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Continuum Percolation

  • Reference work entry
Encyclopedia of Complexity and Systems Science

Definition of the Subject

In this review we will describe the two main features that are very different in latticepercolation and in continuum percolation and which bring about new concepts that are notencountered in the “more traditional” and well developed theory of percolation inlattices [121]. The origin of thedifference is that in continuum percolation we are dealing with “real” objectsthat have sizes and shapes, and that are randomly distributed in space, while in lattices weare dealing with abstract mathematical objects such as dots and line segments in a prioridefined locations. The result is that in the continuum we have to use different quantities todescribe the percolation threshold and we obtain a much richer variability of thecorresponding behaviors of the dynamical properties, such as the electrical transport. Toequip the reader with the basic concepts needed for the discussion of continuum percolation wealso include a short introduction to the relevant principles of...

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Abbreviations

Site percolation:

Assuming a lattice in which the sites can be occupied, with a probability \( { p^\mathrm{s} } \), two sites are assumed locally connected if they are nearest neighbors and occupied. A group of sites in which each of its' sites is occupied and locally connected to at least one other site in the group is called a connected cluster. If there is a cluster that connects the edges of an “infinite” lattice, the cluster is called the percolation cluster. The lowest \( { p^\mathrm{s} } \) for which such a cluster is found is called the percolation threshold, \( { p^\mathrm{s}_\mathrm{c} } \).

Bond percolation:

Assuming a lattice, the segment between nearest neighbor sites is called a bond. A bond can be occupied with a probability \( { p^\mathrm{b} } \). Two bonds are locally connected if they are both occupied and have a common site. A group of bonds in which each is connected to at least one other bond is called a connected cluster of bonds. If the span of this cluster is infinite we have a “percolation cluster ” and the lowest \( { p^\mathrm{b} } \) for which such a cluster is formed is called the bonds percolation threshold, \( { p^\mathrm{b}_\mathrm{c} } \).

Critical behavior:

Percolation can be modeled as a phase transition and thus various properties have a power-low dependence on \( { |p-p_\mathrm{c}| } \) for small values of \( { |p-p_\mathrm{c}|/p_\mathrm{c} } \). The exponent that describes this dependence is called the critical exponent.

Universal and non universal behavior:

If the critical exponent depends only on the dimensionality of the system, we say that the critical behavior is universal. If the exponent depends on other parameters of the system, we say that the behavior is non universal.

Continuum percolation:

In a system of objects one can define a local connectivity criterion such as the overlap of pores in porous media. The objects are assumed to be randomly distributed in space, they may have various and variable shapes and sizes and they may interact with each other. A group of objects such that each object is locally connected to at least one other object in the group is a connected cluster. The cluster that has an infinite span is the “percolating” cluster in the continuum and the lowest concentration of objects that yield such a cluster defines the percolation threshold.

Critical fractional volume:

The content of the objects at the onset of global connectivity, i. e. the percolation threshold, is usually characterized in the continuum by the measurable fractional volume content of the objects of interest in the system. This volume at the percolation threshold is the critical fractional volume.

Excluded volume:

In the case where the local connectivity is determined by a partial overlap of the volumes of two equal objects, the volume in space in which the two centers of the two objects can be, and must be, in order to have such an overlap is defined as the excluded volume of the object. If the objects are not equal one has to define a corresponding average.

Average bonds per object:

The average number of objects that are locally connected to a given object in the system. This quantity, at the percolation threshold, \( { B_\mathrm{c} } \), is the quantity that characterizes topologically the onset of global connectivity i. e. the onset of percolation.

“Pointedness ”:

The parameter that can be defined qualitatively as the degree of deviation from sphericity of a given object. The manifestation of this property for different objects, that have the same excluded volume, is that with its increase the value of \( { B_\mathrm{c} } \) decreases.

Critical behavior of dynamical properties:

The critical behavior of a property that has to do with flow in the system, such as electrical conductivity and fluid permeability. In the continuum, it is related to the distribution of the local values of the dynamical parameter associated with a given bond. If the average of this parameter effects the global critical behavior of the system, its contribution is added to the “universal” critical behavior that is found in lattices.

The random void and the inverted random void systems:

A system of pores or topologically similar systems which are reminiscent of the “Swiss Cheese”. In the random void (RV) case one is concerned with the network excluding the pores. The mirror image of this system, i. e. when one is concerned with the network that consists of the pores (or particles that can coalesce) is called the inverted random void (IRV) system. In these systems the “neck” formed by the separation (RV) or the overlap (IRV) of two adjacent pores determines the local dynamical property.

Tunneling percolation:

The conduction process in a system where the “local connectivity” is not determined by a geometrical contact but is determined electronically. This is in particular by inter‐object tunneling. The corresponding connected system exhibits a percolation‐like critical behavior of the dynamical properties.

“Physically controlled percolation”:

When an externally applied quantity, such as an electric field or mechanical pressure, affects the parameters that characterize the percolation‐like behavior, one may call the corresponding phenomena “physically controlled percolation”.

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Acknowledgments

The present review could not have been written without the stimulation and the intensivecollaboration that I had with the many colleagues and students, whose papers that wereco‐authored with me, are cited in this review. In particular, I would like to thankDr. C. Grimaldi and Dr. N. Wagner for the on going collaborationand for the critical reading of the present manuscript. This work was supported by the IsraelScience Foundation (ISF). The author holds the Enrique Berman chair in Solar Energy Researchat the Hebrew University.

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  1. The Racah Institute of Physics, The Hebrew University, Jerusalem, Israel

    Isaac Balberg

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  1. Isaac Balberg
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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© 2009 Springer-Verlag

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Balberg, I. (2009). Continuum Percolation . In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_95

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