The Immunity of Polymer-Microemulsion Networks

The concept of network immunity, i.e., the robustness of the network connectivity after a random deletion of edges or vertices, has been investigated in biological or communication networks. We apply this concept to a self-assembling, physical network of microemulsion droplets connected by telechelic polymers, where more than one polymer can connect a pair of droplets. The gel phase of this system has higher immunity if it is more likely to survive (i.e., maintain a macroscopic, connected component) when some of the polymers are randomly degraded. We consider the distribution $p(\sigma)$ of the number of polymers between a pair of droplets, and show that gel immunity decreases as the variance of $p(\sigma)$ increases. Repulsive interactions between the polymers decrease the variance, while attractive interactions increase the variance, and may result in a bimodal $p(\sigma)$.