  We present a simple reaction kinetics model to describe the polymer synthesis
used by Lusignan et al. (PRE, 60, 5657, 1999) to produce randomly branched
polymers in the vulcanization class. Numerical solution of the rate equations
gives probabilities for different connections in the final product, which we
use to generate a numerical ensemble of representative molecules. All
structural quantities probed by Lusignan et al. are in quantitative agreement
with our results for the entire range of molecular weights considered. However,
with detailed topological information available in our calculations, our
estimate of the `rheologically relevant' linear segment length is smaller than
that estimated by them. We use a numerical method based on tube model of
polymer melts to calculate the rheological properties of such molecules.
Results are in good agreement with experiment, except that in the case of the
largest molecular weight samples our estimate of the zero-shear viscosity is
significantly lower than the experimental findings. Using acid concentration as
an indicator for closeness to the gelation transition, we show that the
high-molecular-weight polymers considered are at the limit of mean-field
behavior - which possibly is the reason for this disagreement. For a truly
mean-field gelation class of model polymers, we numerically calculate the
rheological properties for a range of segment lengths. Our calculations show
that the tube theory with dynamical dilation predicts that, very close to the
gelation limit, contribution to viscosity for this class of polymers is
dominated by the contribution from constraint-release Rouse motion and the
final viscosity exponent approaches Rouse-like value.
