Bethe lattices, percolation

Bethe lattices, percolation, 39 10 Bethe tree model, 39 26 BET method, in heterogeneous catalysis, 17 15-17... 

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. 

In general, percolation is one of the principal tools to analyze disordered media. It has been used extensively to study, for example, random electrical networks, diffusion in disordered media, or phase transitions. Percolation models usually require approximate solution methods such as Monte Carlo simulations, series expansions, and phenomenological renormalization [16]. While some exact results are known (for the Bethe lattice, for instance), they are very rare because of the complexity of the problem. Monte Carlo simulations are very versatile but lack the accuracy of the other methods. The above solution methods were employed in determining the critical exponents given in the following section. 

This results in a value of d = 2.5 for bond percolation on a 3-dimensional lattice. The fractal dimension of the Bethe lattice (Flory-Stockmayer theory) is... 

The percolation processes were first developed by Flory [235] and Stockmayer [236] to describe polymerization process, which result in gelation, that is, the formation of very large networks of molecules connected by chemical bonds. But, their theory was developed only for a special kind of network, namely, the Bethe lattice, an infinite branching structure without any closed loops. Broadbent and Hammersley have developed a more general theory and have introduced it into the... 

These powers a, (3, 7, p and i/ are called the critical exponents. These exponents are observed to be universal in the sense that although Pc de-pends on the details of the models or lattice considered, these exponents depend the only on the lattice dimensionality (see Table 1.2). It is also observed that these exponent values converge to the mean field values (obtained for the loopless Bethe lattice) for lattice dimensions at and above six. This suggests the upper critical dimension for percolation to be six. 

The Bethe lattice (or tree) is a lattice containing no closed loops (see, e.g., Fig. 6). The latter results in a simple analytical solution of the bond and site problems for these lattices. In addition, the general percolation properties for Bethe and regular lattices are often close. For these reasons, the Bethe lattices are rather popular in applied science publications, although these lattices have no physical sense. 

Finally, it is reasonable to present the equations describing percolation on the Bethe lattices (Fig. 6). The percolation probability can be calculated exactly for this model and is the same for bond and site problems 10). For example, the probability that all open walks from a chosen site are of the finite length, 1 — 9 b, can be represented as... 

Comparing Eqs. (8) and (3), one can conclude that the Bethe model is appropriate for describing the three-dimensional lattices only at z = 3 and 4. If z 3= 5, the percolation probability for the Bethe lattice differs considerably from that for regular lattices. 

At a glance, it can be understood that the intermolecular reaction on lattices has no dimension-dependence it is a function of functionality/alone, while the cyclization rate, Eq. (68), depends strongly on d according to the decreasing function of d. Thus, as the dimensionality increases, the cyclization rate alone naively declines, resulting in the known behavior of the percolation model p rmgj—>0 (Bethe lattice) as d—... 

The percolation model differs from real branching reactions in two points (1) intermolecular reaction of the percolation model has no dimension dependence (2) concentration in the ordinary chemical sense is absent in the percolation model. These differences arise from only the one fact that molecules are fixed on lattices, and give rise to the opposing dimensionality as d— °°, in real systems cyclization becomes predominant, whereas in the percolation model it is suppressed entirely (the Bethe lattice). In these respects, the percolation model is not commensurate with the general features of the branching processes as chemical reactions. 

The regular lattice constructed in this way is called a Bethe lattice (see Fig. 6.13). The mean-field model of gelation corresponds to percolation on a Bethe lattice (see Section 6.4). The infinite Bethe lattice does not fit into the space of any finite dimension. Construction of progressively larger randomly branched polymers on such a lattice would eventually lead to a congestion crisis in three-dimensional space similar to the one encountered here for dendrimers. 

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... 

A unique feature of percolation on a Bethe lattice is that there are many infinite polymers present in the same system above the gel point. The easiest way to understand this result is to start with a single infinite network... 

Similar relations with different coefficients and different exponents hold in all percolating systems near the transition point, not just in the mean-field case of percolation on a Bethe lattice, as will be shown in Section 6.5. 

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-... 

The central result of percolation theory is the existence of the critical probability pc, the percolation threshold at which a cluster of infinite size appears. In a one-dimensional lattice, the percolation threshold is obviously equal to one. For higher dimensions, pc will be smaller than one. To illustrate this central result, we consider the Bethe lattice (also called the Cayley tree). 

Figure 3.15 Evolution of the probability P(p) that a site belongs to the infinite cluster as a function of the occupation probability p for the Bethe lattice with three branches. The percolation threshold is pc = 0.5.

Lattices other than the Bethe lattice were also studied [101]. Table 3.4 summarises the value of the percolation threshold for some other lattices. 

For Bethe lattices (see Chapter 4), the relationship of accessible porosity to total porosity can be analytically derived. For site percolation on a Bethe lattice with coordination number C, the accessible porosity is given by [45, 46] ... 

The simulation of catalyst deactivation by coke formation using a 3-dimensional site-bond-site network model is highly attractive, especially for zeolites, as the processes occurring in cavities (also referred to as voids or intersections) and in channels (also referred as necks, capillaries or arcs) can be readily distinguished. This model is flexible and the cormectivity of pores as well as the local homogeneity of the catalyst can be readily altered. Further, a percolation theory is available for site-bond-site models. In the particular case of Bethe lattices, approximated analytical solutions for the percolation probabilities have been derived[7]. 

Although there are probably other universality classes, this transition was successfully modeled by bond percolation [6]. Generally, bond percolation on a lattice has each bond (line connecting two neighboring lattice sites) present randomly with probability p and absent with probability 1-p. Clusters are groups of sites connected by present bonds. For p > Pc zn infinite cluster is formed. Percolation theory (in a Bethe lattice approximation) was invented by Flory (1941) to describe gelation for three-functional polymers. 

Linke et al. (1983) and Franz (1984, 1986) developed a different version of the lattice-gas concept. These models emphasize the experimental observation (Secs. 3.4 and 4.4) that thermal expansion of liquids is mainly achieved by a reduction of the average near-neighbor coordination number. A given structure such as the artificial, but mathematically convenient Cayley Tree or Bethe lattices can, when partially populated, be viewed as a crystalline alloy of atoms and vacancies. Tight-binding methods then permit calculation of the electronic structure, in particular the density of electronic states. Franz made use of quantum percolation theory to model the DC conductivity. A more recent model (Tara-zona et al., 1996) employs a body-centered-cubic lattice which, when fully occupied, provides a reasonable approximation to the local structure of liquid metals near the melting point. 

Bethe lattice, an example of percolation model to account for gelation... 

The square lattice is only one of a myriad possible representations of space. Every lattice has an associated coordination number, z, which describes the number of bonds emanating from each site for example, the square lattice in Figure 4 has a coordination number of 4. In addition there are lattices which have no obvious dimensionality, like the Bethe lattice. The Bethe lattice is a homogeneous tree structure, the number of sites on the surface of the tree increases without bound as the size of the tree grows. The coordination number of the Bethe lattice can be from 2 to oo. There are also lattice representations that are irregular each site does not have the same characteristic shape. Voronoi lattices, both two- and three-dimensional, are constructed by placing points randomly in space and tessellating around these points to construct an internal surface [39, 40]. Some relevant properties of each lattice-dimensionality D, coordination number z, critical probability p for site and bond percolation--are listed in Table 1. 

By contrast, since Bethe lattices are tree structures-containing no loops or closed paths within them-they are more easily analyzed than other, less regular, lattice structures. Analytical expressions have been derived for the percolation probability, cluster size distribution function [42], and effective conductivity [43] of Bethe lattices. The properties of these special Bethe lattices are quantitatively similar to regular... 

---