Rotation-Time Symmetry Breaking in Frustrated Chiral Nematic Driven by a Pulse-Train Waveform

We report waveform-induced rotation-time symmetry breaking in liquid crystal director motion. Homeotropic cells filled with a negative dielectric anisotropy chiral nematic exhibit persistent and visually observable waves of director orientation with a time period of at least 30 driving field cycles. Their existence in the space of driving waveform parameters is explored. The possibility of utilizing this system, which exhibits both spatial and temporal long-range order, as a modeling tool for experimental studies on discrete time crystals is discussed.

Figure 1

(a) Definition of the waveform (thick blue line) parameters: signal $\mathrm{period}=t_{\text{ON}}+t_{\text{OFF}}$; phase ${\psi }_{\mathrm{sq}}$ of the square wave (dashed magenta line, ${\psi }_{\mathrm{s}\mathrm{q}}=30°$) is set at the beginning of each $t_{\text{ON}}$ time interval. When the square wave frequency $f_{\text{ON}}$ is an integral multiple of $1/t_{\text{ON}}$, the average dc is zero. (b) Coordinates describing LC director ($\mathit{n}$) motion ($\dots –0–1–2–3–4–\dots$): the $z$ axis is normal to the substrates; $\mathit{c}$, projection of $\mathit{n}$ on sample plane $xy$; $\theta$, polar angle; and $\phi$, azimuthal angle. The sphere is an eye guide. During $t_{\text{ON}}$, the electric field is applied along the $z$ axis; because $\mathrm{\Delta }\epsilon <0$, $\mathit{n}$ starts tilting away from the vertical position, following red paths (0–1, 2–3, etc.). During $t_{\text{OFF}}$, the elastic forces return the director to a nearly (limited by finite $t_{\text{OFF}}$) vertical alignment state following different paths (blue, 1–2, 3–4, etc.). (c) Conoscopic observation: the difference between ON (curved magenta arrow) and OFF (straight teal arrow) motion paths of the melatope is visible. A minimum tilt of $\sim 1°$ always remains (dotted circle). (d) Traveling wave pattern in a LC cell between crossed polarizers. Orange arrow depicts the wave propagation direction. The white and black lines designate $\mathit{c}$ parallel and/or perpendicular or at $\pm 45°$ to the polarizer. The red square represents the light-emitting diode (LED) projection on the sample [Fig. 2]. (e) Temporal evolution of the pattern (Supplemental Material [14], “WavesVideo”). Every third cycle of input waveform is shown. $A_{\text{ON}}=6.0\mathrm{V}$, ${\psi }_{\mathrm{s}\mathrm{q}}=0°$, $t_{\text{ON}}=10\mathrm{ms}$, $t_{\text{OFF}}=23.3\mathrm{ms}$, $f_{\text{ON}}=100\mathrm{Hz}$. The red arrows and conoscopic images in the insets depict azimuthal angle evolution at a single point.

Figure 2

(a) Schematic diagram of the setup for time periodicity studies of patterns. (b) Fragment of a typical signal. The red arrows indicate the change in the tilt direction with wave propagation. (c) FFT spectrum of the transmittance for $A_{\text{ON}}=5.2\mathrm{V}$, ${\psi }_{\mathrm{sq}}=0°$, $t_{\text{ON}}=10\mathrm{ms}$, $t_{\text{OFF}}=20\mathrm{ms}$, $f_{\text{ON}}=100\mathrm{Hz}$. Inset: magnified peak at $4\times f_{360}$.

Figure 3

(a) Dependence of the rotation frequency $f_{360}$ of the $\mathbf{c}$ director on the amplitude of the applied voltage $A_{\text{ON}}$ and OFF time $t_{\text{OFF}}$. Experimental parameters are $t_{\text{ON}}=4\mathrm{ms}$, $f_{\text{ON}}=250\mathrm{Hz}$, ${\psi }_{\mathrm{sq}}=0°$, settling $\mathrm{time}=100\mathrm{s}/\mathrm{point}$, acquisition $\mathrm{time}=250\mathrm{s}/\mathrm{point}$. (b) Cross-sectional profiles corresponding to three values of $A_{\text{ON}}$ [dash-dotted lines of corresponding colors on (a)], near the maximum of $f_{360}$. The frequency is converted into a variation in the azimuthal angle $\mathrm{\Delta }\phi$ per cycle of applied waveform.

Figure 4

Dependence of the rotation frequency $f_{360}$ the $\mathbf{c}$ director on the ${\psi }_{\mathrm{sq}}$ parameter of applied pulse train and square wave frequency $f_{\text{ON}}$. $t_{\text{ON}}=8\mathrm{ms}$, $t_{\text{OFF}}=22\mathrm{ms}$, $A_{\text{ON}}=6.2\mathrm{V}$, settling $\mathrm{time}=200\mathrm{s}/\mathrm{point}$, acquisition $\mathrm{time}=250\mathrm{s}/\mathrm{point}$. A linear decline in the frequency for most of the ${\psi }_{\mathrm{sq}}$ range [green line (a)] and regions with no movement (b) was observed. Crossing ${\psi }_{\mathrm{sq}}=180°$ results in change of the movement direction (orange arrows on the insets).