Mean-field model of gelation

The total number density of molecules is the sum of number densities of all finite-size polymers and unreacted monomers. The number density of all finite molecules ntot(p) is the zeroth moment of the number density distribution function  

The number-average degree of polymerization [see Eq. (1.27)] is the average number of monomers per finite-size polymer  

The weight-average degree of polymerization is the ratio of second and first moments [recall Eq. (1.31)]  

Note that the sums are understood to run only over finite-size molecules. Beyond the gel point, the gel fraction is excluded from such sums. 

A convenient way of presenting the mean-field model (though it is not the way it was originally defined) is by placing monomers at the sites of an infinite Bethe lattice, a small part of which is shown in Fig. 6.13. The Bethe lattice has the advantage of directly taking into account the functionality of the monomers/by adopting this functionality for the lattice. For trifunc- 

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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. 

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... 

Theories of Gelation. The classical or mean field theory of polymeri2ation (4) is useful for visuali2ing the conditions for gelation. This model yields a degree of reaction, of one-third at the time of gelation for chemical species having functionaUty equal to four. Two-thirds of the possible... 

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... 

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. 

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. 

So far we have discussed the distribution of degrees of polymerization of molecules both below and above the gel point. In the present section, we will describe their spacial sizes in the polymerization reactor. The Bethe lattice introduced above for the mean-field gelation model properly describes the connectivity of monomers into trcc-like branched molecules, but is not... 

The experimental results presented here, below and beyond the gel point, show clearly that the formation of polyurethane gel by polycondensation is a critical phenomenon which cannot be described by the mean-field theory. The good agreement of exponent values found experimentally and calculated by Monte Carlo simulations shows that this type of polycondensation process can be described by the percolation model. Recent experimental results " performed on different kinds of gelation process seem to indicate that, more generally, percolation and gelation, with permanent crosslinks, belong to the same class of universality. 

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