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

Abstract

We develop an explicit and tractable representation of a twist-grain-boundary phase of a smectic-A liquid crystal. This allows us to calculate the interaction energy between grain boundaries and the relative contributions from the bending and compression deformations. We discuss the special stability of the π/2 grain boundaries and discuss the relation of this structure to the Schwarz D surface.PACS numbers: 61.30. Jf, 61.72.Mm, 61.72.Bb, 11.10.Lm Topological defects are often the essential degrees of freedom. They are the focus in the study of some phase transitions [1,2], high-temperature superconductors [3], and liquid crystalline phases [4]. In the latter, there is a necessary connection between the topology of the defects and their geometry which is on the one hand theoretically challenging while, on the other hand, experimentally accessible through real-space, freeze-fracture imaging [5]. The twist grain boundary (TGB) phase of smectic-A liquid crystals [6], has an arrangement of screw dislocations which alter the geometry of the uniform, flat layers into a discretely rotating layered structure. Though topology constrains the geometry, it does not specify it. 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]. In order to reconcile our understanding of the linear theory with the nonlinear elasticity and in light of the recently observed [11] large angle grain boundaries, here we develop a full nonlinear theory of the largest angle grain boundaries allowed, with rotations of π/2. To do this we explicitly sum the topological defects to render a closed-form expression which is an exact solution of the linear elasticity theory. Unlike parametric representations of surfaces, our surface is given as a multi-valued height function which allows us to directly calculate the compression energy and allows tractable comparison with real space images. We directly calculate the energetics of space-filling TGB structures analytically and find that grain boundaries interact exponentially with separation.Smectic order is characterized by a periodic mass density ρ = ρ 0 + ρ 1 cos [2πΦ(x)/a], where ρ 0 and ρ 1 are constant amplitudes, set at the nematic-to-smectic phase transition, and a is the equilibrium layer spacing. The smectic layers are defined via the level sets of Φ through Φ(x)/a = n ∈ Z, and the elastic free energy has two terms, FIG. 1: Schnerk's first surface, with charge +2 and −2 screw dislocations. Note that the −2 dislocations lie at the center of the rectangle made by the adjacent +2 dislocations. We choose θ = ψ and k 2 ≈ −0.03033 so that K ′ (k)/K(k) = 2 − i. We are reminded that ". . . imaginary things are often easier to see than real ones." [12]

Geometry and the nonlinear elasticity of defects in smectic liquid crystals

“…The resulting layers, dubbed Schnerk's first surface, is shown in figure 5 [31]. Schnerk's first surface is necessarily achiral since it is developed from a set of defects with neutral total topological charge.…”

“…Here, c should be chosen to minimize the compression energy, but a suitable ansatz is to choose it so that the compression energy vanishes along lines midway between adjacent grain boundaries [31]. This sum can be performed exactly to give…”

Smectic pores and defect cores

“…In fact, the layers themselves have no mean curvature, although the defect core may be more complex. We have found it useful to construct smectic textures by viewing the layers as the Riemann surface of a meromorphic function on the plane encoding the two-dimensional arrangement of three-dimensional line defects [4][5][6]. The one-dimensional periodicity arises from the multiple sheets formed by logarithmic branch points.…”

Smectic Liquid Crystals: Materials with One-dimensional, Periodic Order

“…Some analytical progress for K = 0 surfaces has also been made by directly summing the phase fields of screw dislocations. In the case of a twist-grain boundary, where the screw dislocations lie along the line x = 0, there is an unexpected connection to minimal surfaces (Kamien and Lubensky 1999;Santangelo and Kamien 2006). Neither sums of screw dislocations, nor their minimal surface counterparts, however, are exact minima of the smectic energy-the true minimum likely lies somewhere in between (Duque and Shick 2000).…”

“…Evaluating (43), we see that the helicoid also is an extremum of the non-linear theory. Multiple screw dislocations are currently under investigation (Kamien and Lubensky 1999;Kamien 2001;Santangelo and Kamien 2006). …”