Entropy communal

Hoover W G and Ree F H 1968 Melting transition and communal entropy for hard spheres J. Chem. Phys. 49 3609-17... 

M. Blander and S. H. Bauer. Private communication. Entropy of association low frequency modes of H bonded dimers. 

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.

The concept of communal entropy has featured within the lattice models of liquids and mixtures. We show in this appendix that this entropy change is due to a combination of assimilation and expansion. 

This is known as the communal entropy. Closer examination reveals that AS in equation (J.2) is a net result of two effects the increase of the accessible volume... 

Figure J.1 A delocalization experiment. Initially, there are N particles of the same kind, each in a cell of volume v. Upon removal of the partitions between the cells, each particle can access the entire volume V-Nv. The entropy change in this process, for an ideal gas, is the so-called communal entropy.

Originally, Hirschfelder et al. (1937) introduced the concept of communal entropy to explain the entropy of melting of solids. They specifically stated that this communal sharing of volume gives rise to an entropy of fusion. This idea was later criticized by Rice (1938) and by Kirkwood (1950) and now the whole concept of communal entropy in the context of the theory of liquids is considered to be obsolete. 

Here we have cited this example to stress the point that the communal entropy in equation (J.2) is not a result of volume change only but a combination of volume change and assimilation. 

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... 

When the infinite cluster is present, the communal entropy is greatly enhanced, since many atoms have the possibility of extending their movement over the entire system. Therefore, the communal entropy is that entropy associated with the accessibility of all the configurational volume within... 

We have thus expressed in terms of p, v, y , a (p), C (p), and P p). The first two quantities depend only on P(v), and in Section VII we develop a method of determining P(v). The a (p) and P ip) depend on Cy ip). Because C, p) enters the free energy through S, (5.6), it should be determined by minimization of the free energy. As we see below, the results are the same as those of a percolation problem somewhat different from the environmental percolation problem described in Section IV. The essential point is that clusters with v>v are favored because they contribute to the communal entropy, while the formal structure of the problem remains that of a percolation problem. 

We now consider the effects of the communal entropy on the features of the percolation problem. Clusters of sizes less than y tend to be suppressed a p) and A p) move toward unity. The problem moves away... 

This expression is different for p>p and for p

p contribute to the communal entropy Sq(p) and are therefore favored. This difference in the dependence of 5i,(p) and the resulting discontinuity at I = is the only difference from the usual percolation theory. The desired cluster distribution function C (p) is porportional to q T)/V, obtained by substituting (5.18) into (5.12) and treating p as a function of temperature T. Thus C Xp) f e same as that already given by (4.4) with T f-a (p) for Fotp

r, so that clusters with... 

Now that we have an expression for the communal entropy, we can derive the probability distribution P(v). We start from the configurational free energy given by... 

The heat capacity is the sum of contributions from the configuration and communal entropy. 

With our earlier estimates of we crudely estimate a to be positive and in the range 0.3-0.6. Note that the usual exponent a for the percolation problem is negative, since the entropy is related to the configuration probability of being in a cluster. Here the communal entropy is present and results in a different expression for a, dependent only on fi. For the critical contribution to C, which is dominant in the vicinity of p, we take... 

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. 

In the free-volume model,/ becomes the dominant structural mode of the relaxation theories. The rearrangement of the cage structure requires diffusion, which is slowly being frozen out. As T approaches 7, p can no longer follow its equilibrium value, but becomes frozen at a value p>p. Since the variation in p no longer contributes to the heat capacity, decreases. The relevant contribution to arises from the communal entropy, which depends primarily on p and ceases to change. However, at temperatures above 7, the structure can equilibrate rapidly compared to the measurement time and both p and approach their equilibrium value. 

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