Percolation theory correlation lengths

An explanation of the observed relaxation transition of the permittivity in carbon black filled composites above the percolation threshold is again provided by percolation theory. Two different polarization mechanisms can be considered (i) polarization of the filler clusters that are assumed to be located in a non polar medium, and (ii) polarization of the polymer matrix between conducting filler clusters. Both concepts predict a critical behavior of the characteristic frequency R similar to Eq. (18). In case (i) it holds that R= , since both transitions are related to the diffusion behavior of the charge carriers on fractal clusters and are controlled by the correlation length of the clusters. Hence, R corresponds to the anomalous diffusion transition, i.e., the cross-over frequency of the conductivity as observed in Fig. 30a. In case (ii), also referred to as random resistor-capacitor model, the polarization transition is affected by the polarization behavior of the polymer matrix and it holds that [128, 136,137]... 

The correlation length and the number of monomers in a characteristic branched polymer N have simple predictions for vulcanization. These predictions can be easily obtained from the mean-field percolation theory [Eqs (6.105) and (6.125) with exponents a=v = jl] by replacing the monomer in the previous treatment by the precursor linear chain of size bN J containing Nq monomers. 

The concentration c is equivalent to the critical point where the crossover phenomenon occurs from randomness to order. It is also equivalent to Pc (the critical probability) in the percolation theory, where the crossover phenomenon occurs from the finite cluster (such as a macromolecule containing a finite number of monomers) to the infinite cluster (such as the network of an entangled macromolecule, which extends from one end to the other). The three regions are characterized by three important quantities the number of statistical elements per chain N, the number of statistical elements per unit volume p (density), and the correlation or screen length... 

The scaling theory of percolation predicts that the correlation length will obey a scaling law of form [2]... 

As noted in Chapters 3 and 4, small-angle X-ray scattering (SAXS) has shown that silicate polymers have a fractal dimension of 2.1 [68,69]. Similar results have been obtained for alumina aggregates grown in dilute solution [62]. These studies probe the clusters on a scale smaller than -10 nm, where the structure is found to agree with that expected for RLCCA, rather than percolation (d( = 2.5) or the classical gelation theory (rff = 4). SAXS also measures the spatial correlation length, if gs(r) is the probability that a monomer is located a distance r away from another monomer, then g is constant for r > (J for r < [Pg.181]

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