D.
Schebarchov
*^{a},
F.
Baletto
^{b} and
D. J.
Wales
^{a}
^{a}University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, UK. E-mail: Dmitri.Schebarchov@gmail.com; dw34@cam.ac.uk
^{b}Department of Physics, King's College London, London WC2R 2LS, UK. E-mail: francesca.baletto@kcl.ac.uk
First published on 15th December 2017
We consider finite-size and temperature effects on the structure of model Au_{N} clusters (30 ≤ N ≤ 147) bound by the Gupta potential. Equilibrium behaviour is examined in the harmonic superposition approximation, and the size-dependent melting temperature is also bracketed using molecular dynamics simulations. We identify structural transitions between distinctly different morphologies, characterised by various defect features. Reentrant behaviour and trends with respect to cluster size and temperature are discussed in detail. For N = 55, 85, and 147 we visualise the topography of the underlying potential energy landscape using disconnectivity graphs, colour-coded by the cluster morphology; and we use discrete path sampling to characterise the rearrangement mechanisms between competing structures separated by high energy barriers (up to 1 eV). The fastest transition pathways generally involve metastable states with multiple fivefold disclinations and/or a high degree of amorphisation, indicative of melting. For N = 55 we find that reoptimising low-lying minima using density functional theory (DFT) alters their energetic ordering and produces a new putative global minimum at the DFT level; however, the equilibrium structure predicted by the Gupta potential at room temperature is consistent with previous experiments.
One of the first and still widely studied gold nanoparticles is the “Schmid Au_{55}” cluster,^{12,13} formulated as Au_{55} (PPh_{3})_{12} Cl_{6}. The Au_{55} core geometry was first characterised as cuboctahedral,^{13,14} illustrated in Fig. 1a, with single-crystal face-centred cubic (fcc) atomic ordering. However, this initial interpretation was criticised by Vogel et al.,^{15} who found that a model with icosahedral Au_{55} core produced a better fit to the available X-ray powder diffraction data. More recently, Pei et al.^{16} used density functional theory to show that many quasi-icosahedral, decahedral, and disordered core structures are energetically favoured over the closed-shell cuboctahedron. To the best of our knowledge, the debate over the Au_{55} core geometry has still not reached a definitive resolution, exacerbating interpretation of the unusual oxidation resistance^{17} and high catalytic activity^{18} of the naked Au_{55} cluster derived from Au_{55} (PPh_{3})_{12} Cl_{6}. The intriguing chemistry of the naked Au_{55} may well be at least in part due to a particular geometry,^{17} but the geometry is unlikely to be cuboctahedral, because previous theoretical calculations^{19,20} show that the expected lifetime of bare Au_{55} cuboctahedra is too short to be observed under an electron microscope, even in the presence of a substrate.^{19}
Fig. 1 The structure of some Au_{55} isomers from Table 1: (a) Cub; (b) 5; (c) 8; (d) 1. |
In another theoretical contribution, Garzón et al.^{21,22} found several amorphous low-energy structures for the naked Au_{55}, with “amorphous” signifying distorted icosahedral order (see Fig. 1b and c), which can exhibit a degree of chirality^{23} and enantioselective properties.^{24} The lowest-lying amorphous isomer (Fig. 1b) was proposed as the global minimum (GM). Indeed, distorted^{25} or amorphized^{10} icosahedra and the partially chiral^{8} Au_{55} structures have been observed under an electron microscope. However, using the same empirical model for Au_{55} as Garzón et al.,^{21} Bao et al.^{26} found a lower energy structure with fcc order, shown in Fig. 1d, which had previously^{27} been identified for a different interatomic potential. These theoretical predictions suggest that microscopy studies may not be accessing the lowest energy structure, perhaps due to finite temperature effects, an underlying substrate,^{28} the electron beam, or some other factors. Recently,^{29} we performed global optimisation of Au_{55} for a different parametrisation^{30} of the model used by Garzón et al.^{21} and Bao et al.,^{26} and again found the GM to be fcc, with amorphous isomers dominating at finite temperatures, suggesting that thermal effects are indeed contributing to the discrepancy. In the present study we take a closer look and find that the mean occupation probability of the chiral isomer observed by Wang and Palmer^{8} is actually comparable to the equilibrium occupation probability of the model bound by the same potential as in ref. 21 and 26.
Prediction of an fcc GM for Au_{55} by different empirical models has not yet been adequately tested at higher levels of theory. While DFT has been used to show that the lowest-lying amorphous^{22} and fcc^{31} isomers are individually more stable than other structures, as far as we know the two isomers have not been compared directly using the same DFT functional. Hence, the question of what is the true ground-state morphology of naked Au_{55} still remains unresolved from a theoretical viewpoint. We address this issue through a more comprehensive exploration of structures and direct comparison between different levels of theory, producing a new putative GM for Au_{55} at DFT level. We also verify that high-symmetry morphologies (such as the Mackay icosahedron,^{32} Ino decahedron,^{33} and cuboctahedron) with closed geometric shells are not as stable as one might expect. Depending on the level of theory used, this destabilisation of high-symmetry structures has been linked to either the range of the interatomic potential^{21,34,35} or relativistic effects.^{31,35}
Note that identifying the GM is necessary for an accurate description of the equilibrium behaviour within a given model, which is why global optimisation of gold (and other metal) clusters remains an active area of research.^{36} However, the GM alone is not sufficient to explain the finite-temperature behaviour and morphological changes observed in experiments^{5,7,8} and simulations.^{29,37} In fact, finite-system analogues of solid–solid phase transitions have been reported for many metals,^{38,39} sometimes well below the size-dependent^{1} melting temperature, but systematic theoretical analysis of this phenomenon for a range of cluster sizes has been performed only for Lennard-Jonesium.^{40,41} This omission is partly due to technical difficulties, because the relatively long time scales associated with morphological rearrangements below the melting temperature range cannot be easily accessed using conventional simulation methods. The energy landscapes^{11} framework, on the other hand, provides a powerful approach to studying such rare events, complementing more conventional methods. This framework combines a variety of optimisation and search techniques, statistical mechanics, unimolecular rate theory, and often exploits the harmonic approximation to describe the global thermodynamics and kinetics of complex systems such as atomic clusters.
In the present contribution we explore the potential energy landscape of model Au_{N} clusters (30 ≤ N ≤ 147), focusing on their equilibrium thermodyamics in the harmonic superposition approximation,^{42} which accounts for configurational and vibrational entropy. We identify a number of solid–solid transitions in morphology, which arise from the competition between close-packed (single-crystal fcc or lamellar twinned), decahedral, and distorted icosahedral motifs. In selected representative cases we also find the fastest transition pathway between competing motifs. These pathways rarely exhibit the highly cooperative rearrangements identified in other systems,^{43,44} but rather involve more localised distortions and are reminiscent of the melt-freeze scenario described by Koga et al.^{7}
The outline of this paper is as follows: in section II we define the Gupta potential, describe the relevant methods from the energy landscape framework (including molecular dynamics simulation), detail our approach to classifying atomic-level structure, and give the methodological details of our DFT calculations. All the results are discussed in detail in section III, where we first focus on the structure and thermodynamics of the naked Au_{55} cluster, then examine other cluster sizes, and in the end elucidate some rearrangement mechanisms for selected cases. A summary of key findings and conclusions is given in section IV. For completeness, in ESI† we provide: (i) a database of local minima on the Gupta potential energy landscape for all the Au_{N} clusters considered, including the coordinates of the putative global minima; (ii) kinetic transition networks for N = 55, 85, and 147; and (iii) the coordinates for the low-lying Au_{55} structures reoptimised using DFT.
(1) |
(2) |
(3) |
(4) |
p_{m}(T) = g_{m}e^{−Em/kBT}/Z(T), | (5) |
(6) |
To shed light on the rearrangement mechanisms between competing structures, we consider the possible connecting pathways in the corresponding network and identify the one with the largest contribution to the steady-state rate constant.^{58,59} This pathway, referred to as the “fastest” for a given temperature, corresponds to the lowest-energy pathway in the limit of zero temperature.
The kinetic lifetime of individual minima is calculated using harmonic transition state theory.^{62} That is, the escape rate from a minimum m via an adjacent transition state s is
(7) |
(8) |
(9) |
(10) |
Fig. 2 Colour-coded disconnectivity graph for the 500 lowest-lying minima of the Au_{55} cluster. Branches leading to minima of icosahedral (ICO) motif are in red, decahedral (DEC) in green, face-centred cubic (FCC) in blue, twinned face-centred cubic (TWI) in magenta, and the remaining ambiguous (AMB) morphologies in black. The node corresponding to isomer 6 in Table 1 is marked by a square, and isomers 7 and 8 are circled. Ball-and-stick representation of the lowest-lying minimum for each motif is shown from two angles, with atoms colour-coded by the local environment: face-centred cubic (fcc) in gold, hexagonal close-packed (hcp) in white, icosahedral (ico) in red, and ambiguous (amb) in grey. The stick “bonds” are defined by a cut-off distance of 3.5 Å. |
More details for the sixteen lowest-lying minima are given in Table 1, confirming the GM found by Bao et al.,^{26} and showing that the lowest-lying AMB minimum found by Garzón et al.^{21,22} is (at best) fifth lowest overall in the given model. The eighth isomer is identified as the chiral structure imaged by Wang and Palmer.^{8} The lowest-lying DEC minimum is fourteenth overall, with the fivefold disclination significantly off centre—in contrast to the Ino^{33} and Marks^{71} decahedra, but reminiscent of some pentagonally twinned structures reported for lead nanoparticles.^{72}
ΔE_{G} (eV) | Motif | PG | (THz) | N _{t.s.} | p _{m} | τ _{m} (s) | ΔE_{DFT} (eV) | |
---|---|---|---|---|---|---|---|---|
1 | 0 | FCC | C _{1} | 2.16602 | 4295 | 0.0026 | 2 × 10^{−12} | 0 |
2 | 0.021294 | FCC | C _{s} | 2.16593 | 435 | 0.0006 | 3 × 10^{−11} | −0.234 |
3 | 0.027287 | TWI | C _{1} | 2.16607 | 202 | 0.0009 | 8 × 10^{−12} | −0.361 |
4 | 0.030151 | TWI | C _{1} | 2.16641 | 160 | 0.0008 | 1 × 10^{−11} | −0.439 |
5 | 0.035735 | AMB | C _{1} | 2.09791 | 664 | 0.1050 | 2 × 10^{−12} | −0.503 |
6 | 0.037537 | TWI | C _{3v} | 2.16414 | 230 | 0.0001 | 9 × 10^{−11} | −0.039 |
7 | 0.039020 | AMB | C _{1} | 2.09591 | 276 | 0.1086 | 1 × 10^{−11} | −0.391 |
8^{†} | 0.039037 | AMB | C _{1} | 2.09336 | 555 | 0.1313 | 1 × 10^{−12} | −0.551 |
9 | 0.043134 | AMB | C _{s} | 2.13989 | 259 | 0.0017 | 5 × 10^{−11} | −0.759 |
10 | 0.055655 | AMB | C _{1} | 2.09451 | 176 | 0.0632 | 1 × 10^{−12} | −0.544 |
11 | 0.056596 | TWI | C _{s} | 2.16562 | 198 | 0.0002 | 3 × 10^{−11} | −0.729 |
12 | 0.066701 | TWI | C _{s} | 2.16574 | 86 | 0.0001 | 2 × 10^{−12} | −0.759 |
13 | 0.069885 | AMB | C _{1} | 2.12300 | 125 | 0.0044 | 8 × 10^{−14} | 0.210 |
14 | 0.073645 | DEC | C _{s} | 2.15466 | 339 | 0.0002 | 2 × 10^{−11} | 0.449 |
15 | 0.088317 | AMB | C _{1} | 2.10861 | 164 | 0.0062 | 9 × 10^{−14} | −0.500 |
16 | 0.093786 | ICO | C _{1} | 2.07067 | 625 | 0.0899 | 2 × 10^{−11} | 0.030 |
Mac | 0.642895 | ICO | I _{h} | 1.98361 | 46 | 0.0000 | 1 × 10^{−10} | 0.644 |
Ino | 1.070310 | DEC | D _{5h} | 2.10213 | 31 | 0.0000 | 4 × 10^{−14} | 1.084 |
Cub | 1.194899 | FCC | O _{h} | 2.10972 | 29 | 0.0000 | 5 × 10^{−19} | 1.748 |
While the GM structure of Au_{55} is fcc with point group C_{1}, the symmetric cuboctahedron is significantly higher in energy (by 1.2 eV). This disparity can be explained by differences in surface packing: the 42 surface atoms in the cuboctahedron form only (100) facets, which are known^{35} to be particularly unfavourable in the present model for gold; but in the GM structure the 45 surface atoms are more close-packed and exhibit mainly (111) character. Also, in agreement with previous^{21,37} reports of distorted icosahedral order in Au_{55}, snapshots of the AMB minima in Fig. 2 exhibit triangular close-packed facets and multiple fivefold disclinations, which outline the tetrahedral units expected in Mackay^{32} icosahedra. The ideal Mackay icosahedron, on the other hand, is not even in the lowest-lying 10^{5} minima (in a database of more around 3 × 10^{5} minima), though it is the most favourable among the closed-shell high-symmetry shapes.
The disconnectivity graphs in Fig. 2 and 3 (discussed below) help us to visualise the landscape topography and to identify the funnels associated with competing motifs, but these graphs do not really show how the competition between different morphologies is manifested under thermal conditions. To examine finite-temperature effects we consider the heat capacity C_{V} and the collective occupation probability P^{XXX}_{occ} of each motif (XXX) as a function of k_{B}T > 0, plotted in Fig. 4.
Fig. 3 Disconnectivity graphs for 10^{3} minima of Au_{85} (top) and 10^{4} minima of Au_{147} (bottom) with ball-and-stick representations of the lowest-lying minimum for each competing motif. The colour-coding and nomenclature are same as in Fig. 2, and only a subset of atoms is highlighted for clarity. |
Fig. 4 Motif occupation probabilities P_{occ} (stacked on top of each other), vibrational heat capacity C_{V}, and time-averaged Lindemann index δ plotted against temperature for Au_{55} (top), Au_{85} (middle), and Au_{147} (bottom). Five motif shares (FCC, AMB, ICO, TWI and DEC) of P_{occ} are colour-coded as in Fig. 2 and 3. The chiral (eighth in Table 1) Au_{55} isomer's occupancy is highlighted in grey, splitting the black segment in two, since contributions from individual minima of a given motif are stacked in the order of increasing potential energy. Net C_{V} is represented by a thick black line, while thinner black lines correspond to subsets of 10, 10^{2}, 10^{3}, 10^{4}, and 10^{5} lowest-lying minima in our database (sorted by potential energy). The Lindemann index δ was calculated from molecular dynamics simulations at temperatures marked by the datapoints, and datasets with the same starting configuration are colour-coded by the initial motif and traced by a dashed line to guide the eye. |
For Au_{55}, P^{FCC}_{occ} is the highest below 70 K (k_{B}T < 6 meV). This high occupancy is associated mainly with the GM structure, while other low-lying FCC and TWI isomers combined have near-zero occupation probability at all temperatures, largely due to their low vibrational entropy. P^{AMB}_{occ} rapidly grows for k_{B}T in the range 6 ± 1 meV (T ≈ 70 K), with several AMB minima reaching comparably high occupation probability, which is illustrated by singling out the contribution from the experimentally observed^{8} chiral isomer (eighth in Table 1 and visualised in Fig. 1c). Recall that a similar transition at a slightly higher temperature (about 90 K) has been identified for a different set of model parameters.^{29} In both cases, when P^{AMB}_{occ} supplants P^{FCC}_{occ} as the highest value, the crossover temperature coincides with a well-defined peak in the heat capacity. Hence, we characterise the finite-system analogue of a solid–solid like phase transition, where the low-energy phase is represented by a single local minimum of FCC motif, and the high-energy phase comprises multiple AMB isomers. Also note that the absence of magenta and green for Au_{55} in Fig. 4 is due to very low occupancy of DEC and TWI motifs for the entire temperature range considered.
Interestingly, the room-temperature occupation probability of the chiral AMB isomer is 0.13, which is comparable to the occurrence frequency of about 0.1 inferred from experimental data of Wang and Palmer.^{8} This agreement between theory and experiment can be rationalised by taking into account the lifetime of individual isomers (see Table 1). The estimated lifetimes are below a typical vibrational period at room temperature, indicating interconversion among multiple isomers on a timescale significantly shorter than the experimental imaging time. Indeed, Wang and Palmer^{8} acknowledge that their images could be a superimposition of multiple isomers, which would explain why the average occurrence frequency of the chiral isomer observed under the microscope is comparable to the equilibrium occupation probability in Table 1. This rapid fluctuation between several different isomers could also be interpreted as quasi-melting.^{9}
Fig. 4 shows that the ICO motif hardly features in Au_{55} at low temperatures, but its occupation probability steadily increases over a temperature range that roughly coincides with a second and more dominant peak in the heat capacity. The time- and atom-averaged Lindemann index δ also increases from below 0.1 to above 0.3 in that temperature range, indicating a finite-system analogue of a solid–liquid phase transition (i.e. melting). The agreement between MD results and the harmonic superposition approximation is encouraging, with the latter formulation revealing interesting changes in the atomic-level structure of the molten Au_{55}. The rise of P^{ICO}_{occ} with k_{B}T in the range 30 meV < k_{B}T < 80 meV, with P^{AMB}_{occ} decreasing yet remaining significant, suggests a gradually growing preference for local (poly)icosahedral order (i.e. increasing fivefold-disclination density) in the melted region.
To cross-check our analysis of the Au_{55} cluster, we reoptimised the structure of Gupta minima in Table 1 using DFT. The relaxation produces fairly minor geometric changes, mainly via uniform expansion or compression of the isomers. However, there is significant re-ordering of the energies: the fcc GM predicted by Gupta is not even in the ten lowest at the DFT level, where the new putative GM of point group C_{s} is obtained by relaxing isomers nine and twelve (in Table 1). The DFT GM structure is shown in Fig. 5, illustrating its amorphous nature and revealing two voids in the subsurface region (see Fig. 5b), consistent with the known propensity of gold clusters to form cage-like structures.^{73} These voids are filled by a nearby surface atom when the geometry is reoptimised for the Gupta potential, as indicated by the two red arrows in Fig. 5c, thus recovering the ninth isomer from Table 1 (with additional and less significant “breathing” of other atoms). Note that, although the two levels of theory do not exactly agree on the GM structure, they both predict the three high-symmetry structures to be very unfavourable, particularly the cuboctahedron. It is also noteworthy that the DFT energy differences (ΔE_{DFT} values in Table 1) are an order of magnitude larger than the Gupta energy differences (ΔE_{G} values), suggesting that the frustrated nature of the Gupta energy landscape may not be preserved at higher levels of theory.
Fig. 5 Au_{55} GM structure at DFT level from two (a, b) different viewpoints along the (vertically oriented) symmetry plain, with the ball-and-stick representation colour-coded as in Fig. 2. In (c), the DFT structure (black atoms) is superimposed over the ninth isomer (white atoms) from Table 1, with red arrows highlighting the main difference between the two geometries. |
To conclude our discussion of the naked “Schmid Au_{55}”, we calculate the HOMO–LUMO energy gap and the partial density of states for isomers 1, 6, 8, 9, 14, Mac, Ino, and Cub in Table 1. The HOMO–LUMO gap (calculated using the ΔSCF method,^{74} comparing the ground state of charge +1 and −1 with the neutral system, including the Makov–Payne correction^{75}) ranges from 2.18 to 2.26 eV among these isomers. The total density of states for the three symmetric isomers (particularly Mac) shows pronounced peaks around specific values, while for the more ambiguous structures the density is more broadly distributed. Unfortunately, partial density of states does not immediately reveal any links between the electronic structure and the cluster geometry, but we hope to explore this issue in more detail as a separate study.
As for Au_{55}, the ideal Au_{147} Mackay icosahedron^{32} is energetically unfavourable, with the disconnectivity graph based on 10^{4} lowest-lying minima featuring hardly any traces of the ICO motif. The fraction of AMB minima is also reduced (compared to Au_{55}) and the existence of two structurally homogeneous funnels is apparent: one is predominantly DEC, and the other is TWI. Note that the GM of Au_{147} is a 146-atom Marks decahedron,^{72} with six atoms along the fivefold disclination (i.e. the decahedral spine), and an extra adatom on one of the peripheral (100) facets. The DEC motif is the most populated for k_{B}T ≤ 40 meV (Fig. 4) and is gradually supplanted by the ICO motif in the range 50 ± 10 meV, over which P^{AMB}_{occ} and P^{TWI}_{occ} also rise up to 0.09 and 0.04, respectively. The onset of the ICO motif is more abrupt than for Au_{55}, still coinciding with a C_{V} peak, but the corresponding temperature range is noticeably above the range over which δ reaches the value of 0.3. This apparent mismatch between molecular dynamics and the harmonic superposition approximation may be due to our database of minima under-representing the melted region, but it could also be due to an harmonic effects.
Au_{85} is in some ways intermediate between Au_{55} and Au_{147}. Its disconnectivity graph (see Fig. 3) shows fairly pronounced FCC, TWI, and DEC funnels, and each funnel exhibits a considerable number of AMB minima. In this particular case the distinction between DEC and AMB minima is marginal, because many AMB structures still exhibit a well-defined decahedral spine, albeit with a higher degree of amorphisation and/or locally (poly)icosahedral order at the surface. As a consequence of our motif definitions, the low-temperature C_{V} peak for Au_{85} in Fig. 4 straddles two crossover temperatures. At k_{B}T = 13 meV the TWI motif is supplanted by the DEC motif as the most populated, and at k_{B}T = 17 meV it is the AMB motif that starts to dominate. The Lindemann index rises at a noticeably lower temperature than the second C_{V} peak, similar to Au_{147}, showing that the discrepancy is not specific to a particular cluster size. Interestingly, the increase in δ for Au_{85} and Au_{147} seems to better align with the onset of ICO isomers, when P^{ICO}_{occ} ceases to be negligible but does not yet dominate, suggesting that the onset of multiple fivefold disclinations and icosahedral cores can be take as an indicator of melting.
From Fig. 4 it is apparent that the maximal value of P^{AMB}_{occ} diminishes with cluster size, which is consistent with Bao et al.^{26} finding amorphous GM only for N < 55. However, our thermodynamic analysis shows that the AMB motif can also dominate in larger clusters at finite temperatures. To explore this avenue further we systematically analysed Au_{N} clusters with N = 30–147, accumulating a database of about 10^{4}–10^{5} low-lying minima for each N. We also determined two crossover temperatures, T_{A} and T_{I}, respectively marking when the AMB and ICO motifs become the most populated. The results are summarised in Table 2, together with an estimate of the melting temperature (T_{m}) range obtained from molecular dynamics simulations using the Lindemann^{66} index defined in eqn (10).
N | E _{GM} (eV) | Motif | PG | T _{A} (K) | T _{I} (K) | T _{m} (K) | N | E _{GM} (eV) | Motif | PG | T _{A} (K) | T _{I} (K) | T _{m} (K) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
a Low-lying isomers of borderline ICO/AMB motif produce reentrant behaviour and a particularly broad heat capacity peak. b The GM motif is first supplanted by the DEC motif, and then by the AMB motif at temperature T_{A}. c The lowest-lying ICO isomer is borderline and has been reclassified as AMB to eliminate artificial reentrant behaviour. | |||||||||||||
30 | −104.743100 | AMB | C _{3v} | 0 | 460 | 290 ± 50 | 60 | −213.536346 | TWI | C _{1} | 142 | 446 | 370 ± 30 |
31 | −108.183974 | AMB | C _{2} | 0 | 402 | 61 | −217.254337 | TWI | C _{3v} | 223 | 585 | ||
32 | −111.794339 | AMB | C _{3} | 0 | 306 | 62 | −220.855915 | TWI | C _{s} | 209 | 504 | ||
33^{a} | −115.404251 | ICO | C _{1} | 193 | 339 | 63^{b} | −224.532317 | TWI | C _{s} | 223 | 316 | ||
34 | −119.082484 | AMB | C _{3} | 0 | 295 | 64 | −228.254019 | DEC | C _{2v} | 299 | 397 | ||
35 | −122.678712 | AMB | C _{s} | 0 | 330 | 260 ± 50 | 65 | −231.868903 | DEC | C _{2v} | 220 | 457 | 370 ± 50 |
36 | −126.240885 | AMB | C _{2} | 0 | 100 | 280 ± 50 | 66 | −235.547173 | DEC | C _{s} | 70 | 476 | |
37 | −129.991492 | AMB | C _{2v} | 0 | 248 | 280 ± 50 | 67 | −239.158600 | DEC | C _{s} | 207 | 555 | |
38 | −133.584814 | AMB | C _{s} | 0 | 309 | 300 ± 50 | 68 | −242.838649 | DEC | C _{2v} | 153 | 935 | |
39 | −137.184370 | FCC | C _{4v} | 12 | 137 | 300 ± 50 | 69 | −246.450465 | DEC | C _{1} | 172 | 982 | |
40 | −140.789863 | AMB | C _{s} | 0 | 35 | 330 ± 50 | 70 | −250.154102 | DEC | C _{s} | 100 | N/A | 370 ± 30 |
41 | −144.403059 | AMB | C _{s} | 0 | 262 | 71 | −253.959033 | DEC | C _{2v} | 339 | N/A | ||
42 | −148.023359 | AMB | C _{1} | 0 | 367 | 72 | −257.571776 | DEC | C _{s} | 239 | N/A | ||
43 | −151.721379 | DEC | C _{2v} | 9 | 339 | 320 ± 50 | 73 | −261.253928 | DEC | C _{s} | 95 | N/A | |
44 | −155.322847 | AMB | C _{s} | 0 | 260 | 74 | −264.922500 | DEC | C _{5v} | 297 | N/A | ||
45 | −158.916745 | DEC | C _{2v} | 9 | 422 | 330 ± 20 | 75 | −268.761948 | DEC | D _{5h} | 397 | N/A | 380 ± 30 |
46 | −162.598451 | AMB | C _{3} | 0 | 487 | 76 | −272.372659 | DEC | C _{2v} | 337 | N/A | ||
47 | −166.245363 | DEC | C _{2v} | 128 | 441 | 340 ± 50 | 77 | −275.982269 | DEC | C _{2v} | 123 | N/A | |
48 | −169.873475 | AMB | C _{1} | 0 | 471 | 78 | −279.588677 | DEC | C _{2v} | 158 | N/A | ||
49 | −173.562095 | DEC | D _{5h} | 111 | 513 | 79 | −283.417486 | FCC | O _{h} | 385 | N/A | 420 ± 30 | |
50 | −177.090944 | TWI | D _{3h} | 35 | 520 | 330 ± 20 | 80 | −287.022207 | FCC | C _{4v} | 265 | N/A | 410 ± 30 |
51 | −180.698557 | AMB | C _{1} | 0 | 457 | 81 | −290.660813 | DEC | C _{2v} | 169 | N/A | ||
52 | −184.431342 | AMB | C _{2v} | 0 | 545 | 82 | −294.269106 | DEC | C _{2v} | 232 | N/A | ||
53 | −188.009110 | AMB | C _{3v} | 0 | 552 | 83 | −297.933795 | TWI | C _{s} | 276 | N/A | ||
54 | −191.686569 | FCC | C _{2v} | 14 | 404 | 84 | −301.598992 | DEC | C _{s} | 181 | N/A | ||
55 | −195.284851 | FCC | C _{1} | 70 | 501 | 350 ± 20 | 85 | −305.259030 | TWI | C _{2v} | 197 | N/A | 420 ± 30 |
56 | −199.015100 | FCC | D _{2h} | 165 | 344 | 86 | −308.936312 | DEC | C _{2v} | 318 | N/A | ||
57^{c} | −202.624795 | TWI | C _{2v} | 176 | 534 | 360 ± 50 | 87 | −312.550788 | DEC | C _{2v} | 367 | N/A | |
58 | −206.216538 | TWI | C _{2} | 42 | 483 | 88 | −316.280622 | FCC | C _{s} | 327 | N/A | ||
59 | −209.852603 | TWI | C _{2v} | 153 | 295 | 89 | −319.884977 | FCC | C _{s} | 443 | N/A | ||
120 | −434.088354 | DEC | C _{s} | 624 | N/A | 470 ± 20 | 90 | −323.613329 | FCC | C _{s} | 436 | N/A | 420 ± 20 |
140 | −508.052209 | DEC | C _{s} | 778 | N/A | 500 ± 30 | 95 | −341.874115 | TWI | C _{s} | 309 | N/A | 420 ± 30 |
147 | −533.942249 | DEC | C _{s} | N/A | 590 | 490 ± 30 | 100 | −360.342412 | FCC | C _{1} | 343 | N/A | 450 ± 30 |
Table 2 shows that the GM structures are predominantly AMB for N < 54, then primarily FCC or TWI in the size range 54 ≤ N < 64, and mostly DEC in the range 64 ≤ N < 88, which includes the closed-shell Marks decahedron^{71} for N = 75. Note that AMB and DEC motifs are in close competition for N between 42 and 49, as discussed in more detail in section IIIC, while the size range 88 ≤ N ≤ 147 shows re-entrance of the FCC/TWI and DEC motifs. As cluster size increases, the GM motif changes in the sequence AMB → FCC/TWI → DEC → FCC/TWI → DEC → …, which differs from the generally expected^{41,76} sequence ICO → DEC → FCC/TWI. Furthermore, although the AMB motif is the GM only for N < 54, we find that T_{A} < T_{m} and T_{A} < T_{I} for most of the cluster sizes considered, and so amorphous structures still dominate for N ≥ 54 at temperatures between T_{A} and min{T_{m}, T_{I}}. Also note that T_{I} > T_{m} or T_{I} is N/A in many cases, which implies general thermodynamic instability of well-defined icosahedral order in the solid state.
In passing we confirm that the GM of Au_{38} is a low-symmetry AMB isomer, as discovered by Garzón et al.,^{22} with the symmetric truncated octahedron (of FCC motif) higher in energy by 6 meV. The ordering is reversed when one extra atom is added: the GM of Au_{39} is FCC, comprising the 38-atom truncated octahedron with the extra atom placed on one of the (100) facets, essentially converting the facet into a vertex. This behaviour is again consistent with energetically unfavourable (100) facets,^{35} making Au_{39} the smallest cluster with a single-crystal fcc GM, which beats the next-lowest AMB isomer by less than 2 meV. Given these small energy differences it is reasonable to expect the ordering to vary for different models and levels of theory.
To check that the range of odd–even behaviour in Table 2 is not an artefact of our motif definitions, but actually has thermodynamic implications, we consider the heat capacity in the size range 42 ≤ N ≤ 49. Fig. 7 shows two well-defined C_{V} peaks featuring for odd N, one near the temperatures T_{A} and the other closer to T_{I}. The low-temperature peak in each case marks the crossover from DEC to AMB motif, while the high temperature peak straddles the melting range and a crossover from the AMB to the ICO motif as the most populated. The low-temperature peaks are absent for even N, consistent with T_{A} = 0. The results for odd N = 43, 47, and 49, on the other hand, show that thermally activated manifestation of seemingly minor local icosahedral order near the surface of a cluster can produce a well-defined peak in the heat capacity.
Fig. 7 Heat capacity (C_{V}) versus temperature (k_{B}T) for Au_{N} clusters with even (top) and odd (bottom) N in the range 42 ≤ N ≤ 49. For Au_{48}, the indicated C_{V} shoulder corresponds to the isomer illustrated in the inset supplanting the GM shown in Fig. 6. |
Fig. 8 Potential energy profiles of the fastest discrete pathway between selected endpoints (isomers Cub, 1, 5 and 8 from Table 1). The horizontal and the vertical axes correspond to the (Gupta) potential energy and the discrete path length, respectively, with the same scaling for all the profiles. Solid lines trace pathways for k_{B}T = 26 meV (room temperature) and dashed for 5 meV. Filled and unfilled symbols correspond to minima and saddles, respectively. The structure of three minima (labelled A, B and C) is shown with the amb and ico atoms highlighted and colour-coded as in Fig. 2. |
Although the lowest-lying AMB isomer is only 36 meV higher in energy than the GM, the lowest overall energy barrier is about 0.5 eV, and the corresponding pathway involves minima with partially disordered geometries, such as snapshot B in Fig. 8. Hence, if this transition is observed, it would most likely resemble partial melting followed by crystallisation into the fcc structure. Interestingly, the maximal degree of amorphisation along the fastest path decreases as the temperature increases, which can be seen by comparing the fastest pathways at k_{B}T = 26 meV and 5 meV. This somewhat counter-intuitive trend is not unexpected considering that most low-lying minima of Au_{55} exhibit fairly amorphous or ambiguous structure (recall Fig. 2). It is also noteworthy that the number of steps in the fastest pathway is more than halved when the temperature is increased from k_{B}T = 5 meV to room temperature (26 meV), showing that the fastest mechanism is temperature dependent.
Fig. 8 also shows the potential energy profile for the optimal pathway between isomers five and eight in Table 1 at two different temperatures. The profiles involve considerably lower energy barriers compared to the other pathways, consistent with the disconnectivity graph in Fig. 2, and showing no significant change with temperature.
Recall that the most competitive motifs for N > 56 at low temperatures are DEC and TWI, so we examine if the fastest pathway between these two motifs follows any particular mechanism, using Au_{85} and Au_{147} at room temperature as our test cases. For Au_{85}, Fig. 9 shows that part of the twin boundary in the lowest-lying TWI isomer is preserved along the pathway to the lowest-lying DEC isomer. However, formation of the fivefold twin axis is accompanied by a significant level of disorder, which would probably be interpreted as partial melting if observed in experiments. A more significant level of disorder occurs in the Au_{147} pathway (see Fig. 9), where (unlike Au_{85}) the amorphous intermediates exhibit multiple well-defined fivefold disclinations, somewhat resembling a distorted icosahedron. Most of these amorphous intermediates are about 1 eV higher in energy than the TWI and DEC endpoints, while among themselves they are separated by relatively small energy barriers, which also suggests that the calculated pathway passes through a liquid-like state.
Fig. 9 Potential energy profiles for the fastest discrete pathway from the lowest-lying TWI minimum (leftmost) to the lowest-lying DEC minimum (rightmost) for Au_{85} (top) and Au_{147} (bottom) at room temperature (k_{B}T = 26 meV). The horizontal and the vertical axes, the meaning of filled/unfilled symbols, and the colour-coding of the atoms in the selected snapshots are as in Fig. 8. |
The global minimum (GM) structure of most clusters in the size range 30 ≤ N ≤ 53 was found to be fairly ambiguous, but with discernible fivefold disclinations. We identified a case of geometric odd–even behaviour in the range 42 ≤ N ≤ 49, where the GM alternates between a structure with just one disclination and a structure with multiple disclinations. Global minima of larger clusters (54 ≤ N ≤ 147) are typically a lamellar-twinned or single-crystal fcc lump, or a decahdral motif with a single fivefold disclination. While this structural change is consistent with previous global optimisation studies,^{26} our thermodynamic analysis further shows that ambiguous motifs with multiple fivefold disclinations can still dominate for N ≤ 85 (except N = 71, 75, 76, and 79) at around room temperature.
In most cases where the GM has at most one disclination, we found thermally-driven morphological transitions below the size-dependent melting temperature. These finite-system analogues of a solid–solid phase transition often coincide with a premelting feature in the heat capacity curve, and they typically correspond to the GM occupation probability dropping below that of multiple ambiguous structures above a system-specific temperature. In certain cases (N = 63, 79, 83, 85, 89, 90) we found two consecutive premelting transitions between distinctly different motifs, with the corresponding crossover temperatures correlating with a smeared shoulder or peak in the heat capacity. In all cases, the higher-energy phase exhibits more fivefold disclinations and higher vibrational entropy.
We calculated the fastest solid–solid transition pathways for Au_{N} with N = 55, 85, 147, where the energy barrier separating some of the competing motifs is up to 1 eV. While such energy barriers are difficult to treat using conventional simulation methods, which makes direct simulation of the rearrangement mechanism unfeasible, discrete path sampling and harmonic transition state theory provide a useful approximation that performs best at low temperatures. The calculated pathways pass through many metastable intermediates with fivefold disclinations and/or a high degree of amorphisation, consistent with the melt-freeze scenario described by Koga et al.^{7} for larger gold clusters.
Finally, we confirmed a previously reported^{26,29,80} fcc GM structure of point group C_{1} for Au_{55}, but our thermodynamic analysis revealed that several distorted icosahedra collectively become more favourable at temperatures above 50 Kelvin. Interestingly, the room-temperature occupation probability of a particular isomer was found to be consistent with electron microscopy observations of Wang and Palmer.^{8} This apparent agreement is surprising, because the Gupta potential energy landscape for Au_{55} clearly disagrees with density functional theory (DFT): relaxing the geometry of sixteen lowest-lying Gupta minima using DFT produced a markedly different energetic ordering, with a new putative GM of point group C_{s} at the DFT level. While the consistency between our Gupta-level calculations and previous experiments may well be fortuitous, it nonetheless highlights the importance of accounting for thermal fluctuations in geometry—something that is often overlooked when comparing empirical potentials with DFT. In future studies it would be interesting to investigate the equilibrium thermodynamics of Au_{55} (and other clusters) at the DFT level, where it is also possible to compute the normal-mode frequencies for the harmonic superposition approximation.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7nr07123j |
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