Molar mass distribution and gel fraction

It is possible to generalize the results, derived in Section 6.4 for the Bethe lattice, to any percolation problem. Of particular interest is gelation (per-colation) in two-dimensional and three-dimensional spaces. Unlike mean- 

Near the gel point, the system consists of a highly polydisperse distribution of polymers. One of the most important features of gelation is that the number density of polymers near the gel point has a power law 

The critical exponent t is the same above and below the gel point and is called the Fisher exponent. The number of monomers N in the characteristic branched polymer increases as the gel point is approached (from either ide) and diverges as a powetM )f the distance from the gel point, char- 

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 

N/N constructs a universal scaling curve that reduces all molar mass distributions at different extents of reaction below the gel point to a single curve. Two such scaling curves are shown in Fig. 6.26. 

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