Of liquidlike cells

Figure 3.03 Sketch of the communal entropy as a function of the fraction of liquidlike cells in the Cohen-Grest percolation formulation of the glass transition.

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 pliquidlike clusters exist, which might imply a glass 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. 

Fig. 11. Sketch of the probability of liquidlike cells p versus T T near 7], for 1.

The segmenting of f(v) in (3.4) enables us to divide the cells into two classes. Those with v>v we call liquidlike, and those with v > have a free volume, which we take as ... 

As stated above,/(u) has two contributions/o(t>) and/i(n), and the latter depends sensitively on the nature of the cell s immediate environment. This dependence is not so crucial for smaller expansions, u < in the quadratic range, but in the linear range v>v it must be taken into account. We therefore decompose f into two corresponding parts fo and f, leave Cg as a constant, and introduce the environment dependence into f,. The system clearly becomes more rigid as the volume decreases is maximal when the system is entirely solidlike. We can characterize the deviation from solidlike behavior through the mean free volume within the liquidlike fraction of the material ... 

The essence of the free-volume theory described in Section III is that the only change in free energy associated with a redistribution of free volume is in the entropy of the probability distribution of the free volume. This arises from the decomposition of the free energy into a sum of terms depending only on the volume of a single cell, the local free energy /(u,), and from the linearity of /(u,) in v, for liquidlike cells. Of the two, the former is the more serious approximation. 

Consider two liquidlike cells that are not nearest neighbors and are individually surrounded by solidlike cells. From the construction of the Voronoi polyhedra defining the cell volume, it is clear that the cell volumes are not... 

A free exchange of free volume can take place only between liquidlike cells that (1) are nearest neighbors and (2) have enough other liquidlike nearest neighbor cells ( > z) to ensure that the volumes of any neighboring solidlike cells are not constrained to change simultaneously. This defines a type of percolation problem." ... 

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... 

In the usual percolation problem with z= 1, all pN liquidlike cells are in clusters if one counts all isolated liquidlike cells as clusters of size one. That is no longer true when z= l. Only a fraction a (p) of thepN liquidlike cells are now in the cluster [a,(/j) = l]. The cluster distribution C (p), v = 1,2,..., is normalized so that" ... 

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. 

Fig. 10. Sketches of the free energy 9 p) as a function of the liquidlike cell fraction p. Curves a, b, and c correspond to those so labeled in Fig. 9. The positions of the solutions of Fig. 9 are indicated by dots. Curve a corresponds to (Pi) (Pi) at the temperature 7, where the first-order phase transition occur. Crosses emphasize the infinite negative slope of S at p, the percolation threshold.

The free-volume model was originally derived to explain the temperature dependence of the viscosity. We have shown that it has a much broader application and can explain many of the outstanding experimental observations. This includes the existence of an entropy catastrophe at 7 and the approximate equality of Tj, and 7], first observed by Angell and coworkers.The relation between ln and 7, measured by Moynihan et al., also follows naturally and quantitatively from the notion that the liquidlike cell fraction p is the important variable that ceases to reach equilibrium when the relaxation rates become longer than the time scale for the measurement. 

These observations were explained in terms of a free-volume treatment that adopts the Grest-Cohen model, in which a system consists of free-volume cells, each having a total hole volume vh. These free-volume cells can be classified as solidlike (n < v c) or liquidlike (w > Vhc), where Vhc is a critical hole volume. Moreover, it is assumed that the free volume associated with a liquidlike cell of the amorphous phase consists of free-volume holes whose size distribution is given by a normal frequency distribution, H vk). This leads to a cumulative distribution function of free-volume hole sizes, r vh), given by... 

In a liquid cluster, an atom or molecule is not confined to a particular cell or cage but can wander over the entire volume of the cluster. The communal entropy of a single cluster of size v>v is then given by /ci ln( - l)fJ, where o, is the average configurational volume of a liquidlike... 

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