When nematic liquid crystals are constrained to a curved surface, the geometry induces distortions in the molecular orientation. The mechanisms of the geometrical frustration involve the intrinsic as well as the extrinsic geometry of the underlying substrate. We show that the nematic elastic energy promotes the alignment of the flux lines of the nematic director towards geodesics and/or lines of curvature of the surface. As a consequence, the influence of the curvature can be tuned through the Frank elastic moduli. To illustrate this effect, we consider the simple case of nematics lying on a cylindrical shell. By combining the curvature effects with external magnetic fields, the molecular alignment can be reoriented or switched between two stable configurations. This enables the manipulation of nematic alignment for the design of new materials and technological devices.

We propose a continuum model to describe the molecular alignment in thin nematic shells. By contrast with previous accounts, the two-dimensional free energy, aimed at describing the physics of thin films of nematics deposited on curved substrates, is not postulated, but it is deduced from the conventional three-dimensional theories of nematic liquid crystals. Both the director and the order-tensor theories are taken into account. The so-obtained surface energies exhibit extra terms compared to earlier models. These terms reflect the coupling of the shell extrinsic curvature with the nematic order parameters. As expected, the shape of the shell plays a key role in the equilibrium configurations of nematics coating it.

We found a mistake in the derivation of the torque tensor in [2]. It does not affect all the other results contained in that paper, such as the derivations of the equilibrium equations and the stress tensor.Equation ( 35) in [2] is wrong. To derive the torque tensor and the equation of balance of torques, one must appeal to the frame indifference of the energy density w = w(ν, L, n, ∇ s n, q, ∇ s q) and not to the invariance under rigid virtual displacements as stated in [2]. Therefore, by following similar arguments as in [1] the invariance of the energy density w under infinitesimal rigid rotations implies thatwhere Ω is an arbitrary skew-symmetric tensor. In view of equation ( 21) in [2] and the representation of the stress tensor reported there, (1) can be rewritten asBecause of the arbitrariness of the skew-symmetric tensor Ω, the tensor between the curly brackets in ( 2) is symmetric and thus its corresponding axial vector vanishes, namely(3) Next, on using the equilibrium equation ( 24) in [2] we obtain Corrigendum

Layered phases are a common pattern of self-organization for several soft materials. These phases undergo buckling instability when subjected to dilatative strain: beyond a critical threshold, layers, initially flat, exhibit a periodical undulation. By using a continuum model, in a finite deformation framework, an expression for the critical threshold is provided, which differs from that predicted by the Helfrich-Hurault theory and yet it reverts to it in a thick specimen limit. With respect to the relevant literature, an analogous disagreement is found in the undulation amplitude expression as well. The obtained results appear particularly relevant when dealing with layered materials whose intrinsic coherence length is comparable to the cell thickness.

We derive the hydrodynamic equations for nematic liquid crystals lying on curved substrates. We invoke the Lagrange-Rayleigh variational principle to adapt the Ericksen-Leslie theory to two-dimensional nematics in which a degenerate anchoring of the molecules on the substrate is enforced. The only constitutive assumptions in this scheme concern the free-energy density, given by the two-dimensional Frank potential, and the density of dissipation which is required to satisfy appropriate invariance requirements. The resulting equations of motion couple the velocity field, the director alignment, and the curvature of the shell. To illustrate our findings, we consider the effect of a simple shear flow on the alignment of a nematic lying on a cylindrical shell.