Geometry, Anisotropy And Heterogeneity -- A Study Of Modulated Nematics, Ecology And Evolution.
Liquid crystalline materials form a whole array of interesting textures and phases. While there are many cholesteric and smectic phases, the discovery of new nematic phases is rare since the space of possible configurations for achiral molecules is small. Here, we look at the novel class of modulated nematic structures -- the twist-bend, splay-bend and splay-twist. While twist-bend nematic phases have been extensively studied, the experimental observation of two dimensional, oscillating splay-bend phases is recent. We consider two theoretical models that have been used to explain the formation of twist-bend phases—flexoelectricity and bond orientational order—as mechanisms to induce splay-bend phases. Flexoelectricity is a viable mechanism, and splay and bend flexoelectric couplings can lead to splay-bend phases with different modulations. We show that while bond orientational order circumvents the need for higher order terms in the free energy, the important role of nematic symmetry and phase chirality rules it out as a basic mechanism.
The Hopf fibration has inspired any number of geometric structures in physical systems, in particular, in chiral liquid crystalline materials. Because the Hopf fibration lives on the three sphere, some method of projection or distortion must be employed to realize textures in flat space. Here, we explore the geodesic preserving gnomonic projection of the Hopf fibration, and show that this could be the basis for a new modulated nematic texture with only splay and twist. We outline the structure and show that it is defined by the tangent vectors along the straight line rulings on a series of hyperboloids. The phase is defined by a lack of bend deformations in the texture, and is reminiscent of the splay-bend and twist-bend nematic phases. We show that domains of this phase may be stabilized through anchoring and saddle-splay.
The second part of this thesis is about ecology and evolution in heterogenous environments. Organisms in nature have to be competent at multiple tasks in order to survive and a given phenotype cannot usually be optimal at all tasks at the same time. Recent studies employ the concept of Pareto optimality from economics and engineering to capture this inherent trade-off. If we associate each task with a different environmental niche, Pareto optimality is a useful framework to capture phenotypic plasticity. We compare Pareto optimal fronts to the well known ecological concept of fitness sets, and show how the shape of Pareto fronts in trait space can be connected to the determination of the optimal strategy in a heterogenous environment. We consider both temporal and spatial heterogeneity.