Phase Winding in Chiral Liquid Crystalline Monolayers due to Lehmann Effects

We model recent experimental results by Tabe and Yokoyama on chiral liquid crystalline monolayers driven by the transmembrane transfer of water molecules. We point out that the target patterns found for the orientation of the in-plane preferred direction can be described as phase winding due to Lehmann effects, which are specific to macroscopically chiral systems. The model presented has only one variable and is driven and dissipative. Including the effects of noise leads to an accurate representation of the patterns observed experimentally. Future experimental tests of the model are suggested.

Figure 1

Time evolution of the simulated phase-winding pattern with noise (noise amplitude ${\xi }_0=10{\omega }_L$) under the reflecting-polarizing microscope. The system is rectangular with aspect ratio 1.3 (according to 650 by 500 grid points); the director is horizontal ($\Phi =0$) initially. We show plots from top (left column) at times $0.01t_0$, $0.03t_0$, $0.05t_0$, and $0.07t_0$, and (right column) $0.09t_0$, $0.11t_0$, $0.13t_0$, and $0.15t_0$. The gray scale in this and the following figures was calculated using the dependence of the reflectivity on the director orientation given in Ref. 7.

Figure 2

The pattern at time $0.1t_0$ for a square cell (500 by 500 grid points) is shown as a function of the applied noise strength ${\xi }_0$, which is 0, $8{\omega }_L$ (left column), $16{\omega }_L$, and $24{\omega }_L$ (right column). Note that ${\xi }_0^2{\Delta }^2{\Delta }_t$ is the relevant parameter for noise; e.g., decreasing $\Delta$ by some factor one must increase ${\xi }_0$ by the same factor for an equal noise effect, provided that $\Delta$ is small compared to all other physical length scales (analogously for ${\Delta }_t$).

Figure 3

A spiral pattern (without noise) at time $0.1t_0$. As an initial condition, a defect of strength $+1$ has been implemented.