Solidlike clusters

Phase transitions in small clusters have been studied extensively using both analytical and numerical (molecular dynamics and Monte Carlo) methods. The results of these studies are the main subject of this chapter and are discussed in detail in the forthcoming sections. Preempting this discussion, we want to draw attention here to one particular feature. In agreement with the prediction of our analytical model and in accord with Hill s picture, the results of our numerical studies " clearly bore out the fact that there exists a finite range of temperatures over which the solidlike and liquidlike... 

Figure 3. Schematic representation of the forms of partition functions for solidlike and liquidlike clusters of fixed composition, if densities of states are qualitatively as depicted in text.

All in all, the conclusions of this stage of the analysis are that even quite simple physical models can account for many of the properties of the coexistence of solidlike and liquidlike clusters. But let us recall that all this was carried out assuming that the two forms could exist in thermodynamic equilibrium. The question was posed by Natanson et al. ° of what would be the necessary and sufficient conditions for that hypothesis to be valid. Answering that question and pursuing its immediate consequences, even at a qualitative or semiquantitative level, constituted the second stage of the analytic study. 

Thus we obtain a picture of the phase equilibrium between solidlike and liquidlike clusters which differs in a fundamental way from bulk equilibrium because of the finite tempierature range over which the solidlike and liquidlike forms may coexist. Clearly the transformation between these phases cannot be the same as a first-order phase transition, although it becomes so in the limit of large N. In fact the transformation of a cluster between solidlike and liquidlike is simply not in any of the traditional categories of first, second, or higher order. 

Davis et al. have since jjerformed very long MC and isothermal MD simulations on Arij, which demonstrate that a coexistence between a solidlike and a liquidlike cluster occurs over a finite range of temperatures. Their results can be used to reconcile previous MC results, but are also important for several other reasons. One of these is direct supjjort of the canonical two-phase model, which is not directly provided—for a system of so few particles —by corresjjonding isoergic simulations. A more complete description of the calculations of Davis et al. follows, but first we discuss in detail the calculations of Etters and Kaelberer, Quirke and Sheng, and Nauchitel and Pertsin. 

Of central interest to the study of clusters is information on both dynamic and equilibrium properties of the systems. Dynamic properties such as diffusion coefficients can be used to investigate solidlike to liquidlike phase transitions as the cluster temperatures are varied. Equilibrium thermodynamic properties are of great utility in identifying cluster stability. Further-... 

We have defined a liquidlike cell to be in a cluster if it has at least z neighbors that are also liquidlike.Within such a liquidlike cluster, cells can exchange their free volume freely without restriction by neighboring solidlike cells. The usual percolation problem has z = l, so that all isolated liquidlike cells would be clusters of size one. Thus we have introduced a new percolation problem, which we call environmental percolation. In... 

We call all clusters liquidlike. However, a cluster for which (5.2) holds is liquid, rather than liquidlike, in the sense that each atom or molecule within it moves in time through the entire cluster. That is, each molecule finds accessible the configuration space of every other molecule in the cluster. We now suppose that exchange of free volume between solidlike and liquidlike cells is so slow compared to exchange between liquidlike cells that we can ignore it in the computation of equilibrium properties. We return to this point later. As we shall see in Section X, the two time scales differ by much more than 2 orders of magnitude. 

Thus with the exception of p, the parameters in (8.7) for v are smooth functions of temperature and show no anomaly at 7. However,/ is a discontinuous function of T at 7 and, as a result, this model predicts a jump discontinuity in v at T. The volume will show a change of slope near Tj, as the contributions from the infinite cluster dominate for / >/ <,. while the solidlike and finite-size clusters dominate forp[Pg.503]

There has been no direct verification of the conceptual structure of the theory. That is, a microscopic determination of the cluster distribution function has not been made, and the effects of percolation have not been seen. Assuming that the structure of the glass is well-defined liquidlike clusters in a denser solidlike background, one might expect to be able to see these clusters by either neutron or X-ray scattering. Since v is probably between 100 and 400 A and r c SO at Tscattered wave vectors on the order of 0.1 A could be used. 

Fig. 10. Calculated free energy barrier for homogeneous crystal nucleation of hard-sphere colloids. The results are shown for three values of the volume fraction. The drawn curves are fits to the CNT-expression Eq. (1). For the identification of solid like particles we used the techniques described before. The cutoff for the local environment was set to Vq = 1.4

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