Liquidlike clusters

The concept of liquidlike clusters seems to have first emerged as a result of the explosion of simulations by the molecular dynamics (MD) method of McGinty, ° Cotterill et al., Damgaard Kristensen et al., and Briant and Burton, and by the Monte Carlo (MC) method of Lee et al. These were followed quickly by further MD simulations and MC simula-... 

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. 

When p is nonzero, there are clusters of liquidlike cells, each one of which has at least z liquidlike neighbors. It is well known that in such situations there is a critical concentration above which there exists an infinite cluster. Thus for p>p, there is an infinite, connected liquidlike cluster, and we can consider the material within it to be liquid. For pglass phase because the fluidity would be reduced. However, percolation theory tells us that just above p the infinite cluster is very stringy or ramified so that bulk liquid properties are not fully developed. 

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 note also that atomic mobility occurs within finite liquidlike clusters that exist below the transition. Thus the fluidity of the system would in principle persist below 7. ... 

The average free volume within a liquidlike cluster is ty, given by (3.6). Thus for diffusive motion to take place within a given cluster, its size p must be at least v /vj. 

We saw in Section IX that the system s falling out of equilibrium at 7 has a profound effect on p, the fraction of liquidlike clusters. Since the decrease of p requires a structural rearrangement of molecules in the dense liquid, it becomes more difficult as T approaches 7, where the value of p is frozen at Pf, >Pcz- This freezing in of liquid clusters affects the low-temperature properties of glasses by giving rise to the tunneling levels, described in Section II. 

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. 

FIGURE 1.2 Free energy versus size of the largest liquidlike cluster in the NPT ensemble (T = 0.741) for different supersaturations (5) in the LJ system. (Reproduced from Bhimalapuram, P. et al., Phys. Rev. Lett., 98, 206104, 2007. With permission.)... 

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