Localized dissipative vortices in chiral nematic liquid crystal cells

Solitary waves and solitons have played a fundamental role in understanding nonlinear phenomena and emergent particle-type behaviors in out-of-equilibrium systems. This type of dynamic phenomenon has not only been essential to comprehend the behavior of fundamental particles but also to establish the possibilities of novel technologies based on optical elements. Dissipative vortices are topological particle-type solutions in vectorial field out-of-equilibrium systems. These states can be extended or localized in space. The topological properties of these states determine the existence, stability properties, and dynamic evolution. Under homeotropic anchoring, chiral nematic liquid crystal cells are a natural habitat for localized vortices or spherulites. However, chiral bubble creation and destruction mechanisms and their respective bifurcation diagrams are unknown. We propose a minimal two-dimensional model based on experimental observations of a temperature-triggered first-order winding/unwinding transition of a cholesteric liquid crystal cell and symmetry arguments, and investigate this system experimentally. This model reveals the main ingredients for the emergence of chiral bubbles and their instabilities. Experimental observations have a quite fair agreement with the theoretical results. Our findings are a starting point to understand the existence, stability, and dynamical behaviors of dissipative particles with topological properties.

Figure 1

Cholesteric liquid crystal cell under homeotropic anchoring and crossed polarizers. (a) Schematic representation of the experimental setup. $TC$ represents the thermal chamber. $O$ stands for the objective of the microscope, with magnifications ranging from $20\times$ to $50\times$. $P$ and $A$ are the polarizer and analyzer, respectively. (b) Director representation $\vec{n}$ of the chiral nematic liquid crystal displaying a TIC phase. $p$ is the cholesteric pitch. (c) The subcritical bifurcation between the (d) unwound state and the (e) TIC phase. The light blue shaded region stands for the bistability region. (f) Cholesteric labyrinth. (g) Loop of CF-1. (h) Chiral bubbles. (i) Fingering instability of the localized vortices.

Figure 2

Phase diagram of model Eq. (2) with $\beta =1$ and $\delta =0.05$. ${\mu }_{lb}=-1/4$ and ${\mu }_{ub}=0$ are the limits of the bistability region between $A_o$ and $A_T$. ${\mu }_{\text{MP}}=-3/16$ is the Maxwell point. The green line accounts for the spatial instability of $A_T$. The red curve is the saddle-node bifurcation of the localized vortices. The yellow line with $\square$ markers stands for the transition between loops and spherulites. The blue line with $\circ$ markers shows the mode-3 instability of chiral bubbles. ${\chi }_o=0.26$ is the minimum value for the existence of spherulites ($☆$). The $▵$ symbol is the triple point of the system. Regions I, II, III, IV, and V account for the stable zone of uniform state $A_o$, TIC phase $A_T$, cholesteric patterns, cholesteric loop, and chiral bubble, respectively. Path VI represents the fingering instability. $\psi =\text{Re}\left(A\right)\text{Im}\left(A\right)$ is the polarization field. The right panels illustrate the states in the respective regions.

Figure 3

Transition from CF-1 loops to chiral bubbles. (a) Experimental bifurcation triggered by decrease the temperature in the sample with 25 wt % of chiral dopant. (b) Numerical transition in model Eq. (2) with $\beta =1$, and $\delta =0.05$, when parameters change from region IV to V in the phase diagram of Fig. 2. $\psi =\text{Re}\left(A\right)\text{Im}\left(A\right)$ is the polarization field. $\varphi$ is the phase field. The middle panels are transient states.

Figure 4

Disappearance of chiral bubbles. (a) Experimental observation of the spherulite loss in the sample with 25 wt % of chiral dopant. The black dots correspond to the maximum intensity peak of the localized structure when the temperature is decreased from $56.3{}^{\circ }\mathrm{C}$ (i) until the disappearance at $T_d\approx 55.3{}^{\circ }\mathrm{C}$. The blue curve is the fit $128{(T-T_d)}^{0.08}$. The insets shows the spherulite changes under circular polarization. (b) Divergence of $A_{ac}$ at ${\chi }_c=0.989$, in Eq. (2) with $\mu =-0.45$, $\beta =1$, $\delta =0.05$, and $t_o=0$. The fit is $A_{ac}=0.024{({\chi }_c-\chi )}^{-1/2}$, where ${\chi }_c=0.9895$.

Figure 5

(a) Chiral bubble observed in model Eq. (2) with ${\mu }_o=-0.21$, $\chi =0.4$, $\delta =0.05$, and $\beta =1$. The radial profile $\left|A\right|=R\left(r\right)$ is characterized by the core $r_f$ of the interface. (b) Phase portrait of Eq. (4). The yellow curve accounts for the case $\chi =0$. The blue curve shows the force $\dot{r_f}$ when $\chi >{\chi }_o$. The open dots represent unstable solutions. The solid black dot is a stable equilibrium ($r_{eq}$). ${\mu }_{sn}$ is the saddle-node critical parameter. ${\mu }_{\text{MP}}$ is the Maxwell point.