Statistical models of elasticity in main chain and smectic liquid ...

Chapter OneIntroductionThe subject of this thesisistheelastic propertiesof twodifferenttypesof polymer networks: main chain nematic elastomers and smectic elastomers.Both of these systems are anisotropic, and the molecular theoriesused to describe them here are in the same vein as previous phantom chainmodels of rubber elasticity.1.1 Classical rubber elasticityTheclassical theory of rubberelasticity, based on a phantom network of Gaussianchains, is surprisingly successful. Modelling the cross-link points of rubberymaterials as deforming affinely with the external strain (R → λ·R) oneobtains the free energy density of a Gaussian phantom network as)f = 1 2(λ µTr T ·λ ,where µ is the shear modulus of the network [1]. For a rubber network theconstraint of incompressibility (det(λ) = 1) is usually imposed. Despite thesuccesses of the phantom network model it neglects, amongst other things,entanglements of the network strands and non-Gaussian nature of the chains,for example their finite extensibility. One way to correct for these failingsis the Mooney-Rivlin approach. The free energy density should be invariantunder rotation of the target and reference states, so can only be a function ofthe rotational invariants of the Cauchy-Green deformation tensor C = λ T ·λ[2]. These invariants can be added together to phenomenologically fit thedeviations from the phantom network model. However, this approach doesnot provide any insight to the microscopic details of the network. Otherattempts have also been made to include entanglements (e.g. [3]), but suchcorrections will be neglected here.1