Eur. Phys. J. E (2004) 13: 79-86
https://doi.org/10.1140/epje/e2004-00042-3

Power law polydispersity and fractal structure of hyperbranched polymers

Polymer IRC, Department of Physics & Astronomy, University of Leeds, LS2 9JT, Leeds, UK

^* e-mail: d.m.a.buzza@leeds.ac.uk

Abstract

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 _{N 0}-B_{f - 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 , 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 , we assume that the number density of hyperbranched polymers with degree of polymerisation N generally obeys the scaling form 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 cuts off the power law sharply for ). We find the upper critical dimension for this system is d _c = 4, so that for the mean field values for the polydispersity exponent and fractal dimension apply: , d _f = 4. For d = 3, mean field theory is still correct for where is the Ginzburg point; for , mean field theory applies on small mass scales N<N _c but breaks down on larger mass scales N>N _c where is a cross-over mass. Within the Ginzburg zone (i.e., d<d _c , ), we show that the hyperbranched chains on mass scales N>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 so that polydispersity is irrelevant for excluded volume screening in hyperbranched polymer melts.