Complex tensors and simple layers: a theory for smectic fluids

This thesis considers complex fluids that possess non-isotropic internal structures; in particular, we will analytically and numerically explore fluids possessing internal layering. In such fluids, the microscopic structure influences phase transitions, physical symmetries, and rheological behaviours. Smectic phases couple anisotropic properties due to particle alignment with a layered structure. These phases introduce complex challenges for modellers, as they are liquid-like within each layer alongside the associated solid-like stacking effects.

In this thesis, we propose a complex tensor to tackle these challenges. We take two key stances: (i) that layering is a fundamental symmetry breaking, that can occur independently from particle alignment, and (ii) the effects we are interested in occur at scales larger than individual layers. Unlike previous theoretical approaches, we can describe layering through the use of a single mathematical object, combining local layer orientation, compression and the extent of ordering, all while respecting the relevant smectic symmetries. Bypassing the effects of individual particles or layers (the microscale) keeps the focus of these models to the complex topological structures and {defects} that make these fluids so interesting.

We construct a Landau theory in terms of this complex tensor E capable of describing the local degree of lamellar ordering, layer displacement, and orientation of the layers. This theory accounts for elastic contributions both parallel and perpendicular to the layers, before we simplify to a 'one constant' form, for which we develop a Landau-Ginzburg theory, to simulate lamellar systems and defects. Our theory reproduces both dislocation and disclination defects, remaining continuous everywhere by regularising singularities within defect cores. Describing these defects allows simulations of arrested configurations and non-trivial structures arising from interactions with inclusions, while serving as a considerable simplification to existing numerical methods, and respecting physical constraints, such as the Peierls--Nabarro energy barrier. We also apply the complex tensor order parameter to bacteria colonies, and develop a method for evaluating order in individual-based systems of any length scale. Our findings highlight the effectiveness of mesoscale modelling of topological systems, and open avenues to simulate flowing lamellar and smectic materials, for which we propose coupling schemes for our E tensor with the nematic order parameter tensor Q.