Vulcanization universality class

At relative extent of reaction = 1, the gel fraction is of order unity Pgei 1, and most of the chains are attached to the gel. The percolation transition is almost complete, with very small sol fraction left Psol C 1 at extent of reactionp pk 2p (where 1). Since the gel point corresponds to an average of one crosslink per chain, the end of the gelation regime corresponds to an 

The correlation length and the number of monomers in a characteristic branched polymer N have simple predictions for vulcanization. These predictions can be easily obtained from the mean-field percolation theory [Eqs (6.105) and (6.125) with exponents a=v = jl] by replacing the monomer in the previous treatment by the precursor linear chain of size bN J containing Nq monomers. 

The number of monomers in the characteristic branched polymer iV and its size are symmetric around the gel point e = 0). The molar mass distribution is in fact similar for the same s above and below the gel point (within the framework of mean-field scaling). 

However, Eq. (6.140) shows that the overlap of the largest polymers diminishes as the gel point is approached. For any precursor chain length 

For g g the mean-field theory describes the crosslinking eaction. For Isj g, (sufficiently close to the gel point), critical percolation describes the actual transition. For very long precursor chains, sq is extremely small and the entire experimentally accessible region of crosslinking is described by mean-field percolation (in practice Ng s 100 is sufficient). The limit of Ao = 1 is polymerization of monomers with functionality greater than 2, and all extents of reaction in the range - 1 e 1 are described by critical percolation. 

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Another realization of this transition is crosslinking of preformed polymer. In this case one starts with a polymer melt or a concentrated polymer solution, whereas ( -functional crosslinks have been added. This process is called vulcanization. It turns out already in the classical theory that both models, gelation and vulcanization, belong to the same universal class of liquid-solid transitions, but there are a few differences, to be reported within this section. 

A percolation model, that has been introduced in connection with vulcanization of chains is the case in which two different species of bonds, say A and B, are placed on a lattice with concentrations Ca and Cb, respectively. Species A has the same properties as the usual bonds in random percolation whereas on species B is imposed the restriction that no more than two bonds of the same species B can be formed on the same site. Thus, species B forms polymer chains while species A acts as a crosslink. In the limit Ca = 0, Cb 0, the system reduces to self-avoiding chains, described by exponents different fi-om percolation. The opposite limit, Ca 0, Cb = 0, is the usual random-bond percolation. In the intermediate case, it was foimd - that percolation in which clusters are composed of sites connected by bonds of either species belongs to the same universality class as random percolation, unless the particular situation is realized in which percolation occurs when the typical size of chains made out of B bonds only diverges. In this case, there is a crossover from random percolation exponents to selfavoiding walk exponents , similar to the situation in lattice-gas correlated percolation. (These chains of B atoms must be distinguished from the sometimes chain-like structures formed randomly in the usual percolation process.)... 

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