Percolation theory probability distribution

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.,... 

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. 

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,... 

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. 

Percolation phenomena deal with the effect of clustering and coimectivity of microscopic elements in a disordered medium [129], Percolation theory represents a random composite material as a network or lattice structure of two or more distinct types of microscopic elements or phase domains, the so-called percolation sites. These elements represent mutually exclusive physical properties, e.g., electrically conducting vs. isolating phase domains, pore space vs. solid matrix, atoms with spin up vs. spin down states. Here, we will refer to black and white elements for definiteness. The network onto which black and white elements of the composite medium are distributed could be continuous (continuum percolation) or discrete (discrete or lattice percolation) it could be a disordered or regular network. With a probability p a randomly chosen percolation site will be... 

Percolation theory rationalizes sizes and distribution of connected black and white domains and the effects of cluster formation on macroscopic properties, for example, electric conductivity of a random composite or diffusion coefficient of a porous rock. A percolation cluster is defined by a set of connected sites of one color (e.g., white ) surrounded by percolation sites of the complementary color (i.e., black ). If p is sufficiently small, the size of any connected cluster is likely to be small compared to the size of the sample. There will be no continuously connected path between the opposite faces of the sample. On the other hand, the network should be entirely connected if p is close to 1. Therefore, at some well-defined intermediate value of p, the percolation threshold, pc, a transition occurs in the topological structure of the percolation network that transforms it from a system of disconnected white clusters to a macroscopically connected system. In an infinite lattice, the site percolation threshold is the smallest occupation probability p of sites, at which an infinite cluster of white sites emerges. 

Site occupation probability in percolation theory (dimensionless) Percolation threshold of site occupation probability (dimensionless) Pdclet number. Equation 1.30 Particle radius distribution function Capillary pressure (Pa)... 

The measured pore size distribution curves are frequently biased towards the small pore sizes due to the hysteresis effect caused by ink bottle shaped pores with narrow necks accessible to the mercury and wide bodies which are not. Meyer [24] attempted to correct for this using probability theory and this altered the distribution of the large pores considerably. Zgrablich et al. [25] studied the relationship between pores and throats (sites and bonds) based on the co-operative percolation effects of a porous network and developed a model to take account of this relationship. The model was tested for agglomerates of spheres, needles, rods and plates. Zhdanov and Fenelonov [26] described the penetration of mercury into pores in terms of percolation theory. 

There has been no direct verification of the conceptual structure of the theory. That is, a microscopic determination of the cluster distribution function has not been made, and the effects of percolation have not been seen. Assuming that the structure of the glass is well-defined liquidlike clusters in a denser solidlike background, one might expect to be able to see these clusters by either neutron or X-ray scattering. Since v is probably between 100 and 400 A and r c SO at Tscattered wave vectors on the order of 0.1 A could be used. 

The theory of random-bond percolation in Sect. C.II. assumes that every site is occupied by a monomer, and bonds between monomers are formed randomly. In a real gel, besides the f-funtional monomers, also solvent molecules are usually present. In order to take this solvent into account in a first approximation, one can allow the sites to be oompied by a monomer with a probability

solvent molecule otherwise, with probability — nearest-neighbor monomers may form a bond with probability p whereas no bonds emanate fi-om or lead to the solvent molecules. The original random-bond percolation model is thus transformed into a random site-bond percolation in which the clusters consist of randomly distributed monomers connected by random bonds. 

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