No abstract

Orientational order parameters can be effectively and economically defined using spherical tensors. However, their definition in terms of Cartesian tensors can sometimes provide a clearer physical intuition. We show that it is possible to build a fully Cartesian theory of the orientational order parameters which is consistent with the traditional spherical tensor approach. The key idea is to build a generalised multi-pole expansion of the orientational probability distribution function in terms of outer products of rotation matrices. Furthermore, we show that the Saupe ordering super-matrix, as found, for example, in the text by de Gennes and Prost [The Physics of Liquid Crystals, 2nd ed. (Oxford University Press, Oxford, UK, 1995)] and which is used to define the Cartesian second-rank orientational order parameters, is not consistent with its spherical tensor counterpart. We then propose a symmetric version of the Saupe super-matrix which is fully consistent with the spherical tensor definition. The proposed definition is important for a correct description of liquid crystal materials composed of low symmetry molecules. (C) 2011 American Institute of Physics. [doi:10.1063/1.3589961

Despite the fact that quantitative experimental data have been available for more than forty years now, nematoacoustics still poses intriguing theoretical and experimental problems. In this paper, we prove that the main observed features of acoustic wave propagation through a nematic liquid crystal cell - namely, the frequency-dependent anisotropy of sound velocity and acoustic attenuation - can be explained by properly accounting for two fundamental features of the nematic response: anisotropy and relaxation. The latter concept - new in liquid crystal modelling - provides the first theoretical explanation of the structural relaxation process hypothesised long ago by Mullen and co-workers [Mullen et al., Phys. Rev. Lett., 1972, 28, 799]. We compare and contrast our proposal with an alternative theory where the liquid crystal is modelled as an anisotropic second-gradient fluid.

Nematic liquid crystals exhibit both crystallike and fluidlike features. In particular, the propagation of an acoustic wave shows an interesting occurrence of some of the solidlike features at the hydrodynamic level, namely, the frequency-dependent anisotropy of sound velocity and acoustic attenuation. The non-Newtonian behavior of nematics also emerges from the frequency-dependent viscosity coefficients. To account for these phenomena, we put forward a viscoelastic model of nematic liquid crystals, and we extend our previous theory to fully include the combined effects of compressibility, anisotropic elasticity, and dynamic relaxation, at any shear rate. The low-frequency limit agrees with the compressible Ericksen-Leslie theory, while at intermediate frequencies the model correctly captures the relaxation mechanisms underlying finite shear and bulk elastic moduli. We show that there are only four relaxation times allowed by the uniaxial symmetry.

A growing or compressed thin elastic sheet adhered to a rigid substrate can exhibit a buckling instability, forming an inward hump. Our study shows that the strip morphology depends on the delicate balance between the compression energy and the bending energy. We find that this instability is a first-order phase transition between the adhered solution and the buckled solution whose main control parameter is related to the sheet stretchability. In the nearly unstretchable regime, we provide an analytic expression for the critical threshold. Compressibility is the key assumption which allows us to resolve the apparent paradox of an unbounded pressure exerted on the external wall by a confined flexible loop.

The growth of an elastic film adhered to a confining substrate might lead to the formation of delamination blisters. Many results have been derived when the substrate is flat. The equilibrium shapes, beyond small deformations, are determined by the interplay between the sheet elastic energy and the adhesion potential due to capillarity. Here, we study a non-trivial generalization to this problem and consider the adhesion of a growing elastic loop to a confining circular substrate. The fundamental equations, i.e., the Euler Elastica equation, the boundary conditions and the transversality condition, are derived from a variational procedure. In contrast to the planar case, the curvature of the delimiting wall appears in the transversality condition, thus acting as a further source of adhesion. We provide the analytic solution to the problem under study in terms of elliptic integrals and perform the numerical and the asymptotic analysis of the characteristic lengths of the blister. Finally, and in contrast to previous studies, we also discuss the mechanics and the internal stresses in the case of vanishing adhesion. Specifically, we give a theoretical explanation to the observed divergence of the mean pressure exerted by the strip on the container in the limit of small excess-length. (C) 2014 Elsevier B.V. All rights reserved

We put forward a hydrodynamic theory of nematic liquid crystals that includes both anisotropic elasticity and dynamic relaxation. Liquid remodeling is encompassed through a continuous update of the shear-stress free configuration. The low-frequency limit of the dynamical theory reproduces the classical Ericksen-Leslie theory, but it predicts two independent identities between the six Leslie viscosity coefficients. One replicates Parodi's relation, while the other-which involves five Leslie viscosities in a nonlinear way-is new. We discuss its significance, and we test its validity against evidence from physical experiments, independent theoretical predictions, and molecular-dynamics simulations.

Abstract. We investigate the invariants of the 25-dimensional real representation of the group SO(3) Z 2 given by the left and right actions of SO(3) on 5 × 5 matrices together with matrix transposition; the action on column vectors is the irreducible 5-dimensional representation of SO(3). The 25-dimensional representation arises naturally in the study of nematic liquid crystals, where the second-rank orientational order parameters of a molecule are represented by a symmetric 3 × 3 traceless symmetric matrix, and where a rigid rotation in R 3 induces a linear transformation of this space of matrices. The entropy contribution to a free energy density function in this context turns out to have SO(3) Z 2 symmetry. Although it is unrealistic to expect to describe the complete algebraic structure of the ring of invariants, we are able to calculate the Molien series giving the number of linearly independent invariants at each homogeneous degree, and to express this as a rational function indicating the degrees of invariant polynomials that constitute a basis of 19 primary invariants. The algebra of invariants up to degree 4 is investigated in detail.