Structural properties of nanoclusters: Energetic, thermodynamic, and kinetic effects

The structural properties of free nanoclusters are reviewed. Special attention is paid to the interplay of energetic, thermodynamic, and kinetic factors in the explanation of cluster structures that are actually observed in experiments. The review starts with a brief summary of the experimental methods for the production of free nanoclusters and then considers theoretical and simulation issues, always discussed in close connection with the experimental results. The energetic properties are treated first, along with methods for modeling elementary constituent interactions and for global optimization on the cluster potential-energy surface. After that, a section on cluster thermodynamics follows. The discussion includes the analysis of solid-solid structural transitions and of melting, with its size dependence. The last section is devoted to the growth kinetics of free nanoclusters and treats the growth of isolated clusters and their coalescence. Several specific systems are analyzed.

Figure 1

High-resolution electron microscopy image of Ag clusters deposited on an inert substrate after being produced in an inert-gas aggregation experiment. The clusters indicated by arrows are identified as being icosahedra. Adapted from 383.

Figure 2

High-resolution electron microscopy image of a $8.6\times 6.3\text{−}{\mathrm{nm}}^2$ truncated decahedral gold particle deposited on amorphous carbon. The particle was produced in an inert-gas aggregation experiment and then deposited and observed. Adapted from 263.

Figure 3

Qualitative behavior of Δ [Eq. (5)] for crystalline, icosahedron, and decahedron clusters.

Figure 4

Face-centered-cubic clusters: (a) octahedron; (b) truncated octadedron; (c) cuboctahedron. Each cluster is shown in four views. (a) An octahedron is made up of two square pyramids sharing a common basis. Its surface consists of eight triangular close-packed (111) facets, but the structure has a high surface/volume ratio. Polyhedra with a lower surface/volume ratio, are obtained by truncating symmetrically the six vertices of an octahedron, thus obtaining square and hexagonal (or triangular, see below) facets. A truncated octahedron can be characterized by two indexes: $n_l$ is the length of the edges of the complete octahedron; $n_{\mathit{\text{cut}}}$ is the number of layers cut at each vertex. In the figure, for the octahedron in (a) $(n_l,N_{\mathit{\text{cut}}})=(7,0)$, the truncated octahedron in (b) $(n_l,N_{\mathit{\text{cut}}})=(7,2)$, and the cuboctahedron in (c) $(n_l,N_{\mathit{\text{cut}}})=(7,3)$. A perfect truncated octahedron has thus a number of atoms, $N_{\mathit{TO}}(n_l,n_{\mathit{\text{cut}}})=\frac{1}{3}(2n_l^3+n_l)-2n_{\mathit{\text{cut}}}^3-3n_{\mathit{\text{cut}}}^2-n_{\mathit{\text{cut}}}.$ This equation defines the series of magic numbers for truncated octahedron structures. The square facets have a (100) symmetry and edges of $n_{\mathit{\text{cut}}}+1$ atoms. The (111) facets are not in general regular hexagons. In fact, three edges of the hexagons are in common with square facets, thus having $n_{\mathit{\text{cut}}}+1$ atoms, while the remaining three edges have $n_l-2n_{\mathit{\text{cut}}}$ atoms. Regular hexagons are thus possible if $n_l=3n_{\mathit{\text{cut}}}+1$; truncated octahedra with regular hexagonal facets are referred to as regular truncated octahedra. When $n_l=2n_{\mathit{\text{cut}}}+1$ the hexagonal facets degenerate to triangles and the cuboctahedron is obtained, which is usually not energetically favored because of its large (100) facets.

Figure 5

Icosahedral clusters. The Mackay icosahedron is a noncrystalline structure organized in shells. An icosahedron with $k$ shells has $N_{\mathit{Ih}}\left(k\right)=\frac{10}{3}{k}^3-5k^2+\frac{1}{3}k-1$ atoms (so that the series of magic numbers is 1, 13, 55, 147,…). The icosahedron in the figure has $k=4$ shells. An icosahedron with $k$ shells has the same number of particles as a cuboctahedron with $n_{\mathit{\text{cut}}}=k-1$. An icosahedron has 20 triangular facets of side $k$ and 12 vertices. Each pair of opposite vertices lie along a fivefold symmetry axis. An icosahedron can be thought of as composed of 20 fcc tetrahedra sharing a common vertex (in the central site). When 20 regular tetrahedra are packed around a common vertex, large interstices remain. To fill these spaces the tetrahedra must be distorted, thus generating a huge strain on the structure. Intershell distances are compressed, while intrashell distances are expanded. The facets of an icosahedron are of distorted (111) symmetry. Adatoms deposited on the facets of a Mackay icosahedron can be placed either on sites of fcc or hcp stacking. Islands of fcc stacking are part of the next Mackay shell, while islands of hcp stacking form a so-called anti-Mackay overlayer.

Figure 6

Decahedral clusters: (a) regular decahedra; (b) Ino truncated decahedra (230); (c) Marks truncated decahedra. A decahedron is made up of two pentagonal pyramids sharing a common basis. It has a single fivefold axis and is formed by five tetrahedra sharing a common edge along the fivefold axis. When five regular tetrahedra are packed, gaps remain, but they are smaller than in the case of the icosahedra. These gaps are filled by distorting the tetrahedra, thus introducing some strain. Regular decahedra (first row) are limited by ten close-packed (111)-like facets, but have a large surface/volume ratio, which can be lowered by truncating the edges around the common basis, thus obtaining the Ino decahedron with five (100)-like facets. An even better structure is the Marks decahedron (309), obtained by introducing reentrances that separate the (100)-like facets (see the third row). A decahedron is characterized by three integer indices $(m,n,p)$, where $m$ and $n$ are the lengths of the sides of the (100) facets, perpendicular and parallel to the axis, respectively; $p$ is the depth of the Marks reentrance. A regular decahedron has indices of the form $(m,1,1)$ [the (5,1,1) decahedron is shown in the top row]; Ino decahedra have indices $(m,n,1)$, with $n>1$ [the (4,2,1) Ino decahedron is shown in the second row], and Marks decahedra have $(m,n,p)$ with $n,p>1$ [the (2,2,2) Marks decahedron is shown in the third row]. A Marks decahedron has $h=m+n+2p-3$ atoms along its symmetry axis and a total number of atoms given by $N_{M-\mathit{Dh}}=(30p^3-135p^2+207p-102)∕6+\{5m^3+(30p-45)m^2+[60(p^2-3p)+136]m\}∕6+\left\{n[15m^2+(60p-75)m+3(10p^2-30p)+66]\right\}∕6-1.$ From this formula it follows that a Ino decahedron has $N_{\mathit{\text{Ino}}}=[5m^3-15m^2+16m+n(15m^2-15m+6)]∕6-1$ atoms. For $n=m$ and $p=1$ [square (100)-like facets] a decahadron has thus the same number of atoms as an icosahedron of $m$ shells and as a cuboctahedron with $n_{\mathit{\text{cut}}}=m-1$. Finally, a regular decahedron has $N_{\mathit{Dh}}=(5m^3+m)∕6$ atoms.

Figure 7

(Color in online edition) Morse potential for ${\rho }_0$ values: dash-dotted line, sodium atoms; dashed line, Lennard-Jones molecules; and solid line, ${\mathrm{C}}_{60}$ molecules. The last is a very sticky interaction.

Figure 8

Phase diagram for Morse clusters: The lines separate domains pertaining to different structural motifs. Figure courtesy of Jonathan Doye.

Figure 9

Mass spectrum of ${\mathrm{Na}}_n$ clusters of size $n$, photoionized with $3.02\text{−}\mathrm{eV}$ photons. Closed-shell clusters are more difficult to ionize, so that they correspond to minima in the spectrum. Two sequences of minima appear in the spectrum. These sequences are at equally spaced $n^{1∕3}$ intervals on the size scale and correspond to an electronic shell sequence and a structural shell sequence. Adapted from 312.

Figure 10

(Color in online edition) Transformation of the potential-energy surface (PES) to a staircase. Figure courtesy of Giulia Rossi.

Figure 11

(Color in online edition) Disconnectivity diagrams: Top panel, schematic of a single-minimum potential-energy surface (PES) with weak noise; center panel, a single-funnel PES. The disconnectivity diagrams of these two panels show at which energetic level the different local minima of a PES can be considered connected. Adapted from 48. The lowest panel gives the disconnectivity diagram for the complete double-funnel PES of the Lennard-Jones cluster of size 38. Figure courtesy of Jonathan Doye.

Figure 12

Lowest-energy isomers of Au clusters according to 60. All clusters are planar up to $N=10$ at least. The binding energy per atom $E_b∕N$ (in eV) is also shown. For isomers with energy difference smaller than $0.1\mathrm{eV}$, both competing structures are shown (see sizes 3, 4, and 7). From 60.

Figure 13

(Color in online edition) Two views of the lowest-energy isomer of Au at $N=55$ as found by 178. This cluster is a strongly rearranged and distorted 55-atom icosahedron, similar to the double-rosette structure discussed in the case of ${\mathrm{Pt}}_{55}$ in Sec. 5A4. Figure courtesy of Ignacio Garzón.

Figure 14

Δ as a function of $N$ for different cluster motifs in Ni, Cu, Pd, Ag, Pt, and Au: 엯, iscosahedra; ◻, decahedra; ▵, trunacted octahedra. Data are taken from 28, except for the data concerning Ni, which are original. For decahedra and truncated octahedra only the most favorable clusters are shown in the different size ranges.

Figure 15

Size dependence of the binding energies of neutral silicon clusters: ◻, binding energies of the two groups of cluster isomers found by calorimetric measurements; ●, data for neutral silicon clusters obtained from the collision-induced dissociation experiments of 236 on the corresponding silicon cluster cations, taking the experimentally determined photoionization potentials into account as from 169; ▴, the Knudsen mass spectrometric measurements of 414, 415). The crossover from elongated to the spherical neutral silicon structures can be estimated from the binding energies to be located around $N=25$. The structure for an elongated ${\mathrm{Si}}_{26}$ cluster is taken from 189. Adapted from 21.

Figure 16

The best isomers of ${\mathrm{Si}}_{20}$ according to 331. Structure E is the lowest in energy according to quantum Monte Carlo calculations. From 331.

Figure 17

Mass spectra of ${\mathrm{C}}_{60}$ clusters: panel (a), clusters produced at low $T$ do not reveal any magic number; panel (b), after mild annealing to evaporate the less bound molecules, a clear series of icosahedral magic numbers emerges; panels (c) and (d), decahedra and close-packed magic numbers appear only after a high-$T$ annealing. Adapted from 71.

Figure 18

Size-dependence of the measured melting temperatures and latent heat of melting for sodium clusters. These quantities are compared to the mass spectrum of the top panel, which shows electronic magic numbers. The correlations between the electronic magic sizes and the peaks in the melting temperature are rather weak, indicating that the melting behavior of small sodium clusters cannot be explained by simple models. Adapted from 413. Reprinted with permission from Nature http://www.nature.com/

Figure 19

(Color in online edition) Schematic representation of the probabilities $p_S$ and $p_{S^{\prime }}$ of the local energy minima $S$ and $S^{\prime }$, with $S^{\prime }$ having higher energy than $S$ vs temperature. In the temperature range close to the transition temperature $T_{S{S}^{\prime }}$ the two structures coexist, being almost equally likely.

Figure 20

Structural phase diagram in the $N$,$T$ plane for Lennard-Jones clusters in the harmonic superposition approximation. From 122.

Figure 21

Schematic representation of the experimental method for measuring $E\left(T\right)$ for free clusters. Adapted from 413. Reprinted with permission from Nature http://www.nature.com/

Figure 22

(Color in online edition) Caloric curves from a molecular-dynamics melting simulation: ●, ${\mathrm{Ag}}_{38}$; ∎, ${\mathrm{Cu}}_{38}$. The heating rate is of $0.1\mathrm{K}∕\mathrm{ns}$. The clusters are modeled by the 394 potential. $E^{*}$ is the average total energy $E$ of the cluster after the subtraction of the global-minimum energy and of the kinetic and harmonic contributions: $E^{*}=E-E_{\mathit{GM}}-3(N-1)k_{B}T$. The jump in the caloric curve is clear for ${\mathrm{Ag}}_{38}$, and less evident for ${\mathrm{Cu}}_{38}$. Figure courtesy of Christine Mottet.

Figure 23

Heat capacity and bond-length fluctuations in the 57-atom Lennard-Jones cluster. Left panel, heat capacity curves. Open circles and solid line represent the results of parallel tempering simulations, while the dotted line and the thin solid line are the results of Metropolis Monte Carlo runs of different length. Right panel, ${\Delta }_{\mathit{DF}}$ calculated in two Monte Carlo simulations of different length (the longer one corresponds to the open circles). The temperature at which ${\Delta }_{\mathit{DF}}$ starts its strong increase corresponds well to the temperature of the heat capacity peak. The heat-capacity $c$ and ${\Delta }_{\mathit{DF}}$ are given in reduced units (see 162). Adapted from 162.

Figure 24

A particle with a solid core of radius $r-\delta$ and a liquid shell of thickness $\delta$.

Figure 25

Melting point of gold clusters from molecular-dynamics simulations (black dots), and comparison with different theoretical results: solid line, Pawlow’s theory; ×, second-order corrections from 79; heavy dashed curves, the liquid-shell model by 406, Eq. (42); thin dashed curves, second-order corrections to Sambles’s formula. Reprinted with permission from 99. Copyright 2001 American Chemical Society.

Figure 26

Comparison of theoretical and experimental melting points of supported tin clusters: ●, experiment; solid line, the fitting by means of Eq. (42). From 272.

Figure 27

(Color in online edition) Percentages of the different structures obtained in molecular-dynamics simulations of Ag cluster freezing at $r_c=1\mathrm{K}∕\mathrm{ns}$ for $130<N<310$: ●, icosahedral structures; ∎, decahedra; ▴, fcc clusters. Adapted from 33.

Figure 28

(Color in online edition) Molecular-dynamics simulation of silver cluster freezing. On the left, we show the initial configurations of Ag liquid nanodroplets at magic icosahedral numbers (147, 309, 561, and 923), while on the right, there are typical final structures obtained by cooling at $1\mathrm{K}∕\mathrm{ns}$: from top to bottom, 147-atom icosahedron; 309-atom icosahedron; a decahedron at 561 [strongly asymmetric and with an island on hcp stacking above (27)]; a fcc polyhedron at 923.

Figure 29

(Color in online edition) Platinum cluster structures at $N=55$. From left to right icosahedron (Ih), single rosette (Sr), and double rosette (Dr). The Ih is the lowest in energy, the Sr and the Dr are, respectively, 0.48 and $0.42\mathrm{eV}$ higher in energy.

Figure 30

Melting and freezing molecular-dynamics simulations of ${\mathrm{Pt}}_{55}$ clusters. In the upper panel we show the melting caloric curves for for three structures: dashed line, icosahedral (Ih); solid line, single rosette (Sr); dash-dotted line, double rosette (Dr). For $600<T<700\mathrm{K}$, all melting curves exhibit several oscillations among the three structures. The lower panel shows two freezing curves. The dashed line ends with an icosahedral structure, while the solid line ends with a single-rosette structure. This is an example of the interplay of entropic and kinetic effects. Due to entropic effects, the Sr structure is the most likely in the temperature range in which the cluster begins to solidify. This structure may then be preserved down to low temperatures by a trapping effect.

Figure 31

Probabilities of the different structures of ${\mathrm{Pt}}_{55}$ vs $T$ in the harmonic superposition approximation: solid line, the probability of the single rosette; dashed line, probability of the icosahedron; dash-dotted line, probability of the double rosette; dotted line (which coincides with the $T$ axis on this scale), probability of the decahedron. The last always have a very small probability, even if they are energetically favored with respect to the single rosette, which is the best structure from the entropic point of view and becomes the most likely at high $T$.

Figure 32

A 146-atom decahedron plus an hcp island above a cap. The black dots indicate fcc adsorption sites, the gray dots the hcp sites. The fcc sites on the edges between facets are not stable for single adatoms. Compared to adsorption on icosahedral clusters, the fcc and hcp sites correspond to Mackay and anti-Mackay sites, respectively.

Figure 33

The solid-state $\text{decahedron}\to \text{icosahedron}$ transformation obtained in growth simulation of clusters of ${\mathrm{C}}_{60}$ molecules. The same kind of transformation is also found in simulations of Ag cluster growth. Ag and ${\mathrm{C}}_{60}$ present completely different interactions, soft and sticky, respectively. Adapted from 24.

Figure 34

Growth simulations of small metallic clusters. Δ as a function of cluster size at different temperatures: low temperatures ($200\mathrm{K}$ for Au and Ag, $300\mathrm{K}$ for Ni and Pt, $400\mathrm{K}$ for Cu and Pd) on the left and high temperatures ($400\mathrm{K}$ for Au and Ag, $600\mathrm{K}$ for Ni, $700\mathrm{K}$ for Pd, Pt, and Cu). Minima in Δ single out the most stable structures.

Figure 35

(Color in online edition) ${\mathrm{Ag}}_{38}$ growth simulation results. In the top row, snapshots from a simulation at $T=200\mathrm{K}$ are shown at different sizes. The structures of the snapshots pertain to the global minima up to $N=33$; then growth is kinetically trapped into defected decahedron structures, which are neither energetically nor entropically favored. The graph shows the structural frequencies $f_{\mathit{TO}}$, $f_{\mathit{Dh}}$, $f_{\mathit{LS}}$ of different structures at $N=38$ and at different growth temperatures: open stars, truncated octahedra; solid stars, decahedra; triangles, low-symmetry structures. The lines are only guides to the eyes. At low temperatures, strong kinetic trapping into decahedral structures occurs. This is followed by an intermediate temperature regime where the three structural motifs are essentially equally likely. At high temperatures, entropic effects favor decahedral and low-sym-metry structures at the expense of the truncated octahedron. Adapted from 35.

Figure 36

(Color in online edition) ${\mathrm{Cu}}_{38}$ growth simulation results. Symbols as in Fig. 35. In the top row, snapshots from a Cu simulation at $T=400\mathrm{K}$ are reported. Again, the snapshots reproduce the global minima up to $N=33$. On the contrary, the growth structure at $N=38$ is not the global minimum but a Dh, which is entropically favored at this temperature. Around $200\mathrm{K}$, there is a strong kinetic trapping into Dh structures formed at smaller sizes. At intermediate temperatures the TO structure is preferentially grown; at high $T$, Dh structures again prevail due to entropic effects. Adapted from 35

Figure 37

(Color in online edition) Snapshots of silver cluster growth at ${\tau }_{\mathit{dep}}=7\mathrm{ns}$ from three simulations at three different temperatures: left column, $400\mathrm{K}$, middle column, $500\mathrm{K}$, right column, $600\mathrm{K}$. In each column, the snapshots are taken at sizes of 55, 105, and 147 atoms from top to bottom. At 75 decahedra are preferentially grown up to $500\mathrm{K}$. Icosahedral structures of 147 atoms are obtained in the simulations at 400 and $600\mathrm{K}$, and a decahedron at $500\mathrm{K}$. At 55 atoms, we obtain an icosahedron at 400 and $500\mathrm{K}$ (the structure at $600\mathrm{K}$ is rapidly fluctuating).

Figure 38

Growth sequences of intermediate-size Ag clusters at different $T$: from left to right, 400, 450, and $600\mathrm{K}$. In the top row, $N\simeq 150$, in the middle row $N\simeq 200$, and in the bottom row $N\simeq 300$.

Figure 39

High-resolution electron microscopy image of a large icosahedral Au cluster grown in an inert-gas aggregation source and then deposited on an inert substrate. Adapted from 263.

Figure 40

Experimental population distributions of icosahedral, decahedral, and fcc Au clusters grown in an inert-gas aggregation experiment. Adapted from 263.

Figure 41

Experimental mass spectrum of aluminum clusters. Adapted from 448.

Figure 42

Growth sequences of ${\mathrm{C}}_{60}$ clusters for ${\tau }_{\mathit{dep}}=100\mathrm{ns}$ and different temperatures. From top to bottom snapshots at $N=13$, 25, 38, 45, and 55 particles are shown. Adapted from 25.

Figure 43

Molecular-dynamics simulation of the coalescence of two 565-atom lead icosahedra initially at $300\mathrm{K}$. The sequence of snapshots at $3.75\text{−}\mathrm{ps}$ intervals shows the early growth of the neck after the initial contact. Adapted from 212.

Figure 44

Evolution of the temperature and of the aspect ratio during the molecular-dynamics simulation of the coalescence of two 565-atom lead icosahedra at the initial temperature of $300\mathrm{K}$. At the point of first contact between clusters (approximately $30\mathrm{ps}$ after the beginning of the simulation) the temperature rises sharply due to the release of surface energy. The inset shows the final cluster structure, which is almost spherical. The spherical shape is quickly reached, since the aspect ratio is very close to one after $1\mathrm{ns}$. Adapted from 212.