The 55-atom Morse Cluster

 In this section we shall illustrate the effects of the range of the potential on cluster thermodynamics in more detail, by presenting results of simulations of M₅₅ at different values of the range parameter, $\rho_0$.As in §3.3 we use STA temperature distributions (e.g. Figure 4.13) to decompose the caloric curve into contributions from different states. The caloric curves for M₅₅ at five different values of $\rho_0$ are shown in Figure 4.14. At $\rho_0$=6, the melting behaviour is very similar to that seen for M₅₅ in §3.3. The caloric curve has a Van der Waals loop, and surface defects of the Mackay icosahedron can be generated at energies just below that required for complete melting. In comparison, the $\rho_0$=9 caloric curve shows a higher melting temperature, a deeper S-bend and a greater role for the defective states. This is a direct result of the increasing energy gap between the Mackay icosahedron and the `liquid-like' band of minima noted in §4.3. At $\rho_0$=4, the melting temperature is lower than for $\rho_0$=6, the latent heat is smaller and defective Mackay icosahedra are not seen. Again this can be related to the correlation diagram Figure 4.4 and is simply a result of the decrease in the energy gap between the Mackay icosahedron and the `liquid-like' band of minima as $\rho_0$ is decreased.

  

Figure 4.13: An example of a multimodal short time-averaged temperature distribution. It is from an MD run for M₅₅ at $\rho_0=6$ at an energy of $-3.483\,\epsilon\, \hbox{atom}^{-1}$.Each temperature peak is labelled with the structure associated with it. MI stands for Mackay icosahedron. The averaging period is 3000 time steps.

  

Figure 4.14: Caloric curves for M₅₅ for $\rho_0=$ (a) 3, (b) 4, (c) 6, (d) 9 and (e) 13. For $\rho_0$=3 and 13 Lindemann's $\delta$ has also been plotted, so that the melting region can be identified. For $\rho_0$=4, 6 and 9 the solid line without error bars is the overall caloric curve, and the dashed lines with error bars are the average values of the temperature for the peaks in the STA temperature distribution. The latter lines are labelled with the structures associated with them. MI stands for Mackay icosahedron, + nd for a structure with n defects, and LL for liquid-like.

The behaviour at $\rho_0$=3 and 13 is significantly different; in particular the melting transitions are closer to a continuous transition than a two-state transition, i.e. between a liquid and a solid. At $\rho_0=3$, the lowest energy minimum that we have found is 55A. This minimum lies at the bottom of the band of `liquid-like' minima, which is approximately continuous in energy. At low temperatures transitions from 55A to higher energy minima begin to occur, because of the small differences in energy. This leads to the fluxional liquid-like behaviour which is exhibited in the rise in Lindemann's $\delta$ (Figure 4.14a). Melting occurs at low temperatures without producing any noticeable feature in the caloric curve--the latent heat is essentially zero.

For $\rho_0$=13, on the other hand, the global minimum is the decahedral structure 55C. The difference in energy between this structure and the `liquid-like' band of minima is now so large that a transition to the latter is only observed at the highest energies ($E\geq -2.4\epsilon\,\hbox{atom}^{-1}$)probed by our simulation. This transition occurs after $\delta$ has risen to a value which indicates that the cluster has already melted, and does not lead to a noticeable feature in the caloric curve. Instead, as energy is added to 55C, the cluster progresses up a ladder of increasingly defective structures (Figure 4.15--the minima bunch into bands that have the same number of nearest neighbours). This climbing up the PES leads to the decreased slope of the caloric curve at higher energies. The melting transition is quasicontinuous and mediated by defect motion. Defective states of 55C are observed at a lower temperature than the defective states of the Mackay icosahedron in the simulations at $\rho_0$=6 and 9 simply because the energy gap between the icosahedron and its defective states is larger; the first defective state of the icosahedron has three fewer nearest-neighbour contacts whereas the first defective state of 55C has one less contact[71].

  

Figure 4.15: Probability distributions of the potential energy for samples of minima for M₅₅ at $\rho_0$=13. The solid line is for a sample of 357 minima obtained by quenching from simulations performed for $\rho_0=$13 at energies below $-2.7\epsilon\,\hbox{atom}^{-1}$. The dashed line is for the $\rho_0=$6 sample of minima after reoptimization at $\rho_0=$13.

The dependence of the thermodynamic behaviour of M₅₅ on its energetic distribution of minima agrees very well with that predicted in a seminal paper by Bixon and Jortner[152], which elucidated this relationship for model PES's. In particular, these authors predicted that significant features in the caloric curve, such as an S-bend, would only be seen when there is a large energy gap between the solid and liquid states (e.g. $\rho_0$=4, 6 and 9), and not when the energetic distribution of minima is quasi-continuous (e.g. $\rho_0$=3 and 13).