Percolation correlation length

As Bergman has pointed out (Bergman and Stroud 1992), and discussed earlier (in Section 1.2.2(c)), this behaviour is expected when the Lifshitz scale is greater than the percolation correlation length. It is not clear however if the Lifshitz scale is given by /max InL or simply by InL near pc-... 

In this limit, we consider clusters of conducting defects and their mean size is the percolation correlation length The voltage Vi between two such neighbouring clusters is given by Vi = V /L. The field between two clusters is increased as the field is zero inside a conducting cluster. The maximum value will be between the clusters with minimal distance between them i.e. one unit cell of the lattice. When the field between two such clusters... 

As discussed in Section 1.2.1, for such systems near the percolation threshold Pc the nearest-neighbour occupied bonds (or sites) form a statistically defined super-lattice , made of tortuous link-bonds (of chemical length Lc) crossing at nodes separated by an average distance the percolation correlation length (see Fig. 1.3 of Chapter 1). The external stress... 

We have studied the the fracture properties of such elastic networks, under large stresses, with initial random voids or cracks of different shapes and sizes given by the percolation statistics. In particular, we have studied the cumulative failure distribution F a) of such a solid and found that it is given by the Gumbel or the Weibull form (3.18), similar to the electrical breakdown cases discussed in the previous chapter. Extensive numerical and experimental studies, as discussed in Section 3.4.2, support the theoretical expectations. Again, similar to the case of electrical breakdown, the nature of the competition between the percolation and extreme statistics (competition between the Lifshitz length scale and the percolation correlation length) is not very clear yet near the percolation threshold of disorder. 

On the basis of random walk arguments, Koplik et al. (1988) show the relations K Pn, K Pn In Pn, and K Pi for (i) a bundle of uniform stream tubes that meet at perfect mixing chambers separated by a characteristic length /, (ii) nonuniform bond transit times, and (iii) the presence of dead-end pores, respectively. The relationship K oc pi is also obtained for percolation networks near the percolation threshold when the characteristic length is the percolation correlation length and the molecular diffusion coefficient is adjusted for the presence of the percolation network. 

Charlaix et al. (1988) also conducted a study of NaCl and dye transport in etched transparent lattices. A fully connected square lattice with a lognormal distribution of channel widths and a partially connected hexagonal lattice (a percolation network) were considered. They concluded that the disorder and heterogeneity of the medium determined the characteristic dispersion length. From experimental data on the percolation network, they showed that this dispersion length was close to the percolation correlation length, p. 

As it has been already shown the adsorption-induced change in the value of percolation electric conductivity of adsorbent and tangent of inclination angle in its pre-relaxation VAC are determined by adsorption-related change in the value of the percolation level and correlation length of the system, i.e. by functions c(0 and f ( c>0. One can readily conclude for 4c t) that when Nt < 4a - 4 co < i-e. at small Nt and... 

The similar law is for the concentration dependence of the correlation length of the percolation cluster (that is the size at which a self-crossing of current paths occurs or the effective radius of the percolation net) ... 

The reduced value of the scaling exponent, observed in Fig. 29 and Fig. 30a for filler concentrations above the percolation threshold, can be related to anomalous diffusion of charge carriers on fractal carbon black clusters. It appears above a characteristic frequency (O when the charge carriers move on parts of the fractal clusters during one period of time. Accordingly, the characteristic frequency for the cross-over of the conductivity from the plateau to the power law regime scales with the correlation length E, of the filler network. 

An explanation of the observed relaxation transition of the permittivity in carbon black filled composites above the percolation threshold is again provided by percolation theory. Two different polarization mechanisms can be considered (i) polarization of the filler clusters that are assumed to be located in a non polar medium, and (ii) polarization of the polymer matrix between conducting filler clusters. Both concepts predict a critical behavior of the characteristic frequency R similar to Eq. (18). In case (i) it holds that R= , since both transitions are related to the diffusion behavior of the charge carriers on fractal clusters and are controlled by the correlation length of the clusters. Hence, R corresponds to the anomalous diffusion transition, i.e., the cross-over frequency of the conductivity as observed in Fig. 30a. In case (ii), also referred to as random resistor-capacitor model, the polarization transition is affected by the polarization behavior of the polymer matrix and it holds that [128, 136,137]... 

For anisotropic particles, the percolation limit is a function of the aspect ratio. For ellipsoids of revolution, the percolation limit for a simple cubic lattice was studied by Boissonade et al. [85]. They found as the aspect ratio increases from 1 (a sphere) to 15 (a fiber), the percolation limit decreased from a volume fraction of 0.31 to 0.06 and the correlation length (i.e., aggregate size) did not change (i.e., it was the same as that of the sphere). 

The properties of the mean shortest path has been studied theoretically near the percolation threshold (Chayes et al 1986, Stinchcombe et al. 1986) and numerically in the whole range of p (Duxbury et aL, unpublished). The important property is the variation of = (no)/T, the mean number of resistances in the shortest path, with p. For p near 1, g decreases from 1 linearly with a slope dg/dp about 3 (for the square lattice). In the vicinity oiPc, g goes to zero as (p —Pc) where i/ is the correlation length exponent. 

In all these four cases, they found that g goes to zero at Pc with an exponent equal to that of the correlation length. For p approaching 0, g goes to unity, but for the three-dimensional regular percolation and for the directed percolation (in both d — 2 and 3) there is a steep decrease of g from unity when p begins to increase. This behaviour is reminiscent of the behaviour of E y but this is not very well understood for the gap. 

Bowman and Stroud (1989) solved numerically the Laplace equation at all the sites of the lattice made of insulators (with probability I — p) and conductors (probability p) and got the values of the potential across each bond. This is solved by taking into account the condition that no potential difference can be maintained within a conducting cluster. They found that Eh (defined as the field giving the breakdown of the first bond) decreases towards 0 as p approaches pc, with an exponent equal to 1.1 0.2 in d = 2 and equal to 0.7 di 0.2 in d = 3 for both site and bond percolation. These values are consistent with the above predictions, namely that the exponent of Eh must be equal to the correlation length exponent z/. We recall that... 

In a disordered solid, as modelled by the percolation model (discussed in Section 1.2) with intact bond concentration p on a lattice, there occur various vacancy clusters (voids), or pre-existing cracks, of different sizes and shapes. The typical crack size in such a percolating solid being of the size of the correlation length (see Section 1.2.1) the application of the Griffith fracture criterion gives (Chakrabarti 1988, Ray and Chakrabarti 1985a,b) from equation (3.4)... 

Percolation media can be characterized not only by the percolation probability but also by other quantities (Table II)—for example, by the correlation length, which is defined as the average distance between two sites belonging to the same cluster. Near the percolation threshold, all these quantities are usually assumed to be described by the power-law equations (Table II). All current available evidence strongly suggests that the critical exponents in these equations depend only on the dimensionality of the lattice rather than on the lattice structure (72). Also, bond and site percolations have the same exponents. 

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]. 

The correlation length defines the connectivity of clusters. It defines the scale range within which percolation clusters behave self-similarly and, consequently, are characterized by a fractal dimension [38,39]. The correlation length E, for a percolation lattice can be defined as... 

Thus, according to Eq. (137) the correlation length l (p) is the average dimension of those clusters that contribute most to the second moment of the distribution of cluster dimensions near the percolation threshold, pc. 

Two linear dimensions, the minimum length Iq (the lattice constant) and the correlation length play a key role in the behavior of a percolation system. For real systems, an intermediate asymptotic region exists such that... 

Figure 15. Composition dependence of the correlation length near the percolation threshold (schematic).

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