Bethe tree models

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

To calculate the percolation probability in Eq. (34), Mason (18-20) and Palar and Yortsos (26,27) have employed the Bethe tree model. This model is known (see Section II) to yield quantitative results only at 3 z 5. In general, however, one can use in Eq. (32) the universal percolation probability calculated for regular lattices (see Section II). 

Mayagoitia et al. (28-32) have analyzed the desorption process from the pore space described by a joint site-bond radius distribution with a correlation function that carries structure information about the network. The integral equations derived (29) are based on the Bethe tree model. 

Harln et al Iref 31] analyzed the data of Dumez and Froment in terms of the Bethe tree network model outlined above. For instan taneoua growth of coke to a size sufficient to block the mesopores but not the macropores, Beeckman and Froment [ref. 20 derived the following equation for the average coke content of the particle ... 

Nicholson et al. [7] introduced a simple network model, and applied it on gas relative permeability [8, 9]. For the gas relative permeabihty, an explicit approximate analytical relation between the relative permeability and the two network parameters, namely the z and the first four moments of the pore size distribution, f(r), has been developed, based on the Effective Medium Approximation (EMA) [10, 11). Bethe trees, which are lattices that do not admit... 

Q. To proceed further at this point one has to specify a pore model for the catalyst, and a model for the active site distribution. Froment and co-workers have examined a variety of cases such as single pore models (single-ended pores and pores open on both sides) with both deterministic and stochastic active site distributions, the bundle of parallel pores model and various tree-like models of the porous structure, which were earlier used by Pismen (40) to describe transport and reaction in porous systems. Such treelike models contain interconnected pores but lack any closed loops and are usually called Bethe networks or lattices. They are completely characterized by their coordination number Z, which is the number of pores connected to the same site of the network. 

The blame for this should be placed on the use of tree-like models. These tree-like, Bethe networks of pores are characterized by a finite ratio of surface pores to those in the bulk (equal to (Z - 2) / (Z -... 

The first theory that attempted to derive the divergences in cluster mass and average radius accompanying gelation is that of Flory [52] and Stockmayer [53]. In their model, bonds are formed at random between adjacent nodes on an infinite Cayley tree or Bethe lattice (see Figure 47.7). The Flory-Stockmayer (FS) model is qualitatively successful because it correctly describes the emergence of an infinite cluster at some critical extent of reaction and... 

The role of positional fluctuations in polymer networks is central to some theories of elasticity, and has been investigated with an MC method based on a modified bond-fluctuation model (265). The simple model used in the simulations gave results close to those calculated from theory for a Bethe lattice (also known as a Cayley tree). More extensive results bearing on the role of fluctuations in polymer networks have been reported by Grest and co-workers (225). They find that entanglements limit fluctuations, giving behavior similar to the description provided by the tube model. 

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. 

Equations (5.43) with (5.51) give the activities as functions of the densities for this model in the Bethe-Guggenheim approximation. If the occupancies of the cells neighbouring any one cell were indeed independent of each other, these relations would be exact. They are therefore exact for the class of lattices called trees, on which there are no closed paths between any two neighbours of one site there is no path other than via that site, so that, once the state of occupancy of the cell at that site is specified, the cells at the neighbouring sites are decoupled from, and independent of, each other. Because the Bethe-Guggenheim approximation is thus exact for one class of models, it is necessarily therm ynami-cally consistent and, indeed, it may be verified by explicit calculation that (5.43) with (5.51) satisfies (5.44). 

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