Percolation model, gelation

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. 

This model leads to A = 0.67 at the gel point, using the zero-frequency values for s and u. Use of the values for s and u calculated by treating the gelation phenomena as a three-dimensional percolation model of a supra-conductor/resistor network (electrical analogy), gives A = 0.72 0.02. 

Domain coalescence (Karplus and Weaver, 1976) is a possible mechanism for protein folding. Zientara et al. (1980) examined the dependence of the coalescence lifetime on the hydration shell. The lifetime depends on the activation barrier contributed by the shell and the extent of the shell. If domains resemble the native protein in hydration, then the minimal extent of the shell and its fluidity favor coalescence. In passing, one notes that the percolation model may apply to folding the coalescence of domains should be analogous to gelation or to diffusion on a partially filled lattice. 

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. 

The most characteristic aspect of the critical point problem is that the three phenomena, cyclization, excluded volume effects, and dimension, intimately interacting with each other, spontaneously appear at the critical point. At the beginning, it was thought that cyclization would make little contribution to such an important question that has remained unsolved for so long in physical science. The author s early conjecture was wrong. As we have seen in the text, cyclization plays a central role in the location of the critical point. For the percolation model, dimension is almost equivalent to cyclization (Sects. 4 and 5) even excluded volume effects seem to manifest themselves as an element of cyclization (Sects. 6 and 7), while dimensionality is in close conjunction with excluded volume effects (Sect. 7). In real gelations, the three effects are deeply connected with one another. 

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. 

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... 

The values of the critical exponents r and a and the cutoff functions /+ (N/N ) and/ (N/N ) depend only on the dimension of space in which gelation takes place. The percolation model has been solved analytically in one dimension (d=, see Sections 1.6.2 and 6.1.2) and critical exponents have been derived for two dimensions (d = 2). The mean-field model of gelation corresponds to percolation in spaces with dimension above the upper critical dimension (d>6). The cutoff function in the mean-field model [see Eq. (6.77)] is approximately a simple exponential function [Eq. (6.79)]. The exponents characterizing mean-field gelation are o — 1/2 and... 

Figure 7. Illustrations of gelation according to the classical Flory-Stockmayer model and the percolation model. In the classical model, cyclic configurations are avoided, so the unphysical situation M R4 results. As illustrated by the 29Si NMR spectrum of an acid-catalyzed TEOS sol (36), cyclic species are quite prominent sol components. (Reproduced with permission from reference 36. Copyright 1988.)...

It is clear that the percolation model is a very crude representation of any gelation processes. We shall now discuss two possible criticisms (i) the monomers are not on a lattice, but are disordered (<7) in many practical cases, the monomers are mixed with a solvent, and this feature is absent in the percolation model. 

A fundamental question is how many crosslinks are required to transform a polymer melt into a full polymer network (gel) that behaves under external action as a uniform strucmre. Gelation is a type of connectivity transition that can be described by braid percolation models [59, 60]. Slightly below the transition (gel point Co), the system consists of a mixmre of polydisperse branched polymers. Slightly beyraid the gel point, the simation is still approximately the same, but at least raie chain percolates through the entire system. Simultaneously, the system, as a whole, acquires a nonzero static shear modulus (response) [61]. The fully developed... 

Modulus-frequency master curves have been constructed by applying appropriate time dependent renormalisation factors to the frequency and modulus individual data. From the scaling of these factors with reaction time, the static scaling exponents t and s have been calculated and observed to be independent of the chemical nature of the midblock, suggesting a unique gelation mechanism. For all the samples, 1.84scalar elasticity percolation model. 

Figure 6.33. Schematic plots of cure time variation of the conductivity measured (a) at constant cure temperature (b) at a constant low frequency at which charge migration dominates. An estimate of the time to reach gelation (fgei) for curing at a relatively low temperature Taae.i), based on the percolation model for the o(fc) function, is shown (see Section 6.7.2.2). For plots with actual experimental data following the dependencies described in this figure the reader is referred to the works of, for example, Eloundou et al. (1998b, 2002) and Nunez-Regueira et al. (2005).

In fact, in the straightforward gelation/percolation analogy, the gel fraction (and thus the density) is associated with the percolation probability, p( P), (the probability for a site to belong to the infinite cluster) and scales with an exponent, p p cx p P) oc (P - Pc), then E oc The experimentally determined, a, in Figures 14-8(a) and (b) would be r/p. Using a P value equal to 0.4 (the theoretical prediction in a 3 dimension model), we find that r is equal to 1.5, far from the predicted exponent. 

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

The percolation models represent many features of a gelling system, but there are obviously many differences. For example, in bond percolation, the lattice is assumed to have a monomer on every site, whereas gelation generally occurs in dilute systems. This deficiency is addressed by site-bond percolation, in which the sites are randomly populated with monomers and solvent molecules. As the concentration of solvent rises from zero, it is found [35] that Pc increases continuously from the bond-percolation to the site-percolation threshold. (See Fig. 13.) The remarkable (and convenient) fact... 

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