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Percolation Thresholds in 2-Dimensional Prefractal Models of Porous Media* - Semantic Scholar
Correspondingly, one can define the percolation threshold by the total occupied volume fraction of the conducting spheres, i. Topologically, then, these two systems are expected to have the same onset of global connectivity. In a large system that is statistically sufficient i. Applying now their above conjecture, i. In Balberg et al. This is in contrast with the systems mentioned so far. Following the B c values in lattices of various dimensions Zallen , their values are intuitively expected to be between 1 and 5.
As such, the quantity V ex is well defined for any permeable or partially permeable object, see below. Naturally, the simple relation we had above for spheres Eq. It turns out that the application of the concepts exhibited by Eqs. We mentioned already that B c is 2. We also know then, as found directly Balberg and Binenbaum b , that the value of B c decreases, in the transition with the increase b , toward a value of 1.
Let us turn now to see how the abovementioned quantities play a role in the determination of the most important parameter in the study of the critical behavior, i. If we define a local direct connectedness criterion e.
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On the other hand, a more successful, albeit biased, approximation has yielded, as mentioned below, very accurate results. For the description of the latter, we start then from Eq. Hence, the aim of the procedure is to determine the, by now, well-understood and well-characterized parameter B c. Having these relations and implementing the fact see Eq. The Basic Physics of the Nonuniversal Behavior of the Conductivity The physics of the critical behavior of the global resistance in continuum systems can be described as follows.
To consider however the behavior of R L in more detail i. Let us list the values of the local conductances in the system in a descending order. Following our basic assumption that there is no correlation between the location of the bond and its attached g value, we can use the fact that any randomly selected subset of p c conductors is, by definition, a percolating cluster and conclude that the above chosen p c subset of the top value conductors constitute a percolation cluster. Of course, the largest conductance value in the system, g 2 , is also the largest possible g value in the so chosen subset.
This is done without loss of generality since all other g values in the system can be normalized accordingly. Then, the lowest g value of the conductors in the above subset p c of p has the corresponding normalized value g c. Above, we have selected a subset of the conductors in the system i. Over this range, i. Hence, as the decrease of p is associated with the decrease of g c , we will get an apparent nonuniversal behavior.
We note that while, strictly speaking, the critical behavior i. Examining the various experimental data in the literature Vionnet-Menot et al. Since both approaches have been given in detail in the literature, we will only outline here the principal steps in their utilization for the determination of the corresponding t values.
There are three principal local configurations that were studied in detail. These three configurations are illustrated in Fig. This neck determines the resistance of the neck that is associated with the two adjacent spheres Halperin et al. Let us derive now these results in a physically more transparent manner Feng et al. The random distribution of the distances of the centers of the nearest neighbor spheres from a given sphere center in the corresponding 1D system is the well-known 1D Hertz distribution Balberg a , b ; Torquato et al.
The first implication of that is that the use of a constant a say, the one that applies to p c in the above equations e. Following our above discussion and the result given in Eq. The results shown in Fig. In addition to this main result, i. As to the meaning of this shift in practice, let us note that for a given type of a system i.
Correspondingly, as apparent from the smaller g values involved in the conduction process, then the decay of H 3 r in Eq. A similar argument can be derived from Eq. In passing, we also note that the parameters that determine the critical behavior can be controlled externally. Acknowledgments The present review could not have been written without the stimulation and the intensive collaboration that I had with the many colleagues and students, whose papers that were coauthored with me are cited in this review.
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Anisotropy of percolation conduction
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