Fractal dimension critical percolation

In two dimensions (2D), the exact value for the hull fractal dimension, dj =, differs from the bulk fractal dimension. Although percolation in 3D is of greater practicd interest, no exact result is known in this case. The aim of the present paper is to review briefly the numerical results obtained so far for the hulls of 3D percolation clusters and to discuss the possibility that the hull and bulk critical behaviors are the same in this case. 

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). 

Lastly, Argyrakis and Kopelman [33] have simulated A + B -4 0 and A + A —> 0 reactions on two- and three-dimensional critical percolation clusters which serve as representative random fractal lattices. (The critical thresholds are known to be pc = 0.5931 and 0.3117 for two and three dimensions respectively.). The expected important feature of these reactions is superuniversality of the kinetics independent on the spatial dimension and... 

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. 

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... 

The final relation uses the fractal dimension of the randomly branched polymer, which is 27 = 2.53 for critical percolation in three dimensions. The fact that randomly branched polymers are fractal means that the... 

Here, the discussion of the viscoelastic exponent n in relation to the assumed gelation model (e.g. electrical analogy percolation or Rouse model), as well as the fractal dimension dfof the critical gel (Muthukumar and Winter 1986 Muthukumar 1989) will be ignored. The reader is referred e.g. to (Adam and Lairez 1996 Martin and Adolf 1991). It has been also shown, that stoichiometry, molecular weight and concentration have an impact on the critical gel properties (Winter and Mours... 

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. 

It is easy to see, that the indicated mode corresponds to the gelation point, where the value D 2.5 was confirmed experimentally [10]. Besides, it is necessary to note the critical value D-2.5 in the gelation point universality, which is the same for both linear and branched polymers. The same value of fractal dimension was obtained practically for all known critical phenomena (percolation, Ising and Potts models and so on), which correspond to the well-known Suzuki mle [6]. The indicated community of the critical dimension of 2.5 is defined by the common cause of its reaching— the excluded voliune effect compensation by the screening effect. 

After having discussed the behavior of SAWs on deterministic fractals, we move on to the second major topic of this chapter, namely the numerical study of SAWs on random fractals, the latter modelled by percolation. As non-trivial changes in the exponents characterizing the structure of SAWs on the incipient percolation cluster (and as a consequence on its backbone) axe only expected at criticality [5], i.e. for probability p of available sites being p = Pc, the following discussion is restricted to this case. A summary of exponents and fractal dimensions characterizing critical percolation is given in Table 1. 

Critical exponents and fractal dimensions of the incipient percolation cluster, for spatial dimensions d = 2, 3, and d > 6. Note that usually d is called df in the fractal literature, but for the sake of consistency with our notation we refer here to this quantity as d. ... 

Vojta and Sknepnek also performed analogous calculations for the quantum percolation transition at p = pp, J < 0.16/ and the multicritical point 2itp=pp,J = 0.16/. A summary of the critical exponents for all three transitions is found in Table 3. The results for the percolation transition are in reasonable agreement with theoretical predictions of a recent general scaling theory of percolation quantum phase transitions P/v = 5/48, y/v = 59/16 and a dynamical exponent oi z = Df = (coinciding with the fractal dimension of the critical percolation cluster). 

The difference between the FS model and percolation model is in the critical phenomenon. As summarized in Table 1, if the statistical values are normalized by the equivalent distance e(= 1 — a/a ) from the gel point (the critical point), there is a significant difference in critical index for flie FS model and percolation model. This difference reflects the difference in size distribution (see Fig. 1 [6]). The difference of the structure in flie model is reflected on the fractal dimension D of the fraction that has a certain degree of polymerization x. If the radius of a sphere that corresponds to the volume of the branched polymer fiaction with the degree of polymerization x is R, the relationship between x and R is fimm the fiactal dimension D... 

Thus if we analyse these results taking into account the swelling effect due to dilution, which modifies the fractal dimension (D = 2 instead of 2.5), we can conclude that gelation is a critical phenomenon of connectivity described by the percolation theory. 

In paper [126] it was shown that universality of the critical indices of the percolation system was connected directly to its fractal dimension. The self-similarity of the percolation system supposes the availability of the number of subsets having order n (n = 1, 2, 4,. ..), which in the case of the structure of amorphous polymers are identified as follows [125]. The first subset (n = 1) is a percolation cluster frame or, as was shown above, a polymer cluster network. The cluster network is immersed into the second loosely packed matrix. The third (n = 4) topological structure is defined for crosslinked polymers as a chemical bonds network. In such a treatment the critical indices P, V and t are given as follows (in three-dimensional Euclidean space) [126] ... 

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