Spatiotemporal patterns in a Langmuir monolayer due to driven molecular precession

Langmuir monolayers of chiral liquid crystals on the surface of water exhibit orientational waves with complex spatiotemporal patterns. These patterns arise from a collective precession of the mesogenic molecules, driven by the evaporation of water through the monolayer. We investigate the behavior of these orientational waves around topological defects in the molecular orientation. Through Brewster angle microscopy, we find that the waves form a reversing spiral pattern, which rotates about the central vortex. With increasing relative humidity, the rotation slows and then stops. We model the system theoretically, and show that predicted patterns are in good agreement with the experiments.

Figure 1

Surface pressure $\left(\pi \right)$–area per molecule $\left(A_m\right)$ isotherm of cholesteric acid (ChA) monolayer at the air-water interface. Kinks in the isotherm are denoted by arrows at $\mathbf{a}$ $(A_m=57{\mathrm{\AA }}^2)$, $\mathbf{b}$ $(A_m=52{\mathrm{\AA }}^2)$, and $\mathbf{c}$ $(A_m=35{\mathrm{\AA }}^2)$. Inset: Chemical structure of the ChA molecule.

Figure 2

BAM images of the ChA monolayer at the air-water interface: (a) Coexistence of dark region and gray domains with stripe patterns $(A_m=87.0{\mathrm{\AA }}^2)$. (b) Stripe patterns on gray background $(A_m=54.7{\mathrm{\AA }}^2)$. Scale $\text{bar}=500\mu \mathrm{m}$.

Figure 3

(a) ChA monolayer, epifluorescence image at $A_m=54{\mathrm{\AA }}^2$. The gray uniform texture represents the uniform $L_1^{\prime }$ phase. Scale $\text{bar}=312\mu \mathrm{m}$. (b) BAM image at the same $A_m$, after $24\mathrm{h}$ at $\pi =2.4\mathrm{mN}∕\mathrm{m}$. Scale $\text{bar}=500\mu \mathrm{m}$.

Figure 4

BAM images at $A_m=54{\mathrm{\AA }}^2$, four minutes after compression to surface $\text{pressure}=2.4\mathrm{mN}∕\mathrm{m}$. Time between $\text{images}=2\sec$; arrows show rotation sense. Scale $\text{bar}=335\mu \mathrm{m}$.

Figure 5

Spiral rotation rate as a function of relative humidity. The points represent the experimental data, and the dashed line is a guide to the eye.

Figure 6

Spiral patterns for a single vortex surrounded by a domain wall: (a) Experimentally observed BAM image (size of full $\text{image}=1100\mu \mathrm{m}$). (b) Theoretical prediction for the preliminary single-Frank-constant approximation, with rigid anchoring on the inner and outer boundaries. (c) Prediction for the case of unequal Frank constants, $K_3∕K_1=1.5$. In case (c), the anisotropy of Frank constants gives an effective pinning of the director field at the vortex core, even without any imposed anchoring at the inner boundary.

Figure 7

(Color online) Plots of the director orientation $\tilde{\phi }\left(r\right)$, for infinite anchoring on the outer boundary and no anchoring on the inner boundary. The elastic anisotropy is $K_3∕K_1=1.5$ and radius is $r_{\max }=5$, with $\gamma {\omega }_L∕K=0–1.4$, in steps of 0.1, from bottom to top.

Figure 8

(Color online) Plots of the total free energy $F_{\text{total}}$ as a function of $\tilde{\phi }\left(r_{\max }\right)$ for several values of the anchoring strength $W$, with $K_3∕K_1=1.5$, $r_{\max }=5$, and $\gamma {\omega }_L∕K=0.5$.

Figure 9

(a) Experimental BAM image (scale $\text{bar}=500\mu \mathrm{m}$) and (b) simulated defect pattern in an extended domain.

Figure 10

Snapshots from a lattice simulation of a domain with finite anchoring energy, showing the creation, motion, and annihilation of four dislocation pairs.