Three Faces of Polymer Entanglements

There are four classes of flexible polymer networks: unentangled networks well-described by the classical phantom model and three classes of networks with qualitatively different types of entanglements: collective, pairwise, or transient. The classical model of polymer entanglements is the confining tube model that represents the topological effects of neighboring polymers on a given one by a confining potential. Since the potential cannot store any energy upon network deformation, the diameter of the confining tube and the conformations of network strands change non-affinely with macroscopic deformation. In contrast, the pairwise entanglements model and the primitive path analysis based on it identify topological restriction between a pair of chains (similar to slide-rings locally holding these two chains together). Such pairwise topological entanglements deform affinely with macroscopic deformation of polymer network and therefore are qualitatively different from confining tube entanglements. We develop a theory connecting these two models of polymer entanglements and show that the molecular weight between entanglements in the confining tube model is smaller than the one obtained from the primitive path analysis. The number of monomers in the affine strand increases upon network swelling due to the dis-interpenetration of network chains and saturates at a value corresponding to the pairwise entanglements determined by the primitive path analysis. This saturation corresponds to the collective to pairwise entanglement transition. Both collective and pairwise entanglements improve mechanical properties, such as the toughness of polymer networks by re-equilibrating tension in network strands. Similar features of entanglements of strands of the second network in the fine mesh of the first network lead to exceptional properties of double-networks. The third type of topological interactions between polymers is observed upon polymer deswelling and is similar to entanglements forcing non-concatenated rings into compact loopy globular conformations. Such transient entanglements formed upon deswelling of polymer gels store the extra length of polymer strands in fractal loopy conformations and allow super-elasticity of de-swollen gels – their very large reversible deformability.