Modeling smectic layers in confined geometries: Order parameter and defects

Abstract

We identify problems with the standard complex order parameter formalism for smectic-A (SmA) liquid crystals and discuss possible alternative descriptions of smectic order. In particular, we suggest an approach based on the real smectic density variation rather than a complex order parameter. This approach gives reasonable numerical results for the smectic layer configuration and director field in sample geometries and can be used to model smectic liquid crystals under nanoscale confinement for technological applications.

Mathematics and liquid crystals

“…We already gave in Section 4.1 a suggestive description of how this might happen for nematics via a coarsegraining procedure. Less clear is the situation for smectics, in some theories of which (for example [44,54,79,105,133,148]) the molecular mass density c(x) plays a role, its variation describing smectic layers. How can we have a macroscopic variable which varies on a microscopic length scale?…”

Particle-resolved topological defects of smectic colloidal liquid crystals in extreme confinement

“…Due to their simultaneous orientational and positional ordering, defects in the smectic phase naturally exhibit an even higher degree of complexity. The main emphasis has been put hitherto on the positional layering [23][24][25][26][27] or orientational textures 19,28,29 alone, as well as, on coarse-grained calculations 1,19,25,[29][30][31][32] and computer simulation of particle models [33][34][35][36] .…”

Discontinuous Order Parameters in Liquid Crystal Theories

“…In seeking a suitable generalization of these models for smectic A which would allow for planar discontinuities in n, we found it easier instead to follow the route proposed in a recent paper of Pevnyi, Selinger & Sluckin [63], who criticize the modelling of smectics in terms of the complex order parameter Ψ and the de Gennes free-energy functional on the two grounds (i) that Ψ cannot be chosen as a single-valued function for index 1 2 defects, for essentially the same reason as discussed in Section 5 that such defects are not orientable (ii) that it does not represent the local free-energy density at the length-scale of the smectic layers themselves (so that, for example, it cannot model dislocations in the smectic layers). (A related criticism to (i), that the de Gennes free-energy density is not invariant to the transformation Ψ → Ψ * , is made in [22].)…”

“…The free energy (6.29) reduces to that in [63] when ψ E (Q) = K 2s 2 |∇Q| 2 and k = 0, though it is expressed in terms of uniaxial Q and ρ rather than n and ρ. We allow subquadratic growth of ψ E (Q, ∇Q) in ∇Q so as to be able to model disclinations as line defects with finite energy.…”