Lehmann rotation of cholesteric droplets: Role of the sample thickness and of the concentration of chiral molecules

We study the role of the sample thickness $d$ and of the concentration $C$ of chiral molecules during the Lehmann rotation of cholesteric droplets of radius $R$ subjected to a temperature gradient $\vec{G}$. Two configurations are studied depending on how the helix is oriented with respect to $\vec{G}$. The first result is that, at fixed $C$ and $R$, the rotation velocity $\omega$ increases with $d$ when the helix is parallel to $\vec{G}$, whereas it is independent of $d$ when the helix is perpendicular to $\vec{G}$. The second result is that, for a given $C,{\omega }_0={lim}_{R\to 0}\omega \left(R\right)$ is the same for the two types of droplets independently of $d$. This suggests that the, as yet unknown, physical mechanism responsible for the droplet rotation is the same in the two types of droplets. The third result is that the Lehmann coefficient $\overline{\nu }$ defined from the Leslie-like relation ${\omega }_0=-\overline{\nu }G/{\gamma }_1$ (with ${\gamma }_1$ the rotational viscosity) is proportional to the equilibrium twist $q$. Last, but not least, the ratio $\overline{R}=\overline{\nu }/q$ depends on the liquid crystal chosen but is independent of the chiral molecule used to dope the liquid crystal.

Figure 1

Two striped droplets of similar radii $(\approx 15\mu \mathrm{m})$ observed under natural light in two samples of different thicknesses. In (a), $d=20.1\mu \mathrm{m}$ and in (b) $d=68.8\mu \mathrm{m}$ (MBBA + 1% R811, $\Delta T=5{}^{\circ }\mathrm{C})$.

Figure 2

Product ${\Theta }_{\perp }\Delta T$ of the period of rotation times the temperature difference as a function of the droplet radius $R$. Each symbol corresponds to a different thickness given in $\mu \mathrm{m}$ on the graph. All the points correspond to banded droplets, except the three points marked by an arrow referring to droplets with a different texture (see the main text).

Figure 3

Two types of droplet observed under natural light in the 6.$8\mu \mathrm{m}$-thick sample. In (a) the bands form a double spiral, and in (b) they form a labyrinth. The bar is $20\mu \mathrm{m}$ long (MBBA+1% R811).

Figure 4

Two droplets of similar radii $(\sim 11.5\mu \mathrm{m})$ observed between crossed polarizers in two samples of different thicknesses. The helix is oriented parallel to the temperature gradient by applying a 10 kHz electric field. In (a), $d=21.6\mu \mathrm{m},V=25$ Vrms and in (b) $d=70\mu \mathrm{m},V=80$ Vrms. (MBBA+1% R811, $\Delta T=10{}^{\circ }\mathrm{C})$.

Figure 5

(a) Product ${\Theta }_{\parallel }\Delta T$ of the period of rotation times the temperature difference as a function of the droplet radius $R$ and their best linear fits. Each symbol corresponds to a different thickness given in $\mu \mathrm{m}$ on the graph.(b) Slope of the straight lines as a function of the sample thickness $d$. The dashed line is only a guide for the eye.

Figure 6

(a) Product ${\Theta }_{\perp }\Delta TC$ of the period of rotation times the temperature difference times the concentration as a function of the dimensionless radius $qR$. The host liquid crystal is MBBA in (a) and 7CB in (b). The chiral molecule is R811.

Figure 7

(a) Product ${\Theta }_{\parallel }\Delta TC$ of the period of rotation times the temperature difference times the concentration as a function of the radius $R$. MBBA+R811, $d=30\mu \mathrm{m}$. The different symbols correspond to different concentrations (from Ref. [12]). (b) Extrapolation at radius $R=0$ of the product ${\Theta }_{\parallel }\Delta TC$ as a function of the thickness $d$. These values are deduced from the data shown in Fig. 5. The dashed line gives the average value, close to 0.6.