Discussion

 In this chapter we have attempted to find the global minimum for Morse clusters as a function of $\rho_0$ and the number of atoms. The global minimum represents the equilibrium structure at zero Kelvin. However, we need to consider how important the global minimum is in determining the structure at finite temperature. To answer this question we have to examine the effects of other local minima on the PES. The thermodynamic properties of a collection of minima can be approximated by superposition of the density of states from each minimum, as described in detail in Chapter 3.

 

decahedral icosahedral

$$\epsilon \epsilon$$ n_nn n_nn

-351.472365 D_5h 319 -351.177041 C₁ 328

-349.500086 C₁ 317 -351.139799 C₁ 328

-349.498552 C₁ 317 -351.135122 C_s 328

-349.498393 C₁ 317 -350.984741 C₁ 328

-349.496058 C₁ 317 -350.912649 C₁ 328

-349.489864 C₁ 317 -350.890122 C₁ 328

-349.275647 C₁ 317 -350.874153 C₁ 328

-348.606586 C₁ 316 -350.686821 C₁ 327

-348.332763 C₁ 316 -350.612539 C_s 329

-347.636104 C₁ 315 -350.464407 C₁ 327

Here we apply this method to 75 at $\rho_0=6$ to illustrate the limitations of finite temperature structural predictions based on knowledge of the global minimum alone. Sets of low energy decahedral and icosahedral minima were obtained by performing minimizations from configurations generated by low energy MD runs starting from icosahedral and decahedral regions of configuration space. The ten lowest energy minima of each sample are given in Table 2.3. For 75, the Marks' decahedron, 75C, is the most stable cluster at $\rho_0=6$. There is an energy gap of $2\epsilon$ to the next lowest energy decahedral isomer (Table 2.3). At this value of $\rho_0$ the lowest energy icosahedral structure is only $0.3\epsilon$ higher in energy than 75C and there are many icosahedral isomers of similar energy (Table 2.3). In fact we found 31 icosahedral isomers which were of lower energy than the second lowest energy decahedral isomer. From this it might be expected that as the energy is increased the density of states of the icosahedral isomers would become larger than the decahedral isomers, and so the structure would change from decahedral to icosahedral with increasing energy or temperature.

We calculated within the harmonic approximation the equilibrium probabilities that the structure would be icosahedral or decahedral as a function of the microcanonical total energy. The results are given in Figure 2.19. From this it can be seen that the equilibrium structure changes from decahedral to icosahedral when the total energy is only $4\,\epsilon$ above the bottom of the lowest well, which corresponds to a reduced temperature of $0.018\,\epsilon\,k^{-1}$. It is likely that such a transition would not be observed until higher energies because of the free energy barrier involved. In fact we once observed this transition in an MD run started from structure 75C. The run was $1\times 10^6$ time steps long and at a total energy of $74\epsilon$ above the lowest well.

  

Figure 2.19: Calculated equilibrium values for the microcanonical probability that an 75 cluster at $\rho_0=6$ has an icosahedral or a decahedral structure as a function of the total energy. The energy is measured with respect to the bottom of the well of the Marks' decahedron, 75C.

This calculation also helps to explain why 75C has not been previously reported as the lowest energy structure for the much-studied LJ potential. Global optimization algorithms are often based upon tracking the free energy global minimum as the temperature is decreased. This is the principle behind the widely-used simulated annealing method[107]. However, such methods will not find the global minimum if a change in the free energy global minimum occurs at a temperature where the rate of isomerization is so low that a transition to the new structure cannot occur on the time scale of the simulation.

We should not conclude from the above calculation, though, that the global minimum is not important in understanding cluster structures. This example is probably an exception rather than the rule. Such a transition is only likely to occur for values of $\rho_0$ near to where the global minimum changes and in the above example the cause is clearly the differences in the energy spectrum of minima for the two morphologies. In general, then, the global minimum provides a reasonable guide to the finite temperature structural properties of a cluster, which can be usefully supplemented by a knowledge of other low energy minima on the PES.

If there is a unique low energy global minimum separated by a large energy gap from higher energy structures it is appropriate to associate the solid-like structure of the cluster with this single minimum. This would be the case, for example, for the icosahedron 55B at intermediate ranges of the potential. For the LJ potential, microcanonical simulations have shown that the 55-atom cluster resides only in the icosahedral potential well up to an energy of $40\epsilon$ (measured with respect to the bottom of the icosahedral well) above which the formation of surface defects is observed and then complete melting[108,109,110,111,112]. However, when there is a set of low energy minima with very similar energies, at all but very low temperatures an equilibrium ensemble of clusters will contain a mixture of these isomers. It is therefore more appropriate to associate the `structure' with this set of minima rather than the global minimum alone. For example, this would be the case for 55C, which has three other low energy isomers with the same n_nn.

In the above discussion we have been considering the equilibrium structural properties. However, we also need to consider the question of kinetic versus thermodynamic products. How important are the low potential energy structures in practice? When will they actually be found in an experiment? Van de Waal has suggested that kinetic factors could be essential in the growth of fcc rare-gas clusters and the nucleation of fcc solid from the LJ liquid[113,114]. Indeed an extremely good fit is obtained to electron diffraction results for argon clusters from non-optimal structures that involve cross-twinning to allow rapid growth[115,116]. As we noted earlier, the effect of increasing the size and decreasing the range of the potential is to increase the complexity of the PES. Therefore, as the range of the potential decreases the formation of glassy or amorphous clusters, rather than the lowest energy structure, becomes more likely. Whether this is the case for the larger clusters we have studied when the potential is short-ranged would require further investigation. The results for bulk C₆₀ may also provide some insight into this question. When formed by vapour phase deposition C₆₀ gives an amorphous soot[117], and only forms the crystalline phase on recrystallization from benzene.

In this chapter we have been considering the effect of the range of attraction of pair interactions on cluster structure. As rare gas clusters and clusters of C₆₀ molecules can be reasonably modelled by pair potentials we would expect the structures we have found at the appropriate $\rho_0$ to be very similar to the actual structures of these clusters. Interestingly, the results lead us to predict that neutral clusters of C₆₀ molecules exhibit decahedral and fcc structures at small sizes because of the short range of the intermolecular potential. This basic conclusion has been confirmed in studies using more realistic potentials[95,118]. However, this prediction for neutral cluster differs from experimental results[58] for charged clusters for which icosahedral structures are observed up to 55 molecules for $({\rm C}_{60})_N^{+}$ and 147 molecules for $({\rm C}_{60})_N^{++}$.Given the exceptionally large polarizability of , though, it is not unreasonable that the introduction of a long range Coulomic term would significantly affect the energetic competition between morphologies. Therefore, it would be very interesting if this dramatic difference in structure between charged and neutral clusters could be observed experimentally, say by electron diffraction.

In contrast, making predictions for metal clusters from our results is much more difficult because the range of the potential is only one factor influencing structure. In particular, many-body terms may also be important[62,63]. These terms may affect the relative surface energies of $\{111\}$ and $\{100\}$ faces, and so alter the energetic competition between icosahedral, decahedral and fcc structures[102]. For example, in a study of lead clusters cuboctahedra are always found to be lower in energy than icosahedra because the surface energies of $\{111\}$ and $\{100\}$ faces are nearly equal[104]. The value of our results is in providing candidate structures for comparison with the increasingly detailed structural information being provided by experiments, such as chemical probe studies[66,67,68], and for theoretical studies with more realistic, but also more computationally expensive, descriptions of the interactions.

Our results also allow us to comment on a global optimization method proposed by Stillinger and Stillinger in which the PES is deformed by increasing the range[92]. We have to echo Chang and Berry's caution[119] that although increasing the range of a potential does lead to a simplification of the PES it can also cause large changes in the relative stability of different minima. The global minimum when a potential is long-ranged may be very different from the global minimum at the conditions of interest.

For mixtures of spherical colloidal particles and non-adsorbing polymer, the range of the attractive interaction between the colloidal particles can be systematically varied by changing the size of the polymer[120]. This effect has been used to study experimentally the phase diagram for such colloidal systems as a function of the range of the interaction[121,122,123]. We know of no physically realizable system for clusters where the range of the potential could be similarly varied to cause range-induced transitions in cluster structure. However, it remains an intriguing possibility.