Percolation theory infinite

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]. 

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. 

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 ... 

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. 

Most of the pore structures (e.g., spongy structures) consist of extensive three-dimensional networks in which there is a profusion of interconnections between voids within the structure. The latter interconnections affect considerably the kinetics of various processes in porous solids. This effect can adequately be described by employing the ideas developed in percolation theory 7-13). In the framework of this theory, the medium is defined as an infinite set of sites interconnected by bonds. Percolation theory can be applied to porous solids via identification of network sites with voids, and bonds with necks. Thus, the theory is applicable primarily to spongy porous structures but in some cases also to corpuscular structures. 

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)... 

In that region, the EMA predicted Pr curve decreases linearly with I s, while the network solution exhibits a non-linear behavior and reaches a higher percolation threshold, Vsc - This is because Ksr predicted by the network model, corresponds to the theoretical fbc predicted by percolation theory ( c 1.5/r, [9]), while Vsc found by EMA corresponds to fhc"=2/z [10]. A similar picture is presented in figure 2b, where Pr is plotted as a function of the fraction of the open pores, //,. It can be seen that for all r, near the percolation threshold, EMA shows a linear decrease of Pr with /. On the other hand, network results indicate that, in the same region, Pr decreases with / according to a power law. For an infinitely large network percolation theory states ... 

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... 

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 agreement with percolation theory, when the fraction of active grain boundaries reaches some critical value, acr, an infinite cluster is formed, and the entire polycrystal contained between the surface that is in contact with... 

When p is nonzero, there are clusters of liquidlike cells, each one of which has at least z liquidlike neighbors. It is well known that in such situations there is a critical concentration above which there exists an infinite cluster. Thus for p>p, there is an infinite, connected liquidlike cluster, and we can consider the material within it to be liquid. For pliquidlike clusters exist, which might imply a glass phase because the fluidity would be reduced. However, percolation theory tells us that just above p the infinite cluster is very stringy or ramified so that bulk liquid properties are not fully developed. 

Percolation Lattices. Percolation theory requires that space be represented as a lattice, often infinite in extent. In Figure 4.19, for example, two-dimensional space is discretized on to a square lattice. The points of... 

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... 

Crete surface to the bulk of the concrete. Permeability is high (Figure 1.6) and transport processes like, e. g., capillary suction of (chloride-containing) water can take place rapidly. With decreasing porosity the capillary pore system loses its connectivity, thus transport processes are controlled by the small gel pores. As a result, water and chlorides will penetrate only a short distance into concrete. This influence of structure (geometry) on transport properties can be described with the percolation theory [8] below a critical porosity, p, the percolation threshold, the capillary pore system is not interconnected (only finite clusters are present) above p the capillary pore system is continuous (infinite clusters). The percolation theory has been used to design numerical experiments and apphed to transport processes in cement paste and mortars [9]. 

More recently, percolation theory and computer simulation of the dissolution process was applied. This latter approach resulted in 2D and 3D percolation thresholds (that is, composition thresholds at which infinite connected paths of the fast dissolving component were formed) as well as in images of the atomic scale disorder induced by dealloying. 3D site percolation thresholds 20 at.% in a fee lattice), leading to an infinite connected cluster of nearest neighbors of less noble atoms, were considered to correlate with the absolute parting limits of alloys with high such as... 

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... 

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. 

Figure 5.1, we see a cluster that connects opposite edges of the system. The latter is called the spanning cluster. In percolation theory, we are concerned mainly with infinite systems and thus we call the spanning duster the percolation or the infinite cluster. The illustration given in Figure 5.1 reflects here a finite (small) portion of the infinite system but it has to be remembered that quantitative characterizations of the various properties are considered meaningful only at the infinite system limit... 

The kind of mechanisms that lead to gelation characterised by infinite clusters are not clear. The infinite cluster contains of course a finite fraction G(t) of the total mass (M(t) + G(t) = 1). Pre-gel and post-gel states separated by a gelation transition can be analysed in terms of a kinetic equation. Sol-gel transitions are similar to phase transition phenomena. It is not surprising that scale invariance principles elaborated in the theory of phase transition can be adopted for polymer systems. Modern percolation theory (see, for example Stauffer (1979)) offer a conceptual framework to treat cluster formation. 

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