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The Localization Model of Rubber Elasticity

While the unusual elastic properties of rubbery polymer materials have been studied for literally centuries, a quantitative theoretical description of these properties in terms of molecular structural parameters remains a challenging task. Classical network elasticity theories are based on the concept of flexible volumeless network chains fixed into a network in which there are no excluded volume, or even topological, interactions between the chains and where the chains explore accessible configurations by Brownian motion. In this type of model, the elasticity of the deformed network derives from the entropic changes in this idealized network arising from a deformation of the network junction positions. The shortcoming of this approach is clear from the observation that unswollen rubbery materials are nearly incompressible, reflecting the existence of strong intermolecular interactions that restrict the polymer chains to an exploration of their local molecular environments. The imposition of a deformation of these solid rubbery materials then necessitates a consideration of how local molecular packing constraints become modified under deformation and the impact of these changes on the macroscopic elasticity of the material as a whole. Many researchers have struggled with this difficult problem, but we focus on the simple 'localization model' of rubber elasticity introduced by Gaylord and Douglas (GD), which provides an attractive and mathematically simple minimal model for the network elasticity of rubbers having strong intermolecular interactions in the dense polymer state. GD assume that the network chain segments are localized at the network junctions, as in classical elasticity theory, but they also consider the chains to be localized along their contours by a local harmonic potential arising from inter-particle packing interactions, as in atoms in a simple harmonic crystal. This is the familiar Edwards-De Gennes 'tube' model of polymers in the condensed state. The existence of the tube means that the entropy of the network chains is reduced relative to chains without this constraint, but the crucial problem is how this entropy change becomes modified as the material becomes deformed. GD approach this problem by first observing that both the volume of the material and the network chains are essentially invariant under deformation so they require that the dimensions of the confining tube to also be a deformation invariant where this constraint is applied at a segmental level. The properties of the resulting 'localization model' of elasticity of dry and swollen rubbers are summarized and some extensions of the model to describe chain finite extensibility and stiff polymer networks are briefly summarized.