Percolation theory probability

The percolation theory [5, 20-23] is the most adequate for the description of an abstract model of the CPCM. As the majority of polymers are typical insulators, the probability of transfer of current carriers between two conductive points isolated from each other by an interlayer of the polymer decreases exponentially with the growth of gap lg (the tunnel effect) and is other than zero only for lg < 100 A. For this reason, the transfer of current through macroscopic (compared to the sample size) distances can be effected via the contacting-particles chains. Calculation of the probability of the formation of such chains is the subject of the percolation theory. It should be noted that the concept of contact is not just for the particles in direct contact with each other but, apparently, implies convergence of the particles to distances at which the probability of transfer of current carriers between them becomes other than zero. 

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]. 

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. 

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 ... 

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)... 

As pointed out above, the desorption process is dependent both on the void- and neck-size distributions,/(r) and C Fig. 13a) and the void and neck arrangements are random (the latter term means that the probability for an arbitrary void or neck to have a given value of the radius does not depend on the sizes of the neighboring voids and necks), the desorption process is mathematically equivalent to the bond problem in percolation theory. In particular, the probability that an arbitrary void is empty at a given value of the Kelvin radius during desorption is equal to the percolation probability 9 b(zo ) for the bond problem. Thus, the volume fraction of emptied voids under desorption [1 — Udes(rp)] can be represented as the product of the fraction of pore volume that may be emptied in principle at a given value of rp [1 - Uad(rp)] by the percolation probability b(zo ), i.e.,... 

Mason (18-20) and Palar and Yortsos (26,27) have employed another way of describing desorption from porous solids. Their approach is based on the assumption that the neck arrangement is random, i.e., the probability for an arbitrary neck to have a given value of the radius does not depend on the sizes of adjacent voids and necks. In this case, one can apply the percolation theory data obtained for the bond problem to all the voids. In particular, the probability for an arbitrary void to be empty during the desorption process is precisely 9 b(zo ), where the parameter z is given by Eq. (23). The latter probability is calculated for all the voids. We, however, know for a fact that voids with r < rp are filled. Thus the probability for a void with r > rp to be empty is just 9, (zoq)/F(rp), where F(rp) is the fraction of voids with r > rp [Eq. (33)]. Then, by analogy with Eq. (20), we derive... 

From the data presented in Fig. 27, we can conclude that the approach based on percolation theory permits one to obtain the neck-size distribution only in a relatively narrow range of radii. This is due to the fact that the percolation probability 9 b(z ) has a threshold (3>b(zq) = 0 at zq < 1.5) and increases from 0 to 1 in a relatively narrow range of 1.5 < z < 2.7. 

Figure 2 also shows that when 62—63% of H bonds are broken, the piece of ice is disintegrated into small separate clusters, and the network of H bonds is completely broken down. This result is slightly different from thaU (60—61%) obtained by assuming equal probability of rupture of all H bonds. Let us note 0 3i 60—61% is also the threshold provided by the percolation theory for the tetrahedral structure. 

Beyne and Froment [ref. 28] applied percolation theory to reaction and deactivation in the real three-dimensional ZSH-5 lattice. The structure of the catalyst enters in the equation for the reduced accessibility of active sites caused by blockage, P in (22) and this quantity is related to the percolation probability for this structure, P It is generally accepted that in zBH-5 the reactions take place at the channel intersections The probability that an intersection of channels (the origin in a network] is connected with an infinite number of open intersections is the percolation probability. It decreases as a growing number of intersections becomes blocked and drops to zero well before they are all blocked One way of relating P to the probability that an intersection is blocked, q, is Honte Carlo simulation. Based upon work by Gaunt and Sykes [ref. 29] on the percolation probability and threshold in diamond, Beyne and Froment derived a polynomial expression for P, However, the probability that a site is... 

A Bethe-tree is a particular case of more general networks considered in percolation theory. Sahimi and Tsotsis [1985] applied percolation theory and Monte Carlo simulation to deactivation in zeolites, approximated by a simple cubic lattice. Beyne and Froment [1990, 1993] applied percolation theory to reaction, diffusion and deactivation in the real ZSM-5 lattice. The finite rate of growth was described in terms of a polymerization mechanism. Pore blockage was reached in this small pore zeolite. It also affects the path followed by the diffusing molecules that becomes more tortuous, so that the effective diffusivity has to be expressed in terms of the blockage probability. 

The chapter consists of three main sections. In Section II the elements of fractal theory are given. In Section III the basis of percolation theory is described moreover, a model of fractal structures conceived by us is described. Fractal growth models, constructed using small square or rectangular generating cells as representative structural elements, are considered. Fractal dimensions of structures generated on various unit cells (2x1, 2x2, 2x3, 2x4, 3x1, 3x2, 3x3, 3x4, 4x1, 4x2, 4x3, 4x4) are calculated. Probability... 

Statistical network models were first developed by Flory (Flory and Rehner, 1943, Flory, 1953) and Stockmayer (1943, 1944), who developed a gelation theory (sometimes referred to as mean-field theory of network formation) that is used to determine the gel-point conversions in systems with relatively low crosslink densities, by the use of probability to determine network parameters. They developed their classical theory of network development by considering the build-up of thermoset networks following this random, percolation theory. 

The central result of percolation theory is the existence of the critical probability pc, the percolation threshold at which a cluster of infinite size appears. In a one-dimensional lattice, the percolation threshold is obviously equal to one. For higher dimensions, pc will be smaller than one. To illustrate this central result, we consider the Bethe lattice (also called the Cayley tree). 

In materials of infinite extent, the above definitions remain valid. As noted previously (Chapter 4), for pore space topologies with a given coordination number, there exists a critical filling probability (porosity). In materials with filling probabilities above this critical value, the size of the largest cluster is comparable to the size of the lattice. The presence of this lattice spanning cluster does not require that the material be finite in extent in fact, most analytical results in percolation theory assume that the lattice is infinite. For... 

Percolation theory (the model supposes that asymmetrically structured carbon-black particles are statistically distributed and results in percolation in accordance with probability laws) [3,27,31,32,33,34,35], Although this theory is the most widespread one, it lacks important experimental fundamentals and cannot describe the multitude of factors affecting percolation behaviour,... 

The simulation of catalyst deactivation by coke formation using a 3-dimensional site-bond-site network model is highly attractive, especially for zeolites, as the processes occurring in cavities (also referred to as voids or intersections) and in channels (also referred as necks, capillaries or arcs) can be readily distinguished. This model is flexible and the cormectivity of pores as well as the local homogeneity of the catalyst can be readily altered. Further, a percolation theory is available for site-bond-site models. In the particular case of Bethe lattices, approximated analytical solutions for the percolation probabilities have been derived[7]. 

An important quantity in percolation theory is the percolation probability P p), which gives the probability that given site belongs to the infinite cluster. One can show that there exists a critical value Pc (also called the percolation threshold) such that P(p) is... 

Carbon black or metal powder containing polymer compounds show a similar behavior when dispersed in a matrix above certain different critical concentrations. The percolation theory is thought to be the best tool for the description of this effect [14]. It is believed that metal powder, having a globular particle shape, is distributed in a statistically even manner and the powder particles will make contacts, governed by statistical laws (probability), whenever enough particles are present and close enough to finally form the first continuous conductive pathways. 

It appears at present that although crossflow and other macro-mixing models give a more correct description of the flow of a dispersed phase, the dispersion model predicts the conversion data satisfactorily, at least for simple reactions. In future, cell models probably will become more attractive, because they are closer to reality, making allowance for a better modeling of fluid-dynamics via the percolation theory [69]. 

There are several exact results available in this model to serve as checks on approximate calculations of and L(E). In the limit of small x or 6, all averaged quantities may be expanded in powers of a small parameter. For arbitrary x and 5, a number of moments of the density of states can be calculated exactly (Velicky, 1968). When 5 > 2, the density of states is split into two separated sub-bands, centered about and e , each of width B. Thus in the limit 5 -> a site containing a B atom is forbidden to an electron with energy near and percolation theory (Frisch, 1963) may be used to determine the probability that such an electron is trapped or free to move across the crystal. When x is greater than x, the criticd value for the onset of percolation, there will be extended states in the A sub-band. Since x, is less than 1/2 for all three-dimensional lattices, we observe that at least one of the strongly split sub-bands will always contain extended states, in contrast to the complete localization observed for r>Fg in the Lorentzian model of the preceding section. 

Although there are probably other universality classes, this transition was successfully modeled by bond percolation [6]. Generally, bond percolation on a lattice has each bond (line connecting two neighboring lattice sites) present randomly with probability p and absent with probability 1-p. Clusters are groups of sites connected by present bonds. For p > Pc zn infinite cluster is formed. Percolation theory (in a Bethe lattice approximation) was invented by Flory (1941) to describe gelation for three-functional polymers. 

To introduce the basic concepts of percolation theory [1-3], let us consider the very simple model of a two-dimensional square lattice as illustrated in Figure 5.1 [11,12]. In this system, we assume a probability, p, that a lattice site is occupied and we denote the occupied sites by closed circles. In the present lattice case, two sites are considered connected only if they are nearest neighbors and if they are both occupied. We define a (connected) cluster as an ensemble of occupied sites such that each of them is connected to at least another occupied site of the ensemble. In... 

Consider an incompatible A/B interface reinforced by an areal density E of compatibilizer chains. The vector percolation theory term, p, the occupational probability of the lattice, will be proportional to the number of chains (E) times their length (L) divided by their thickness (X) ... 

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