Percolation problem

A more rigorous approach consists of considering that electron hopping between fixed redox sites is fundamentally a percolation problem, each redox center being able to undergo a bounded diffusion motion.16 If these are fast enough, a mean-field behavior is reached in which (4.24) applies replacing d2 by d2 + 3 Ad2, where Adr is the mean displacement of a redox molecule out of its equilibrium position. 

In this model, proton transport in the membrane is mapped on a percolation problem, wherein randomly distributed sites represent pores of variable sizes and fhus variable conductance. The distinction of pores of differenf color (red or blue) corresponds to interfacial or by bulk-like proton transport. Water uptake by wet pores controls the transition between these mechanisms. The chemical structure of the membrane is factored in at the subordinate structural levels, as discussed in the previous subsections. 

As discussed above, hysteresis loops can appear in sorption isotherms as result of different adsorption and desorption mechanisms arising in single pores. A porous material is usually built up of interconnected pores of irregular size and geometry. Even if the adsorption mechanism is reversible, hysteresis can still occur because of network effects which are now widely accepted as being a percolation problem [21, 81] associated with specific pore connectivities. Percolation theory for the description of connectivity-related phenomena was first introduced by Broad-bent et al. [88]. Following this approach, Seaton [89] has proposed a method for the determination of connectivity parameters from nitrogen sorption measurements. 

Physical gels, as exemplified by gelatin gels, exhibit many common features with chemical gels. Among them, we found the topological disorder of the network formed by the polymer chains, and the formal similarity of the process of gelation with a percolation problem. 

Carlo Tests of the Universality in a Correlated-Site Percolation Problem. 

In this paper, we will consider only the dynamic aspects of this percolation problem, i.e., the stochastic distribution of velocities between the flow structures. To analyze a percolation process, it is useful to represent the scattering medium (i.e. the packed bed) by a lattice as depicted in Figure 2. The sites of the lattice correspond to the contact points between the particles whereas the bonds correspond to the pores connecting two neighbour contact points. The walls of these pores are delimited by the external surface of the particles. The percolation process is... 

For the dynamic case, the percolation problem can be considered in hyperspace, where a temporal coordinate is introduced complementary to the Euclidean spatial coordinates. Using this approach, we shall obtain the scaling relationships in the case of dynamic percolation and derive the dynamic HSR. 

Both problems share the common property that P(p) is of measure zero for p < Pc, with the critical threshold value pc as a function of the type of lattice considered. Few exact formulas exist for P(p) or even pc. There are, however, a number of empirical rules. For instance, for the bond percolation problem,... 

Fig. 1.1. A random conducting network with the conducting blocks (denoted by black squares) with concentration above the percolation threshold. If one assumes the conducting clusters to be formed when the blocks are connected by the nearest-neighbour sites (not by the marginally touching corners), the percolation problem is a random site problem. The current I through the network decreases to zero if the conducting block concentration p falls below the percolation threshold Pc, as shown in the figure on the right side.

Similarly, one can study the growth of the elastic constants (say the rigidity modulus) of a randomly formed elastic network, near the percolation point. The central force elastic problem (for networks formed out of linear springs only) belongs however to a different class of percolation problem, known as elastic percolation or central force percolation, and is discussed separately later (see Section 1.2.1(f)). 

Substituting Eq. (70) into Eq. (72), one arrives at the asymptotic solution of the ring distribution function for the bond percolation problem ... 

The result of the renormalization group method as applied to percolation problems consists of the following a physical state characterized, for instance, by the parameters To evolving via a set of equations Y forms a continuous sequence of new effective equations T(/), characterized by the new parameters... 

It is possible to generalize the results, derived in Section 6.4 for the Bethe lattice, to any percolation problem. Of particular interest is gelation (per-colation) in two-dimensional and three-dimensional spaces. Unlike mean-... 

Figure 2 The basic percolation problem (for clarity, the percolating cluster is shown in black and nonpercolating clusters in gray). (A) p

An alternative to the site percolation problem described above assumes that all sites are occupied and that nearest neighbors are connected to each other by either open or closed bonds. In this case, p is the probability that a randomly selected bond is open (and thus, —p is the probability that it is closed). Sites connected to each other by open bonds belong to the same cluster (this definition makes sense if one recalls the coffee percolator water can only flow through open pores). Since the conclusions drawn from such bond percolation may be understood by using site percolation, we will here focus on the latter. 

Blumberg, R.L., Shlifer, G., and Stanley, H.E. Monte Carlo tests of universality in a correlated-site percolation problem, /. Phys. A Math. Gen., 13, L147,1980. 

By changing the site site Mayer function, this formalism can be applied to a number of different physical situations. For example, if the site-site Mayer functions are chosen as those of an interaction site model of water, then the formalism could be used to study hydrogen bonding as a percolation problem, as has been done in the lattice context by Stanley and Texeira. Another possibility is to choose the Mayer functions to model polyfunctional chemical bonding. The formalism could then form the basis of treatment of gelation in polymer networks. 

A free exchange of free volume can take place only between liquidlike cells that (1) are nearest neighbors and (2) have enough other liquidlike nearest neighbor cells ( > z) to ensure that the volumes of any neighboring solidlike cells are not constrained to change simultaneously. This defines a type of percolation problem." ... 

We have defined a liquidlike cell to be in a cluster if it has at least z neighbors that are also liquidlike.Within such a liquidlike cluster, cells can exchange their free volume freely without restriction by neighboring solidlike cells. The usual percolation problem has z = l, so that all isolated liquidlike cells would be clusters of size one. Thus we have introduced a new percolation problem, which we call environmental percolation. In... 

In the usual percolation problem with z= 1, all pN liquidlike cells are in clusters if one counts all isolated liquidlike cells as clusters of size one. That is no longer true when z= l. Only a fraction a (p) of thepN liquidlike cells are now in the cluster [a,(/j) = l]. The cluster distribution C (p), v = 1,2,..., is normalized so that" ... 

The reader is referred to recent reviews" of percolation theory for z = 1 for a more complete study. Here we summarize some important results, which we expect to carry over to the environmental percolation problem with z. oxp>p and I> z P) assumed to have a scaling form,... 

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