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 fo rmation that occurs when flexible AB(f) monomers undergo step growth has be en simulated with a three-dimensional lattice model in which the monomers a re mapped onto several lattice sites. To explore the effect of functionalit y we have performed studies with f = 2 and 4. The growth is initially fract al, 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 on e 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 th e f = 4 system, and the corresponding values of the degree of polymerizatio n, < x >(n,infinity), are 10.7 and 7.5. Whatever the value of f, the incide nce 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 m ean degree of polymerization is given exactly by < x >(n) = 1/(1 - p(e)), w here p(e) includes only the extent of reaction between the molecules. The n umber 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 corr esponding weight distribution function is -0.50, a value that cannot persis t 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 t he 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 t he double logarithmic plots, and are so even at the end for N-x and W-x whe n x > 12. The initial values of gamma(1) and of chi(n) are 2.8 and 1.29 res pectively, and measure the greater ease of cycle formation and of scope for growth when f = 4. The eventual deviation from the early trends may reflec t the exclusion from the neighborhood of the A groups at the roots of trees of other fractals, thus promoting cyclizations intramolecularly, and it oc curs 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 s ystems, and the extra isomers among the AB(4) oligomers identified. Mean ex tent of reaction Vectors and Kirchhoff matrices were obtained for these and the higher oligomers, so that the patterns of molecular structure are demo nstrated in both systems at three stages of the reaction: some structural c haracteristics are similar as in fractals, as the size varies. The molecule s grow as hyperbranches, but as they flourish a route is eventualy found ba ck to the A group at the tree, so terminating growth and limiting objects t o a finite size.