Fractals percolation systems

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. 

Ardyralds [12] showed that at the study of ohemieal reactions on fractal objects the cor-rections on small clusters availability in the system were necessary. Just such corrections require the usage in theoretical estimations not generally accepted spectral (fracton) dimension ds [13], but its effective value. For percolation system two cases are possible [12] ... 

The results are typical for percolating systems , liquation (9.9) shows that the occurrence of percolation depends on the value of Nq used in the simulation. Elquation (9.7) indicates that the upper length scale of fractal behavior does not depend on Nq for... 

The concept of diffusion-limited cluster-cluster aggregation (DLCA) is very useful and applied in many simulations. In this type of simulation process, particles are placed in a box and subjected to Browni m (random walk) movements. Aggregation (clustering) may occur when two or more particles/clusters come within the vicinity of each other and the combined cluster continues the random walk. The simulation is stopped at the gelation point (percolating system) or when all particles are combined in one final aggregate. The fractal dimension of the DLCA aggregates is approximately 1.8. 

As it is known [39], structures, which behave themselves as fractal ones on small length scales and as homogeneous ones - on large ones, are named homogeneous fractals. Percolation clusters near percolation threshold are such fractals [1]. As it will be shown lower, cluster structure is a percolation system and in virtue of the said above - homogeneous fractal. In other words, local order availability in polymers condensed state testifies to there structure fractality [21]. 

The percolation system fractal dimension can be expressed as follows [39]... 

The other assumption which is of great interest from the theoretical point cS view is whether there exists the concentration threshoM, similar to the percolation threshold for electroconductivity of filled nanocomposites. In other words there is the range magnetically ordered filler concentration in which the transition from superforomagnetic to the superparamr etic state occurs at a certain temperature. Aimther question arising is which rt excitations exist in such fractal-dimensional systems what is the characta- ot spin density wave in that case, and could there arise the nonlinear effects which are characteristic of one-dimensional systems. These questions are still open at present. 

As it is known [39], structures that are fractal on small scales and homogeneous on large scales are called homogeneous fractals. Such fractals are percolation clusters at the percolation threshold. As it was shown in papers [40, 41], the cluster structure is a percolation system and by virtue of the above is a homogeneous fractal. In other words, the availability of local order in polymers, in a condensed state, is defined by their structure fractality. 

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] ... 

The authors [17] showed that the E value for fractal objects, which are HDPE/EP nanocomposites (see Figure 8.1), was given by the percolation Relationship 6.6 (for more details see Section 6.2). At the same time the cluster structure of the polymer amorphous state presents itself as a percolation system [6,18,19], for which the sum K + (pj should be accepted as p, where is the relative fraction of the local orders domains (clusters). In turn, for such a system it can be written [20] ... 

The other method of calculation of the theoretical dependence E DJ for natural nanocomposites (polymers) was given in paper [21]. The authors [22] have shown that the elasticity modulus E value for fractal objects, which polymers are, is given by the percolation Relationship 6.6. As it is known [4], a cluster nanostructure of polymers represents the percolation system, for which p = (p, p = 0.34 [23] and further it can be written ... 

The approaches considered allow modeling of the primary texture of PS and the processes, limited by individual PBUs that mainly correspond to level III and partially to level IV in the hierarchical system of models (see Section 9.6.3). PBUs are identical in regular PSs, and simulation of numerous processes may be reduced to analysis of a process in a single PBU/C or PBU/P. An accurate modeling of the processes in irregular PSs requires the studies of the properties of structure and properties of the ensembles (clusters) of particles and pores (level IV of the system of models) and the lattices of such clusters (levels V to VII of the system of models). Let us consider the composition of clusters on the basis of fractal [127], and the lattices on the basis of percolation [8] theories. 

Leuenberger, H. and Kocova El-Arini, S., Solubilization systems-The impact of percolation theory and Fractal Geometry. InVWter-lnsoluble Drug Formulation, R. Liu (Ed.), Interpharm Press, Denver, Colorado, Chapter 16, 569-608 (2000). 

The considered model of a straight line of M nanoparticles illustrates only general features of dielectric losses caused by an M nanoparticle cluster in polymer matrix. Actually such cluster is a complex fractal system. Analysis of dielectric relaxation parameters of this process allowed the determination of fractal properties of the percolation cluster [104], The dielectric response for this process in the time domain can be described by the Kohlrausch-Williams-Watts (KWW) expression... 

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. 

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). 

S. HavUn and A. Bunde, Percolation II, in Fractals in Disordered Systems, edited by A. Bunde and S. HavUn, Springer-Verlag Berlin, Heidelberg, 1996. 

Beyond that, fractal geometry is of direct relevance for the transport in inhomogeneous systems, since the percolation cluster (immediately at the percolation threshold, c.f. Fig. 77) assumes a fractal shape. In two-phase mixtures of an insulating and a conducting phase this predicts a power law dependence of the conductivity on q>— q>c (qx. volume fraction).281 A relevant example in the context of our considerations is the conduction behavior of AgCl a-AgI composites.118... 

Figure 77. Selfsimilarity of a large percolation cluster at the critical concentration. The windows indicate the section which is enlarged in the respective succeeding figure.281 (Reprinted from A. Bunde and S. Havlin, Percolation I , in Fractals and Disordered Systems. Ed. by A. Bunde and S. Havlin, Springer-Verlag, Berlin. Copyright 1996 with permission from Springer-Verlag GmbH Co. KG.)...

The coupled dipole equations (CDE) have been used in calculating the optical properties of composite media, including larger particles, where the dipoles are arranged to mimic a more complicated system, such as those used in DDA [38], [39], as well as fractal structures [40], which could be applied to model aggregation, surface composition, or percolation. The general nature of the solution allows for calculation of optical properties, as well as enhanced Raman and electric fields at any point in space. 

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. 

At concentrations near f,., the structure of the interconnected fibrillar network appears to be self-similar i.e. it looks the same at any degree of magnification [61,279]. The appearance of self-similarity is consistent with the well-known result that all systems are fractal near the percolation threshold on length scales below the correlation length of the percolating cluster [277]. 

We mention some other systems that have fractal structures. For example, using sputtering regimes that correspond to the diffusional aggregation model [82], thin films consisting of metallic fractal clusters can be obtained. Fractal structures are also characteristic of percolation clusters near the percolation threshold, as well as certain binary solutions and polymer solutions. The dielectric properties of all these systems can be predicted using the above fractal model. 

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