Percolation critical concentration

The percolation critical concentration of surface defects is always several times higher that the bulk one as anticipated from the percolation theory [49]. Fortunately the concentration of defects at the surface may be much higher than far from it. We can estimate the increase of the vacancies concentration at the surface in comparison with a bulk of material using the results of the density functional calculations... 

The configuration of a system in which percolation may occur is classically treated as one in which the ingredients do not move. Considerable work has been devoted to these static models, leading to numerical solutions of the critical concentrations and cluster sizes associated with a percolation threshold. 

The main conclusion of the percolation theory is that there exists a critical concentration of the conductive fraction (percolation threshold, c0), below which the ion (charge) transport is very difficult because of a lack of pathways between conductive islands. Above and near the threshold, the conductivity can be expressed as ... 

Celzard A, McRae E, Deleuze C, Dufort M, Furdin G, Mareche JF. Critical concentration in percolating systems containing a high-aspect-ratio filler. Physical Review B. 1996 Mar 1 53(10) 6209-14. 

At the time of writing, the only evidence for critical fluctuations near the consolute point known to us comes from the work of Damay (1973). The thermopower of Na-NH3 plotted against T at the critical concentration is shown in Fig. 10.21. We conjecture that this behaviour is due to long-range fluctuations between two metallic concentrations, and that near the critical point, where the fluctuations are wide enough to allow the use of classical percolation theory, the... 

Fig. 23. Illustration of the nonmetal-to-metal transition according to the percolation picture of Cohen and Jortner (see text). Solvent regions of high metal concentration are shaded. Metallic regions grow with increase of metal concentration, (a) Below the percolation threshold there are isolated metallic regions and no conductance, (b) Above the percolation threshold a metallic path crosses the material and conduction occurs, (c) Above a certain critical concentration, the insulating regions are disjoint.

The numerical solution of the system (4.86), by a procedure of the Newton-Raphson type with two variables, requires the calculation of the derivatives dtjds and dt /5Wc . The results we obtained for a square lattice are similar to those by Yonezawa and Odagaki180 for a cubic lattice. The most striking feature is the existence, at low concentration, of a gap in the density of states,179 which isolates the zero energy on which a 3 peak builds up. Thus the HCPA produces a forbidden region of energy for the transport the gap and the 3 peak disappear at a critical concentration, analogous to the percolation threshold of the mean-field of resistances. 

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

Figure 77. Selfsimilarity of a large percolation cluster at the critical concentration. The windows indicate the section which is enlarged in the respective succeeding figure.281 (Reprinted from A. Bunde and S. Havlin, Percolation I , in Fractals and Disordered Systems. Ed. by A. Bunde and S. Havlin, Springer-Verlag, Berlin. Copyright 1996 with permission from Springer-Verlag GmbH Co. KG.)...

For a system where the donors are distributed at random it can be shown mathematically that there exists a critical concentration C (also called "percolation concentration") below which the percofation (edge-to-edge connectivity) has a probability of zero and above which the percolation probability (P ) rises sharply with donor concentration (C). A mathematical relation ( ), for a substitutionally disordered binary lattice, is ... 

Critical Concentrations for Bond and Site (pc"") Percolation on Lattices ... 

Percolation describes the geometrical transition between disconnected and connected phases as the concentration of bonds in a lattice increases. It is the foundation for the physical properties of many disordered systems and has been applied to gelation phenomena (de Gennes, 1979 Stauffer et al., 1982). At just above gelation threshold, denoting the fraction of reacted bonds as p and p=Pc + A/ , pc the critical concentration (infinite cluster), the scaling laws (critical exponents) for gel fraction (5oo) and modulus E) are ... 

In these electrodes, the silica sol-gel is mixed with graphite or carbon black powder. At a critical concentration of carbon within the sol-gel (beyond the percolation threshold), there becomes a continuous path(s) of conductive carbon throughout the system. The sol-gel system used was a methyl silicate network (by using methyltrimethoxysilane as the sol-gel precursor, the resulting network has exposed SiCH3 groups), which imparts hydrophobic properties to the electrode... 

The electric properties of soft ferromagnetic nanoparticles in an insulating matrix strongly depend on the concentration of a metallic filler x and are varied between properties of the matrix and those of the filler. In binary nanocomposites a critical concentration (percolation threshold Xq) is reached when a continuous current-conducting cluster of the filler particles is formed through out the sample. 

In order to disperse low concentrations of CPs in thermoplastics and elastomers, it is necessary to prepare highly structured fine aggregates to permit percolation in the polymer blend. With better polymerization techniques and new blending methods, the percolation threshold decreases. Indeed, a critical concentration as low as 1.5vol% and a saturation... 

The critical volume fraction of the filler has a different application in the case of conductive materials. As the amount of conductive filler is increased, the material reaches a percolation threshold. Below the percolation threshold concentration, the electric conductivity is similar to that of matrix. Above the percolation threshold conductivity rapidly increases. Above the critical volume fraction of filler which is, in turn, a concentration above the percolation threshold, there is a rapid increase in conductivity. " The critical volume fraction depends on the type of filler and its particles size. For example, for silver powder, it ranges from 5 to 20 vol% for... 

The onset of ferromagnetism in this system has usually been interpreted in terms of an environmental model. Fe atoms which occupy Al sites are surrounded by 8 Fe nearest neighbours and it is suggested that the percolation of such atoms (with pFc = 2.2pB) on Al sites eventually gives rise to ferromagnetism. The critical concentration does not appear to be significantly different for the two conditions. The forced volume magnetostriction peaks at 4 X 10 9 Oe-1 at 70% Fe for... 

Percolation A condition in a dispersed system in which a property such as conductivity increases strongly at a critical concentration, termed the percolation threshold, as a result of the formation of continuous conducting paths, termed infinite clusters. 

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. 

Criticality in Concentration. The sharp downturn of the disentanglement time vs. concentration curve for the flare effect (Figure 8) indicates that the flare sets in within a narrow concentration range as C is increased. This is the critical concentration already identified as C. The postulation of the existence of a critical lower concentration is consistent with the concept of a network, because for a network to arise there needs to be a pathway of contact across the whole macroscopic system, the establishment of which, as common with all such percolation problems, is a critical phenomenon. In our case the minimum contact criterion is clearly the existence of coil overlap. 

The frequency dependence of the dielectric constant for different levels of conductivities can also be analysed with regard to percolation theory. In fact this theory developed by Stauffer [141] shows that percolation aggregates can be described with seven critical exponents of power laws of (p — Pc) where p and pc are respectively the concentration in inclusions and the critical concentration at percolation threshold. It is shown that the seven critical exponents are linked through five scaling laws having only two exponents are independent variables to be fitted by experiment and not predicted by scaling theory. 

Even two-step non-linearities have been found in filled polymer blends. Not only are percolation theoretical approaches unable to predict this phenomenon, they are also unable to explain it once it has been seen. This is because percolation theory does not allow a structural change at the critical volume concentration—but the non-linear density increase leads to the assumption that a structural change, or phase transition, occurs at the critical concentration, in contradiction to all topological theories. 

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