Cluster melting

The power sjjectrum is merely the Fourier transform of the VAF via the Wiener-Khintchine theorem. The integration is carried out as a discrete sum over the jjeriod of time in which the VAF decays to a zero value. This quantity gives the number of oscillators at a given frequency and is a very informative indicator of the transition from rigid, quasiperiodic motion to nonrigid, chaotic motion. Note that I(co = 0) is proportional to the diffusion constant. This quantity was calculated by Dickey and Paskin - in the study of phonon frequencies in solids and also by Kristensen et al. in simulations of cluster melting. 

From (6.54) it follows that the ensemble average of the cluster radius grows at N > N t, the critical size N (( being finite at T < T. On the other hand, each cluster melts with the overwhelming probability at T> Tm. The fact that T > Tm suggests at first sight a paradox situation, namely, in the temperature range the liquid is metastable with respect to the ensemble of clusters... 

It will be recalled that in the dynamic theory of solidification [6.90, 91], the liquid - glass transition is related to the appearance of infinite relaxation time of the liquid. It must be noted that in our case the infinite relaxation times appear near Tt (here slow transformations of the metastable liquid into a stable one take place) and near T1, where the relaxation processes are connected with cluster melting and reclusterizations which occur with overcoming large energetic barriers. 

It is known from studies using model potentials that even for a cluster as small as (H20)e, Monte Carlo simulations at temperatures between 50 and 200 K (the range typically considered to examine the issue of cluster melting) need to be carried out for on the order of 10 moves to achieve convergence. Such simulations at the MP2 level would... 

Figure 2.29 shows the mole fraction of monomers that are engaged in the /-cluster, jX(j), as a function of /. As expected, at low temperatures the distribution is very broad, but as the temperature increases the larger clusters melt down and the distribution becomes sharper, and eventually all the molecules become monomeric, i.e. / = 1. 

The cluster sizes 55, 57 and 61 have a higher melting temperature than those in the 10 to 10" range. This is a totally unexpected and not understood result. In fact, it seems that the mean cluster melting temperature increases with decreasing cluster site for n below 90 atoms. 

It is safe to assume, that cluster growth passes through coalescence of small nucleating seeds, melting with a rise in substrate temperature. In the framework of the same BOLS model, it is possible to determine the cluster melting temperature, knowing its size and m parameter by the following equation ... 

Fig. 19. Dependence between cluster melting temperature and its radius, curve 1 calculation, 2 - the same dependence shifted in energy by 35 meV, black circles -experimental points.

The accuracy of Lindemann s criterion depends also on the steepness of its variation with temperature at phase transition and on the specification of its percentage increase which should be adequate to discriminate surface (partial) from all-cluster melting. In the present case, we assign to the temperature at which Lindemann s index starts increasing. This allows us... 

It is surprising, however, that a freestanding nanosolid at the lower end of the size limit, or clusters containing 10-50 atoms of Ga" or IV-A elements, melts at temperatures that are 10-100 % or even higher than the bulk T m(oo) [29, 97-100]. For example, Ga 9 4Q clusters melt at about 550 K, while a Ga+n cluster does not melt even up to 700 K compared with the 7] (oo) of 303 K [97]. Small Sn clusters with 10-30 atoms melt at least 50 K above the 7 (00) of 505 K [22]. Numerical optimizations suggest that Ga" i3 and Ga" i7 clusters melt at 1,400 and 650 K [98]... 

Abundance anomalies as discussed above relate to unique atomic structures. Molten clusters will not exhibit magic numbers. This has been utilized to determine melting temperatures of clusters in the gas phase. If sodium clusters are thermalized in warm helium and then allowed to effuse into vacuum, no magic numbers related to atomic structure are observed. If the gas is cooled to 307 K (83% of the bulk melting temperature), magic numbers appear for clusters of size n > 6000. They extend to smaller clusters if the temperature is lowered further. Smaller clusters melt at lower temperatures. Molten sodium clusters show a different sequence of magic numbers as explained in the following section. 

Small particles are also observed to melt in computer simulations at temperatures well below bulk melting. Rare gas clusters as small as 13 particles show apparent phase transitions, although of course at small sizes these are rounded rather than sharp transitions and resemble conformational equilibria more than collective many-body processes. A simulation of gold particles revealed a three-step process for the melting of the larger particles onset of surface diffusion, formation of liquid patches at the surface, and abrupt melting of the whole cluster. The 477-atom cluster melted at 800 K and the 219-atom cluster below 600 K. 

---