Fractals percolating

The first one is the Katz and Thompson s model (1986) which interprets transports within pore solids in terms of these percolation ideas [2]. From that, the authors introduced a fractal percolation model to predict the permeability of a disordered porous media. In invasion percolation, a non-wetting fluid can have access to the first connection from one face of the sample to the other only when the driving pressure is sufficient to penetrate the smallest pore-throat of radius rc in the most efficient conducting pathway. So, the permeability of rocks saturated with a single liquid phase is given from the following relationship ... 

The success of fractal models applied to the physics of disordered media may be explained first of all by the fact that fractal forms are characteristic of a huge number of processes and structures because many diverse models of the formation and growth of disordered objects of disparate nature may ultimately be reduced to a transition model—namely a connected set and an unconnected set—and to a limited diffusive aggregation [1-6]. In the first case a fractal percolation cluster is formed in the second case a fractal aggregate is formed. 

Sahimi, M., Flow phenomena in rocks—From continuum models to fractals, percolation, cellular-automata, and simulated annealing. Rev. Mod. Phys. 65,1393 (1993). 

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. 

Another kind of Flory-type formula is suggested in [30], where it was argued that the spectral dimension dg of the fractal percolation cluster must be an intrinsic property ... 

In the fluence range that corresponds to the appearance and growth of the fractal percolation cluster in the implanted layer, the temperature dependence of conductivity usually follows... 

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]. 

It is easier to understand if the heterogeneous structure of gels is divided into two kinds of heterogeneity [21]. The fractal heterogeneous gel structure model is the case where multiple fractal percolation clusters connect and form networks as shown in Fig. 7(b). In this model, there are... 

Juarez-Maldonado, R., Chdvez-Rojo, M. A., Ramtrez-Gonzdlez, P. E., Yeomans-Reyna, L., and Medina-Noyola, M. 2007. Simplified self-consistent theory of colloid dynamics. Phys. Rev. E 76 062502. Sahimi, M. 1993. Flow phenomena in rocks From continuum models to fractals, percolation, cellular automata, and simulating annealing. Rev. Mod. Phys. 65 1393. 

The percolation network may also have fractal geometry. Charge carrier transport on a fractal percolating cluster is anomalous and the frequency dependence of ac conductivity above some critical frequency is... 

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