Symmetries and Elasticity of Nematic Gels

A nematic liquid-crystal gel is a macroscopically homogeneous elastic medium with the rotational symmetry of a nematic liquid crystal. In this paper, we develop a general approach to the study of these gels that incorporates all underlying symmetries. After reviewing traditional elasticity and clarifying the role of broken rotational symmetries in both the reference space of points in the undistorted medium and the target space into which these points are mapped, we explore the unusual properties of nematic gels from a number of perspectives. We show how symmetries of nematic gels formed via spontaneous symmetry breaking from an isotropic gel enforce soft elastic response characterized by the vanishing of a shear modulus and the vanishing of stress up to a critical value of strain along certain directions. We also study the phase transition from isotropic to nematic gels. In addition to being fully consistent with approaches to nematic gels based on rubber elasticity, our description has the important advantages of being independent of a microscopic model, of emphasizing and clarifying the role of broken symmetries in determining elastic response, and of permitting easy incorporation of spatial variations, thermal fluctuations, and gel heterogeneity, thereby allowing a full statistical-mechanical treatment of these novel materials.Comment: 21 pages, 4 eps figure

Fluctuating Nematic Elastomer Membranes: a New Universality Class

We study the flat phase of nematic elastomer membranes with rotational symmetry spontaneously broken by in-plane nematic order. Such state is characterized by a vanishing elastic modulus for simple shear and soft transverse phonons. At harmonic level, in-plane orientational (nematic) order is stable to thermal fluctuations, that lead to short-range in-plane translational (phonon) correlations. To treat thermal fluctuations and relevant elastic nonlinearities, we introduce two generalizations of two-dimensional membranes in a three dimensional space to arbitrary D-dimensional membranes embedded in a d-dimensional space, and analyze their anomalous elasticities in an expansion about D=4. We find a new stable fixed point, that controls long-scale properties of nematic elastomer membranes. It is characterized by singular in-plane elastic moduli that vanish as a power-law eta_lambda=4-D of a relevant inverse length scale (e.g., wavevector) and a finite bending rigidity. Our predictions are asymptotically exact near 4 dimensions.Comment: 18 pages, 4 eps figures. submitted to PR

Non-Uniform Deformations in Liquid Crystalline Elastomers

We develop a molecular model of non-uniform deformations within the framework of liquid crystal rubber elasticity. The result of this work is the general expression for the free energy of deformations which combines the effects of large non-symmetric affine strains in the rubbery network and gradients of curvature deformations of the director field, $F\sim \lambda^{\rm T}\{\nabla{\bf n}\}^2\lambda$. We derive the molecular expressions for the elastic constants governing non-uniform directors in the presence of elastic strains. These constants depend on the polymer step length anisotropy $\ell_{||}/\ell_\perp$ and - most strikingly - have an overall negative sense. We therefore predict that in some circumstances, especially at large elastic deformations $\lambda$, these new terms may overpower the usual, positive Frank elastic moduli of the underlying nematic structure, and also in some cases the coupling of uniform relative rotations of the director to the elastic matrix. In this event highly distorted polydomain textures n(r) would be favoured