Filaments in the twist-grain-boundary smectic-$A$ phase

A model of filaments of the twist-grain-boundary smectic-$A$ phase (TGBA) arising from the homeotropic smectic-$A$ phase and nucleating on the sample surface is proposed. The model is based on the concept of finite blocks of parallel smectic layers forming a helical structure. The blocks are surrounded by dislocation loops. The model describes the filament structure near the sample surface and the observed inclination of the filament axis with respect to the easy direction of the molecular anchoring on the surface. The model is based on the observations of filament textures of the TGBA phase in a chiral liquid crystalline compound, but can be applied for forming of TGBA filaments in any compound. The compression modulus of the compound has been estimated using such parameters as anchoring energy, estimated from the field necessary to transform the structure into the homeotropic smectic-$A$.

Figure 1

Nucleation of the TGBA phase in PHB(S) compound when cooling down the sample. (a) TGBA phase exhibits blue color at the temperature $33{}^{\circ }\mathrm{C}$, (b) the color of TGBA phase changes to red at the temperature $32.2{}^{\circ }\mathrm{C}$. The width of every micrograph is about $120\mu \mathrm{m}$.

Figure 2

The growth of filaments of the TGBA phase in the homeotropic state of PHB(S) compound after switching off the electric field, temperature $32.2{}^{\circ }\mathrm{C}$. (a) Individual filaments grow with time in the direction making an angle between ${45}^{\circ }$ and ${50}^{\circ }$. with the easy direction at the sample surface. The easy direction is parallel to the edge of the electrode parallel with the edge of the image. (b) The density of filaments increases with time; sometimes also filaments perpendicular to the primary system of filaments occur. (c) When filaments cover the whole sample, an individual filament can coalesce making wider areas of TGBA phase. The width of every micrograph is about $150\mu \mathrm{m}$.

Figure 3

Schematic representation of a block with the smectic-$A$ layers inclined by an angle $\mathrm{\Omega }$ with respect to layers parallel to the sample surface (thick line). The smectic layers are represented by thin lines with molecules shown as short line segments. The full dots show cross sections of edge dislocations, which are connected with the sample surface by a system of screw dislocations (dot-and-dash lines). The open arrow shows the position of a borderline of the inclined block at the surface, the line being parallel to the $x$ axis. Dimensions of the block are the height $h$ and width $l$ determining the characteristic length $L$ as $L=\sqrt{h^2+l^2}$. Distance between edge dislocations is denoted as $d$ and the smectic layer thickness is $b$. The $y$ and $z$ axes are indicated; the $x$ axis is perpendicular to the plane of figure.

Figure 4

Schematic representation of two neighboring blocks shown (a) in full (block in the front) and dashed lines (block behind) and in perspective (b). The drawing corresponds to the positions at $x\sim 0$ and $x\sim P/2$ along the filament. (a) Blocks are inclined by an angle $\frac{\omega }{2}$ and $-\frac{\omega }{2}$ with respect to the parallel smectic layers and thus mutually inclined by $\omega$. The height, $h$, of both blocks is equal. (b) Depicted smectic layers constitute an envelope of the neighbor blocks only. For simplicity curvature deformations of layers in blocks near the sample surface are omitted.

Figure 5

Schematic representation of two neighboring blocks shown (a) in full (in the front) and dashed lines (behind) and in perspective (b). The drawing corresponds to the positions for $x$ between $x=0$ and $x=P/4$. The height of the block in front, $h_f$, differs from the height of the block behind, $h_b$. The relative displacement of the borderlines of the blocks is denoted as $u$.

Figure 6

Schematic representation of two neighboring blocks shown (a) in full (in the front) and dashed lines (behind) and in perspective (b). The drawing corresponds to the position at $x=P/4$ along the filament. Blocks are inclined by an angle $\frac{\pi -\omega }{2}$ with respect to the parallel smectic layers and thus relatively inclined by $\omega$. The height, $h$, of both blocks is equal. The same blocks as in (a) are shown in perspective in (b). Depicted smectic layers constitute an envelope of the neighbor blocks only. For simplicity curvature deformations of layers in blocks near the sample surface are omitted.

Figure 7

3D drawing of the smectic layers which terminate on the sample surface, tracing there a borderline. Depicted smectic layers constitute an envelope of the filament only. The presented simplified view demonstrates, namely, a relative displacement of neighboring blocks on the sample surface caused by chirality induced relative block rotation. The filament segment presented has the length of $P/2$. Note that blocks in the lower part of the figure, corresponding to the interval $0\le x\le P/4$, are continuously reconnected with parallel smectic layers on the left while blocks in the upper part of the figure $(P/4\le x\le P/2)$ are continuously reconnected with parallel smectic layers on the right.

Figure 8

Scheme drawing showing the position of the rotation axis $z_a$ of neighbor filament blocks related to the displacement $u$ of block borderlines. Enveloping smectic layers are relatively inclined by an angle ω while neighbor blocks are inclined with respect to the sample surface by angles $\mathrm{\Omega }\left(x\right)$ and $\mathrm{\Omega }\left(x\right)+\omega$, respectively. It is a simplified situation not taking into account the layer deformation near the surface due to the anchoring.

Figure 9

Schematic drawing of the filament seen along the normal to the sample surface. The filament segment is shown over the length of pitch $P$. Block rotations lead to the inclination of molecules from the sample surface normal. Inclined molecules are represented by nails, the points of which are oriented toward the observer. The length of the nails corresponds to the projection of the inclined molecule to the sample plane. Relative block rotations are illustrated by nails of changing lengths in neighboring blocks. The helix axis is parallel to the $x$ axis. Along the helix axis the molecules rotate by $2\pi$. Filament blocks projection onto the $xy$ plane is limited by $y_D$ and $y_H$ which define the shape of the filament. Mean directions of borderlines which limit filament segments are denoted by dot-and-dash lines. In intervals of the length $P/4$ borderlines are either $y_D$ or $y_H$. The mean filament axis (dashed line) is inclined with respect to the $x$ axis by an inclination angle α. The easy direction of the surface anchoring is parallel to the $y$ axis.