Smectic layering: Landau theory for a complex-tensor order parameter

Abstract

Composed of microscopic layers that stack along one direction while maintaining fluid-like positional disorder within layers, smectics are excellent systems for exploring topology, defects and geometric memory in complex confining geometries. However, the coexistence of crystalline-like characteristics in one direction and fluid-like disorder within layers makes lamellar liquid crystals notoriously difficult to model—especially in the presence of defects and large distortions. Nematic properties of smectics can be comprehensively described by the Q-tensor but to capture the features of the smectic layering alone, we develop a phenomenological Landau theory for a complex-tensor order parameter E, which is capable of describing the local degree of lamellar ordering, layer displacement, and orientation of the layers. This theory can account for both parallel and perpendicular elastic contributions. In addition to resolving the potential ambiguities inherent to complex scalar order parameter models, this model reduces to previous employed models of simple smectics, and opens new possibilities for numerical studies on smectics possessing many defects, within complex geometries and under extreme confinement.

Topological fine structure of smectic grain boundaries and tetratic disclination lines within three-dimensional smectic liquid crystals

“…Throughout this paper, we have in particular shed light on planar grain boundaries which split into two tetratic disclination lines. Our insight will thus be useful for the interpretation of future computational, 104,105 theoretical, 99–104,106 and experimental 79,98,107–109 research on confined smectic structures in three dimensions. To this end, our topological picture can be extended to study more complex geometries and topologies in 3d, e.g.…”

“…To get further insight, the confined systems considered in this work could therefore be investigated using density functionals for hard spherocylinders. 99,100 Alternatively, density functional theory also allows to determine elastic parameters of fluids of hard spherocylinders 101 which could then be used to fix the parameters in phenomenological elasticity theories 100,102,103 for the smectic phase.…”

Defect interactions in a two-dimensional sheared lamellar mesophase

“…Recently, a model has been proposed by Paget et al 23,24 based on an order parameter, which is a complex secondorder tensor of the form E ¼ nn À 1 d I jcðx; tÞj expðifÞ. Here,…”

“…Recently, a model has been proposed by Paget et al 23,24 based on an order parameter, which is a complex second-order tensor of the form . Here, , d is the dimension, n is the unit normal to the layers, I is the identity tensor, and ψ and ϕ are slowly varying functions of space and time.…”

“…Therefore, the concentration variations on the scale of the layer spacing are not directly resolved in the formulation of ref. 23 and 24, and the free-energy functional does not explicitly contain the layer spacing. The model is similar to, but more advanced than, the model based on the layer displacement field for nearly well-aligned smectics.…”