# Stochastic model of randomly end-linked polymer network micro-regions

## TL;DR

This paper develops a stochastic model for the formation of polymer networks, focusing on the variability of micro-region compositions during gelation, providing insights into network dynamics and heterogeneity.

## Contribution

It introduces a master equation approach to model the stochastic formation and variability of micro-regions in polymer networks, including effects of annealing and cooperative binding.

## Key findings

- Model captures the evolution of micro-region configurations over time.
- Quantifies variability of network micro-regions at different reaction extents.
- Highlights the impact of cooperative binding on network formation.

## Abstract

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.