Percolation theory cluster distribution function

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... 

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. 

We now define some statistical quantities of interest in percolation theory. Let ns p) denote the number of clusters (per lattice site) of size s. In fact, a detailed knowledge of ns p) would give us a lot of information on the percolation statistics, as most of the quantities of interest can be extracted from various moments of the cluster size distribution n. Although, in general, we do not have any analytic knowledge about this distribution function ns p) near Pc, we can utilise the powerful observation of scaling behaviour of ns p) near Pc (see the next section). 

To test this theory, the room temperature conductivity of "Nafion" perfluorinated resins was measured as a function of electrolyte uptake by a standard a.c. technique for liquid electrolytes (15). The data obey the percolation prediction very well. Figure 9 is a log-log plot of the measured conductivity against the excell volume fraction of electrolyte (c-c ). The principal experimental uncertainty was in the determination of c as shown by the horizontal error bars. The dashed line is a non-linear least square law to the data points. The best fit value for the threshold c is 10% which is less than the ideal value of 15% for a completely random system. This observation is consistent with a bimodal cluster distribution required by the cluster-network model. In accord with the theoretical prediction, the critical exponent n as determined from the slope of... 

---