Point defects in nematic gels: The case for hedgehogs

We address the question of whether a nematic gel is capable of sustaining a radially-symmetric point defect (or, hedgehog). We consider the special case of a gel cross-linked in a state where the mesogens are randomly aligned, and study the behavior of a spherical specimen with boundary subjected to a uniform radial displacement. For simplicity, we allow only for distortions in which the chain conformation is uniaxial with constant chain anisotropy and, thus, is determined by a unit director field. Further, we use the particular free-energy density function arising from the neo-classical molecular-statistical description of nematic gels. We find that the potential energy of the specimen is a nonconvex function of the boundary displacement and chain anisotropy. In particular, whenever the displacement of the specimen boundary involves a relative radial expansion in excess of 0.35, which is reasonably mild for gel-like substances, the theory predicts an energetic preference for states involving a hedgehog at the center of the specimen. Under such conditions, states in which the chain anisotropy is either oblate or prolate have total free-energy less than that of an isotropic comparison state. However, the oblate alternative always provides the global minimum of the total free-energy. The Cauchy stress associated with an energetically-preferred hedgehog is found to vanish in a relatively large region surrounding the hedgehog. The radial component of Cauchy stress is tensile and exhibits a non-monotonic character with a peak value an order of magnitude less than what would be observed in a specimen consisting of a comparable isotropic gel. The hoop component of Cauchy stress is also non-monotonic, but, as opposed to being purely tensile, goes between a tensile maximum to a compressive minimum at the specimen boundary.