Complex-tensor theory of simple smectics

Abstract

Matter self-assembling into layers generates unique properties, including structures of stacked surfaces, directed transport, and compact area maximization that can be highly functionalized in biology and technology. Smectics represent the paradigm of such lamellar materials — they are a state between fluids and solids, characterized by both orientational and partial positional ordering in one layering direction, making them notoriously difficult to model, particularly in confining geometries. We propose a complex tensor order parameter to describe the local degree of lamellar ordering, layer displacement and orientation of the layers for simple, lamellar smectics. The theory accounts for both dislocations and disclinations, by regularizing singularities within defect cores and so remaining continuous everywhere. The ability to describe disclinations and dislocation allows this theory to simulate arrested configurations and inclusion-induced local ordering. This tensorial theory for simple smectics considerably simplifies numerics, facilitating studies on the mesoscopic structure of topologically complex systems.

Dynamical theory of topological defects II: universal aspects of defect motion

“…Further extensions of the formalism presented in this work can include taking into account the roles of backflow [71], fluctuations [73], or curved geometry [74]. Another interesting extension of the method we propose would be to study the motion of dislocations [54] by using a field theoretical treatment of elasticity [75], or more sophisticated descriptions [76,77]. While some of these developments may be technically challenging, they should not entail any additional conceptual difficulty.…”

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.…”

Lock-key microfluidics: simulating nematic colloid advection along wavy-walled channels

“…9 Liquid crystalline materials are alluring for microfluidic transport because they are highly responsive to flows, [10][11][12] suspended inclusions 13,14 and confining surfaces. [15][16][17] When colloidal particles are dispersed in liquid crystalline fluids, the anisotropic nature of liquid crystals gives rise to emergent properties [18][19][20][21] and imposed anchoring at colloidal surfaces results in topological defects in the vicinity of the colloids to ensure topological charge neutrality. Strong homeotropic or planar anchoring endows the surface with a topological charge of +1 and necessitates the existence of an accompanying À1 charge in the bulk fluid, either as two À1/2 point defects in 2D, or in 3D as defect loops (Saturn rings), À1 point defect (hyperbolic hedgehog), or surface defects (boojum defects).…”