Qualitative chirality effects on the Casimir-Lifshitz torque with liquid crystals

We model a cholesteric liquid crystal as a planar birefringent multilayer system, where the orientation of each layer differs slightly from that of the adjacent one. This allows us to analytically simplify the otherwise acutely complicated calculation of the Casimir-Lifshitz torque. Numerical results differ appreciably from the case of nematic liquid crystals, which can be treated like bloc birefringent media. In particular, we find that the torque deviates considerably from its usual sinusoidal behavior as a function of the misalignment angle. In the case of a birefringent crystal faced with a cholesteric liquid one, the Casimir-Lifshitz torque decreases more slowly as a function of distance than in the nematic case. In the case of two cholesteric liquid crystals, either in the homochiral or in the heterochiral configuration, the angular dependence changes qualitatively as a function of distance. In all considered cases, finite pitch length effects are most pronounced at distances of about 10 nm.

Figure 1

One-dimensional stratification of $N$ finite thickness anisotropic dielectric slabs representing an arbitrary planar multilayer stack. The stratified media, $i=1,N$, with generally anisotropic layers are embedded in an isotropic bathing medium. Each layer has an arbitrary material composition, thickness, and orientation.

Figure 2

Illustration of the continuum approximation used here to model a liquid crystal. It consists of layers with identical thicknesses $d$ but each with a slightly different orientation, each with identical increments $\delta \ll 1$. Consequently, the multilayer structure manifests as a continuous inhomogeneity in the direction perpendicular to the plane of reflection (defined as the $z$-direction), where the orientation effectively behaves as a helix with pitch length $L$.

Figure 3

Dielectric functions of pure, nematic, and doped, cholesteric 5CB. The dielectric function of nematic 5CB as proposed in Ref. [33] (indicated by the black squares and blue triangles) is modified to account for the presence of a chiral dopant. This affects the static values (see the magenta crosses and the red circles), which were obtained from Ref. [57].

Figure 4

The Casimir torque between a birefringent (${\mathrm{BaTiO}}_3$) crystal and a cholesteric liquid crystal. (a) A left-handed crystal facing a birefringent one. (b) A right-handed crystal facing a birefringent one. (c) The Casimir torque as a function of misalignment angle at a distance of $a=10$ nm. The different curves are as follows. The triangles pointing right show the usual result for a nematic liquid crystal. Red circles: left-handed crystal, $\delta =0$. Blue triangles pointing left: left-handed crystal, $\delta \ne 0$. Magenta crosses: right-handed crystal, $\delta =0$. Black squares: right-handed crystal, $\delta \ne 0$. (d): The amplitude of the Casimir torque as a function of separation. The triangles pointing right show the nematic case. The cholesteric case is shown by the other two curves, of which the red circles represents $\delta =0$ and the black squares show the perturbed case. The inset of panel (d) shows the Casimir torque for the cases (a) (red circles) and (b) (blue crosses) at distances between 380 and 800 nm.

Figure 5

The Casimir torque between two cholesteric liquid crystals of identical chirality. (a) Homochiral configuration of two left-handed crystals (b) Homochiral configuration of two right-handed crystals. (c) Casimir torque as a function of misalignment angle at a distance of $a=10$ nm. The different curves are as follows. Triangles pointing right: two nematic liquid crystals. Black crosses: unperturbed, case (a). Blue squares: perturbative contribution to case (a). Red circles: unperturbed case (b). Orange circles: perturbative contribution to case (b). (d) The Casimir torque as a function of misalignment angle for distances between 380 an 800 nm. The blue triangles represent case (a) and the red circles represent case (b).

Figure 6

The Casimir torque for the heterochiral case. (a) The left-handed crystal is placed on the left, and the right-handed crystal is on the right. (b) The reverse of panel (a). (c) Several different contributions at 10-nm distance. Black crosses: difference between case (a) and case (b), including only lowest-order terms. Red asterisks: idem, including higher-order terms. Orange circles: perturbative contribution to case (a), blue squares: perturbative contribution to case (b). (d) Casimir torque at distances between 380 and 800 nm, both for case (a) (black asterisks), and case (b) (red circles).