Power law polydispersity and fractal structure of hyperbranched polymers

Using the complementary approaches of Flory theory and the overlap function, we study the molecular weight distribution and conformation of hyperbranched polymers formed by the melt polycondensation of A-R-N0-Bf - 1 monomers in their reaction bath close to the mean field gel point p(A) = 1, where p(A) is the fraction of reacted A groups. Here f greater than or equal to 3, N-0 is the degree of polymerisation of the linear spacer linking the A group and the f-1 B groups and condensation occurs exclusively between the A and B groups. For epsilon = (1-p(A)) <<1, we assume that the number density of hyperbranched polymers with degree of polymerisation N generally obeys the scaling form n(N) = N(-tau)f(N/N-t) and we explicitly show that this scaling assumption is correct in the mean field regime (here N-l is the largest characteristic degree of polymerisation and the function f(N/N-t) cuts off the power law sharply for N>N-t). We find the upper critical dimension for this system is d(c) = 4, so that for dgreater than or equal tod(e) the mean field values for the polydispersity exponent and fractal dimension apply: tau = 3/2, d(f) = 4. For d = 3, mean field theory is still correct for epsilon > epsilon(G) where epsilon(G) congruent to N-0(-1) is the Ginzburg point; for epsilon<ε(G), mean field theory applies on small mass scales N&LT;N-c but breaks down on larger mass scales N&GT;N-c where N-c ≅ N-0(3) is a cross-over mass. Within the Ginzburg zone (i.e., d<d(c), epsilon<ε(G)), we show that the hyperbranched chains on mass scales N&GT;N-c are non-Gaussian with fractal dimension given by d(f) = d (for d = 2,3,4). Our results are qualitatively different from those of the percolation model and indicate that the polycondensation of AB(f-1), unlike polymer gelation, is not described by percolation theory. Instead many of our results are similar to those for a monodisperse melt of randomly branched polymers, a consequence of the fact that τ < 2 so that polydispersity is irrelevant for excluded volume screening in hyperbranched polymer melts.