Random percolation

The cluster properties of the reactants in the MM model at criticality have been studied by Ziff and Fichthorn [89]. Evidence is given that the cluster size distribution is a hyperbolic function which decays with exponent r = 2.05 0.02 and that the fractal dimension (Z)p) of the clusters is Dp = 1.90 0.03. This figure is similar to that of random percolation clusters in two dimensions [37], However, clusters of the reactants appear to be more solid and with fewer holes (at least on the small-scale length of the simulations, L = 1024 sites). 

The situation becomes quite different in heterogeneous systems, such as a fluid filling a porous medium. Restrictions by pore walls and the pore space microstructure become relevant if the root mean squared displacement approaches the pore dimension. The fact that spatial restrictions affect the echo attenuation curves permits one to derive structural information about the pore space [18]. This was demonstrated in the form of diffraction-like patterns in samples with micrometer pores [19]. Moreover, subdiffusive mean squared displacement laws [20], (r2) oc tY with y < 1, can be expected in random percolation clusters in the so-called scaling window,... 

Time intervals permitting displacement values in the scaling window a< )tortuous flow as a result of random positions of the obstacles in the percolation model [4]. Hydrodynamic dispersion then becomes effective. For random percolation clusters, an anomalous, i.e., time dependent dispersion coefficient is expected according to... 

A. Klemm, H. P. Muller, R. Kimmich 1997, (NMR microscopy of pore-space backbones in rock, sponge, and sand in comparison with random percolation model objects), Phys. Rev. E 55, 4413. 

In random bond percolation, which is most widely used to describe gelation, monomers, occupy sites of a periodic lattice. The network formation is simulated by the formation of bonds (with a certain probability, p) between nearest neighbors of lattice sites, Fig. 7b. Since these bonds are randomly placed between the lattice nodes, intramolecular reactions are allowed. Other types of percolation are, for example, random site percolation (sites on a regular lattice are randomly occupied with a probability p) or random random percolation (also known as continuum percolation the sites do not form a periodic lattice but are distributed randomly throughout the percolation space). While the... 

Kantor, Y., and Webman, I. (1984). Elastic properties of random percolating systems. Phys. Rev. Lett 52, 1891-1894. 

The percolation model suggests that it may not be necessary to have a rigid geometry and definite pathway for conduction, as implied by the proton-wire model of membrane transport (Nagle and Mille, 1981). For proton pumps the fluctuating random percolation networks would serve for diffusion of the ion across the water-poor protein surface, to where the active site would apply a vectorial kick. In this view the special nonrandom structure of the active site would be limited in size to a dimension commensurate with that found for active sites of proteins such as enzymes. Control is possible conduction could be switched on or off by the addition or subtraction of a few elements, shifting the fractional occupancy up or down across the percolation threshold. Statistical assemblies of conducting elements need only partially fill a surface or volume to obtain conduction. For a surface the percolation threshold is at half-saturation of the sites. For a three-dimensional pore only one-sixth of the sites need be filled. 

In cases in which a sharp question can be formulated and so suggest a test of a mechanism of solvent participation, site-directed mutagenesis provides an experimental tool, perhaps appropriately guided by computer simulation of the protein—solvent system. The participation of fixed or random (percolative) network structures in proton movement is... 

In an earlier section, we have discussed the statistics of clusters of defects, produced for example in the random percolation processes. We have also discussed there how some of the linear responses, like the elastic moduli or... 

The second thing we need to characterise is the type of disorder we shall consider. As mentioned before, a major part of this book will be concerned with the lattice models of disorder, with the statistics governed by that of random percolation. The simplest physical picture of this kind of disorder... 

Other kinds of disorder distribution of the failure threshold Until now, we supposed that the disorder was of the random percolation kind. We describe here another kind of disorder in the case of the resistor lattice. Only the two-dimensional case has been studied. The disorder comes from the fact that the resistors do not have the same failure current or failure voltage. Although in the fuse problem the current is the relevant quantity, Kahng et al (1988) developed this model considering only a distribution of threshold voltage. 

We will discuss these fracture properties of disordered solids, modelled by the random percolation models, and concentrate on their statistics, given by the cumulative failure strength distribution F a) under stress a, and the most probable fracture strength erf of such samples. We will discuss separately the cases for weak disorder p 1) and strong disorder p Pc)-The scaling properties of <7f near p pc and the nature of the competition between the percolation and extreme statistics here, will be discussed in detail. 

This distribution appears whenever g a) is given by a power law in (j, coming from the power law variation of the density of linear cracks g l) with their length 1. In the random percolation model considered here, this does not normally occur (except at the percolation threshold p = Pc)- However, for various correlated disorder models, applicable to realistic disorders in rocks, composite materials, etc., one can have such power law distribution for clusters, which may give rise to a Weibull distribution for their fracture strength. We will discuss such cases later, and concentrate on the random percolation model in this section. 

Kantor, Y., and I. Webman, Elastic Properties of Random Percolating Systems, Ibid. 52 1891-1894 (1984). 

Numerical studies [167] of planar (d = 2) elastic random percolation networks have shown that if their linear dimension L < 0.2c (c is the correlation length), then Poisson s ratio for the system is negative, and if L > 0.2J , Poisson s ratio is positive In this case, if L/c —> oo, the limiting value of Poisson s ratio is vp = 0.08 0.04 and is a universal constant that is, it does not depend on the relative values of the local elastic characteristics If... 

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 use of a homogeneous matrix material leads, in the ideal case, to the preparation of random percolated filler networks. More complex matrix systems have been addressed in an attempt to direct or manipulate the formation of such percolated networks. The use of immiscible polymer blends to create restricted percolation networks has been applied to many types of fillers. To effectively manipulate the network formation in a selected volume fraction of the blend, a phase-separated system must be achieved and, preferably, a distribution of the filler within either of the phases or at the interface must be favored. Using conductive fillers the "directed" network could be electrically connected at a lower overall concentration of filler, - which is favorable for economic reasons and processability. 

Different types of percolation models have been developed. They can be divided into random and not random percolation models. In the last ones—correlated percolation, directed percolation, etc.—the occupation probability or the probability of formation of a cluster is conditioned by external factors. 

The more important percolation models are the random percolation models. This is especially true when looking to their application in pharmaceutical technology, where for the moment, only random percolation models as site, bond, site-bond or continuum percolation models, have been applied. 

The models described above, biased reptation and lattice chains, assume a homogeneous geL An opposite approach, in which gel inhomogeneities play a dominant role, was recently proposed by Zimm [139]. In this model, the tube is represented by a series of lakes in which the chmn can expand, and straits in which it is strongly confined. Finally, simulations of dectophoretic drift in 3D random percolation clustem of obstacles, have started [173]. 

Aoki, K. (1991) Nernst Equation Complicated by Electric Random Percolation at Conducting Polymer-Coated Electrodes.. Electroanal. Chem. Vol.310, No.1-2, 0uly 1991), pp. 1-12, ISSN 1572-6657... 

Along the whole percolation line Pc(T) the critical exponents are the same as for random percolation, ao rding to theory and the Monte Carlo experiments except for the spedal point p = 1/2, T = Tc in two dimensions, where percolation and aitical point coincide. At this point, the following inequalities between perrolation exponents and lattice gas exponents have been proved ) ... 

Percolation in the two-dimensional lattice gas is very instructive in connection with the universality concept If percolation occurs at finite lattice-gas correlation length, the critical exponents are the same as for random percolation. This can be easily understood if one takes into account that near the percolation threshold the lattice-gas correlation length I is much smaller than the typical duster radius thus, the large clusters average over the effects of correlation. This argument breaks down only at the critical point of the two-dimensional lattice-gas where both and vary simultaneously in fact, we have seen that at this point some exponents do change. 

The main difference between this model and the one described in Sect. C. V. 5 is that in the previous case the length describing the correlations between different sites diverges (with exponent v) at the Potts critical point whereas in the present case the correlations extend over at most one lattice spacing. Accordingly, one would expect that this type of polychromatic percolation belongs always to the same universality class as random percolation, and Monte Carlo calculations as well as renormalization group methods have confirmed this expectation. 

A percolation model, that has been introduced in connection with vulcanization of chains is the case in which two different species of bonds, say A and B, are placed on a lattice with concentrations Ca and Cb, respectively. Species A has the same properties as the usual bonds in random percolation whereas on species B is imposed the restriction that no more than two bonds of the same species B can be formed on the same site. Thus, species B forms polymer chains while species A acts as a crosslink. In the limit Ca = 0, Cb 0, the system reduces to self-avoiding chains, described by exponents different fi-om percolation. The opposite limit, Ca 0, Cb = 0, is the usual random-bond percolation. In the intermediate case, it was foimd - that percolation in which clusters are composed of sites connected by bonds of either species belongs to the same universality class as random percolation, unless the particular situation is realized in which percolation occurs when the typical size of chains made out of B bonds only diverges. In this case, there is a crossover from random percolation exponents to selfavoiding walk exponents , similar to the situation in lattice-gas correlated percolation. (These chains of B atoms must be distinguished from the sometimes chain-like structures formed randomly in the usual percolation process.)... 

To describe steric hindrance effects in gelation one may study percolation on a lattice in which bonds are restricted in a way that no more than v bonds can enamate from the same site, or no site may have more than v nearest neighbors . Similarly, valence saturation may occur for the monomers in the gelation process. The case v = 2 is similar to self-avoiding walks , while for larger v one expects random percolation exponents, as confirmed by the Monte Carlo methods in two and three dimensions. Then, on a large... 

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