Zig-zag wall lattice in a nematic liquid crystal with an in-plane switching configuration

Liquid crystals displays with tailoring electrodes exhibit complex spatiotemporal dynamics when a large voltage is applied. We report experimental observations of the appearance of a programmable zig-zag lattice using an in-plane-switching cell filled with a nematic liquid crystal. Applying a small voltage to a wide range of frequencies, the system exhibits an Ising wall lattice. Increasing the voltage, this lattice presents a spatial instability generating an undulating wall lattice, and to higher voltages it becomes zig-zag type. Experimentally, we characterize the bifurcations and phase diagram of the wall lattice. Theoretically, we develop, from first principles, a descriptive model. This model has a good qualitative agreement with experimental observations.

Figure 1

Sketch of the experimental setup, which represents an in-plane-switching cell connected to a generator and observed by a microscope (top), with $40\times$ magnification, in white light (bottom). Thickness between the two glass plates, $d=8.8\pm 0.2$ $\mu \mathrm{m}$. Thickness of a glass plate, $g=1$ mm. Active zone, $l\times l=1$ ${\mathrm{cm}}^2$. Gap between two electrodes, $e=15\mu \mathrm{m}$. Parallel polarizers to the molecules anchoring, $P_{\perp }$ (following the $x$ axis).

Figure 2

Wall lattice, experimental snapshots without voltage (top left panel) and for $T=20$ ${\mathrm{V}}_{\mathrm{pp}}$ with different frequencies: top right panel, $f=10$ Hz, Ising wall lattice; bottom left panel, $f=15$ Hz, undulating wall lattice; bottom right panel, $f=100$ Hz, zig-zag wall lattice.

Figure 3

(a) The electric field representation inside the liquid crystal sample using formula (4). (b) The molecules orientation in the liquid crystal sample.

Figure 4

Experimental characterization of the undulation wall. Amplitude of the undulation wall vs (a) frequency for $V=20{\mathrm{V}}_{\mathrm{pp}}$; (b) amplitude for $f=100$ Hz. $⧫$ experimental points; the dashed and solid lines, respectively, represent the analytical expectation value $|a_{\max }|$ of Eq. (1) without and with noise, respectively. (a) $\alpha =0.53$, $f_0=14.7$ Hz; $\alpha =0.53$, $f_0=14.7$ Hz, $\eta =0.005$. (b) ${\alpha }^{\prime }=1.92$, $V_0=15.7$ V; ${\alpha }^{\prime }=2.02$, $V_0=15.7$ V, $\eta =0.002$.

Figure 5

Phase diagram of the wall lattice in the frequency and voltage domain. The dark and the light zones account for the regions of Ising and zig-zag wall lattice, respectively. Dashed curve accounts for the modulational instability. $⧫$ stands for experimental points of the spatial bifurcations.

Figure 6

Spatiotemporal diagram (smoothed) of a Ising wall. (a) Experimental spatiotemporal of the pattern, and (b) plot of the last line of the spatiotemporal diagram. $V=17$ ${\mathrm{V}}_{\mathrm{pp}}$ and $f=1$ kHz.

Figure 7

(a) Molecules orientation in the liquid crystal sample, showing the two boundary layers in green with a thickness $\delta$ and the bulk in white with a thickness $d$. (b) Horizontal component of the electric field representation (continuous line) and its approximation (dashed curve).

Figure 8

Interface dynamics. (a) Growth rate relation, Eq. (17), for three values of $\lambda$. Experimental and numerical interface profiles for the three previous cases: (a) $f$ = 10 Hz (exp), ${\lambda }_1=0.30$ (num); (b) $f$ = 15 Hz (exp), ${\lambda }_1=0.24$ (num); (c) $f$ = 100 Hz (exp), ${\lambda }_1=0.0001$ (num). $V=20$ ${\mathrm{V}}_{\mathrm{pp}}$ (exp), $D=-1$ (num), ${\lambda }_c=D^2/4=0.25$.

Figure 9

Spatiotemporal diagram of an Ising wall obtained from numerical simulations of the wall equation (16), $D=-1$ and $\lambda =0.2$. (a) Spatiotemporal evolution of the wall; (b) plot of the last line of the spatiotemporal diagram.

Figure 10

Bifurcation diagram of the amplitude of the wall, model (16), as a function of the bifurcation parameter $\Delta =D^2/4-\lambda$. The points represent the results obtained by numerical simulations of the model (16). The solid curve is obtained by using the formula (21).