Bond percolation model

In addition to specific applications, the dynamic bond percolation model has been extended to focus on the importance of lattice considerations. The role of correlations among different renewal processes and the... 

It may be useful to inspect the critical point of the bond percolation model. As the critical point is approached, the total number, of chances of cycli-zation should diverge. From Eq. (66), it follows that... 

Gelation is a connectivity transition that can be described by a bond percolation model. Imagine that we start with a container full of monomers, which occupy the sites of a lattice (as sketched in Fig. 6.14). In a simple bond percolation model, all sites of the lattice are assumed to be occupied by monomers. The chemical reaction between monomers is modelled by randomly connecting monomers on neighbouring sites by bonds. The fraction of all possible bonds that are formed at any point in the reaction is called the extent of reaction p, which increases from zero to unity as the reaction proceeds. A polymer in this model is represented by a cluster of monomers (sites) connected by bonds. When all possible bonds are formed (all monomers are connected into one macroscopic polymer) the reaction is completed (/> = 1) and the polymer is a fully developed network. Such fully developed networks will be the subject of Chapter 7, while in this chapter we focus on the gelation transition. 

In a bond percolation model on a Bethe lattice, we assume that all lattice sites are occupied by monomers and the possible bonds between neighbouring monomers are either formed with probability p or left unreacted with probability 1 —/ . In the simplest version, called the random... 

Random branching and gelation bond percolation model, the probability p of forming each bond is assumed to be independent of any other bonds in the system. The basic assumptions of the mean-field model are implicit in the... 

Figure 18. A bond percolation model of gas flow in a pore network containing foam lamellae blocking pore throats. Line intersections represent pore bodies. (Reproduced with permission from reference 14. Copyright 1993.)...

T. Odagaki and M. Lax [1980] AC Hopping Conductivity of a One-dimensional Bond-Percolation Model. Phys. Rev. Lett. 10, 847-850. 

In contrast, in the bond percolation model (Figure 8.11(b)), the connection bonds are randomly placed on the lines between the nearest neighboring (n.n.) sites on the lattice. Bonds sharing the same lattice point are regarded as connected. The fraction p in this... 

Historically, the first one was the bond percolation model, developed in the works of Broadbent and Hammersley. Nevertheless, it was soon observed that the site percolation model provides a more general point of view, so that every situation described by the bond percolation model can also be described by a site percolation model. For this reason, the site percolation model has been the one more exhaustively studied [71]. 

Figure 4.6 Square lattices for site and bond percolation models.

The brief discussion above shows that the structure of a polymer electrolyte and the ion conduction mechanism are complex. Furthermore, the polymer is a weak electrolyte, whose ions form ion pairs, triple ions, and multidentate ions after its ionic dissociation. Currently, there are several important models that attempt to describe the ion conduction mechanisms in polymer electrolytes Arrhenius theory, the Vogel-Tammann-Fulcher (VTF) equation, the Williams-Landel-Ferry (WLF) equation, free volume model, dynamic bond percolation model (DBPM), the Meyer-Neldel (MN) law, effective medium theory (EMT), and the Nernst-Einstein equation [1]. 

Let us consider the bond-percolation model in a one-dimensional chain in which s,s, distributed according to the following function ... 

A summary of these ten examples shows how the random percolation problem can be modified The critical exponents change only if the modification introduced can be seen on a scale which may become infinitely large, as in particular at the critical consolute point of phase separation. Otherwise, the modification concerns only inessential details and does not change the critical exponents. In some sense, the correlated site-bond percolation model described in Chapter D is only a further generalization of modifications 1 and 2 above providing similar results for the critical exponents. 

The theory of random-bond percolation in Sect. C.II. assumes that every site is occupied by a monomer, and bonds between monomers are formed randomly. In a real gel, besides the f-funtional monomers, also solvent molecules are usually present. In order to take this solvent into account in a first approximation, one can allow the sites to be oompied by a monomer with a probability

solvent molecule otherwise, with probability — nearest-neighbor monomers may form a bond with probability p whereas no bonds emanate fi-om or lead to the solvent molecules. The original random-bond percolation model is thus transformed into a random site-bond percolation in which the clusters consist of randomly distributed monomers connected by random bonds. 

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