Percolation fractal dimensions

The cluster properties of the reactants in the MM model at criticality have been studied by Ziff and Fichthorn [89]. Evidence is given that the cluster size distribution is a hyperbolic function which decays with exponent r = 2.05 0.02 and that the fractal dimension (Z)p) of the clusters is Dp = 1.90 0.03. This figure is similar to that of random percolation clusters in two dimensions [37], However, clusters of the reactants appear to be more solid and with fewer holes (at least on the small-scale length of the simulations, L = 1024 sites). 

This results in a value of d = 2.5 for bond percolation on a 3-dimensional lattice. The fractal dimension of the Bethe lattice (Flory-Stockmayer theory) is... 

The power-law variation of the dynamic moduli at the gel point has led to theories suggesting that the cross-linking clusters at the gel point are self-similar or fractal in nature (22). Percolation models have predicted that at the percolation threshold, where a cluster expands through the whole sample (i.e. gel point), this infinite cluster is self-similar (22). The cluster is characterized by a fractal dimension, df, which relates the molecular weight of the polymer to its spatial size R, such that... 

Lenormand R, Zarcone C (1985) Invasion percolation in an etched network Measurement of fractal dimension. Phys Rev Lett 54 2226-2229... 

The fractal dimensions of the excitation paths in samples D, F, and G lie between 2 and 3. Thus, percolation of the charge carriers (protons) is also moving through the Si02 matrix because of the availability of an ultra-small porous structure that occurs after special chemical and temperature treatment of the initial glasses [156]. 

Thus, the non-Debye dielectric behavior in silica glasses and PS is similar. These systems exhibit an intermediate temperature percolation process associated with the transfer of the electric excitations through the random structures of fractal paths. It was shown that at the mesoscale range the fractal dimension of the complex material morphology (Dr for porous glasses and porous silicon) coincides with the fractal dimension Dp of the path structure. This value can be obtained by fitting the experimental DCF to the stretched-exponential relaxation law (64). 

In the context of the SLSP model the relationship between the fractal dimension Ds of the maximal percolating cluster, the value of its size sm, and the linear lattice size L is determined by the asymptotic scaling law [152,213,220]. 

Note that the dynamic fractal dimension obtained on the basis of the temporal scaling law should not necessarily have a value equal to that of the static percolation. We shall show here that in order to establish a relationship between the static and dynamic fractal dimensions, we must go beyond relationships (83) and (84) for the scaling exponents. 

In order to establish the relationship between the static and dynamic fractal dimensions, the initial conditions of the classical static percolation model must be considered for the solution of differential equation (89) which can be written as 0 = Qs = 1 for D = Ds. Here the notation s corresponds to the static percolation model, and the condition s = 1 is fulfilled for an isotropic cubic hyperlattice. The solution of (89) with the above-mentioned initial conditions may be written as... 

Taking into account that for dynamic percolation D = 1/a = Dd, we can easily obtain the relationship between the dynamic and static fractal dimensions, namely,... 

Figure 34 shows the temperature dependencies of the static fractal dimensions of the maximal cluster. Note that at percolation temperature the value of the static fractal dimension Ds is extremely close to the classical value 2.53 for a three-dimensional lattice in the static site percolation model [152]. Moreover, the temperature dependence of the stretch parameter v (see Fig. 34) confirms the validity of our previous result [see (62)] Ds = 3v obtained for the regular fractal model of the percolation cluster [47]. 

Figure 34. The temperature dependence of the static Ds (A) fractal dimensions and the product 3v ( ). At the percolation threshold temperature (Tp = 26.5°C) Ds = 3v. (Reproduced with permission from Ref. 2. Copyright 2002, Elsevier Science B.V.)...

In the rest of this chapter, we will discuss briefly the theoretical ideas and the models employed for the study of failure of disordered solids, and other dynamical systems. In particular, we give a very brief summary of the percolation theory and the models (both lattice and continuum). The various lattice statistical exponents and the (fractal) dimensions are introduced here. We then give brief introduction to the concept of stress concentration around a sharp edge of a void or impurity cluster in a stressed solid. The concept is then extended to derive the extreme statistics of failure of randomly disordered solids. Here, we also discuss the competition between the percolation and the extreme statistics in determining the breakdown statistics of disordered solids. Finally, we discuss the self-organised criticality and some models showing such critical behaviour. 

In fact, if one measures the total number of bonds (sites) on the infinite cluster at the percolation threshold (pc) in a (large) box of linear size L, then this number or the mass of the infinite cluster will be seen to scale with L as where die (< d) is called the fractal dimension of the infinite cluster at the percolation threshold. Similar measurements for the backbone (excluding the dangling ends of the infinite cluster) give the backbone mass scaling as, de < die, where dfi is called the backbone (fractal) dimension. In fact, die can be very easily related to the embedding Euclidean dimension d of the cluster by... 

A brief introduction to some theoretical ideas and models Table 1.3 Fractal dimensions of the percolation clusters... 

Anomalous subdiffusion occurs on percolation clusters or on objects that in a statistical sense can be described as fractal, by which we mean that selfsimilarity describes simply the scaling of mass with length. Connections between v, the fractal dimension of the cluster, D, and the spectral dimension, d, have been established, relations that were originally derived by Alexander and Orbach [35], who developed a theory of vibrational excitations on fractal objects which they called fractons. An elegant scaling argument by Rammal and Toulouse [140] also leads to these relations, and we summarize their results. 

The chapter consists of three main sections. In Section II the elements of fractal theory are given. In Section III the basis of percolation theory is described moreover, a model of fractal structures conceived by us is described. Fractal growth models, constructed using small square or rectangular generating cells as representative structural elements, are considered. Fractal dimensions of structures generated on various unit cells (2x1, 2x2, 2x3, 2x4, 3x1, 3x2, 3x3, 3x4, 4x1, 4x2, 4x3, 4x4) are calculated. Probability... 

The correlation length defines the connectivity of clusters. It defines the scale range within which percolation clusters behave self-similarly and, consequently, are characterized by a fractal dimension [38,39]. The correlation length E, for a percolation lattice can be defined as... 

Probability functions Y lx,ly,p) for fractal ensembles grown on several lattices (of the generating cells lx x ly where 2 < lx < 4,1 < ly < 4) are presented in the Appendix, while calculated values of the percolation threshold Pc, fractal dimension of the ensemble at p = 1, d (lxIy ), mean fractal dimension at p = pc df), and critical indices p(/v, ly) and v(/v, ly) are listed in Table IX. The index ai in this table is calculated from... 

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