Modeling dipolar and quadrupolar defect structures generated by chiral islands in freely suspended liquid crystal films

We report a detailed theoretical analysis of quadrupolar interactions observed between islands, which are disklike inclusions of extra layers, floating in thin, freely suspended smectic-$C$ liquid crystal films. Strong tangential anchoring at the island boundaries results in a strength $+1$ chiral defect in each island and a companion $-1$ defect in the film—these forming a topological dipole. While islands of the same handedness form linear chains with the topological dipoles pointing in the same direction, as reported in the literature, islands with different handedness form compact quadrupolar structures with the associated dipoles pointing in opposite directions. The interaction between such heterochiral-island-defect pairs is complex, with the defects moving to minimize the director field distortion as the distance between the islands changes. The details of the interisland potential and the trajectories of the $-1$ defects depend strongly on the elastic anisotropy of the liquid crystal, which can be modified in the experiments by varying the material chirality of the liquid crystal. A Landau model that describes the energetics of freely mobile defects is solved numerically to find equilibrium configurations for a wide range of parameters.

Figure 1

(Color online) Chiral islands in a freely suspended smectic film of achiral liquid crystal. Depolarized reflected light microscope (DRLM) images of a smectic-$C$ film of racemic MX8068 showing (a) two islands containing $+1$ defects with the same handedness and (b) two islands with $+1$ defects of opposite handedness. Each $+1$ defect is accompanied by a $-1$ defect in the film. The defects in (a) form a dipolar chain, while those in (b) form a topological quadrupole. The inset in (a) shows a single island with a $+1$ defect inside and a matching $-1$ defect in the background film, the pair forming a dipole. The laser illumination (shown by the arrow) is obliquely incident in the $x\text{−}z$ plane, allowing us to differentiate between left- and right-handed islands by comparing the brightness of the brushes in their upper and lower halves. The equilibrium $\mathbf{c}$-director field $(⊢)$ is shown around two islands with (c) the same handedness and (d) opposite handedness, with the head of the $⊢$ indicating the end of the molecule closer to the viewer.

Figure 2

Textures of heterochiral islands interacting on a film of 25% chirally doped MX8068. (a) The quadrupolar structure is in equilibrium when the islands almost touch. (b) The equilibrium separation between $-1$ defects increases as the islands are separated using optical tweezers. (c) When the separation is sufficiently large, the quadrupolar symmetry is broken. (d) When the islands are forced even further apart, the quadrupole evolves into two separate dipoles.

Figure 3

(Color online) Equilibrium vertical separation $S$ of the $-1$ defects as a function of the island center-to-center separation $D$ in the quadrupolar configuration regime, for racemic and 25% chirally doped films of MX8068, compared with the results of numerical calculations for systems with elastic anisotropies $\kappa =0.2$, 1.0, and 2.4. The straight lines are linear fits of the experimental data. The dashed lines are fits to the numerical data shown here to guide the eye. The value of $\kappa =2.4$ was chosen to reproduce the slope of the linear fit for the chiral mixture. Experimental textures [32, 40] indicate that the chiral mixture should correspond to $\kappa \approx 1.0$. Note, however, that the theoretical results for this $\kappa$ underestimate the value of $S/R$ at contact $\left(D/R=2\right)$. Fluctuations are expected to increase the average defect separation, $S$, and account, at least in part, for this discrepancy. The average slope of the experimental data for the racemate is high, indicating very small values of $\kappa$, a limit which is numerically inaccessible in our simulations.

Figure 4

(a) Total free energy as a function of homochiral island separation $D$, for different $\kappa$. When $\kappa ={\mathcal{K}}_b/{\mathcal{K}}_s=1$, the free energy exhibits a minimum at a separation $D_{\text{min}}=2\sqrt{2}R$. (b) Equilibrium separation $D_{\text{min}}$ as a function of the elastic anisotropy $\kappa$ of the system.

Figure 5

Equilibrium director configurations for different heterochiral-island separations with $\kappa =1$. (a) $D=2.2R$: the defects are equally shared by the islands, in a symmetric, quadrupolar configuration. (b) $D=3.0R$: the symmetry is broken; here the defects are found in a metastable configuration where they are both closer to one island. (c) $D=3.5R$: each defect follows a different island. (d) $D=4.0R$: even at this separation, the positions of the defects are still strongly influenced by the presence of the other island-defect pair, with configurations typical of isolated dipoles occurring at larger separations.

Figure 6

Equilibrium director configurations for different heterochiral-island separations with $\kappa =0.2$. The order parameter around the $-1$ defects has no longer a circular symmetry as in Fig. 5. This is reflected in the narrowness of the white brushes that come out of the $-1$ defects. (a) $D=2.2R$. (b) $D=3.0R$. (c) $D=3.5R$. (d) $D=4.0R$.

Figure 7

(a) Total free energy as a function of heterochiral-island separation $D$, for $\kappa ={\mathcal{K}}_b/{\mathcal{K}}_s=1$, 2, and 3. All systems have a well-defined equilibrium separation $D_{\text{min}}$ that is smaller than in the homochiral case and decreases monotonically with increasing $\kappa$ (b).

Figure 8

(a) Computed horizontal position of the right defect $x_d$ as a function of island separation $D$, for $\kappa =1.8$, 3.0, and 4.0. The reference system has its origin at the midpoint between the islands. (b) The quadrupolar symmetry of the island-defect pair is generally broken at a critical island separation $D_c$, shown here as a function of $\kappa$. The noisiness of the data when $\kappa <1$ reflects the numerical difficulties encountered in modeling this regime.

Figure 9

(a) Computed vertical displacement $y_d$ of the upper $-1$ defect as a function of island separation $D$, for $\kappa =1.4$, 2.4, and 4.5 [$y_d$ is half the vertical defect separation $S$ plotted in Fig. 3.] For small $D$ (close to equilibrium), the displacement of the defect increases linearly with a well-defined slope $m$, shown in (b) as a function of the elastic anisotropy $\kappa$.