Percolation classical theory

The classical theory predicts values for the dynamic exponents of s = 0 and z = 3. Since s = 0, the viscosity diverges at most logarithmically at the gel point. Using Eq. 1-14, a relaxation exponent of n = 1 can be attributed to classical theory [34], Dynamic scaling based on percolation theory [34,40] does not yield unique results for the dynamic exponents as it does for the static exponents. Several models can be found that result in different values for n, s and z. These models use either Rouse and Zimm limits of hydrodynamic interactions or Electrical Network analogies. The following values were reported [34,39] (Rouse, no hydrodynamic interactions) n = 0.66, s = 1.35, and z = 2.7, (Zimm, hydrodynamic interactions accounted for) n = 1, s = 0, and z = 2.7, and (Electrical Network) n = 0.71, s = 0.75 and z = 1.94. 

Value t — 2 for the granular metals has been confirmed experimentally in several papers (see, for example, Ref. [1]) however, for Nix(Si02)i A nanocomposite with granules of nanometer size it was found t x 2.7, g x 2 [65]. It rather essentially differs from the classical theory predictions. Also, the noticeable differences of the experimental values of critical indexes from the theoretical ones have been found in papers [66,67]. Authors of these papers attributed the discrepancy between the experimental data and results of the classical percolation theory to the quantum effects, which lead to the Anderson localization of charge carriers [57,58]. 

One of the advantages of the proposed model versus classical theories is its ability to explain the changes in the release behavior of the matrices by means of a change in the critical points of the system (related to the drug and excipient percolation thresholds), which can be experimentally calculated, providing a scientific basis for the optimization of these dosage forms. 

Pc, at which a spanning cluster occurs is called the percolation threshold. There are two versions of percolation, site and bond. With bond percolation, the sites are initially filled and the bonds are added to connect the sites. With site percolation, a grid placed over a region is gradually filled with spheres. The percolation threshold is lower for bond percolation than for site percolation because a bond is attached to two sites while a site is connected to a maximum of z bonds. Taking the coordination number, z, around the sites, the threshold for bond percolation is seen to be close to that of the classical theory [21]... 

The percolation probability has different values based on the classical theory site or bond percolation for different structures, as shown in Table 12.2. This critical percolation volume fraction, <, is calculated from the percolation threshold and the space filling factor. The volume fraction for site percolation for various structures is essentially the same as follows. In three dimensions, the site percolation threshold occurs at —16% volume. Near the percolation threshold the average cluster size diverges as does the spanning length of clusters. 

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. 

A number of authors have used concentration of the polymer, c, in place ofp. Theoretical values for the exponents from the percolation theory are fi = 0.39, f 1.7-1.9 also, t > fi because of dangling chains. In contrast, the classical theory s predictions for the same quantities are ... 

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 ... 

In the classical theory, however, the neglect of loops significantly affects the size distribution and other properties of the clusters as one approaches the gel point. Some of the critical exponents that describe these properties in the classical theory and in percolation theory near p Pc are compiled in Table 5-1 (Martin and Adolf 1991). 

Statistical network models were first developed by Flory (Flory and Rehner, 1943, Flory, 1953) and Stockmayer (1943, 1944), who developed a gelation theory (sometimes referred to as mean-field theory of network formation) that is used to determine the gel-point conversions in systems with relatively low crosslink densities, by the use of probability to determine network parameters. They developed their classical theory of network development by considering the build-up of thermoset networks following this random, percolation theory. 

The gel point in this classical theory is the percolation threshold (p ) given by... 

All the scaling laws derived from the percolation model and quoted above are very different from the predictions of classical theory (the tree s proximation) - ... 

Before continuing the discussion of percolation theory, we should note that Isaacson and Lubensky [25] and de Gennes [26] have extended the classical theory by taking account of excluded volume effects (i.e., the... 

Although this method removes the uncertainty about the location of the critical point, it does not necessarily help to distinguish between theories. For example, the exponent in Eq. 51 has the value y/v = 2 for both percolation and the classical theory. (See values in Table 3.) Fortunately, this is not the case for every pair of properties, so the general approach can be successfully applied, as we shall see. 

Percolation Theory avoids the unrealistic assumptions of the classical theory and makes predictions about the properties of gelling systems that are in good accord with experimental observations. Unfortunately, only a few results can be obtained analytically, so these models must generally be... 

Percolation theory offers a description of gelation that does not exclude the formation of closed loops and so does not predict a divergent density for large clusters. The disadvantage of the theory is that it generally does not lead to analytical solutions for such properties as the percolation threshold or the size distribution of polymers. However, these features can be determined with great accuracy from computer simulations, and the results are often quite different from the predictions of the classical theory. Excellent reviews of percolation theory and its relation to gelation have been written by Zallen [19] and Stauffer et al. [24]. 

Table 2 shows the values of / <. found for several lattices in 1,2, 3, and higher dimensions. (Computer experiments are not limited to three dimensions as we mortals are ) In 1-d, the trivial result is that percolation requires every site to be filled. In higher dimensions, the value of / <, depends on the shape (e.g., simple cubic, face-centered cubic) of the lattice. This may seem unsatisfying, since real gelation does not occur on a lattice at all, but the situation is better than it appears. Taking the coordination number (number of bonds surrounding a site) of the lattice as equivalent to the functionality of the monomer, the threshold for bond percolation is seen to be close to that of the classical theory. 

This chapter describes the main polymers employed in the manufacturing of pharmaceutical matrix systems. Furthermore, the principal factors affecting drug release from polymeric matrices are analyzed from the point of view of the classical theories as well as the percolation theory. 

However, most authors reported that the effect of particle size seems to disappear for matrices containing high polymer concentrations. Classical theories fall to explain the dependence between the influence of the particle size and the polymer concentration [60]. The explanation provided by the percolation theory will be discussed in Sections... 

Polymeric excipients are showing amazing growth in the field of pharmaceutical technology. In this chapter, the behavior and characteristics of the main polymers employed for the preparation of matrix systems that constitute the most popular type of prolonged drug delivery systems have been studied, from both the point of view of classical theories, as well as from the new perspective offered by a theory coming from Statistical Physics and named the percolation theory. 

Furthermore, the explanation based on the percolation approach is more complete than that provided by the classical theories. Thus it supposes an advance in the concept of Quality by Design of the pharmaceutical formulations. Therefore the knowledge of the critical points of the pharmaceutical formulations is essential for medicine manufacturing in the current regulatory environment. 

Some experimental results are reviewed to check whether the percolation theory agrees with reality no clear answer has been found so tea, due to experimental dSficulties. For instance, for the viscosity a power law (pc - p) , which agrees with cme of the percolation ideas has been established in several experiments the shear modulus of the gel vani s roughly as (p - pj in some experiments, which agrees better with the classical theory. 

M. Gordon suggested some time ago that the behavior of gels at the sol-gel phase transition should be investigated more closely. And indeed shortly thereafter theoretical predictions were published according to which the critical exponents for these phase transitions should differ drastically from those of the widely accepted classical theories These speculations were based on the analogy with other phase transitions like the liquid-gas critical point, and in particular with the percolation problem and its recent advances. 

Thus, this review explains critical exponents and percolation theories and compares these theories, preferred by physicists, with the classical approach (Flory-Stockmayer type theory used by chemists, and summarizes experimental evidence both in favor and against the theoretical predictions. Since most readers are well acquainted with classical theories we emphasize here recent developments of percolation theories. Due to the rapid development of the situation since 1979, some earlier reviews are partly outdated now. We hope that the same can be said soon about the present article, too. 

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