Anisotropic wave propagation in nematic liquid crystals

Chiral phase-coexistence in compressed double-twist elastomers

We adapt the theory of anisotropic rubber elasticity to model cross-linked double-twist liquid crystal cylinders such as exhibited in biological systems. In mechanical extension we recover strain-straightening, but with an exact expression in the small twist-angle limit. In compression, we observe coexistence between high and low twist phases. Coexistence begins at small compressive strains and is robustly observed for any anisotropic cross-links and for general double-twist functions - but disappears at large twist angles. Within the coexistence region, significant compression of double-twist cylinders is allowed at constant stress. Our results are qualitatively consistent with previous observations of swollen or compressed collagen fibrils, indicating that this phenomenon may be readily accessible experimentally.

Two-shape-tensor model for tumbling in nematic polymers and liquid crystals

Most, but not all, liquid crystals tend to align when subject to shear flow, while most nematic polymeric liquid crystals undergo a tumbling instability, where the director rotates with the flow. The reasons of this instability remain elusive, as it is possible to find similar molecules exhibiting opposite behaviors. We propose a continuum theory suitable for describing a wide range of material behaviors, ranging form nematic elastomers to nematic polymers and nematic liquid crystals, where the material parameters have meaningful physical interpretations and the conditions for tumbling emerge clearly. There are two possible ways to relax the internal stress in a nematic material. The first is the reorganization of the polymer network, the second is the alignment of the network natural axis with respect to the principal direction of the effective strain. We show that tumbling occurs whenever the second mechanism is less efficient than the first. Furthermore, we provide a justification of the experimental fact that at high temperatures, in an isotropic phase, only flow alignment is observed and no tumbling is possible, even in polymers.

Active nematic gels as active relaxing solids

I propose a continuum theory for active nematic gels, defined as fluids or suspensions of orientable rodlike objects endowed with active dynamics, that is based on symmetry arguments and compatibility with thermodynamics. The starting point is our recent theory that models (passive) nematic liquid crystals as relaxing nematic elastomers. The interplay between viscoelastic response and active dynamics of the microscopic constituents is naturally taken into account. By contrast with standard theories, activity is not introduced as an additional term of the stress tensor, but it is added as an external remodeling force that competes with the passive relaxation dynamics and drags the system out of equilibrium. In a simple one-dimensional channel geometry, we show that the interaction between nonuniform nematic order and activity results in either a spontaneous flow of particles or a self-organization into subchannels flowing in opposite directions.

Generalization of the Ericksen-Leslie theory

We generalize the Ericksen-Leslie theory for the dynamics of nematic liquid crystals by including the director-density coupling energy [Formula: see text]. It corresponds to the cost in energy due to the interaction of spatially varying mass density [Formula: see text] and the liquid crystal director [Formula: see text]. A striking confirmation of the theory is achieved by confronting it with experimental data on acoustical Fréedericksz transition.

Viscoelastic nematodynamics

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.

Hydrodynamic theory for nematic shells: The interplay among curvature, flow, and alignment

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.

Liquid relaxation: A new Parodi-like relation for nematic liquid crystals

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.