  Polymerization and formation of crosslinked polymer networks are important
processes in manufacturing, materials fabrication, and in the case of hydrated
polymer networks, synthesis of biomedical materials, drug delivery, and tissue
engineering. While considerable research has been devoted to the modeling of
polymer networks to determine averaged, mean-field, global properties, there
are fewer studies that specifically examine the variance of the composition
across "micro-regions" (composed of a large, but finite, number of polymer
network strands) within the larger polymer network.Here, we mathematically
model the stochastic formation of polymer networks comprised of linear
homobifunctional network strands that undergo an end-linking gelation process.
We introduce a master equation that describes the evolution of the
probabilities of possible network micro-region configurations as a function of
time and extent of reaction. We specifically focus on the dynamics of network
formation and the statistical variability of the gel micro-regions,
particularly at intermediate extents of reaction. We also consider possible
annealing effects and study how cooperative binding between the two end-groups
on a single network-strand affects network formation. Our results allow for a
more detailed and thorough understanding of polymer network dynamics and
variability of network properties.
