Confinement and magnetic-field effect on chiral ferroelectric nematic liquid crystals in Grandjean-Cano wedge cells

We explore the structure and magnetic-field response of edge dislocations in Grandjean-Cano wedge cells filled with chiral mixtures of the ferroelectric nematic mesogen DIO. Upon cooling, the ordering changes from paraelectric in the cholesteric phase ${\mathrm{N}}^{*}$ to antiferroelectric in the smectic ${\mathrm{SmZ}}_{\mathrm{A}}^{*}$ and to ferroelectric in the cholesteric ${\mathrm{N}}_{\mathrm{F}}^{*}$. Dislocations of the Burgers vector $b$ equal to the helicoidal pitch $\mathcal{P}$ are stable in all three phases, while dislocations with $b=\mathcal{P}/2$ exist only in the ${\mathrm{N}}^{*}$ and ${\mathrm{SmZ}}_{\mathrm{A}}^{*}$. The $b=\mathcal{P}/2$ dislocations split into pairs of ${\tau }^{-1/2}{\lambda }^{+1/2}$ disclinations, while the thick dislocations $b=\mathcal{P}$ are pairs of nonsingular ${\lambda }^{-1/2}{\lambda }^{+1/2}$ disclinations. The polar order makes the ${\tau }^{-1/2}$ disclinations unstable in the ${\mathrm{N}}_{\mathrm{F}}^{*}$ phase, as they should be connected to singular walls in the polarization field. We propose a model of transformation of the composite ${\tau }^{-1/2}$ line-wall defect into a nonsingular ${\lambda }^{-1/2}$ disclination, which is paired up with a ${\lambda }^{+1/2}$ line to form a $b=\mathcal{P}$ dislocation. The ${\mathrm{SmZ}}_{\mathrm{A}}^{*}$ behavior in the in-plane magnetic field is different from that of the ${\mathrm{N}}_{\mathrm{F}}^{*}$ and ${\mathrm{N}}^{*}$: the dislocations show no zigzag instability, and the pitch remains unchanged in the magnetic fields up to 1 T. The behavior is associated with the finite compressibility of smectic layers.

Figure 1

Structural organization of (a) an apolar chiral nematic ${\mathrm{N}}^{*}$ with a helicoidal axis $\widehat{\mathit{\chi }}$, local director $\widehat{\mathit{\lambda }}$, orthogonal direction $\widehat{\mathit{\tau }}$, and period $\mathcal{P}/2$; (b) the ${\mathrm{N}}^{*}$ director field in a Grandjean-Cano wedge with thin $b=\mathcal{P}/2$ and thick $b=\mathcal{P}$ edge dislocations; (c) the chiral ferroelectric nematic ${\mathrm{N}}_{\mathrm{F}}^{*}$; the structural period is $\mathcal{P}$ rather than $\mathcal{P}/2$.

Figure 2

Measurements of the pitch: (a) lattice of edge dislocations in a parallel buffed wedge cell ($\alpha = 1.{24}^{\circ }$) in the ${\mathrm{N}}^{*}, {\mathrm{SmZ}}_{\mathrm{A}}^{*}$, and ${\mathrm{N}}_{\mathrm{F}}^{*}$ phases; (b) temperature dependence of the helical pitch in M0.2.

Figure 3

Birefringence fringes in M0.04 wedge cells with parallel, (a)–(f), and antiparallel, (g), assembly: (a) thin $b=\mathcal{P}$/2 and thick $b=\mathcal{P}$ dislocations in the ${\mathrm{N}}^{*}$; (b) a high-magnification image of a thin $b=\mathcal{P}$/2 core in the ${\mathrm{N}}^{*}$; the inset shows the transmitted light intensity along the white dotted line; (c) the same $b=\mathcal{P}$/2 core in the ${\mathrm{SmZ}}_{\mathrm{A}}^{*}$; (d) a high-magnification image of a thick $b=\mathcal{P}$ core in the ${\mathrm{N}}^{*}$; the inset shows the transmitted intensity along the white dotted line; (e) the same $b=\mathcal{P}$ core in the ${\mathrm{SmZ}}_{\mathrm{A}}^{*}$ and (f) in the ${\mathrm{N}}_{\mathrm{F}}^{*}$; (g) $b=\mathcal{P}$ in the ${\mathrm{N}}_{\mathrm{F}}^{*}$ wedge with antiparallel assembly. Dihedral angle $\alpha =3.{39}^{\circ }$ in (a)–(f) and $\alpha =3.{36}^{\circ }$ in (g).

Figure 4

Dislocations in the magnetic field $\mathbf{B}$: (a) parallel assembly ${\mathrm{N}}^{*}$ wedge ($\alpha =1.{51}^{\circ }$), $b=\mathcal{P}/2$ dislocations remain straight, $b=\mathcal{P}$ dislocations adopt zigzag shapes above $B_{zz}$; (b) the same wedge, all dislocations in the ${\mathrm{SmZ}}_{\mathrm{A}}^{*}$ remain straight; (c) the same wedge, all $b=\mathcal{P}$ dislocations in the ${\mathrm{N}}_{\mathrm{F}}^{*}$ change to zigzag shapes above $B_{zz}$; the $2\pi$ disclination near the edge of the wedge remains straight; (d) antiparallel assembly wedge ($\alpha =1.{79}^{\circ }$) in the ${\mathrm{N}}_{\mathrm{F}}^{*}$; $b=\mathcal{P}$ dislocations change to zigzag shapes above $B_{zz}$; note the arched shape of the first disclination in the absence of the field, which is attributed to the splay-cancellation mechanism illustrated in; (e), see text for details; (f) period of zigzag undulations in parallel rubbed ${\mathrm{N}}_{\mathrm{F}}^{*}$ wedge cells increases with the dihedral angle $\alpha$; $B=0.6$ T. Mixture M0.1.

Figure 5

Magnetic-field dependence of helical pitch in parallel buffed wedge cell ($\alpha =1.{51}^{\circ }$) in the ${\mathrm{N}}^{*}, {\mathrm{SmZ}}_{\mathrm{A}}^{*}$, and ${\mathrm{N}}_{\mathrm{F}}^{*}$ phases of M0.1.

Figure 6

Structural schemes of the Grandjean-Cano wedge with (a) parallel assembly of buffed polyimide alignment layers and (b) antiparallel assembly. The dislocations represent split pairs of nonsingular ${\lambda }^{-1/2}{\lambda }^{+1/2}$ disclinations.

Figure 7

Transformations of thin-edge dislocations $b=\mathcal{P}/2$ in the thin part of the wedge on cooling in (a) parallel assembly, $\alpha =0.{74}^{\circ }$ and (b) antiparallel assembly, $\alpha =0.{77}^{\circ }$. The edge dislocations preserve their structure upon the ${\mathrm{N}}^{*}$ to ${\mathrm{SmZ}}_{\mathrm{A}}^{*}$ transition but increase the separation distances since the pitch increases. In the ${\mathrm{N}}_{\mathrm{F}}^{*}$, the number of dislocations is reduced by a factor of 2 and all of them are of the Burgers vector $b=\mathcal{P}$; (c) optical retardance of the untwisted regions shown in (a) grows linearly with distance $d$ along the thickness gradient in the ${\mathrm{N}}^{*}$ ($100{}^{\circ }\mathrm{C}$), line AB, and the ${\mathrm{N}}_{\mathrm{F}}^{*}$ phase ($55{}^{\circ }\mathrm{C}$), line AC. Mixture M0.2.

Figure 8

Nonsingular disclinations in the ${\mathrm{N}}_{\mathrm{F}}^{*}$. (a)–(d) Restructuring of (a) ${\tau }^{-1/2}{\lambda }^{+1/2}$ into (c),(d) ${\lambda }^{-1/2}{\lambda }^{+1/2}$ disclination pair through the creation of a singular wall of polarization (b). The restructuring amounts to an addition of a half-pitch $\mathcal{P}/2$ to the Burgers vector of the dislocation, which eliminates Grandjean zones with an odd number of $\pi$ twists in a parallel assembly of a wedge; (e) Lehmann cluster $b=0$ comprising of two ${\lambda }^{-1/2}$ and two ${\lambda }^{+1/2}$ disclinations is fully compatible with the polar ordering of the ${\mathrm{N}}_{\mathrm{F}}^{*}$.

Figure 9

Edge dislocations in the wedge heated from the ${\mathrm{N}}_{\mathrm{F}}^{*}$ to ${\mathrm{SmZ}}_{\mathrm{A}}^{*}$ and ${\mathrm{N}}^{*}$; (a) parallel assembly wedge, $\alpha =0.{80}^{\circ }$; (b) antiparallel assembly, $\alpha =0.{77}^{\circ }$. The thick $b=\mathcal{P}$ dislocations restructure and split into two thin-edge dislocations $b=\mathcal{P}/2$. Mixture M0.2.

Figure 10

Unwinding of a $\pi$-twisted (a)–(c) ${\mathrm{N}}^{*}$, (d)–(f) ${\mathrm{SmZ}}_{\mathrm{A}}^{*}$ in planar cells with parallel assembly by an in-plane electric field applied along the buffing direction. Mixture M0.2, $h= 4.1\mu \mathrm{m}$.