Step Growth of Two Flexible AB f 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 = ∞ the most likely cycle has m = 1 residue, with 〈m〉n, ∞ = 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, ∞, 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 -γ 1 , with p a the extent of reaction. γ 1, being 2.714(±0.005) for the AB 2 system, and C o = N o 〈x〉n, ∞ /ζ(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, ∞ x -1.5 for 0 < x < 48. The exponent, −χ 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 γ 1 and of χ 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 pr