Electron microscope images

Figure 1 shows examples of HAADF STEM images of Au 561 clusters on amorphous silicon nitride and corresponding multi-slice simulations from a simulation atlas19. Figure 1a and b was recorded at 20 °C. Figure 1a shows an fcc cluster and Fig. 1b shows a decahedral cluster. Figure 1c and d was recorded at 500 °C. Figure 1c shows an fcc cluster and Fig. 1d an on-axis decahedron. Both decahedra in Fig. 1b and d show some Marks reentrant features. In comparison of experimental and simulated images, we concentrate on the core atomic structure because this is where the signal-to-noise levels are the highest, so that we can compare them with simulations of perfect cuboctahedra and Ino-decahedra. HAADF STEM images matched to the cuboctahedron simulations are denoted face-centred-cubic (fcc), which allows for variation in the exact surface truncation; similarly images matched to the Ino-decahedron are denoted decahedra (Dh). Clusters that display ‘ring-dot’ features in the images, a characteristic of an icosahedron, are denoted simply as icosahedra (Ih).

Fig. 1 HAADF STEM images of Au 561 clusters at 20 °C and 500 °C. a–d HAADF STEM images of Au 561 clusters and e–h matching multi-slice electron scattering simulations of the cuboctahedron and Ino-decahedron at different orientations. a, b Experimental images recorded at 20 °C; c, d Images recorded at 500 °C. i Rotation angle of the cuboctahedron and Ino-decahedron geometries Full size image

Proportions of different isomers

Figure 2a is a plot of the proportions of structural isomers, extracted from the fits to the experimental data, for Au 561 clusters on amorphous silicon nitride at temperatures ranging from 20 °C to 500 °C. The same sample was used for all measurements so that formation conditions would not affect the results25. Cluster structures are identified as either fcc, Dh, Ih or unidentified/amorphous (UI/A). The error bars on the proportions of structural isomers are statistical counting errors and the error on the temperature is 5%, due to the heating chip calibration. At all temperatures investigated the most abundant isomer is fcc, followed by Dh, while Ih has a very low abundance (0–3%). We find that the clusters still provide a good match with the simulated structures at high temperature and there is no evidence of melting in the temperature range explored here, as can be seen from Fig. 1. The percentage of unidentified or unknown (UI/A) structures—clusters that are amorphous or cannot be identified using simulation atlases for the Ino-decahedron, cuboctahedron or icosahedron—is fairly constant across the temperature range. One explanation for such images is that only single-shot data was taken (to minimise the electron dose), and clusters often rotate during scanning.

Fig. 2 The proportion of structural isomers versus temperature. a The proportion of structural isomers for Au 561 clusters on amorphous silicon nitride at ten temperatures: 20 °C, 50 °C, 75 °C, 100 °C, 125 °C, 150 °C, 200 °C, 300 °C, 400 °C and 500 °C. The clusters are classified as face-centred-cubic (circles), decahedral (diamonds), icosahedral (squares) or unidentified/amorphous (triangles). The numbers of experimental images recorded at each temperature are 133, 161, 128, 126, 151, 141, 132, 191, 167 and 143 respectively. Poisson error bars, derived from these statistics, are shown for the isomer proportions. b The ratio of Dh to fcc clusters versus temperature. The low-temperature regime (20–125 °C) is in diamond markers and the high-temperature regime (125–500 °C) in circle markers. Lines between points plotted are simply a guide to the eye. Error bars, derived according to the error propagation law, are shown for the Dh:fcc ratio Full size image

Figure 2b shows a plot of the ratio of the two most abundant ordered isomers, Dh and fcc, versus temperature. Two distinct temperature regimes are clearly visible. Between 20 °C and 125 °C the Dh:fcc ratio decreases from 0.81 to 0.24, whereas between 125 °C and 500 °C the Dh:fcc ratio increases from 0.24 to 0.45. The underlying and associated errors are derived from Fig. 2a. Between 20 °C and 150 °C the increase in temperature results in an increase in the abundance of the fcc isomer, but at temperatures ≥150 °C the proportion of fcc gradually decreases again. Complementary to this, between 20 °C and 125 °C, the proportion of Dh decreases, whereas at temperature ≥125 °C there is a slight increase in Dh as temperature rises.

The increase in the proportion of fcc clusters from 20 °C to 125 °C, and the corresponding decrease in the proportion of Dh, can be explained in terms of the release of trapped metastable Dh structures to a lower free energy fcc structure. We previously reported that Au 561 clusters undergo a one-way transition from Dh to fcc when continuously exposed to the STEM electron beam at very high magnification20, which corresponds to moderate heating of the sample. However, the behaviour we observe takes on a new character above 125 °C with the ratio of Dh to fcc increasing again.

This repopulation behaviour can be understood if the fcc structure is a lower free-energy structure than the Dh. Then, beyond the release of kinetically trapped Dh clusters by annealing at temperatures from 20 °C to 125 °C, we may expect that an equilibrium distribution of isomers will be established at higher temperatures. A proportion of the clusters (based on Boltzmann statistics) will be excited from the fcc to the higher energy Dh structure26. In fact, if we assume equilibrium between isomers of energy E Dh and E fcc , we obtain the ratio between the probabilities p Dh and p fcc of the corresponding structures given by (see the Supplementary Note 1 for a derivation of this formula)

$${\mathrm{ln}}\left( {p_{{\mathrm{Dh}}}/p_{{\mathrm{fcc}}}} \right) = \beta (E_{{\mathrm{fcc}}} - E_{{\mathrm{Dh}}}) + c$$ (1)

where β = (k B T)−1.

In this system the Ih must have much higher energy, as we do not see repopulation of this isomer even at 500 °C; this is in agreement with experimental observations of Ih Au 923 clusters under the electron beam, which transformed to Dh or fcc structures after very short exposure times19. If the increase in the proportion of Dh clusters in the high-temperature region is a result of thermal repopulation of this excited state, the energy difference between the Dh and fcc structural isomers can be derived, as we show below. A second hypothetical explanation for the change in ratio is that, as the temperature of the clusters increase, atoms are lost through sublimation resulting in a smaller cluster size at higher temperatures where the decahedron might in principle be more stable. However, based on analysis of the diameters of the clusters at 500 °C (Fig. 1), we are confident that no major loss of atoms has occurred.

Figure 3 shows a plot of the natural log of the ratio of the Dh and fcc abundances as a function of the reciprocal of the absolute temperature. From Eq. (1), the slope of the line in the higher temperature equilibrium regime gives the energy difference between the local minima of the two competing isomers, whereas the intercept gives the entropy difference (see Supplementary Note 1 for detailed explanation). This does not apply to the low temperature, kinetic regime. The dashed line shows a weighted linear least squares fit to the high-temperature region (398–773 K) of the plot. The gradient of this line is −510 ± 240 K, which corresponds to a value of 0.040 ± 0.020 eV (E = k B T) for the energy difference between Dh and the lower lying fcc isomers (ΔE Dh–fcc ). The intercept c = −0.2 ± 0.4 is the entropy difference in units of k B (Supplementary Note 1), which indicates a negligible entropy difference between these structures.