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Critical percolation index

Thus, the percolation critical indices P and V are border values for P, indicating which structural component of an epoxy polymer defines its behaviour. At P = P nanoclusters or, more precisely, the percolation cluster network, identified with the nanocluster network, are such a component. At P < P < v epoxy polymer behaviour is defined by the combined influence of the nanoclusters and the loosely packed matrix. At P = V the loosely packed matrix will be a structural component, defining epoxy polymer behaviour. The estimations according to Equations 5.4 and 5.5 have shown that in the considered case the average d value is equal to 2.644 and then according to Equations 9.43 and 9.44 let us obtain p = 0.38, v = 0.76. Then, using Equation 5.49, P values for the considered epoxy polymers can be calculated [66]. [Pg.458]

Value t — 2 for the granular metals has been confirmed experimentally in several papers (see, for example, Ref. [1]) however, for Nix(Si02)i A nanocomposite with granules of nanometer size it was found t x 2.7, g x 2 [65]. It rather essentially differs from the classical theory predictions. Also, the noticeable differences of the experimental values of critical indexes from the theoretical ones have been found in papers [66,67]. Authors of these papers attributed the discrepancy between the experimental data and results of the classical percolation theory to the quantum effects, which lead to the Anderson localization of charge carriers [57,58]. [Pg.610]

The increase of infinite cluster density (p) near the percolation threshold p, (Ap/p c lj is characterized by a critical index (1... [Pg.134]

Another value characterizing a percolation system is the average cluster dimension S p). The critical behavior near the percolation threshold is defined by the critical index y [1] ... [Pg.135]

This simple analysis confirms the Mott-CFO model in every detail. There is a Gaussian tail of localized states associated with density fluctuations, a mobility edge at = -0.52, channel and resonant extended states just above e, and ordinary extended states further above e. The mobility is of course zero below e. and positive above it. A final comment is in order regarding the behavior of ju (E) for E just above E. Figure 3.7b shows a linear increase which comes from an assumed linear increase in the percolation probability, liowever, we believe it more likely that the percolation probability, and (X)nsequently the classical mobility, would have the critical index behavior of Eq. (3.17a). [Pg.121]

Critical index values in the problem of bond percolation (Obukhov, 1985)... [Pg.407]

Let us consider the critical index Vp physical significance. In Ref. [18] it has been shown that percolation cluster is a fiactal object with dimension for which the following relationship is valid ... [Pg.101]

As it has been shown in Ref. [70], the value p.j. in the general case is a function molecular mobility level of polymer and p.j. > P, where P is the corresponding critical index of percolation cluster, the fomation oiwhich is controlled by geometrical interactions only [31]. The equality p.. = P is reached only in the case of completely inhibited molecular mobility., that is, in the case of quasiequilibrium state. [Pg.226]

The difference between the FS model and percolation model is in the critical phenomenon. As summarized in Table 1, if the statistical values are normalized by the equivalent distance e(= 1 — a/a ) from the gel point (the critical point), there is a significant difference in critical index for flie FS model and percolation model. This difference reflects the difference in size distribution (see Fig. 1 [6]). The difference of the structure in flie model is reflected on the fractal dimension D of the fraction that has a certain degree of polymerization x. If the radius of a sphere that corresponds to the volume of the branched polymer fiaction with the degree of polymerization x is R, the relationship between x and R is fimm the fiactal dimension D... [Pg.128]

In this range, the connecting set is a fractal that is, it is geometrically similar to a percolating cluster, and its properties depend on the linear scale. Therefore, both the correlation length and the P s of the connecting set (the upper index oo means that the limit / —> oo is taken) should scale with distance from the critical point (i.e., percolation threshold pc = p ) as... [Pg.150]

Probability functions Y lx,ly,p) for fractal ensembles grown on several lattices (of the generating cells lx x ly where 2 < lx < 4,1 < ly < 4) are presented in the Appendix, while calculated values of the percolation threshold Pc, fractal dimension of the ensemble at p = 1, d (lxIy ), mean fractal dimension at p = pc df), and critical indices p(/v, ly) and v(/v, ly) are listed in Table IX. The index ai in this table is calculated from... [Pg.153]

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. [Pg.290]

Table 8.2 summarizes the theoretical critical indices and their experimental values. The index s characterizes the divergence of the viscosity, and t characterizes the appearance of the elastic modulus. The index t is larger than because not all the chains in the connected network are elastically effective. There are many free ends near the percolation... [Pg.273]

The value of the powers, superscripts s and /, in Eqs. (6) and (7) are 1.3 and 1.8, respectively, by replacing p with the polymer concentration. There are many careful measurements reported on the modulus near the gel point. Tokita and Hikichi [187] measured the modulus of dilute agarose aqueous solution as a function of temperature and determined the value of / to be 1.9 and the critical temperature, = 80°C. The concentration dependence of the agarose aqueous solution at 20°C was also measured and obtained as = 0.0137 g/lOOmL, /=1.93. Gauthier-Manuel et al. [188] measured the dynamic viscoelasticity if = 10 Hz) of silica particle suspension as a function of reaction time and obtained / = 2. The experimental value is closer to the index 1.9 of the 3D percolation theory than is / = 3 of the Flory-Stockmayer... [Pg.328]

Figure 9.35 The dependence of the thermal cluster order parameter index on the relative fraction of nanoclusters for epoxy polymers EP-1 (1) EP-1-200 (2) EP-2 (3) and EP-2-200 (4). Horizontal dashed lines give the values of critical percolation indices P (5) and v (6) [66]... Figure 9.35 The dependence of the thermal cluster order parameter index on the relative fraction of nanoclusters for epoxy polymers EP-1 (1) EP-1-200 (2) EP-2 (3) and EP-2-200 (4). Horizontal dashed lines give the values of critical percolation indices P (5) and v (6) [66]...

See other pages where Critical percolation index is mentioned: [Pg.273]    [Pg.129]    [Pg.54]    [Pg.208]    [Pg.273]    [Pg.210]    [Pg.107]    [Pg.108]    [Pg.128]    [Pg.250]    [Pg.457]    [Pg.267]    [Pg.575]    [Pg.403]    [Pg.209]    [Pg.432]    [Pg.16]    [Pg.283]   
See also in sourсe #XX -- [ Pg.132 , Pg.133 , Pg.134 , Pg.135 , Pg.136 ]

See also in sourсe #XX -- [ Pg.132 , Pg.133 , Pg.134 , Pg.135 , Pg.136 ]




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