Networks percolation, fractal dimension

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

It has been possible to directly image the percolation network at the surface of a CB-polymer composite. An early report is that of Viswanathan and Heaney [24] on CB in HOPE in which it was shown that there are three regions of conductivity as a function of the length L, used as a metric for the image analysis. Below I = 0.6pm, the fractal dimension D of the CB aggregates is 1.9 0.1. Between 0.8 and 2 pm, the data exhibit D = 2.6 0.1 while above 3 pm, D = 3 corresponding to homogeneous behavior. Theory predicts D = 2.53. It is not obvious that the carbon black-polymer system should be explainable in terms of standard percolation theory, or that it should be in the same universality class as three-dimensional lattice percolation problems [24]. Subsequent experiments of this kind were made by Carmona [25, 26]. 

Of the several mechanisms investigated, the most commonly adopted is based on the filler network breakage [48, 49]. Kraus [7, 50] proposed a phenomenological model of the Payne effect based on this interpretation. In this model, under dynamic deformation, filler-filler contacts are continuously broken and reformed. The Kraus model considers filler-filler interactions but the loss modulus and effect of temperature were not taken into account. In the model of Huber and Vilgis [9, 50, 51] the existence of dynamic processes of breakage and reformation of the filler network is explained. In this model, the Payne effect is related to the fractal nature of the filler surface. At sufficiently high volume fractions of filler, percolation occurs and a continuous filler network is formed, characterized by its fractal dimension and its... 

In low-humidity tetragonal crystal with the partial density of lysozyme of about 0.80 g/cm, approximately 120 water molecules are in the first hydration shell of lysozyme molecule. In order to explore a wide range of hydration level up to monolayer coverage (about 300 water molecules), partial density of lysozyme in powder should be < 0.80 g/cm. In Ref. [401], two models for protein powder were studied densely packed powder with the density of dry protein 0.66 g/cm and loosely packed powder with a density 0.44 g/cm. In loosely packed powder, the percolation transition of water was noticeably (by a factor of two) shifted to higher hydration levels compared with experiment. The fractal dimension of the water network at the percolation threshold as well as other properties evidenced that the percolation transition of water in this model was not two dimensional. The spanning water network consists of the 2D sheets at the protein surface as well as of the 3D water domains, formed due to the capiUaiy condensation of water in hydrophilic cavities. The latter effect causes essential distortion of various distribution functions of water clusters in loosely packed powder. Therefore, below we present an overview of the results obtained for the densely packed model powder. 

Herein, ds is the space dimension, i.e. 2, and df is the fractal dimension (Hausdorff dimension) of the 2D aggregated particle cluster. Typical values for Hausdorff dimensions are e.g. 1.44 for diffusion limited cluster aggregation (DLCA) and 1.55 for reaction limited cluster aggregation (RLCA) [22, 31], Assuming isolated 2D aggregates which were formed by interfacial particle-particle aggregation, an exponent of about —6 is estimated. Close to the two- dimensional sol-gel transition, the system should behave like a percolated network and the corresponding exponent is determined to —9.5 [31],... 

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] ... 

DfIzAF and Df the fractal dimension of the percolating network, respectively. Prom eq. (2), the fracton dimension dAF of antiferromagnetic fractons is given by... 

In conclusion, the hierarchical structures in the swollen pellicles are composed of (1) the dendritic network (Figure 12 (b-1)) composed of the percolated bundles with mass fractal dimension of Dm = 2.5 and the upper and lower cutoff lengths of Ls4 30pm and Ls3 2pm, respectively (2) the bundles in turn are composed of the ribbons with Dm =1.0 and the upper and lower cutoff lengths of 1 3 2 pm and Ls2 600nm, respectively (3) the ribbons are further composed of the mass fractally arranged MFs oriented parallel to the ribbon axis with the lateral mass fractal dimension Dmj.= 1.35 and the upper and lower cutoff lengths of Ls2 600 nm and Lsi 6 nm, respectively. 

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