Clusters, random fractal

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... 

Let us consider statistical TV-mer (TV —> oo) as the random fractal with dimensionality Df. Such an approach can be compared to the accepted practice of the construction of fractal clusters by the Monte Carlo method of random addition of new units the only difference is the statistical polymer is automatically random and contains all possible structures of randomly constructed clusters (of course, if they contain the same number TV of units). We may assume that the statistical TV-mer can be considered as the averaged structure obtained after the infinite number of operations of constructions of TV-meric clusters. 

To characterize the dynamic movement of particles on a fractal object, one needs two additional parameters the spectral or fracton dimension ds and the random-walk dimension dw. Both terms are quite important when diffusion phenomena are studied in disordered systems. This is so since the path of a particle or a molecule undergoing Brownian motion is a random fractal. A typical example of a random fractal is the percolation cluster shown in Figure 1.5. 

In our studies, we consider several types of aggregated structures such as bispheres, linear chains, plane arrays on a plane rectangular lattice, compact and porous body-centered clusters embedded on the cubic lattice (bcc clusters, the porosity was simulated by random elimination of monomers), and random fractal aggregates (RF clusters). To generate RE clusters, a three-dimensional lattice model with Brownian or linear trajectories of both single particles and intermediate clusters was employed for computer simulations of aggregation process. At the initial time moment, = 50,000 particles are generated at... 

FIGURE 2 Two examples of linear fractals Vicsek snowflake (a) and percolation cluster (b), representing a deterministic fractal and random fractal, respectively. 

In the case of random fractals, the paradigm and most commonly studied model is percolation (see also Chapter 1 by Chakrabarti). It is important to note that non-trivial modifications of the scaling behavior of SAWs on percolation are believed to occur only at the percolation threshold [5]. Indeed, it is only at criticality, that the infinite critical (the so-called incipient) percolation cluster spans self-simil u structures on all length scales. 

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. 

On the incipient percolation cluster, being a paradigm example of random fractals, SAWs do not obey the standard des Cloizeaux relation, in the form of Eq. (6). Rather, numerical results are consistent with the generalized relation, Eq. (43) [74]. The latter is based on two main features, one is the underlying multifractal nature of SAWs on such random fractals, reflected by the presence of the first-moment 71, and the second the effects of the structural disorder on the probability that the end-to-end SAW distance... 

Figure 7.14. The small-angle diffraction pattern of a silica aerogel (left) that consists of clusters of size R built up of dense smooth particles of size r. Within a size range bounded by r and R the clusters are fractal. At right the scattering pattern of Vycor glass is shown. From D. W. Schaefer, A. J. Hurd, and A. M. Glines. In Random Fluctuations and Pattern Growth Experiments and Models p. 62. With kind permission from Kluwer Academic Publishers.

P — where is the dynamical scaling exponent. In order to introduce the idea of polymeric fractals we have to extend the considerations of ordinary fractals, i.e. those created on a lattice. It has been shown that at least three fractal dimensions are necessary to characterize crudely a fractal object. Indeed in random fractals like percolation clusters or diffusion-limited... 

It is important to note that we assume the random fracture approximation (RPA) is applicable. This assumption has certain implications, the most important of which is that it bypasses the real evolutionary details of the highly complex process of the lattice bond stress distribution a) creating bond rupture events, which influence other bond rupture events, redistribution of 0(microvoid formation, propagation, coalescence, etc., and finally, macroscopic failure. We have made real lattice fracture calculations by computer simulations but typically, the lattice size is not large enough to be within percolation criteria before the calculations become excessive. However, the fractal nature of the distributed damage clusters is always evident and the RPA, while providing an easy solution to an extremely complex process, remains physically realistic. 

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). 

Simulation of structure formation on a lattice [7,100] demonstrated that randomly formed branched clusters also fulfill self-similarity conditions and gave fractal dimensions of [7,104,105] ... 

Kosmidis et al. [87] reexamined the random release of particles from fractal polymer matrices using the percolation cluster at the critical point, Figure 4.11, following the same procedure as proposed by Bunde at al. [84]. The intent of the study was to derive the details of the release problem, which can be used... 

The release problem can be seen as a study of the kinetic reaction A+B —> B where the A particles are mobile, the B particles are static, and the scheme describes the well-known trapping problem [88]. For the case of a Euclidean matrix the entire boundary (i.e., the periphery) is made of the trap sites, while for the present case of a fractal matrix only the portions of the boundary that are part of the fractal cluster constitute the trap sites, Figure 4.11. The difference between the release problem and the general trapping problem is that in release, the traps are not randomly distributed inside the medium but are located only at the medium boundaries. This difference has an important impact in real problems for two reasons ... 

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]... 

Note that the fractal dimensions discussed here are the fractal dimensions of the excitation transfer paths connecting the hydration centers located on the inner surface of the pores. Due to the low humidity, all of the water molecules absorbed by the materials are bound to these centers. The paths of the excitation transfer span along the fractal pore surface and depict the backbone of clusters formed by the pores on a scale that is larger than the characteristic distance between the hydration centers on the pore surface. Thus the fractal dimension of the paths Dp approximates the real surface fractal dimension in the considered scale interval. For random porous structures, Dp can be also associated with the fractal dimension D, of the porous space Dp = Dr. Therefore, the fractal dimension Dp can be used for porosity calculations in the framework of the fractal models of the porosity. 

The temperature at which the phase transition occurs is called the critical temperature or Tg. Most, but not all, magnetic phase transitions are continuous , sometimes called second order . From a microscopic point of view, such phase transitions follow a scenario in which, upon cooling from high temperature, finite size, spin-correlated, fractal like, clusters develop from the random, paramagnetic state at temperatures above Tg, the so-called critical regime . As T Tg from above, the clusters grow in size until at least one cluster becomes infinite (i.e. it extends, uninterrapted, throughout the sample) in size at Tg. As the temperature decreases more clusters become associated with the infinite cluster until at T = 0 K all spins are completely correlated. 

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