Percolation cluster formation

Several unifying conclusions may be based upon the order parameter results illustrated here for microstructural transitions driven by three different field variables, (1) disperse phase volume fraction, (2) temperature, and (3) chemical potential. It appears that the onset of percolating cluster formation may be experimentally and quantitatively distinguished from the onset of irregular bicontinuous structure formation. It also appears that... 

Thus, the offered in Refs. [29, 30] treatment assumes percolation not by a sample cross-section area, but by a line - sample width B. Such treatment reason will be considered below. Besides it is supposed [30], that percolation cluster formation beginning (percolation threshold) corresponds to any small amount of local shear zone (shear bands network) formation in deformation process and therefore, to the first approximation the percolation... 

Electrochemical redox studies of electroactive species solubilized in the water core of reverse microemulsions of water, toluene, cosurfactant, and AOT [28,29] have illustrated a percolation phenomenon in faradaic electron transfer. This phenomenon was observed when the cosurfactant used was acrylamide or other primary amide [28,30]. The oxidation or reduction chemistry appeared to switch on when cosurfactant chemical potential was raised above a certain threshold value. This switching phenomenon was later confirmed to coincide with percolation in electrical conductivity [31], as suggested by earlier work from the group of Francoise Candau [32]. The explanations for this amide-cosurfactant-induced percolation center around increases in interfacial flexibility [32] and increased disorder in surfactant chain packing [33]. These increases in flexibility and disorder appear to lead to increased interdroplet attraction, coalescence, and cluster formation. 

The order parameter values calculated from the data of Fig. 4 are illustrated in Fig. 5. The data there suggest the existence of two continuous transitions, one at a = 0.85 and another at a = 0.7. The first transition at a = 0.85, denoted by the arrow labeled a in Fig. 5, is assigned to the formation of percolating clusters and aggregates of reverse micelles. The onset of electrical percolation and the onset of water proton self-diffusion increase at this same value of a (0.85) as illustrated in Figs. 2 and 3, respectively, are qualitative markers for this transition. This order parameter allows one to quantify how much water is in these percolating clusters. As a decreases from 0.85 to 0.7, this quantity increases to about 2-3% of the water. 

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

According to this model the tunnel current arises due to formation of the infinite percolation cluster of contacting external spheres with i d. The... 

The origins of percolation theory are usually attributed to Flory and Stock-mayer [5-8], who published the first studies of polymerization of multifunctional units (monomers). The polymerization process of the multifunctional monomers leads to a continuous formation of bonds between the monomers, and the final ensemble of the branched polymer is a network of chemical bonds. The polymerization reaction is usually considered in terms of a lattice, where each site (square) represents a monomer and the branched intermediate polymers represent clusters (neighboring occupied sites), Figure 1.4 A. When the entire network of the polymer, i.e., the cluster, spans two opposite sides of the lattice, it is called a percolating cluster, Figure 1.4 B. 

The success of fractal models applied to the physics of disordered media may be explained first of all by the fact that fractal forms are characteristic of a huge number of processes and structures because many diverse models of the formation and growth of disordered objects of disparate nature may ultimately be reduced to a transition model—namely a connected set and an unconnected set—and to a limited diffusive aggregation [1-6]. In the first case a fractal percolation cluster is formed in the second case a fractal aggregate is formed. 

A more strict approach to the characterisation of the effect under consideration is the concept of the unscreened perimeter of any object introduced in [34], for a filler. This perimeter is a measure of the accessibility for the formation of adhesion bonds [34]. The particulate-filled polymers at a definite values of (pf may be considered as percolation clusters with the first percolation threshold (pf = 0.15-0.17 [35]. Because the maximum value of (pf for the composites considered in this paper is equal to 0.176, one can believe these systems are below the percolation threshold and from them the following relationship is valid [34] ... 

Since the successful S-S thermodynamic theory assumes fee or hep packing with the coordination number z = 12, the most probable value of Pc is 0.120. However, if the cluster formation starts at Tc, from Tc/Tg=l/ 1 -pc)= 1.15 - 1.35, then pc = 0.13 to 0.26, which suggest fee hep or bcc packing. It is noteworthy that the hindered molecular dynamics at Tg occurs at the percolation threshold similar in magnitude to the values found for formation of percolative phase co-continuity in polymer blends (i.e., Pc = 0.15 to 0.21) [Lyngaae-Jprgensen and Utracki, 1991]. 

The kind of mechanisms that lead to gelation characterised by infinite clusters are not clear. The infinite cluster contains of course a finite fraction G(t) of the total mass (M(t) + G(t) = 1). Pre-gel and post-gel states separated by a gelation transition can be analysed in terms of a kinetic equation. Sol-gel transitions are similar to phase transition phenomena. It is not surprising that scale invariance principles elaborated in the theory of phase transition can be adopted for polymer systems. Modern percolation theory (see, for example Stauffer (1979)) offer a conceptual framework to treat cluster formation. 

Key words Acrylamide - AOT -alkylamide - cluster formation -continuous transition - cosurfactant -order parameter - percolation -reverse microemulsion... 

This differential increase in partitioning into the continuous phase over the subthreshold value of x = 0.013 suggests the formation of a third pseudo-phase assigned to percolating clusters of droplets. In this three-pseudo-phase model, if was assumed [58] that the molar ratio of water to AOT in the clusters is the same as that of the isolated swollen micelles, and it may then be concluded that this excess mole fraction derived for water in the continuous pseudophase represents the volume fraction of the percolating pseudo-phase ((Pe ). An order parameter (S) for the disperse pseudo-phase (clusters and isolated swollen micelles) was defined as [58, 59]... 

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