Dislocation geometry in theTGBAphase:  Linear theory

Abstract

We demonstrate that an arbitrary system of screw dislocations in a smectic-A liquid crystal may be consistently treated within harmonic elasticity theory, provided that the angles between dislocations are sufficiently small. Using this theory, we calculate the ground state configuration of the TGBA phase. We obtain an estimate of the twist-grain-boundary spacing and screw dislocation spacing in a boundary in terms of the macroscopic parameters, in reasonable agreement with experimental results. I. INTRODUCTIONCondensed matter systems offer a vast stage for the intricate interplay between order and disorder. A beautiful example of such interplay is the twist-grain-boundary phase (TGB) of chiral smectics, which is the liquidcrystalline analog of the Abrikosov vortex state of type II superconductors [1,2]. Morphologically, the phase consists of blocks of pure smectic (which can be either smectic-A or smectic-C) separated by parallel, regularly spaced twist grain boundaries, where each boundary is formed by a periodic array of screw dislocations. The direction of dislocation lines rotates by a constant angle from one grain boundary to the next. Such a dislocation arrangement causes the smectic blocks to rotate about the axis perpendicular to the grain boundaries dragging the nematic director along. Thus the TGB structure combines the properties of smectics and cholesterics: the nematic director twists on average as in cholesterics while the lamellar structure of a smectic is preserved. In this paper only the TGB A phase will be considered. As suggested by its name, the smectic blocks in TGB A are smectic-A. The analogy between the TGB A phase and the Abrikosov vortex lattice is based on the mathematical similarity of the Gibbs free energies for metals in a magnetic field and for chiral smectics [1][2][3][4]. Their respective forms, known as the Ginzburg-Landau free energy and the de Gennes free energy, areandwhere ψ is a complex order parameter, A the vector potential, H the magnetic intensity, h a chiral field determined by molecular structure [5], and n the unit director, often decomposed as n = n 0 + δn. The Ginzburg-Landau free energy (1.1) defines two characteristic length scales: the order parameter coherence length ξ = (h 2 /2m * |r|) 1/2 and the magnetic field penetration depth λ = (m * c 2 g/4πµ e * 2 |r|) 1/2 . Their ratio κ = λ/ξ, called the Ginzburg parameter, controls the phase diagram as a function of temperature and external magnetic field. When the Ginzburg parameter is less than the critical value κ c = 1/ √ 2, the system is type I, and there is a first order transition between a normal metal in a field and the Meissner phase with ψ = constant = 0 and magnetic field B = 0. When κ > κ c , the system becomes type II, and a new phase intervenes between the normalmetal state and the Meissner state. This new phase, the Abrikosov flux phase, is characterized by a proliferation of linear topological defects of the complex order parameter field ψ. The defects are magnetic flux lines, and they form a two-dim...

Nonlinear effects in the TGB A phase

“…This implies that the lattice structure is completely determined by the elastic energy cost of the smectic distortions created by the screw dislocations. We demonstrated in [2] that these distortion fields can be consistently treated within the harmonic approximation to the elastic free energy. In this approximation, we express the elastic free energy in terms of the (Eulerian) layer displacement field u and the deviation δn of the nematic director from the z-axis:…”

“…In an earlier paper [2], we considered the screw dislocation lattice in the TGB A phase near the transition to the nematic phase. In this regime the dislocation density is low, so the interaction between dislocation cores is negligible.…”

“…As with screw defects and vortices, the TGB phase is the analog of the Abrikosov vortex lattice phase in superconductors [1,5], although the lattice structure of defects in the TGB phase is significantly more complicated due to the relative rotation of dislocations in different grain boundaries. We showed in [2] that the linearized interaction energy of two grain boundaries is…”

“…The TGB phase balances the competing interactions by forming a lattice of screw dislocations arranged in twist grain boundaries. In earlier work [2] we investigated the geometry of the TGB phase within a harmonic free energy in which topological defects interact via screened exponential potentials. However, it was argued in [3] and [4] that defects in the same grain boundary interact via powerlaw potentials when those nonlinearities required by rotational invariance were included in the free energy.…”

Elliptic Phases: A Study of the Nonlinear Elasticity of Twist-Grain Boundaries

“…Rather, the free energy of the deformed smectic layers sets the periodicity of the lattice. Prior analysis [7] relied on linear elasticity to study small angle grain boundaries. However, it has been shown that when the angles and deformations are large, the energetics of the rotationally-invariant nonlinear theory are not only quantitatively, but qualitatively different than in the linear theory [8,9,10].…”