Step growth of two flexible ABf monomers:: The self-return of random branching walks eventually frustrates fractal formation

The competition between the growth of hyperbranched structures and cycle formation that occurs when flexible AB(f) monomers undergo step growth has been simulated with a three-dimensional lattice model in which the monomers are mapped onto several lattice sites. To explore the effect of functionality we have performed studies with f = 2 and 4. The growth is initially fractal, for molecules and branches are self-similar, but it becomes controlled by the formation of intramolecular bonds, a possibility enhanced by growth, for the A group at the root of the growing Cayley tree might react with one of the B groups on the tips of the developing branches. Ultimately every molecule contains a cycle. At t = infinity the most likely cycle has m = 1 residue, with < m >(n,infinity) = 1.65 for the f = 2 system and 1.39 for the f = 4 system, and the corresponding values of the degree of polymerization, < x >(n,infinity), are 10.7 and 7.5. Whatever the value of f, the incidence of cycles throughout the reaction of the two AB(f) monomers follows the relationships R-m = C(o)p(a)(m) m(-)/1, with p(a) the extent of reaction. gamma(1), being 2.714(+/-0.005) for the AB(2) system, and C-o = N-o < x >(n,infinity) /zeta(2.714), where N-o is the initial number of monomers. The mean degree of polymerization is given exactly by < x >(n) = 1/(1 - p(e)), where p(e) includes only the extent of reaction between the molecules. The number of oligomers of size x follows the Flory distribution expression just to start with, and later only if the expedient is adopted of replacing p(a) with p(e), but at the end-when f = 2-a second power series is found: N-x = N-1,N-infinity x(-1.5) for 0 < x < 48. The exponent, -chi(w), in the corresponding weight distribution function is -0.50, a value that cannot persist to high values of x, since the sum of that series is not bounded, so N-x and W-x must fall faster at higher x. These power laws are independent of the manner in which the AB(2) molecule is mapped onto the lattice. In the AB(4) system again rings form, but both their distribution at moderate values of m and the number and weight distributions, N-x and W-x, are curved on the double logarithmic plots, and are so even at the end for N-x and W-x when x > 12. The initial values of gamma(1) and of chi(n) are 2.8 and 1.29 respectively, and measure the greater ease of cycle formation and of scope for growth when f = 4. The eventual deviation from the early trends may reflect the exclusion from the neighborhood of the A groups at the roots of trees of other fractals, thus promoting cyclizations intramolecularly, and it occurs sooner, as m or x rises, when f is doubled. The populations of all the structural isomers among the lower oligomers have been obtained for both systems, and the extra isomers among the AB(4) oligomers identified. Mean extent of reaction Vectors and Kirchhoff matrices were obtained for these and the higher oligomers, so that the patterns of molecular structure are demonstrated in both systems at three stages of the reaction: some structural characteristics are similar as in fractals, as the size varies. The molecules grow as hyperbranches, but as they flourish a route is eventualy found back to the A group at the tree, so terminating growth and limiting objects to a finite size.