Layer-Scale Optical Chirality of Liquid-Crystalline Phases

We present a model for the optical chirality of layered liquid-crystalline phases. The model demonstrates that uniform stacking of chiral layers can lead to significant collective optical rotation, even in the absence of a superlayer helix. We predict the optical rotation of the $B2$ phases of bent-core liquid crystals, which can have optical rotation as large as 1000 times the molecular optical activity.

Figure 1

Schematic of the three distinct structural regimes of condensed phases that produce optical rotation: (a) isotropic phases of chiral molecules, (b),(c) chiral superlayer structures, (d)–(g) layer optical chirality (LOC). (a) Isotropic liquid of chiral molecules, which produces small visible wavelength optical rotation ($1\mathrm{deg}/\mathrm{cm}$). The optical rotation can be modeled by introducing gyrotropy into the molecular polarizability tensor. (b) A $\mathrm{Sm}{C}^{*}$ supermolecular helix, with rotation of the molecular tilt (magenta/gray straight $\mathbf{T}$ symbols) and polarization (arrows) about the layer normal $z$. (c) Projection representation of the local achiral layer structure as a biaxial dielectric tensor. The colossal optical rotation ($100\mathrm{deg}/\mu \mathrm{m}$) for wavelengths near a Bragg reflection can be accounted for by the helical winding of such achiral elements, i.e., without local optical chirality. (d) Representation of the inherently chiral local layer structure by deformed $\mathbf{T}$ symbols. (e) Uniformly stacked chiral elements that form multilayer synclinic synpolar layers ($\mathrm{Sm}{C}_S{P}_S$). (f),(g) Optical model of LOC, motivated by the molecular structure of a typical bent-core mesogen (W508). The molecule is sketched in $x\mathrm{\text{−}}z$ and $y\mathrm{\text{−}}z$ projections, where the envelope around the molecular structure is thicker in regions that project towards the reader. The core region (light background) consists of a pair of homogeneous achiral uniaxial slabs [light gray (yellow) cylinders], consistent with the $C2$ symmetry of the molecules. The structure is made chiral by the addition of distinct tail regions (dark background), which are taken to be isotropic [dark gray (cyan) circles], as the core birefringence dominates in bent-core molecules. Alternatively, the tail region could consist of uniaxial slabs [dark gray (cyan) cylinders] with orientation different from the core regions. This structure produces large optical rotation ($1\mathrm{deg}/\mu \mathrm{m}$), intermediate between the optical chirality of individual molecules and superlayer helices.

Figure 2

(a) Three-dimensional polar plot of optical rotation, $\varphi$, vs optical propagation direction, $\mathbf{k}$, in the $\mathrm{Sm}{C}_A{P}_S$ (inset). This structurally achiral phase is optically chiral when $\mathbf{k}$ breaks the structural mirror or glide symmetries. (b) OR, $\varphi$, vs molecular core tilt angle, $\theta$, for on-axis propagation in the $\mathrm{Sm}{C}_S{P}_S$ parallel to $x$ (solid line) and $y$ (dashed line). The phase is generally chiral for $\theta \ne 0$. (c) OR, $\varphi$, vs $\theta$ for 45° off-axis propagation in the $x\mathrm{\text{−}}z$ plane (blue arrow), shown for the $\mathrm{Sm}{C}_A{P}_S$ (dashed line) and $\mathrm{Sm}{C}_S{P}_S$ (solid line). The $\mathrm{Sm}{A}_P$ phase ($\theta =0$) has optical rotation for $\mathbf{k}$ off axis.