The contribution to rubber elasticity of topological entanglements

Abstract

A microscopic explanation of the departures from the statistical kinetic theory of rubber elasticity is given using a topological description of entanglements. This is accomplished in terms of linking numbers which can be topologically defined between pairs of entangled polymer molecules. A statistical mechanical treatment of the linking numbers is presented and an expression derived for the elastic modulus of an entangled coil. Our results show the expected deformation softening in both uniaxial extension and compression. We identify the dependence of this behaviour on the density, the entangling ability of the molecule and on the details of fabrication of the network. We also show that, with the framework of the concepts developed in the paper, the deformation softening behaviour implies that the system is under-entangled with respect to an intrinsic degree of entanglement characteristic of the system. This intrinsic degree of entanglement is defined in the paper and for states of entanglement exceeding this quantity a deformation hardening is predicted.