Percolation backbone

So, how does the invaded cluster algorithm work Basically, it is just a variation of the Swendsen-Wang algorithm in which the temperature is continually adjusted to look for the critical point. The algorithm finds the fraction of links at which a percolating backbone of spins first forms across the lattice and uses this fraction to make successively better approximations to the critical temperature at each Monte Carlo step. In detail, here s how it goes ... 

Incipient percolation backbone [57] (using Hoshen-Kopelman algorithm [27]). 

Incipient percolation backbone [74] (using Leath [26] and burning algorithm [64]). 

Fig. 12. Tensile strength a, vs. number of backbone bonds per monomer 1 /a, reported for a range of polymers by Vincent [75]. The solid line is the theoretical line for vector percolation analysis of strength discussed herein.

A. Klemm, H. P. Muller, R. Kimmich 1997, (NMR microscopy of pore-space backbones in rock, sponge, and sand in comparison with random percolation model objects), Phys. Rev. E 55, 4413. 

In the case of redox sites covalently bound to a polymer backbone, when only Dg contributes to charge transport. Equation 2.12 has systematically failed to explain the dependence of D pp with the concentration of redox sites. Blauch and Saveant have shown that for completely immobile centers, charge transport is basically a percolation process random distribution of isolated clusters of electrochemically coimected sites [33,40]. Only by dynamic rearrangements can these clusters become in contact and charge transport occur, giving rise to the concept of bound diffusion where each... 

Armand (1994) has briefly summarised the history of polymer electrolytes. A more extensive account can be found in Gray (1991). Wakihara and Yamamoto (1998) describe the development of lithium ion batteries. Sahimi (1994) discusses applications of percolation theory. Early work on conductive composites has been covered by Norman (1970). Subsequent edited volumes by Sichel (1982) and Bhattacharya (1986) deal with carbon- and metal-filled materials respectively. Donnet et al. (1993) cover the science and technology of carbon blacks including their use in composites. GuF (1996) presents a detailed account of conductive polymer composites up to the mid-1990s. Borsenberger and Weiss (1998) discuss semiconductive polymers with non-conjugated backbones in the context of xerography. Bassler (1983) reviews transport in these materials. 

Fig. 1.2. Portion of a random bond percolating cluster backbone, connecting the points A and B. Here, the thick black lines represent the singly connected bonds or red bonds which, if cut, will disconnect the connection between A and B. The bonds in the blob portions are indicated by dotted lines. The dangling bonds are indicated by thin black lines (cf. StauflPer and Aharony 1992).

In fact, if one measures the total number of bonds (sites) on the infinite cluster at the percolation threshold (pc) in a (large) box of linear size L, then this number or the mass of the infinite cluster will be seen to scale with L as where die (< d) is called the fractal dimension of the infinite cluster at the percolation threshold. Similar measurements for the backbone (excluding the dangling ends of the infinite cluster) give the backbone mass scaling as, de < die, where dfi is called the backbone (fractal) dimension. In fact, die can be very easily related to the embedding Euclidean dimension d of the cluster by... 

Figure 8. A two-dimensional illustration for the concept of percolation. The shaded and crossed areas correspond, respectively, to sites that were previously occupied and sites that have just been occupied. Those marked L in (b) are empty sites that must be occupied before the onset of ion transport. The percentage of occupancy of the grid are 18,31, 45, and 53% for Cases a to d, respectively. In this context, the empty and occupied sites would represent the fluorocarbon backbone and the electrolyte phase, respectively.

Luxmoore and Ferrand (1993) pointed out that pores that belong to the sample spanning cluster but not the backbone can be thought of as containing stagnant backwater zones. Thus, empirical determination of the proportions of backbone and backwater porosity in random and nonrandom pore percolation networks could be quite useful. They anticipated that percolation modeling would play an important role in understanding the effects of transient pore scale processes on solute transport. 

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