Dislocation geometry in the TGB(A) phase: linear theory

Phase transitions and separations in a distorted liquid crystalline mixture

A theoretical method is proposed for modelling phase transitions and phase ranges in a multi-component liquid crystalline mixture where the liquid crystal structure is distorted and defects are formed. This method employs the Maier-Saupe and Kobayashi-McMillan theories of liquid crystalline ordering and the Flory-Huggins theory of mixtures. It builds on previous work on mixed systems that can form smectic-A and nematic phases by incorporating "distortion factors" into the expression for the local free energy of the mixture, which account for the effects of a deviation of the liquid crystal structure from the uniform nematic and smectic-A states. The method allows a simple description of chiral defect phases such as the blue phase and the twist grain boundary phase. In a previous work, it was shown that a model of the blue phase along these lines could effectively explain the observed effect whereby an added guest compound can stabilize the phase by separating into the high energy defect regions of the structure. It is shown here that with the correct choice of guest material a similar effect could be observed for the twist grain boundary phase.

Triply periodic smectic liquid crystals

Twist-grain-boundary phases in smectics are the geometrical analogs of the Abrikosov flux lattice in superconductors. At large twist angles, the nonlinear elasticity is important in evaluating their energetics. We analytically construct the height function of a pi2 twist-grain-boundary phase in smectic-A liquid crystals, known as Schnerk's first surface. This construction, utilizing elliptic functions, allows us to compute the energy of the structure analytically. By identifying a set of heretofore unknown defects along the pitch axis of the structure, we study the necessary topological structure of grain boundaries at other angles, concluding that there exist a set of privileged angles and that the pi2 and pi3 grain boundary structures are particularly simple.

Numerical simulation of the twist-grain-boundary phase of chiral liquid crystals

We study the structure of the twist-grain-boundary phase of chiral liquid crystals by numerically minimizing the Landau-de Gennes free energy. We analyze the morphology of layers at the grain boundary, to better understand the mechanism of frustration between the smectic layer order and chirality. As the chirality increases, the layer compression energy strongly increases while the effective layer bending rigidity is reduced due to unlocking of the layer orientation and the director. This results in large deviation of the layer morphology from that of Scherk's first minimal surface and linear stack of screw dislocations.

Elliptic phases: a study of the nonlinear elasticity of twist-grain boundaries

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 pi/2 grain boundaries and discuss the relation of this structure to the Schwarz D surface.

Giant-block twist grain boundary smectic phases

Study of a diverse set of chiral smectic materials, each of which has twist grain boundary (TGB) phases over a broad temperature range and exhibits grid patterns in the Grandjean textures of the TGB helix, shows that these features arise from a common structure: "giant" smectic blocks of planar layers of thickness l(b) > 200 nm terminated by GBs that are sharp, mediating large angular jumps in layer orientation between blocks (60 degrees < Delta < 90 degrees ), and lubricating the thermal contraction of the smectic layers within the blocks. This phenomenology is well described by basic theoretical models applicable in the limit that the ratio of molecular tilt penetration length-to-layer coherence length is large, and featuring GBs in which smectic ordering is weak, approaching thin, melted (nematic-like) walls. In this limit the energy cost of change of the block size is small, leading to a wide variation of block dimension, depending on preparation conditions. The models also account for the temperature dependence of the TGB helix pitch.

Tracer diffusion in a dislocated lamellar system

Many lamellar systems exhibit strongly anisotropic diffusion. When the diffusion across the lamellae is slow, an alternative mechanism for transverse transport becomes important. A tracer particle can propagate across the lamellae by encircling a screw dislocation. We calculate the statistical properties of this mode of transverse transport. When either positive or negative dislocations are in excess, transport across the lamellae is ballistic. When the average dislocation charge is zero, the mean square of the normal displacement grows like TlogT for large times. To obtain this result, the trajectory of the tracer must be smoothed over distances of order of the dislocation core size.