The 55-atom Lennard-Jones Cluster

 

  

Figure 3.1: Probability distributions of STA temperatures for 55 from MD runs at (a) $54.243\,\epsilon$ and (b) $62.273\,\epsilon$ above the energy of the Mackay icosahedron, $-279.248\,\epsilon$.

55 has been the focus of much study as an archetype for the melting of small clusters. It exhibits a clear Van der Waals loop and a large peak in the heat capacity[135]. If the potential energy is used as an order parameter, there is a range of energy for which two Landau free energy minima occur corresponding to the solid-like and liquid-like states[133]. If short time averaging is used more structure is revealed in the solid-like branch of the caloric curve. Before melting, atoms can pop out of the surface layer creating a vacancy and a mobile adatom[108,110,111,149]. This defect creation leads to enhanced diffusion in the surface layer, whilst the core of the cluster remains rigid. Kunz and Berry observed that the cluster would spend a period of time in one region of configuration space, say with a single surface vacancy, and then jump to a region with a different number of vacancies and a different characteristic potential energy. As a result, multimodality is seen in the probability distributions for the microcanonical STA temperature. An example is shown in Figure 3.1a where three coexisting states are seen. As the STA temperature reflects the minimum energy of the wells with which the cluster is associated (high temperatures correspond to low energy minima), we expect that the STA temperature peaks will be related to features in the energy distribution of minima (Figure 3.2).

  

Figure 3.2: Probability distribution of the potential energy for a sample of 989 minima of 55.

To test this hypothesis, quenches were performed from configurations generated by an MD run at the energy of Figure 3.1a if they had an STA temperature near to the maxima of any of the three peaks. The resulting minima were then assigned to one of the six energy ranges given in Table 3.1. The results, given in Table 3.2, unequivocally show the correspondence between the STA temperature peaks and the peaks in the potential energy probability distribution for minima. Therefore the STA temperature can be used to elucidate the connection between the observed behaviour and local minima on the PES.

  

Table 3.1: Division of the of 55 PES into energy ranges based upon the features in the distribution of minima (Figure 3.2). There are no minima with energies between regions I and II, and II and III.

 

Lower Upper Number of

Temperature Temperature quenches I II III IV and V

0.3047 0.3050 110 110

0.2909 0.2912 72 2 70

0.2756 0.2785 86 1 9 74 2

  

Figure 3.3: A selection of representative minima of 55 from energy ranges II, III and IV. Energies and point groups are given in Table 3.3.

The highest temperature peak corresponds to the Mackay icosahedral global minimum, but to assign sets of structures to the other peaks, some of the minima in energy ranges II and III were examined further. Range II consists of icosahedral-based structures with a vacancy in a five-coordinate vertex site and a single adatom. There are eleven such structures, four with the adatom in the centre of a triangular face (Figure 3.3a) and seven with the adatom in an off-centre site (Figure 3.3b). These structures have been reported previously in a study using a more sophisticated potential to represent Ar[108]. Range III consists of structures with two vertex vacancies and two adatoms (Figure 3.3c-f and h-j), and structures with a vacancy in a six-coordinate edge site and a single adatom (Figure 3.3g). For the lower energy structures in this range, the two adatoms form a pair, either on the same face (Figure 3.3c and d) or straddling two faces (Figure 3.3e and f). We also examined some minima corresponding to range IV. We found structures with three adatoms, as well as some with adjacent edge and vertex vacancies and two adatoms (Figure 3.3k) and more complicated distorted icosahedra (Figure 3.3l).

 

$$\epsilon$$ Label Description PG

a vertex vacancy + centred adatom C_s -276.59676

b vertex vacancy + off-centred adatom C₁ -276.19935

c 2 vertex vacancies + adatom pair C_s -274.28136

d 2 vertex vacancies + adatom pair C₁ -274.25058

e 2 vertex vacancies + adatom pair C₁ -274.21687

f 2 vertex vacancies + adatom pair C₁ -274.13672

g edge vacancy + centred adatom C_s -274.08950

h 2 vertex vacancies + 2 adatoms C₁ -273.83234

i 2 vertex vacancies + 2 adatoms C₁ -273.62356

j 2 vertex vacancies + 2 adatoms C₁ -273.21257

k fused vertex and edge vacancy + adatom pair C₁ -272.82603

l distorted icosahedron C_s -272.58659

To map out the microcanonical caloric curve MD runs each of $2.5\times 10^6$ time steps were performed at 23 energies around the coexistence region. As expected the resulting curve has a Van der Waals loop. The STA temperature distributions were calculated for each energy. At low energies the distribution is unimodal, because the cluster remains in the Mackay icosahedral well. As it becomes possible to create surface vacancies, the distributions first become bimodal (at $48.193\,\epsilon$), and then trimodal (at $49.183\,\epsilon$). At $60.238\,\epsilon$ a much broader peak appears at lower temperature. Figure 3.1b shows this bimodal behaviour; it is due to dynamic coexistence between the liquid-like and solid-like states. The shoulders of the high-temperature peak are the remnants of the trimodal behaviour seen at lower energies. The peaks have merged because the greater vibrational energy has broadened the temperature distributions for each state. The trimodal character of the high temperature peak would reappear if a longer averaging period was used.

The STA temperature distributions were partitioned according to their peaks, and the mean and standard deviation of the temperature were calculated for each, along with the integrated probabilities. This allows the caloric curves for restricted regions of the configuration space to be obtained (Figure 3.4a). Regions 2 and 3 have a small effect on the caloric curve; they cause it to bend gradually away from the temperature corresponding to the Mackay icosahedron alone. From the probabilities (Figure 3.4b) it can be seen that, despite their length, some of the MD runs have not achieved ergodicity in the energy range $47-55\,\epsilon$. This is because the free energy barriers between the different states can be large, and so passage between the different states can be infrequent on the time scale of the run.

  

Figure 3.4: (a) Plot of the temperature against energy for 55. The caloric curve has been split into contributions from regions of configuration space corresponding to peaks in the STA temperature distribution. The overall caloric curve is represented by the dashed line without any error bars. (b) Plot of the probability of 55 being in regions of phase space I, II, III and VI characterized by their different STA temperatures. Above 60$\,\epsilon$ regions I-III were indistinguishable. The energy is measured relative to the energy of the global minimum icosahedron.