  The universality of the swelling of the radius of gyration of a homopolymer
relative to its value in the $\theta$ state, independent of polymer-solvent
chemistry, in the crossover regime between $\theta$ and athermal solvent
conditions, is well known. Here we study, by Brownian dynamics, a polymer model
where a subset of monomers is labelled as "stickers". The mutual interaction of
the stickers is more attractive than those of the other ("backbone") monomers,
and has the additional important characteristic of "functionality" $\varphi$,
i.e., the maximum number of stickers that can locally bind to a given sticker.
A saturated bond formed in this manner remains bound until it breaks due to
thermal fluctuations, a requirement which can be viewed as an additional
Boolean degree of freedom that describes the bonding. This, in turn, makes the
question of the order of the collapse transition a non-trivial one.
Nevertheless, for the parameters that we have studied (in particular,
$\varphi=1$), we find a standard second-order $\theta$ collapse, using a
renormalised solvent quality parameter that takes into account the increased
average attraction due to the presence of stickers. We examine the swelling of
the radius of gyration of such a sticky polymer relative to its value in the
altered $\theta$ state, using a novel potential to model the various excluded
volume interactions that occur between the monomers on the chain. We find that
the swelling of such sticky polymers is identical to the universal swelling of
homopolymers in the thermal crossover regime. Additionally, for our model, the
Kuhn segment length under $\theta$ conditions is found to be the same for
chains with and without stickers.
