Solidlike cells

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... 

The first-order transition is a direct result of the divergence of d /dp as p- Pc for P<. The latter is caused by the rigid and arbitrary manner in which the cells were divided into liquid- and solidlike cells. We discuss below the consequences of eliminating this unphysical feature of the model. 

The theory now proceeds as developed in Sections V and VI, essentially unchanged. For example, P v) will have the same bimodal structure as shown in Fig. 14, but will now be continuous. Similar smoothing of all artificially introduced discontinuities will not affect the theory in any essential way. The loss of a sharp distinction between liquid- and solidlike cells could vitiate use of the percolation theory. The nonanalyticity in S will certainly be lost, leading to a communal entropy for which 9S/9p is always less than infinity. However, the first-order phase transition should be preserved, just as it was for most of the parameter space even when )3> 1. The discontinuity in p and v would be reduced, as would be the latent heat. One important effect of this smearing will be the appearance of a critical end point for the liquid, a temperature below which the liquid phase is no longer even metastable. The second-order transition, which is only a small region of parameter space for /8> 1, is now wiped out completely by the restoration of analyticity. Our theory thus leads to a first-order phase transition or no transition at all. However, the entropy catastrophe can be resolved within our theory only if a transition occurs. 

Immobilized water. This is meant to be water that does not leak out of a solidlike food. It is present in closed cells, in open pores in a solid matrix (like a sponge), or between chains of coiled polymers. Binding sites for water need not be present for the water to be held, and bound water clearly is a misnomer. Better names are held, trapped, or imbibed water. Its amount can be large in several gels, oneg of polymer can readily hold 100 g of water. Actually, it is generally an aqueous solution that is held, rather than pure water. 

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 ... 

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, quantitative PM IRRAS has been employed to investigate the orientation and the field-induced transformations of a monolayer [82] and a bilayer [83] of n-octadecanol at a gold electrode surface. The -octadecanol forms 2D solidlike films. These studies illustrate how to use PM IRRAS to determine not only the tilt angle of adsorbed molecules but also the packing and orientation of molecules into a unit cell of the 2D lattice of a solid film. 

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... 

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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