Percolation theory clusters

Lagues et al. [17] found that the percolation theory for hard spheres could be used to describe dramatic increases in electrical conductivity in reverse microemulsions as the volume fraction of water was increased. They also showed how certain scaling theoretical tools were applicable to the analysis of such percolation phenomena. Cazabat et al. [18] also examined percolation in reverse microemulsions with increasing disperse phase volume fraction. They reasoned the percolation came about as a result of formation of clusters of reverse microemulsion droplets. They envisioned increased transport as arising from a transformation of linear droplet clusters to tubular microstructures, to form wormlike reverse microemulsion tubules. 

Figure 2.9.3 shows typical maps [31] recorded with proton spin density diffusometry in a model object fabricated based on a computer generated percolation cluster (for descriptions of the so-called percolation theory see Refs. [6, 32, 33]).The pore space model is a two-dimensional site percolation cluster sites on a square lattice were occupied with a probability p (also called porosity ). Neighboring occupied sites are thought to be connected by a pore. With increasing p, clusters of neighboring occupied sites, that is pore networks, begin to form. At a critical probability pc, the so-called percolation threshold, an infinite cluster appears. On a finite system, the infinite cluster connects opposite sides of the lattice, so that transport across the pore network becomes possible. For two-dimensional site percolation clusters on a square lattice, pc was numerically found to be 0.592746 [6]. 

Diffusion of cations in a Nation membrane can formally be treated as in other polymers swollen with an electrolyte solution (Eq. (2.6.21). Particularly illustrative here is the percolation theory, since the conductive sites can easily be identified with the electrolyte clusters, dispersed in the non-conductive environment of hydrophobic fluorocarbon chains (cf. Eq. (2.6.20)). The experimental diffusion coefficients of cations in a Nation membrane are typically 2-4 orders of magnitude lower than in aqueous solution. 

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

The scaling of the relaxation modulus G(t) with time (Eq. 1-1) at the LST was first detected experimentally [5-7]. Subsequently, dynamic scaling based on percolation theory used the relation between diffusion coefficient and longest relaxation time of a single cluster to calculate a relaxation time spectrum for the sum of all clusters [39], This resulted in the same scaling relation for G(t) with an exponent n following Eq. 1-14. 

The percolation model, which can be applied to any disordered system, is used for an explanation of the charge transfer in semiconductors with various potential barriers [4, 14]. The percolation threshold is realized when the minimum molar concentration of the other phase is sufficient for the creation of an infinite impurity cluster. The classical percolation model deals with the percolation ways and is not concerned with the lifetime of the carriers. In real systems the lifetime defines the charge transfer distance and maximum value of the possible jumps. Dynamic percolation theory deals with such case. The nonlinear percolation model can be applied when the statistical disorder of the system leads to the dependence of the system s parameters on the electrical field strength. 

Objective criteria which clearly distinguish cluster structures from cellular and mesh structures have been developed by S. F. Shandarin (1983) [65] on the basis of percolation theory, whose usefulness for differentiating structures was conjectured by Ya.B. The percolation of cellular- or mesh-structure objects on a union of r-neighborhoods begins to occur at smaller r than in the case of a system of independent, randomly located points or clusters with the same (average) density. 

The initial porous texture of a catalyst pellet and the change in texture caused by metal deposition in it can be described using the percolation theory. In the percolation approach the pellet is constructed as a binary interdispersion of void space and (deposited) solid material. In this binary interdispersion, the void space can exist as (1) isolated clusters surrounded by solid material or (2) sample overspanning void space that allows mass transport from one side to the other. The total void space c can be split into the sum of the volume fraction of isolated clusters t1 and the volume fraction of accessible void space tA, If is below a critical value, called the percolation threshold all the void space is distributed as isolated clusters and transport is impossible through the pellet. 

The origins of percolation theory are usually attributed to Flory and Stock-mayer [5-8], who published the first studies of polymerization of multifunctional units (monomers). The polymerization process of the multifunctional monomers leads to a continuous formation of bonds between the monomers, and the final ensemble of the branched polymer is a network of chemical bonds. The polymerization reaction is usually considered in terms of a lattice, where each site (square) represents a monomer and the branched intermediate polymers represent clusters (neighboring occupied sites), Figure 1.4 A. When the entire network of the polymer, i.e., the cluster, spans two opposite sides of the lattice, it is called a percolating cluster, Figure 1.4 B. 

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]... 

It should be noted that, on one hand, an approach such as this is sufficiently closely related to the fluctuation theory of disperse systems developed in Shishkin s works [73], and on the other hand, it reduces to one of the variants of the flow problems in the percolation theory [78, 79] according to which the probability of the existence of an infinite liquid-like cluster depends on the value of the difference (P — Pcr), where Pcr is the flow threshold. At P < Pcr, only liquid-like clusters of finite dimensions exist which ensure the glassy state of liquid. It is assumed that at P > Pcr and (P — Pcr) 1 the flow probability is of the following scaling form ... 

The authors of the cluster theory draw the conclusion that the theory affords a sufficiently rigorous theoretical derivation of Doolittle s equation (72). Verification of the free volume theory advanced by Cohen and Grest was carried out by Hiwatari using computer simulation [97], showed that glass transition in liquids can really be described in terms of the percolation theory, the value of Pcr in this case being close to 0.2. Unlike Cohen and Grest s assumptions, however, this transition is not accompanied by a drastic change in the fluidity of the liquid near Per-... 

Percolation theory is a statistical theory that studies disordered or chaotic systems where the components are randomly distributed in a lattice. A cluster is defined as a group of neighboring occupied sites in the lattice, being considered an infinite or percolating cluster when it extends from one side to the rest of the sides of the lattice, that is, percolates the whole system [38],... 

Application of the percolation theory allows explanation of the changes in the release and hydration kinetics of swellable matrix-type controlled delivery systems. According to this theory, the critical points observed in dissolution and water uptake studies can be attributed to the excipient percolation threshold. Knowledge of these thresholds is important in order to optimize the design of swellable matrix tablets. Above the excipient percolation threshold an infinite cluster of this component is formed which is able to control the hydration and release rate. Below this threshold the excipient does not percolate the system and drug release is not controlled. 

Basically, the process of tablet compression starts with the rearrangement of particles within the die cavity and initial elimination of voids. As tablet formulation is a multicomponent system, its ability to form a good compact is dictated by the compressibility and compactibility characteristics of each component. Compressibility of a powder is defined as its ability to decrease in volume under pressure, and compactibility is the ability of the powdered material to be compressed into a tablet of specific tensile strength [1,2], One emerging approach to understand the mechanism of powder consolidation and compression is known as percolation theory. In a simple way, the process of compaction can be considered a combination of site and bond percolation phenomena [5]. Percolation theory is based on the formation of clusters and the existence of a site or bond percolation phenomenon. It is possible to apply percolation theory if a system can be sufficiently well described by a lattice in which the spaces are occupied at random or all sites are already occupied and bonds between neighboring sites are formed at random. 

In the rest of this chapter, we will discuss briefly the theoretical ideas and the models employed for the study of failure of disordered solids, and other dynamical systems. In particular, we give a very brief summary of the percolation theory and the models (both lattice and continuum). The various lattice statistical exponents and the (fractal) dimensions are introduced here. We then give brief introduction to the concept of stress concentration around a sharp edge of a void or impurity cluster in a stressed solid. The concept is then extended to derive the extreme statistics of failure of randomly disordered solids. Here, we also discuss the competition between the percolation and the extreme statistics in determining the breakdown statistics of disordered solids. Finally, we discuss the self-organised criticality and some models showing such critical behaviour. 

As mentioned before, the disordered solids will be mostly modelled in this book using randomly diluted site or bond lattice models. A knowledge of percolation cluster statistics will therefore be necessary and widely employed. Although this lattice percolation kind of disorder will not be the only kind of disorder used to model such solids, as can be seen later in this book, the widely established results for percolation statistics have been employed successsfully to understand and formulate analytically various breakdown properties of disordered solids. We therefore give here a very brief introduction to the percolation theory. For details, see the book by Stauffer and Aharony (1992). 

We now define some statistical quantities of interest in percolation theory. Let ns p) denote the number of clusters (per lattice site) of size s. In fact, a detailed knowledge of ns p) would give us a lot of information on the percolation statistics, as most of the quantities of interest can be extracted from various moments of the cluster size distribution n. Although, in general, we do not have any analytic knowledge about this distribution function ns p) near Pc, we can utilise the powerful observation of scaling behaviour of ns p) near Pc (see the next section). 

As discussed earlier, the percolation theory or similar theories of the cluster statistics in disordered solids can give us the probability density g l) of the defect clusters of linear size 1 ... 

The percolation probability (q) for the lattice models is defined as the probability that a given site (or bond) belongs to an infinite open cluster (47). It is fundamental to percolation theory that there exists a critical value qc of q such that 9(q) = 0 3t q < qc, and (q) > 0 if > qc. The value qc is called the critical probability or the percolation threshold. Mathematical methods of calculating this threshold are so far restricted to two dimensions, consistent with the experience in the field of phase transitions that three-dimensional problems in general cannot be solved exactly (12,13). Almost all quantitative information available on the percolation properties of specific lattices has come from Monte Carlo calculations on finite specimens (8,11,12). In particular. Table I summarizes exactly and approximately known percolation thresholds for the most important two- and three-dimensional lattices. For the bond problem, the data presented in Table I support the following well-known empirical invariant (8)... 

Application of percolation theory to describing the desorption process from porous solids is based on the identification of network sites with voids, and bonds with necks. A bond is considered to be unblocked if the neck radius r > Kp. Unblocked sites belonging to the percolation cluster correspond to voids containing nitrogen vapor. 

In the classical theory, however, the neglect of loops significantly affects the size distribution and other properties of the clusters as one approaches the gel point. Some of the critical exponents that describe these properties in the classical theory and in percolation theory near p Pc are compiled in Table 5-1 (Martin and Adolf 1991). 

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