Filler networking fractal nature

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 paper [19] it has been shown that the universality of the critical indices of the percolation system is connected directly with the fractal dimension of this system. The self-similarity of the percolation system assumes the availability of a number of subsets having the order ( = 1,2,4), which in the case of the structure of polymeric materials are identified as follows. The percolation cluster network or matrix physical entanglement cluster network is the first subset n = 1) in the polymer matrix. The loosely packed matrix, into which the cluster network is immersed, is the second one (n = 2). For polymer composites, the filler particles network, which is naturally absent in epoxy polymers, is the third subset (n = 4). In such a treatment the percolation cluster critical indices P and v are given as follows (in three-dimensional Euclidean space) [19] ... 

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