Liquidlike cells free volume

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 vliquidlike cells with t> > 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. 

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... 

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. 

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... 

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