Cluster infinite

In [220] it was shown, for spheres with a radius tending to zero, that there must be a critical value of Bcr = 4 3 " 1ma3N in the system which is conductive to an infinite cluster of bonded sites (here a is site spacing N is volumetric concentration). 

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 power-law variation of the dynamic moduli at the gel point has led to theories suggesting that the cross-linking clusters at the gel point are self-similar or fractal in nature (22). Percolation models have predicted that at the percolation threshold, where a cluster expands through the whole sample (i.e. gel point), this infinite cluster is self-similar (22). The cluster is characterized by a fractal dimension, df, which relates the molecular weight of the polymer to its spatial size R, such that... 

Solving the problem of sites is similar to that of bonds. One can also calculate the percolation probability, Q q), of formation of an infinite cluster of sites of one type, for example, B, depending on their numerical part q = B/(A+B), which is proportional to the part of the 3D volume occupied with phase B. The invariants of the problem of sites for 3D lattices of contiguous monospheres are [3,238]... 

Thus, phase B, which occupies the volumetric part a(B) < 0.16 0.02 of a PS, is distributed in the form of separated clusters with no permeability through this phase, but with a(B) > 0.32 0.2 the whole phase B is practically interlinked. A gap corresponds to coexistence of interlinked clusters of phase A and phase B. The thresholds in 2D lattices correspond to a degree of surface coverage. The whole phase B is interlinked into one cluster under a(B) > 0.7, but coexistence of two infinite clusters of both phases is impossible because this contradicts one of the topology theorems [8],... 

We study here the A + 5B2 —> 0 reaction upon a disordered square lattice on which only a certain fraction S of lattice sites can be accessed by the particles (the so-called active sites). We study the system behaviour as a function of the mole fractions of A and B in the gas phase and as a function of a new parameter S. We obtain reactive states for S > Sq where Sq is the kinetically defined percolation threshold which means existence of an infinite cluster of active sites. For S < Sq we obtain only finite clusters of active sites exist. On such a lattice all active sites are covered by A and B and no reaction takes place as t —> 00. 

We have studied the system (9.1.39) to (9.1.41) by means of the Monte Carlo method on a disordered surface where the active sites form a percolation cluster built at the percolation threshold and also above this threshold [25]. Finite clusters of active sites were removed from the surface to study only the effect of the ramification of the infinite cluster. The phase transition points show strong dependence on the fraction of active sites and on the... 

For S > Sc we obtain an infinite cluster for which in principle a reactive state exists. We use this fact to define the percolation threshold in a kinetic way for the particular reaction at hand as the transition point from the reactive (Rco2 > 0) to the non-reactive (Rco2 — 0) state. As we have shown above, this transition happens in such a way that the kinetic phase transition points of 2/1 and are approaching each other if S —> Sc [25]. At S = Sc the... 

We have studied above a model for the surface reaction A + 5B2 -> 0 on a disordered surface. For the case when the density of active sites S is smaller than the kinetically defined percolation threshold So, a system has no reactive state, the production rate is zero and all sites are covered by A or B particles. This is quite understandable because the active sites form finite clusters which can be completely covered by one-kind species. Due to the natural boundaries of the clusters of active sites and the irreversible character of the studied system (no desorption) the system cannot escape from this case. If one allows desorption of the A particles a reactive state arises, it exists also for the case S > Sq. Here an infinite cluster of active sites exists from which a reactive state of the system can be obtained. If S approaches So from above we observe a smooth change of the values of the phase-transition points which approach each other. At S = So the phase transition points coincide (y 1 = t/2) and no reactive state occurs. This condition defines kinetically the percolation threshold for the present reaction (which is found to be 0.63). The difference with the percolation threshold of Sc = 0.59275 is attributed to the reduced adsorption probability of the B2 particles on percolation clusters compared to the square lattice arising from the two site requirement for adsorption, to balance this effect more compact clusters are needed which means So exceeds Sc. The correlation functions reveal the strong correlations in the reactive state as well as segregation effects. 

Fig. 11.18. Temperature dependence of the metallic conductance of the infinite cluster GidT) deduced from the difference between the experimental conductance and that corresponding to the hopping regime (see text) 6 K (7 ) = [Gexp(r) — Gho] T)]. The dashed lines are guides for eyes [85],...

The most interesting result of Refs. [58, 59] concerns the crossover regime between dilute and semidilute regions of polymer 0-solution. The author shows that in this crossover regime there exists the critical concentration c corresponding to the appearance of an infinite cluster of entangled with each other macromolecules. It is also shown that near this critical concentration the relative viscosity t r of the 0-solution has a scaling form ... 

This chapter will focus on infinitely-extended suspensions in which potential complications introduced by the presence of walls are avoided. The only wall-effect case that can be treated with relative ease is the interaction of a sphere with a plane wall (Goldman et ai, 1967a,b). The presence of walls can lead to relevant suspension rheological effects (Tozeren and Skalak, 1977 Brunn, 1981), which result from the existence of particle depeletion boundary layers (Cox and Brenner, 1971) in the proximity of the walls arising from the finite size of the suspended spheres. Going beyond the dilute and semidilute regions considered by the authors just mentioned is the ad hoc percolation approach, in which an infinite cluster—assumed to occur above some threshold particle concentration—necessarily interacts with the walls (cf. Section VI). 

Values ascribed to the critical exponent s in the literature are somewhat controversial. Results (Derrida and Vannimenus, 1982) indicate that s is close to 1.28. Below pc the bulk conductance is zero, since the network fails to form an infinite cluster whose existence would permit current to be conducted through the system. 

By analogy to percolation arguments, a critical concentration x the fraction of spheres belonging to an infinite cluster, the percolation analogy with Eq. (6.2) suggests that... 

Formation of such an infinite cluster appears likely to change the suspension hydrodynamics drastically. For example, the classical parabolic Poiseuille velocity profile existing within a circular tube is likely to be severely blunted, a phenomenon observed experimentally by Karnis et al. (1966). 

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. 

The temperature at which the phase transition occurs is called the critical temperature or Tg. Most, but not all, magnetic phase transitions are continuous , sometimes called second order . From a microscopic point of view, such phase transitions follow a scenario in which, upon cooling from high temperature, finite size, spin-correlated, fractal like, clusters develop from the random, paramagnetic state at temperatures above Tg, the so-called critical regime . As T Tg from above, the clusters grow in size until at least one cluster becomes infinite (i.e. it extends, uninterrapted, throughout the sample) in size at Tg. As the temperature decreases more clusters become associated with the infinite cluster until at T = 0 K all spins are completely correlated. 

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