  A mean field rate theory description of the homo- and co-polymerization of
$f$-functional molecules is developed, which contains the formation of short
cyclic structures inside the network. The predictions of this model are
compared with Monte-Carlo simulations of cross-linking of star polymers in
solution. We find that homo-polymerizations are well captured by mean-field
models at concentrations larger than one quarter of the geometrical overlap
concentration. All simulation data can be fit using a single geometric
parameter for cyclization. The simulation data reveal that within the range of
parameters of the present study correlations among multiply connected molecules
can be neglected. Thus, mean-field treatments of homopolymerizations are
reasonable approximations, if short cycles are properly addressed.
Co-polymerization is considered in the case of strict A-B reactions, where all
reactive groups of individual molecules are either of type A or B. For these
systems we find a clear influence of the local intermixing of A and B groups
for all concentrations investigated. In consequence, mean-field models are less
appropriate to describe the simulation data. The lack of ring structures
containing an odd number of molecules as compared to homopolymerizations at the
same extent of reaction allows for the formation of stable AB networks at
concentrations one order of magnitude below the geometrical overlap
concentration.
