**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

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.

*© EDP Sciences, Società Italiana di Fisica, and Springer-Verlag, 2004*