Defects in the twist-bend nematic phase: Stabilities and instabilities of focal conic domains and related topics

This article provides an explanation for the instability of toric focal conic domains and the stability of parabolic focal conic domains in the twist-bend nematic ($N_{\mathrm{TB}}$) phase, by reason of the presence (the first case) or absence (the second case) of defect densities attached to the conics, which are disclinations. This result is obtained after a reminder of the theory of attached defects (the extended Volterra process) and is followed by its usage in other cases (dislocations, disclinations, dispirations, double helices) in the $N_{\mathrm{TB}}$ phase. The notion of spin disclination is introduced.

Figure 1

Imperfect toric focal conic formations in the $N_{\mathrm{TB}}$ phase of a $20\text{−}\mu \mathrm{m}$-thick CB7CB layer in a ${90}^{\circ }$-twist cell. (a) 8.1 V, 10 kHz and (b) after switch off from 15 V, 1 kHz. Defect regions are encircled. The appearance of these defects is a first step in the transformation of TFCDs into PFCDs. Notice the small grainy defects all along circles. Crossed polarizers with their axes along ($x,y$). Scale bar: $10\mu \mathrm{m}$.

Figure 2

Focal conics in the $N_{\mathrm{TB}}$ phase of CB7CB, close to its setting point: (a) PFCDs and FCDs (see later for eccentricities of the numbered conics here), (b) electrically generated (8 V, 1 kHz) PFCDs, arranged along the substrate rubbing direction. The precursor $N$ was: (a) aligned along $x$, (b) homeotropic. Single polarizer along $x$ in (a), crossed polarizers along $x$ and $y$ in (b). Scale bar: $20\mu \mathrm{m}$. $\text{Sample}\text{thickness}=20\mu \mathrm{m}$.

Figure 3

(a) Single-helical (SH) and double-helical (DH) defects formed in the $N_{\mathrm{TB}}$ phase of the mixture of CB7CB and 7OCB (1:1 by weight) held in a ${90}^{\circ }$-twist cell at ${31}^{\circ }\mathrm{C}$. The dark rails parallel to the DHs of the right upper part are bunches of screw dislocations of small Burgers vectors, whose total Burgers vector is probably of the same order as the giant Burgers vector of the DHs. The central diagonal region is oppositely twisted relative to outer regions. The disclinations separating the oppositely twisted regions in the precursor $N$ phase develop into SH defects in the $N_{\mathrm{TB}}$ phase. The DHs run in the direction of midplane nematic director. The helices also form by coalescence of drifting TFCDs that evolve during the transition from electric homeotropic Freedericksz state to the planar state (b)–(d). Scale: $5\mu \mathrm{m}$ each small div. Double arrows represent polarizers. Sample thickness: (a) $5\mu \mathrm{m}$, (b) $20\mu \mathrm{m}$.

Figure 4

Attached infinitesimal defects (dislocations, disclinations) along a disclination line $L$. The Frank vector $\mathbf{f}$ varies along the line in direction. $\mathbf{P}$ and ${\mathbf{P}}^{\prime }$ are close points.

Figure 5

Cut through two $k=1/2$ wedge disclination lines A and B. The attached infinitesimal disclinations are restricted to the ribbon between A and B, which ribbon can be made to vanish, AB = 0, yielding a $k=1$ wedge line without attached defects.

Figure 6

Along the ellipse of a FCD. Intersections of two cyclides at a distance multiple of $p$. It is assumed that there is a $\mathbf{\tau }$ direction along these intersections; $\mathbf{\tau }$ rotates helically about the normal $\mathbf{\chi }$ to these cyclides.

Figure 7

Quasiparallel FCDs in a $20\text{−}\mu \mathrm{m}$-thick layer of CB7CB in a planar cell at $0.6^{\circ }\mathrm{C}$ below the setting point and subjected to a 30 V, 1 kHz field. The FCD ellipses are connected by a set of attached quantized dislocations approximately parallel to the minor axes. P and A denote polarizer and analyzer.

Figure 8

Solitary PFCD in a $20\text{−}\mu \mathrm{m}$-thick ${\mathrm{N}}_{\mathrm{TB}}$ layer of CB7CB in a planar cell. Scale: $2\mu \mathrm{m}$ each small div. Polarizer along $x$.

Figure 9

Curve fittings, see text.