Percolation theory finite

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 above considerations referred to the practically important examples of more or less ordered heterogeneities. If we face random distribution, usually effective medium and percolation theory have to be referred to in order to evaluate the inhomogeneous situations properly. However, attention has to be paid to the fact that they often require nonrealistic approximations. For more details see Ref.300 In such cases numerical calculations, e.g., via finite element methods are more reliable. 

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 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)... 

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 single most important concept in percolation theory is the percolation threshold, which may be understood as follows. First, assume that p (the probabihty of occupation) is very small. In this case, very few sites are occupied, and the clusters are all of finite size (each cluster is surrounded by many empty sites). Then assume that p is close to unity, so that almost all sites are occupied. Reversing the argument, it is realized that a few holes of finite size exist in an otherwise unbounded cluster that spans the... 

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. 

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]. 

Reiser expanded the diffusion model for dissolution of novolac 13-24) using percolation theory (25, 2d) as a theoretical framework. Percolation theory describes the macroscopic event, the dissolution of resist into the developer, without necessarily understanding the microscopic interactions that dictate the resist behavior. Reiser views the resist as an amphiphilic material a hydrophobic solid in which is embedded a finite number of hydrophilic active sites (the phenolic hydrogens). When applied to a thin film of resist, developer diffuses into the film by moving from active site to active site. When the hydroxide ion approaches an active site, it deprotonates the phenol generating an ionic form of the polymer. In Reiser s model, the rate of dissolution of the resin. .. is predicated on the deprotonation process [and] is controlled by the diffusion of developer into the polymer matrix (27). 

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... 

Fracture by disentanglement occurs in a finite molecular weight range, McPercolation theory predicts that the critical draw ratio. 

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. 

The concentration c is equivalent to the critical point where the crossover phenomenon occurs from randomness to order. It is also equivalent to Pc (the critical probability) in the percolation theory, where the crossover phenomenon occurs from the finite cluster (such as a macromolecule containing a finite number of monomers) to the infinite cluster (such as the network of an entangled macromolecule, which extends from one end to the other). The three regions are characterized by three important quantities the number of statistical elements per chain N, the number of statistical elements per unit volume p (density), and the correlation or screen length... 

Scalar percolation theory deals with the connectivity of a component randomly dispersed in another [7-8]. Examples of percolation are gelation during a polymerization of monomers with multifunctional linkages and the onset of conductivity in blends of conducting and non-conducting materials [9-10]. The percolation threshold p for a finite-sized object is defined as the minimum concentration (of the percolating medium) at which connectivity is established between the top and bottom surface, is different for lattices of different geometry [7]. For Id site percolation, p = 59.20%, while = 31.17% for a cubic lattice. 

Different studies have investigated the influence of the particle size of the components of a formulation on their percolation threshold in inert matrices. Initially, the existence of a linear relationship between the drug particle size and the drug percolation threshold was demonstrated, but later studies showed that what really determines the drug percolation threshold is the relative drug particle size, i.e., the ratio between the mean drug and excipient particle size [86,87]. This fact can be explained according to the percolation theory since an increase in the particle size of all the components of a finite system is equivalent to a decrease in the size of the system. Therefore, in a binary... 

Gelation is a critical phenomenon of connectivity and as such we will use the percolation theory to describe it. As percolation theory was described in detail, here we recall the behaviors of measurable quantities which experimental results are given hereafter. Below the gelation threshold the system is composed of finite size polymers branched in the 3-dimensions of space. We shall call those polymers "polymers clusters" in order to distinguish them from other branched polymers as stars or combed polymers. Below the gelation threshold, the system is viscous at zero frequency. At the gelation threshold, there appears a giant polymer clus-... 

Gel phase fraction was measured on samples (see Ref. 14 for more details) formed beyond the gel point. One can see in Fig. 8 that G is a linear function of the stoichiometric ratio. Concerning this figure two comments have to be made. The p value corresponding to the stoichiometric ratio at which the gel phase is nul (p = 0.5639) is very different from the ifferent batches of monomers were used for the preparation of the two series of samples. A small but finite gel phase was measured below p (G = 1.6 10 at p = 0.5632) this could be due to either experimental imprecision on G or the fact that this sample prepared near the gel point contains very large polymer clusters of finite size which could not pass through the membrane used for the sol extraction. This result, G P is a linear function of p with a P-exponent value deduced from x and y exponent values measured below the gel point, indicates that below and beyond the gelation threshold, the percolation theory is well adapted to describe the properties linked to connectivity of polymer clusters formed by polycondensation. 

As emphasized in the previous sections, in the context of percolation theory, at sufficientiy diiute concentrations such that there are no connected paths, the conductivity would be zero. As the concentration of conducting polymer is increased above the percolation threshold, the conductivity would become finite and increase as the connectivity (i.e. the number of conducting paths) increases. In contrast, the data presented in Figure V.2 show no indication of a weli-defined percolation threshold. Attempts to estimate a... 

Non-mean field corrections can be treated by renormalization group theory,which is not discussed here. In order to leave the tree approximation we turn to percolation description and model the microgels as finite clusters. The percolation theory itself is not essential for this description, but only the fractal character of the clusters on their scale of extension. Polymeric fractals have been discussed already in Section 8.2.6 and we use the properties here as well. Indeed most of the results may be applied here. In Ref. 123 the dynamics of the sol phase is discussed extensively, but we do not want to go in these details here. 

Introduction of the complex conductivity into Eq. (5.262) (a instead of capacitive effects. This will be treated in more detail in Chapter 7. Here it will suffice to mention that the relevant, high-conductivity paths are parallel to the buLk. More complex distributions must be dealt with using percolation theory, effective medium theory or finite element calculations [265,266]. The result for the... 

S. Greenspoon Finite-size effects in one-dimensional percolation a verification of scaling theory. Canadian J. Phys. 57, 550-552 (1979)... 

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