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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.

Primary Literature Abeles B Granular metal films. Phys Rev Lett — Google Scholar. Andrade JS et al Flow between two sites on a percolation cluster. Archie GE The electrical resistivity log as an aid in determining some reservoir characteristics. Azulay D et al Electrical-thermal switching in carbon black-polymer composites as a local effect. Balberg I a Tunneling and nonuniversal conductivity in composite materials. Balberg I b Recent developments in continuum percolation. Balberg I New limits on the continuum-percolation transport exponents.

### Anisotropy of percolation conduction

Balberg I A comprehensive picture of the electrical transport phenomena in carbon black-polymer composites. Carbon — CrossRef Google Scholar. Balberg I, Binenbaum N A computer study of the percolation threshold in a two-dimensional anisotropic system of conducting sticks. Balberg I, Binenbaum N Cluster structure and conductivity of three-dimensional continuum systems. Balberg I, Binenbaum N a Invariant properties of the percolation thresholds in the soft core-hard core transition.

Balberg I, Blanc J Capacitive noise spectra of a disordered material. Balberg I, Bozowski S Percolation in composites of random stick-like conducting particles.

## Continuum Percolation

Balberg I et al b Excluded volume and its relation to the onset of percolation. Balberg I et al Critical behavior of the electrical resistance and its noise in inverted random-void systems. Balberg I et al Tunneling and percolation behavior in granular metals. Balberg I et al Percolation and tunneling in composite materials. Balberg I et al Fundamental transport processes in ensembles of silicon quantum dots. Phys Rev B — Google Scholar. Bergman DJ Exact relation between critical exponents for elastic stiffness and electrical conductivity of percolation systems.

Bergman DJ et al Critical behavior of the low-field hall conductivity at a percolation threshold. Phys Rev E — Google Scholar. Berkowitz B, Balberg I Percolation approach to the problem of hydraulic conductivity in porous media. Berkowitz B, Balberg I Percolation theory and its application to groundwater hydrology. Bug ALR et al b Do interactions raise or lower a percolation threshold?

Cametti C et al Theory and experiment of electrical conductivity and percolation locus in water. Chatterjee AP Continuum percolation in macromolecular fluids. Clerc JP et al The ac electrical conductivity of binary-disordered systems, percolation clusters, fractals and related models. Dalmas F et al Carbon nanotube-filled polymer composites. Numerical simulations of electrical conductivity in three-dimensional entangled fibrous networks.

Day AR et al Spectral representation of the electrical properties of layered materials. Drory A a Theory of continuum percolation. General formalism. Drory A b Theory of continuum percolation. Mean field theory. Drory A Exact solution of a one-dimensional continuum percolation model. Drory A et al Analytic derivation of percolation thresholds in anisotropic systems of permeable objects.

Drory A, Balberg I, Berkowitz B Application of the central-particle potential approximation for percolation in interacting systems. Drory A et al Theory of continuum percolation. Low-density expansion. Du F et al Nanotube networks in polymer nanocomposites: rheology and electrical conductivity. Flory PM Molecular size distribution in three dimensional polymers. Introduction and relation to other models. Foygel M et al Theoretical computational studies of carbon nanotube composites and suspensions: electrical and thermal conductivity.

Phys Rev B Google Scholar.

Gawlinski ET, Redner S Monte Carlo renormalization group for continuum percolation with excluded-volume interactions. Grassberger P Critical percolation in high dimensions. Grest GS et al Dynamic percolation in microemulsions. Grimaldi C, Balberg I Tunneling and non-universality in continuum percolation systems. Grimaldi C et al Segregated tunneling-percolation model for transport nonuniversality.

Isichenko MB Percolation, statistical topography, and transport in random media. Johner N et al Transport exponent in a three-dimensional continuum tunneling-percolation model. Kirkpatrick S Percolation and conduction.

- Introduction.
- Percolation Thresholds in 2-Dimensional Prefractal Models of Porous Media*.
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Knite M et al Electric and elastic properties of conductive polymer nanocomposites on macro- and nanoscales. Kogut PM, Straley J Distribution-induced non-universality of the percolation conductivity exponents. Laria D, Vericat F Percolation behavior of long permeable objects: a reference interaction-site-model study. Mandal P et al Temperature and magnetic field dependence of the resistivity of carbon-black composites.

McCarthy JF Continuum percolation of disks and the random lattice. Miller A, Abrahams E Impurity conduction in low concentrations. Neda Z, Florian R, Brechet Y Reconsideration of continuum percolation of isotropically oriented sticks in three dimensions.

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Scher H, Zallen R Critical density in percolation processes. Schrijver CJ et al Patterns in the photospheric magnetic-field and percolation theory. Shimoni N et al Tomographic-like reconstruction of the percolation cluster as a phase transition. Sov Phys Semicond — Google Scholar. Smart JS Effective field theories of magnetism. Sanders, Philadelphia Google Scholar. Stanley HE Introduction to phase transitions and critical phenomena.

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Stauffer D, Aharony A Introduction to percolation theory. Stinchcombe RB Conductivity and spin-wave stiffness in disordered systems-an exactly soluble model. Stockmayer WH Theory of molecular size distribution and gel formation in branched-chin polymers. Toker D et al Tunneling and percolation in metal-insulator composite materials. Vionnet-Menot S et al Tunneling-percolation origin of nonuniversality: theory and experiments. Vyssotsky VA et al Critical percolation probabilities bond problem.

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