Some geometrical aspects of packing of bent-core molecules in the triclinic smectic-$C_g$ liquid crystals and simple models of the modulated $\mathrm{Sm}{C}_{gmod}$ phase

Organic compounds with bent-core (BC) molecules usually form the layered smectic liquid crystals with tilted molecules and polarization (P) which lies in the plane of the layer. A few such compounds have been found in which P itself tilts out of the plane of the layer, and the medium with general tilt ($\mathrm{Sm}{C}_g$) of the molecules has the low chiral triclinic symmetry. We discuss the geometric constraints of molecular packing in this structure to show that projecting groups attached to one of the arms of the BC molecules favors the formation of the $\mathrm{Sm}{C}_g$ phase. We also extend our model for the modulated phases exhibited by BC molecules to show that the stripe structure made of bilayers shown by a few BC compounds, which is a signature of $\mathrm{Sm}{C}_g$ layers, is preferentially formed by rotation of the BC molecules about their long axes, rather than about the layer normal. The theoretical results based on the unified model which describes all the modulated phases exhibited by the BC molecules broadly reflect experimental trends.

Figure 1

Orientation of a bent-core molecule ABC in a $\mathrm{Sm}{C}_g$ layer whose normal is along the ζ axis. The long axis $\mathit{BC}$ which lies in the ηζ plane makes the tilt angle θ with ζ. The arrow axis $\mathit{OA}$ makes the angle ψ with the ξ axis. In the $\mathrm{Sm}\mathit{CP}$ layer of the $B_2$ liquid crystal, $\psi =0$.

Figure 2

Structural formula of the compound NORABOW which has been shown to form $\mathrm{Sm}{C}_g$ layers in Refs. [9, 14]. If the hydroxyl groups are absent, only the $B_2$ phase is formed.

Figure 3

Structural formula of the compound $E_{Bm}$ with $m=6$. The homologs with $m=10–14$ exhibit modulated bilayer ($\mathrm{Sm}C{G}_{2\mathrm{mod}}$) phases [16].

Figure 4

Geometry of packing of BC molecules in $\mathrm{Sm}{C}_g$ layers. Top row: $\psi ={90}^{\circ }$. (a) The molecules pack well side by side. (b) The molecular packing in the plane containing the arrow axes, such that the layer spacing maintained is less dense. Dipolar groups which protrude from the upper arm (shown symbolically by spherical sections) can favor this packing compared to that in the usual SmCP layer with $\psi =0$. Bottom row: When 0 < ψ < 90 °, the packing conditions are broadly similar to those at $\psi ={90}^{\circ }$.

Figure 5

Schematic figures illustrating the arrangement of the BC molecules in a stripe occurring between nonpolar domains (which are not shown) of (a) $B_7$ liquid crystal, made of a $\mathrm{Sm}\mathit{CP}$ layer with $\psi =0$, and (b) $\mathrm{Sm}{C}_g$ LC, with $\psi \ne 0$, the splay distortion of P resulting from rotation of the BC molecules about the layer normal. The widths of the upper and lower surfaces of the stripe are equal in the $B_7$ stripe, and unequal in the $\mathrm{Sm}{C}_g$ stripe.

Figure 6

The $\mathrm{Sm}{C}_g$ stripe shown in Fig. 5 can lower the energy by the formation of bilayers such that the surface widths of adjacent layers match, resulting in an achiral structure in which the net polarization along the layer normal also cancels.

Figure 7

The orientation of the BC molecule in a stripe caused by the polarization splay in a $\mathrm{Sm}{C}_g$ layer. ζ is along the layer normal of the Cartesian coordinates ξηζ fixed to the layer. The coordinate system ABN is fixed to the BC molecule, only the upper half of which is shown to avoid clutter. The $N$ axis, which is orthogonal to the plane AOB of the molecule is also not shown. The long axis $\mathit{OB}$ tilts at an angle θ from the ζ axis. $\mathit{OD}$ is the line of intersection of the ξη and $\mathit{AN}$ planes. If the curvature distortion of m occurs by rotation about the layer normal ($\mathit{LN}$), the Eulerian angle ψ between $\mathit{OD}$ and $\mathit{OA}$ is fixed, while the Eulerian angle φ(η) varies between the two walls. If the distortion occurs by a rotation about the long axis ($\mathit{LA}$), φ has a fixed value, and ψ(η) varies to form the stripe.

Figure 8

Schematic diagram showing the disposition of neighboring stripes made of bilayers shown in Fig. 6, forming a two-dimensional rectangular lattice. The shaded areas are made of less bent excited state conformers which rotate freely about their long axes, giving rise to nonpolar walls. The trapeziums symbolically represent the polarization splay stripes made of the more bent ground state conformers (see Figs. 5 and 6).

Figure 9

Schematic diagram showing the disposition of molecules in a polarization splay stripe formed by the rotation of the molecules about their long axes which are confined to the ηζ plane, corresponding to $\phi =0$ in Fig. 7. The rotation angle $\psi =0$ at the center of the stripe, and varies from −π/2 rad at the left edge to +π/2 rad at the right edge, exploring all values in between. The location along ζ of the arrow axis ($A$ in Figs. 1 and 7) also varies across the stripe. The upper and lower surfaces of the stripe have equal widths unlike in the structure shown in Fig. 5, but relatively displaced along the η axis.

Figure 10

The mutual interaction energy between successive $\mathit{LA}$ type stripes can be lowered if they form bilayers as shown in the figure, with the BC molecules at the left end gaining entropically by easy fluctuation between the two layers, and those at the right end by van der Waals interactions, and a combination of the two in between. Both the net chirality and the component of the polarization along the layer normal ζ of the bilayer structure are 0.

Figure 11

Schematic diagram of a two-dimensional oblique lattice formed by the bilayers shown in Fig. 10. The vertical shaded regions are nonpolar walls made of freely rotating ES conformers. The parallelepipeds in between the walls represent the stripes made of GS conformers. If neighboring stripes tilt in opposite directions, a rectangular lattice results [5, 7].

Figure 12

Schematic diagram showing a bilayer made of adjacent layers in which polarization splay stripes have formed by rotation of the BC molecules about their long axes, which are confined to planes parallel to the ζξ plane, corresponding to $\phi =\pi /2$ rad in Fig. 7. The angle $\psi =\pi /2$ rad at the center, and varies from 0 to π rad between the two edges of the stripe.

Figure 13

Tilt angle dependencies of (a) the threshold field $E_{\mathrm{th}}$ and (b) the stripe width at threshold as functions of the tilt angle θ at $T^{*}–T_{\mathrm{tr}}=75\mathrm{K}$, with the parameters given in the first paragraph of this section, for the stripe having $\mathit{LA}$ structure with $\phi ={90}^{\circ }$. In the absence of the field, the stripe width is 35.08 nm independent of θ.

Figure 14

General structural formula of an asymmetric bent-core molecule with the left side arm connected by a hydrogen bond to a chemical group which has a few different moieties made of siloxanes and indicated by the symbol SiOG at the end of undecyloxy chain. The line between the hydroxyl (OH) group and the nitrogen (N) atom stands for the hydrogen bond between them. Compounds with this structure exhibit bilayer stripe phases when SiOG is a short linear moiety oriented either along or orthogonal to the axis of the arm, or even a relatively bulky group [21, 22]. Both the SiOG group and the H bond are necessary for the formation of stripes, which are not exhibited in the absence of either of them.

Figure 15

Schematic diagram showing a possible origin of the stripe phases exhibited by the H-bonded BC molecules shown in Fig. 14. The fragments of the molecules in which the hydrogen bonds are broken are assumed to collect in the wall region shown by the shaded part, and form nonpolar layers. The unbroken molecules form the stripes made of bilayers (see Fig. 12) with a spacing equal to $d_{\mathrm{st}}$ and large (>50 °) tilt angles. The layer spacing of the wall region ($d_{\mathrm{wall}}$) can be expected to be smaller than $d_{\mathrm{st}}$ unlike in the usual BC molecules, and the wall itself has to tilt at an angle $\delta =\mathrm{co}{\mathrm{s}}^{–1}\left(d_{\mathrm{wall}}/d_{\mathrm{st}}\right)$, to fill space. The stripes can form either an oblique or a rectangular 2D lattice, depending on the relative location of the second neighbor stripe.