Critical indices percolation clusters

The increase of infinite cluster density (p) near the percolation threshold p, (Ap/p c lj is characterized by a critical index (1... 

Another value characterizing a percolation system is the average cluster dimension S p). The critical behavior near the percolation threshold is defined by the critical index y [1] ... 

In this range, the connecting set is a fractal that is, it is geometrically similar to a percolating cluster, and its properties depend on the linear scale. Therefore, both the correlation length and the P s of the connecting set (the upper index oo means that the limit / —> oo is taken) should scale with distance from the critical point (i.e., percolation threshold pc = p ) as... 

The critical indices estimated from these relations fall into the admissible ranges of variation P = 0.39-0.40, V = 0.8-0.9, and t = 1.6-1.8, determined in terms of the percolation model for three-dimensional systems. The researchers [7] noted that not only numerical values but also the meanings of these values coincide. Thus the index P characterises the chain structure of a percolation cluster. The 1/p value, which serves as the index of the first subset of the fractal percolation cluster in the model considered [7], also determines the chain structure of the cluster. The index v is related to the cellular texture of the percolation cluster. The 2/df index of the second subset of the fractal percolation cluster is also associated with the cellular structure. By analogy, the index t defines the large-cellular skeleton of the fractal percolation cluster. The relationship between the critical percolation indices and the fractal dimension of the percolation cluster for three-dimensional systems and examples of determination of these values for filled polymers are considered in more detail in the book cited [7]. Thus, these critical indices are universal and significant for analysis of complex systems, the behaviour of which can be interpreted in terms of the percolation theory. 

Let us consider the critical index Vp physical significance. In Ref. [18] it has been shown that percolation cluster is a fiactal object with dimension for which the following relationship is valid ... 

As it has been shown in Ref. [70], the value p.j. in the general case is a function molecular mobility level of polymer and p.j. > P, where P is the corresponding critical index of percolation cluster, the fomation oiwhich is controlled by geometrical interactions only [31]. The equality p.. = P is reached only in the case of completely inhibited molecular mobility., that is, in the case of quasiequilibrium state. 

Figure 9.35 The dependence of the thermal cluster order parameter index on the relative fraction of nanoclusters for epoxy polymers EP-1 (1) EP-1-200 (2) EP-2 (3) and EP-2-200 (4). Horizontal dashed lines give the values of critical percolation indices P (5) and v (6) [66]...

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