Elasticity of polydomain liquid crystal elastomers

Liquid crystal elastomers are rubbery networks of entropically dominated polymer chains that exhibit mobile liquid crystalline order. These materials have been of recent interest for the soft behavior (large deformations at relatively small stresses) observed in monodomain specimens where the director is the same at every point in the relaxed elastomer. This paper concerns the soft behavior of polydomain specimens where the director points in different directions at different points in the relaxed elastomer. We show that there is a significant difference between polydomains cross-linked in homogeneous high symmetry states then cooled to low symmetry polydomain states and those cross-linked directly in the low symmetry polydomain state. Specifically, elastomers cross-linked in the isotropic state then cooled to a nematic polydomain will, in the ideal limit, be perfectly soft, and with the introduction of non-ideality, will deform at very low stress until they are macroscopically aligned. In fact, we expect these samples to exhibit elasticity significantly softer than monodomain samples, as has recently been observed by Urayama et al. Further, the director patterns will be fine-scale structures that are macroscopically isotropic and not schlieren textures. In contrast, polydomains cross-linked in the nematic polydomain state will be mechanically harder and contain characteristic schlieren director patterns. The models we use for polydomain elastomers are spatially heterogeneous extensions of the neo-classical soft and semi-soft free energies used successfully to describe monodomain samples. We elucidate the effective behavior by bounding the energies using Taylor-like (compatible test strain fields) and Sachs-like (equilibrated stress field) bounds, both valid to large deformations. Good agreement is found with experiments. We also analyze smectic polydomain elastomers and propose that polydomain SmC⁎ elastomers cross-linked in the SmA monodomain state are promising candidates for low field electrical actuation.