Mesoscale models for soft layered materials: the role of curvatures in topological defect motion, flows and instabilities

Abstract

Curvature driven phenomena in soft matter involves both complex geometry at small scales and anisotropies associated with material symmetries. In particular, the class of soft modulated materials present molecules that are organized in layers, so that material properties significantly vary between the direction normal to the layer and those of the layers. This is the case of smectic liquid crystals, which behave as a solid in the direction normal to the molecular layers, while each of their layers behave as a two-dimensional fluid. Under appropriate boundary conditions, smectic layers are known to bend and form focal conic defects, whose curvatures significantly increase in magnitude as the tip of the cone is approached. Intriguingly, experiments on smectic films presenting arrays of focal conics have shown that these materials undergo unexpected morphological transitions, which are not explained by classical local equilibrium thermodynamics. For example, annealing of focal conic domains can lead to conical pyramids, changing the sign of both mean and Gaussian curvatures and exposing smectic layers at the interface. In order to understand the role played by high order curvature terms on the stability and evolution of a smectic film interface, we propose a phase-field model for a smectic-isotropic system. Through an asymptotic analysis, we generalize the classical condition of local equilibrium, the Gibbs-Thomson equation, to include contributions from surface bending and torsion and a dependence on the layer orientation at the interface. Numerical results for a diffusive evolution of the interface reproduce the focal conic to conical pyramid transition in smectic films, and we show that such morphologies can be explained in light of the derived interface equations. We then generalize this model to include flows and to allow each phase to have a different density. We derive both a quasi-incompressible and a weakly compressible smectic-isotropic model from this approach, explaining their applicability and limitations. Finally, we investigate the role of flows and defect interactions in two-dimensional active smectics, known as the spiral defect chaos state in Rayleigh-Bénard convection.