Percolation models/theory

More detailed theoretical approaches which have merit are the configurational entropy model of Gibbs et al. [65, 66] and dynamic bond percolation (DBP) theory [67], a microscopic model specifically adapted by Ratner and co-workers to describe long-range ion transport in polymer electrolytes. 

The power-law variation of the dynamic moduli at the gel point has led to theories suggesting that the cross-linking clusters at the gel point are self-similar or fractal in nature (22). Percolation models have predicted that at the percolation threshold, where a cluster expands through the whole sample (i.e. gel point), this infinite cluster is self-similar (22). The cluster is characterized by a fractal dimension, df, which relates the molecular weight of the polymer to its spatial size R, such that... 

The approaches considered allow modeling of the primary texture of PS and the processes, limited by individual PBUs that mainly correspond to level III and partially to level IV in the hierarchical system of models (see Section 9.6.3). PBUs are identical in regular PSs, and simulation of numerous processes may be reduced to analysis of a process in a single PBU/C or PBU/P. An accurate modeling of the processes in irregular PSs requires the studies of the properties of structure and properties of the ensembles (clusters) of particles and pores (level IV of the system of models) and the lattices of such clusters (levels V to VII of the system of models). Let us consider the composition of clusters on the basis of fractal [127], and the lattices on the basis of percolation [8] theories. 

The percolation model, which can be applied to any disordered system, is used for an explanation of the charge transfer in semiconductors with various potential barriers [4, 14]. The percolation threshold is realized when the minimum molar concentration of the other phase is sufficient for the creation of an infinite impurity cluster. The classical percolation model deals with the percolation ways and is not concerned with the lifetime of the carriers. In real systems the lifetime defines the charge transfer distance and maximum value of the possible jumps. Dynamic percolation theory deals with such case. The nonlinear percolation model can be applied when the statistical disorder of the system leads to the dependence of the system s parameters on the electrical field strength. 

Both the Flory-Stockmayer mean-field theory and the percolation model provide scaling relations for the divergence of static properties of the polymer species at the gelation threshold. 

A lot of work has been directed to the modelling of the gelling process. The percolation model offers the most widely accepted theory. A review of the model is beyond the scope of this book. The reader is referred to the reviews of Zallen,22 Staufer et al.23 and Brinker and Scherer.12... 

Another refinement of the VRH model consists in assuming that the charges are delocalized over segments of length L, instead of being strictly localized on point sites [40]. This is indeed a more realistic picture, leading to better fits with the data, but it has the drawback that an extra parameter has been added. Note that the temperature dependence, log o- -T y, can be found by other approaches, such as the percolation model, the effective medium approximation (EMA), the extended pair approximation (EPA) [41], the random walk theory, and so on. 

Percolation theory is helpful for analyzing disorder-induced M-NM transitions (recall the classical percolation model that was used to describe grain-boundary transport phenomena in Chapter 2). In this model, the M-NM transition corresponds to the percolation threshold. Perhaps the most important result comes from the very influential work by Abrahams (Abrahams et al., 1979), based on scaling arguments from quantum percolation theory. This is the prediction that no percolation occurs in a one-dimensional or two-dimensional system with nonzero disorder concentration at 0 K in the absence of a magnetic field. It has been confirmed in a mathematically rigorous way that all states will be localized in the case of disordered one-dimensional transport systems (i.e. chain structures). 

The all or nothing feature of metal powder composites is very much a feature of conductive composite systems. In order to understand this behaviour, most theories borrow from percolation theory (Broadbent and Hamersley, 1957), which was originally developed as a model for predicting fluid permeation through porous media. The percolation model is based on having a medium... 

Careri et al. (1986), using the framework of percolation theory, analyzed the explosive growth of the capacitance with increasing hydration above a critical water content (Fig. 14). The threshold for onset of the dielectric response was found to he 0.15 h for free lysozyme and 0.23 h for the lysozyme—substrate complex. In the percolation model the thresh-... 

The use of the percolation model to analyze the d.c. conductivity in hydrated lysozyme powders (Careri et al., 1986, 1988) and in purple membrane (Rupley et al, 1988) introduces a viewpoint from statistical physics that is relevant to a wide range of problems originating in disordered systems. Percolation theory is described in the appendix to this article, for readers unfamiliar with it. Here, we discuss the significance of percolation specihcally for protein hydration and function. 

In contrast to many other modern research fronts, percolation theory is a problem that is, in principle, easy to define (8-13). In general, a percolation model is a collection of points (or sites) distributed in space, certain... 

Theoretical and experimental treatments of gels go hand-in-hand. The former are covered first because they will help us understand gel point and other concepts. Two main theories have been used to interpret results of experimental studies on gels the classical theory based on branching models developed developed by Floiy and Stockmayer, and the percolation model credited to de Gennes. Gelation theories predict a critical point at which an infinite cluster first appears. As with other critical points, the sol-gel transition can be in general characterized in terms of a set of generally applicable (universal) critical exponents. 

According to the percolation model, the chains in a swollen gel need not be Gaussian. For a true gel when a the stress (tr) versus strain (X) relationships from the percolation model (Equation 6.14) and the classical theory (Equation 6.15) are, respectively ... 

Verification of the values of the exponents in Equations 6.14 and 6.15 continue to be the subjects of considerable debate between proponents of the two theories. The viscoelastic properties of the system that characterize the sol-gel transition are also important features of the percolation model. 

A great number of studies have been published to deal with relation of transport properties to structural characteristics. Pore network models [12,13,14] are engaged in determination of pore network connectivity that is known to have a crucial influence on the transport properties of a porous material. McGreavy and co-workers [15] developed model based on the equivalent pore network conceptualisation to account for diffusion and reaction processes in catalytic pore structures. Percolation models [16,17] are based on the use of percolation theory to analyse sorption hysteresis also the application of the effective medium approximation (EMA) [18,19,20] is widely used. 

For the percolation model, the situation is rather different, where the numerical calculation is easily accessible with the aid of the probability theory combined with a computational calculation. As a matter of course, the same methodology cannot be applied to real branching reactions, since real molecules are not fixed on lattices, and one cannot define the number of configurations that corresponds to the coordination number, q, of the percolation model. 

Series Expansion Method (Percolation Model). Like the polymer theory [6], the percolation theory is based on the principle of the equireactivity of FU. In the percolation model, all mathematical complexity arising from ring formation and excluded volume effects is simply replaced with a set of random bonds. Let a site correspond to a monomer unit, then generate at random bonds between them, and we have an ensemble of bond animals (Fig. 15). 

A method of using GCMC simulation in conjunction with percolation theory [74,75] has been suggested for simultaneous determination of the PSD and network connectivity of a porous solid [76]. In this method, isotherms are measured for a battery of adsorbate probe molecules of different sizes, e.g., CH4, CF4, and SFg. As illustrated in Fig. 9a, the smaller probe molecules are able to access regions of the pore volume that exclude the larger adsorbates. Consequently, each adsorbate samples a different portion of the adsorbent PSD, as shown in Fig. 9b. By combining the PSD results for the individual probe gases with a percolation model, an estimate of the mean connectivity number of the network can be obtained [76]. 

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