Differential rotation in cholesteric pillars under a temperature gradient

Steady rotation is induced in cholesteric droplets dispersed in a specific liquid solvent under a temperature gradient. In this phenomenon, two rotational modes have been considered: (1) collective rotation of the local director field and (2) rigid-body rotation of the whole droplet structure. However, here we present another rotational mode induced in a pillar-shaped cholesteric droplet confined between substrates under a temperature gradient, that is, a differential rotation where the angular velocity varies as a function of the radial coordinate in the pillar. A detailed flow field analysis revealed that every pillar under a temperature gradient involves a double convection roll. These results suggested that the differential rotation in the cholesteric pillars was driven by the inhomogeneous material flow induced by a temperature gradient. The present experimental study indicates that the coupling between the flow and the director motion plays a key role in the rotation of the cholesteric droplets under the temperature gradient.


Caption for videos
Video 1. Movie of the rotational motion in the Ch LC pillar under a temperature gradient (SI1.mp4). The timescale is 5 times faster than real. The experimental condition is the same with Fig.1.
Video 2. Movie of the Ch LC pillar with planar anchoring under a temperature gradient (SI2.mp4).
The timescale is 5 times faster than real. The optical set up is the same with that in Fig.1. The cell thickness was 20m, and averaged temperature was 35℃. The direction of the temperature gradient T was parallel to the paper, and the applied heat flow was ~14.8mW/mm 2 . No rotational motion was observed because of the director field was fixed by the planar anchoring.
Video 3. Schematic representation of director rotation and rotational flow driven by unidirectional flow in Ch LC (SI3(a).mp4 and SI3(b).mp4). In these movies, the situation of Eqs. (4) and (5) is described: a linear flow along the helical axis (z axis) of a left-hand single helix structure is assumed. Owing to the flow, each molecule moves along z axis as time passes. The central movie shows the cross-section parallel to z axis. Both left and right-side movies are the cross-section perpendicular to z axis, while the left one is shown in the coordinate system moving with the same velocity with the flow, and the right is in the spatially fixed coordinate. In SI3(a), the director is not fixed, and each molecule translates with their orientations kept. In this case, the director rotation is induced as shown in the right-side movie. On the other hand, in SI3(b) the director is fixed. In this case, each molecule should move with rotating its orientation as shown in the leftside movie. Owing to this rotational motion, the rotational flow is induced. It should be noted that the directions of the director rotation in (a) and the rotational flow in (b) are opposite from each other.

Supplementary Method. Detailed description about fluorescence photo-bleaching method.
For the flow-field analysis with the photo-bleaching method, we used a commercial microscope (BX61, Olympus) and a CCD camera (Retiga 4000R Fast 1394, Qimaging). The schematics of the experimental system is shown in Fig. S1. When the sample was irradiated with a strong blue light from the mercury lamp during 3 seconds, the fluorescent dyes were photo-bleached. Here, using a photomask, we bleached the dyes as shown in Fig. S2(a). After the excitation beam was turned off, the whole sample was weakly illuminated by the blue light from the mercury lamp through ND filters. Here, the sample was irradiated through the objective lens just above the cell, and we set the light to be focused on just below the upper cell substrate. In this experiment, we used the cells with thickness 20 or 50m, and the bleached dyes diffused to the region with radius ~30m during the measurements as shown in Fig. S2(d). When the sample with the thickness of 50m was irradiated, the dyes near the upper cell substrate was more bleached rather than those near the lower substrate. The measurement was made under the temperature gradient. Hence, when the upper substrate was cooled or heated, the flow field in the low or the high-temperature side was measured, respectively.
In this study, we bleached the samples with a lattice or a line pattern. The analysis procedure for the former case is similar to that described in ref. 13; thus, we show the procedure for the latter case. As shown in Figs. 2(a), 3(a) and 4(a) the samples are bleached with a line pattern just after the photo-bleaching. We set the direction perpendicular and parallel to the line pattern x and y axis respectively. In this measurement, we can obtain the flow velocity component along x direction vx.
In the measurement we obtained the time evolution of the 2-dimensional (2D) fluorescence intensity profiles after the photo-bleaching (I). Normalizing them by the profiles before the photo    Fig.2 (a). Using the intensity profiles in the area framed with the blue box, we obtained the 1D intensity profiles along the x direction as shown in (b). The data in (b) are well fitted with the Gaussian of Eq. (S1), and the time evolution of the Gaussian centre is obtained as shown in (c). Here, the slope of the graph in (c) indicates vx. (d) is a time evolution of the half width of Gaussian, (see, Eq. (S1)). From the fitting with the linear function, the diffusion constant D is obtained as 7.6m 2 /sec. In addition, it can be estimated the dyes diffuses to the region with the radius ~30m during the measurements.
Setting the coordinate origin to be the pillar centre, we considered that the observed 2D flow would be the linear combination of the radial and rotational flows. Thus, vx can be described as cos sin where  is azimuthal coordinate, and vr and v are the flow velocity component of the radial and azimuthal direction respectively. When the centre of the bleached line pattern goes through the origin, the radial flow is not observed because 2

 
. In this case, the angular velocity of the rotational flow can be obtained by the relation: When the rotational flow can be neglected in Eq. (S2), vr is obtained by the relation:

Supplementary Note.
Calculation of dissipation function with a slightly generalized model. In the dissipation function W in Eq. (6), the calculation was performed under the assumption of no radial flow ( 0 r v  ). In this note, we calculate W without this assumption. Using a cylindrical coordinate ( , , ) rz  , the flow field is assumed to be ( , , ) Equation (