Role of anchoring energy on the texture of cholesteric droplets: Finite-element simulations and experiments

We present a numerical method to compute defect-free textures inside cholesteric domains of arbitrary shape. This method has two interesting properties, namely a robust and fast quadratic convergence to a local minimum of the Frank free energy, thanks to a trust region strategy. We apply this algorithm to study the texture of cholesteric droplets in coexistence with their isotropic liquid in two cases: when the anchoring is planar and when it is tilted. In the first case, we show how to determine the anchoring energy at the cholesteric-isotropic interface from a study of the optical properties of droplets of different sizes oriented with an electric field. This method is applied to the case of the liquid crystal CCN-37. In the second case, we come back to the issue of the textural transition as a function of the droplet radius between the double-twist droplets and the banded droplets, observed for instance in cyanobiphenyl liquid crystals. We show that, even if this transition is dominated by the saddle-splay Gauss constant $K_4$, as was recently recognized by Yoshioka et al. [Soft Matter 12, 2400 (2016)], the anchoring energy does also play an important role that cannot be neglected.

Figure 1

Two configurations of a cholesteric droplet when the anchoring is planar at the surface and the LC is of negative dielectric anisotropy. In panel (a), there is no electric field and the droplet adopts a bipolar structure with two virtual point defects diametrically opposed (defining the polar axis). Only the director field lines on the surface of the droplet are represented (orange lines). In panel (b), an electric field $E$ is applied vertically. In this case, the two point defects are stretched into two virtual surface disclination lines (thick solid and dashed lines). The director field lines are the orange solid lines. By convention, we will call “polar axis” the axis drawn in blue solid line that joins the two disclination lines in the equatorial plane. For the sake of simplicity, the configuration represented here corresponds to the limit $W_a\to \infty$ and $E\to \infty$.

Figure 2

The setup used to measure the transmission of the droplets. Observations were made with a SCMOS camera through the objective (O). Thanks to the beam splitter (SR), the sample was illuminated either globally with the lamp (TL) or locally by focusing with the condenser (C) the Gaussian beam of the laser (FL) cleaned by the spatial filter (SF). The orientations of the polarizer (P) and the analyzer (A) can be set independently. At last, the ovens and the sample can be moved horizontally with an $XY$ translation stage.

Figure 3

Minimum (a) and maximum (b) of the transmission signal as a function of the droplet radius. Crosses are experimental points and dashed and solid lines are numerical curves.

Figure 4

(a) Three cuts of the director field for $R=9\mu \mathrm{m}$ along the $xy,xz$ and $yz$ planes. The orientation of the droplet with respect to the three axes is given in the bottom right corner. Tilted molecules are represented by nails, proportional in length to the director projection in the plane of the drawing. The nail point is oriented towards the reader. (b) Numerical and experimental images of the same droplet than in (a). In all these images, the polar axis (as defined in Fig. 1) is vertical and aligned with the $y$-axis.

Figure 5

(a) Equatorial cut of the director field in the DT case (left) and ST case (right). (b) Topology of the surface virtual defects (black dots) and disclination lines (black lines) in a DT droplet (left) and ST droplet (right).

Figure 6

Value of the order parameter $\psi$ as a function of the dimensionless radius $qR$ for three values of the dimensionless anchoring length $q{l}_a$. Inset: First-order derivative of the dimensionless energy as a function of the dimensionless radius. The break in the slope is the hallmark of a second-order transition. All these curves were computed by taking $K_4/K_1=0.78$.

Figure 7

(a) Dimensionless critical radius $q{R}_c$ as a function of the ratio $K_4/K_1$ for two values of the dimensionless anchoring length $q{l}_a$ (dashed and dotted lines). The solid line was calculated from the Yoshioka et al. model. (b) Same quantity as a function of $q{l}_a$ for three values of the ratio $K_4/K_1$.

Figure 8

Coordinate and angle conventions for the director field inside a DT droplet. The position is parametrized by the polar coordinates $(r,\theta ,z)$ by taking the $z$ axis along the revolution axis of the droplet. The director field is parametrized by the Euler angles $(\alpha ,\beta )$ defined in the local frame $({\vec{e}}_{\theta },{\vec{e}}_z{\vec{e}}_r)$.

Figure 9

Euler angles $\alpha$ (a) and $\beta$ (b) as a function of $r/R$ for different values of $z$. The director field was computed by taking $q{l}_a=100,qR=2.7$ and $K_4/K_1=0.78$.