Cycles within micronets and at the gel point

The formation of cycles during the reaction of an A(2) monomer with an NB3 monomer has been modeled in a new manner on a lattice, each component of the residues being placed upon different lattice sites and the functional groups being allowed to react by chance with their neighbors. The Monte Carlo method differs from previous percolation models in providing smooth distributions of cycles of different size in quantities similar to those found by experiment without any adjustable parameter, and differs from mean field treatments in placing the different and nonphantomlike structures properly within three-dimensional space. Not only may cycles form in competition with branching growth, but if the statistics require it, segments of one cycle may be shared with those of any number of others, as Houwink indicated in 1935. After a small number of simple cycle-containing micronet molecules are considered, a new measure of cycle number, C, is introduced to cope with issues intractable with cycle rank, c, alone. For an equimolar mixture of A(2) and B-3 monomers a simulation found that 22% of the two-node molecules at the end of a reaction contained cycles, and reported their proportions, to show that in these the cycles were remote from each other. At the gel point identified by the extent of reactions between molecules an analytical method identified all the cycles present in the system up to a certain Size, and found that when the number of cycles of m nodes, R-m, is examined in terms of the relationship R-m = Km(-k), k has the initial value of 2.50 as for difunctional monomers, but k then falls, an effect permitted by the trifunctional residues and attributed to cycles sharing segments to an increasing extent as m increases. By the time m rises to about 8, k has fallen to 2.00, which is a critical value, as the total weight of C cycles in the system, K'zeta(k - 1), is then unbounded if that trend persists-from the property of the Riemann xi function, zeta(1.00)-for m may rise to infinity in a gel. It appears that at the gel point this critical behavior applies and that it is enhanced, as k falls further to about zero, when m congruent to 24. The periodic boundaries of the model were not large enough to provide the exact behavior of the cycle number as m becomes larger, but an explosion is certainly indicated by k becoming negative. A resilient product is predicted at the gel point.