Percolation exponent

A rigid microporous morphology, which does not reorganize upon water uptake, corresponds to a simple linear relation x w) = yw. In this limiting case, the model resembles the archetypal problem of percolation in bicontin-uous random media. Due to deviations of swelling from fhis law, universal percolation exponents in relations between conductivity and water content are not warranted. 

Percolation Exponents for Two and Three Dimensions, and Corresponding Percolation Quantity ... 

Again several attempts have been made to relate the elastic exponents Tg and Sg with the conductivity exponents tc and Sc and other percolation exponents. Identifying the elastic energy E F /Y of the... 

Remark. This cross-over from percolation exponents to mean field exponents may occur in situations other than vulcanization. Consider for instance a condensation reaction between a difimctional unit AA and a trifunctional unit BB B, in a case where one of the groups (B ) is much less reactive than the two others (BB). Then the early polymerization products are linear chains ... 

Here, <5 is the percolation exponent (we take 6 = 2.5), and is the percolation threshold for the hard phase. The percolation exponent, 6, typically ranges between 1.5 and 2 (see, e.g., ref. [52]), depending on the type of the system and the property described by a percolation model (modulus, conductivity, etc.). There are instances, however, when the percolation exponent could be larger than 2 (see, e.g., ref. [53]). Various models (e.., double percolation -see ref. [54]) have been proposed to explain these high percolation exponents. In our analysis, we refrain firom ascribmg any specific meaning to exponent 5 = 2.5, and treat it simply as an adjustable parameter that is found fi-om the best fit to experimental data. 

It is important to note that Eq. (2.7) should be used only in the vicinity of the hard phase percolation. Indeed, the meaning of the percolation exponent, 6, is to eiccomit for the fact that the percolated pathways are not linear but have some complicated morphology thus, the efficiency of the reinforcement ( fraction of elastically active hard phase elements ) is much less than 100%. 

Percolation exponent values are given in Table 1, they are calculated from Monte Carlo simulations performed on a lattice at a space dimensionality d. Exponent values are very sensitive to d (the mean-field values are reached for d > 6) but insensitive to the exact geometry of the lattice (number of first neighbour) for a given dimensionality. 

The interaction parameter B, proportional to the intrinsic viscosity, is a function of the distance to the gelation threshold. This is an evidence of the swelling of the clusters. If in the expression of B(Eqs. 8 and 16) we insert the D and o (= (3-x)/y = 0.47 + 0.04) values determined experimentally, we find D = 2.42 + 0.15 in perfect agreement with the percolation theory (D = 2.5). This fractal dimension, which corresponds to the unswollen state is larger than that measured in the swollen state (D = 1.98 + 0.03). The e independence of the ratio [ql/B, means that the fractal dimension D of clusters of polyurethane is identical in the two solvents used (THF and dioxane). Due to this swelling, exponent v linking size to the distance to the gel point cannot be directly compared with the percolation exponent value v = 0.88. 

Augstin E. Gonzalez (National University of Mexico) I am puzzled by one thing In the percolation model one assumes that the monomers are fixed in space while the bonds are being formed, while here the clusters formed should migrate, more like in a cluster-cluster aggregation model. Why should we expect percolation exponents instead of cluster-cluster aggregation exponents ... 

A. Geiger, H. E. Stanley, Tests of universality of percolation exponents for a three-dimensional continuum system of interacting waterlike particles, Phys. Rev. Lett. 49 (1982) 1895-1898. 

Wu J, McLachlan D S (1997), Percolation exponents and thresholds obtained from the nearly ideal continuum percolation system graphite-boron nitride , Phys Rev B 56, 1236. 

Table 3. Qimparison of gelation (percolation) exponents (left part) with exponents in thermal phase transitions (right part) for both dassical and modem theories...

Comparing the results in Table 1, we see that the classical exponents typically differ from three-dimensional percolation exponents by a factor of about two. Even experiments of moderate accuracy may distinguish between two such drastically different pre-... 

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... 

As Fig. 6 shows, for any concentration 0 of monomers above the site percolation threshold (which is 0.312 in the simple cubic lattice, as opposed to the bond percolation threshold Pc = 0.248), there is a percolation threshold Pc(0) for the bond formation probability For p above Pc(0), an infinite network of bonds between mtmomers exists. Thus, one has a whole percolation line in a 0 - p diagram, which ends for p = 1 at the site-percolation threshold for 0, and ends for 0 = 1 at the bond-percolation threshold for p. There is strong evidence that the whole percolation line is described by the usual random-percolation exponents . Note that even for p = 1 not all f bonds of afl monomers are formed since the solvent molecules remain inert and may isolate the monomers. 

In contrast, different forms of the renormalization group theory show that random-percolation exponents are obtained along the entire gelation line except at the critical consolute point, if the latter is also the end point of the gelation line. In the latter case, the critical exponents are given by the lattice-gas exponents, i.e. the weight average... 

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