a). This is expected, as cluster isomerization can occur without accessing the higher potential energy configurations associated with the melted cluster. Furthermore, T_iso does not have the same “shelf”-like behavior as T_melt (Fig.<ref>b); instead, T_iso only has sharp peaks for the closed-shell magic numbers (Fe_n, n = 13, 19, 23, etc.). Although T_m,start is closer on average to T_iso, T_m,start remains far higher for smaller near-closed-shell clusters (e.g., n = 12, 14, 20, 24, etc.).





§ DISCUSSION



As mentioned above, the solid-liquid phase transition in Fe nanoclusters differs from melting in bulk Fe: the nanocluster version of melting initially consists of a crystalline cluster with atoms vibrating in place. Upon heating, stronger vibrations initiate atoms swapping with each other within the solid structure. Eventually, the cluster forms different structures (isomers) with higher potential energies and shorter lifetimes. As the cluster heats further, it enters a regime of dynamic co-existence, where higher potential energy isomers become longer-lived, eventually converting to a melted cluster with no memory of any crystalline configuration.



 §.§ Cluster classes

The isomerization process that takes a cluster from solid to liquid also varies by cluster size. The clusters can be separated into 4 categories in terms of melting behavior observed from their caloric curves (energy vs. temperature, Fig.<ref>) and their Lindemann index transition indicating isomerization (Fig.<ref>).


  * Closed-shell clusters are magic number clusters with highly symmetric (icosahedral or interpenetrated icosahedral) structures that resist the onset of isomerization until high temperatures (high T_melt and T_iso) and exhibit first-order-like phase transitions with large latent heats of melting.

  * Near-closed-shell clusters are magic number clusters 1 to a few atoms away from closed-shell structures and isomerize at low temperatures (low T_iso). Early isomerization may delocalize the extra atoms or dangling bonds to enhance local stability. Near-closed-shell clusters still exhibit a first-order-like phase transition at a higher temperature (high T_melt) with large latent heat.

  * Far-from-closed-shell clusters have low resistance to isomerization (low T_iso), rapidly lose their memory of the solid structure at low temperature, and undergo a short, kink-shaped second-order-like phase transition (low T_melt) with no latent heat.

  * Anomalous clusters isomerize at very low temperatures (low T_iso) and begin visiting higher energy isomers at low temperatures, starting a stretched-out second-order-like phase transition (high T_melt). Instead of a kink-shaped phase transition, anomalous clusters have a slow, continuous increase in slope from C_v,solid to C_v,liquid, with no latent heat. Note: Despite the distinct appearance of their caloric curves, anomalous clusters may be misclassified near-closed-shell clusters with smaller first-order phase transitions not identified by the sensitive fitting procedure.
 



 §.§ Cluster melting points and latent heats

Stark size-to-size variation in Fe cluster melting points and latent heat is broadly attributed to geometric magic number effects rooted in structural differences in the optimal cluster configuration. While the symmetry of optimal cluster configuration is suspected to underlie the variation in their melting characteristics, these structural differences don't obviously explain the larger-size ends of the latent heat “shelves” (Fig.<ref>). More subtle changes in structure may be responsible for the drops in latent heat.

Nonetheless, magic number effects in melting behavior may imply size-dependent disorder in real catalytic nanocluster systems at temperatures where some clusters are melted and others are not. Melted clusters with more disorder may have different catalytic activity from those that are solid (or more solid). For example, differences in disorder can influence carbon adsorption, diffusion, and dissolution in CNT growth, leading to different growth rates or modes of CNT growth (tangential vs. perpendicular growth<cit.>).

The presence of other species (e.g. carbon, hydrogen, sulfur, etc.) may also influence the melting behavior of clusters, potentially even increasing the melting point beyond that of the bulk (superheating<cit.>). Furthermore, electronic magic numbers (and electronic or magnetic properties, not modeled in this EAM-FS interatomic potential) may play a role in nanocluster melting behavior. Further investigation using molecular simulation in these systems would be of interest.

 

 §.§ Isomerization temperature

There are key differences between the calculated melting points: T_melt, derived from the MD caloric curves, and T_iso, derived from the Lindemann index jump (Fig.<ref>). First, the melting points quantitatively disagree even for closed-shell magic number clusters. This is expected, since T_melt corresponds to an increase in potential energy from longer lifetimes of higher potential-energy isomers in the cluster isomerization phase, while T_iso is a measure of the onset of isomerization. Closed-shell magic number clusters with first-order-like phase transitions melt over a range of temperatures, with T_melt, start as the range's start. So T_iso should be close to T_melt, start. For most cluster sizes, T_iso lies near T_melt, start.

This trend is violated for small near-closed-shell clusters (e.g., Fe_n for n = 12, 14, 20, 24, etc.). Melting and isomerization behave qualitatively differently, with T_iso much lower than both T_melt and T_melt, start. The structural similarity of near-closed-shell clusters to closed-shell clusters may confer a deep PE well that requires a higher temperature to escape, leading to a high T_melt. Near-closed-shell clusters isomerize at lower temperatures, potentially delocalizing extra atoms or defects (dangling bonds).

T_melt and T_iso also differ qualitatively for larger clusters where “shelves” of high T_melt are observed. This may be because larger clusters have more atomic degrees of freedom and therefore a lower barrier to isomerization.

Both T_melt and T_iso may affect the catalytic activity of Fe nanoclusters. In CNT growth, mobilization of Fe atoms (T_iso) could accelerate the diffusion of C atoms on the NP surface, increasing the CNT growth rate. However, an increase in potential energy (T_melt) associated with longer lifetimes of new isomers with different structures – and therefore different crystal facets or disorder in the outer layer – may change both the adsorption energy of other species and their transport characteristics, likely influencing the rate of CNT growth. Investigation of adsorption energies and diffusion characteristics would be useful in determining the direction and magnitude of the effect on CNT growth rate.



§ CONCLUSION


Classical molecular dynamics calculations in the microcanonical ensemble (NVE) were used to calculate the melting properties of Fe nanoclusters up to 100 atoms (1.2 nm) in size. Cluster-to-cluster variations (magic number effects) were observed. The key takeaways from this work are the following:

    
  * Addition of one to two atoms in a cluster can cause strong variations in the melting point, latent heat of melting, and onset temperature of isomerization. There may be implications for enhanced or suppressed catalytic activity through changes in species adsorption or transport on the cluster surface.
    
  * Three types of nanoclusters with qualitatively distinct caloric curves are identified. Most clusters can be categorized as closed-shell, near-closed-shell, or far-from-closed-shell clusters.
    
  * Cluster caloric curves revealed first-order-like phase transitions for closed-shell or near-closed-shell cluster sizes. Far-from-closed-shell sizes had second-order-like phase transitions. Second-order-like phase transitions in clusters have been reported before, but only in systems with concurrent transitions in electronic behavior<cit.> not modeled in the classical interatomic potential used in this work.
    
  * The temperatures at which cluster isomerization begins (T_iso) and where potential energy increases (T_melt,start) differed strongly in near-closed-shell clusters.
    
  * Geometric magic number effects alone conferred a deviation from the Gibbs-Thomson melting point depression scaling followed by nanoparticles.




§ SUPPLEMENTARY MATERIALS


Additional materials such as data used in method benchmarking can be found in SI.pdf. Caloric curves for all cluster sizes, Lindemann index curves for all cluster sizes, and snapshots of minimum energy configurations for most cluster sizes can be found in the caloric-curves, lindemann-curves, and cluster-snapshots folders, respectively.



§ ACKNOWLEDGEMENTS


The support of Princeton University's Andlinger Center for Energy and the Environment, and the Program in Plasma Science and Technology at the Princeton Plasma Physics Laboratory is gratefully acknowledged. In addition, this research utilized computing resources on the Princeton University Della and Stellar clusters.


COI statement: The authors have no conflicts of interest to disclose.


Data availability statement: The data are contained within the article and supplementary material.


Author contribution statement: 

Louis E. S. Hoffenberg: conceptualization (equal); formal analysis (lead); writing – original draft preparation (lead). Alexander Khrabry: conceptualization (equal); formal analysis (supporting); review and editing (equal). Yuri Barsukov: conceptualization (equal); review and editing (equal). Igor D. Kaganovich: funding acquisition (supporting); conceptualization (equal); supervision (equal). David B. Graves: funding acquisition (lead); conceptualization (equal); review and editing (equal); supervision (equal).




