From Vulcanization to Isotropic and Nematic Rubber Elasticity

– A Landau theory is constructed for the vulcanization transition in cross-linked polymer systems with spontaneous nematic ordering. The neo-classical theory of the elasticity of nematic elastomers is derived via the minimization of this Landau free energy; this neoclassical theory contains the classical theory of rubber elasticity as its isotropic limit. Our work not only reveals the statistical-mechanical roots of these elasticity theories, but also demonstrates that they are applicable to a wide class of random solids. It also constitutes a starting-point for the investigation of sample-to-sample fluctuations in various forms of vulcanized matter. The classical theory of rubber elasticity [1] has been remarkably successful. A blend of phenomenology and molecular-level reasoning, it is based on a few simple assumptions, and bears great predictive and descriptive power. By modeling rubbery materials (i.e. elastomers) as incompressible networks of entropic Gaussian chains, it gives their elastic free energy density f at temperature T as f = μ 2 TrΛΛ, (1) for all uniform deformations r → Λ · r that conserve the volume (i.e. detΛ ≡ 1). For most rubber-like materials the assumption of incompressibility is well satisfied.() The shear modulus μ is given by nc T , where the constant nc is usually referred as “the density of effective chains in the network.” The classical theory [i.e. Eq. (1) and associated arguments] explain many essential features of rubbery materials, such as their stress-strain curves (at least for deformations that are not too large), and the striking temperature dependence of their shear moduli, as well as their strain-birefringence (i.e. the stress-optical effect). ()This assumption may break down for swollen rubbers and gels. In this case, a finite-compressibility version of Eq. (1) remains valid. In fact, this is what we will derive in the paper.