Phase transition dynamics and stochastic resonance in topologically confined nematic liquid crystals

Topological defects resulting from boundary constraints in confined liquid crystals have attracted extensive research interest. In this paper, we use numerical simulation to study the phase transition dynamics in the context of stochastic resonance in a bistable liquid crystal device containing defects. This device is made of nematic liquid crystals confined in a shallow square well and is described by the planar Lebwohl-Lasher model. The stochastic phase transition processes of the system in the presence of a weak oscillating potential is simulated using an overdamped Langevin dynamics. Our simulation results reveal that, depending on system size, the phase transition may follow two distinct pathways: In small systems the preexisting defect structures at the corners hold until the last stage and there is no newly formed defect point in the bulk during the phase transition and in large systems new defect points appear spontaneously in the bulk and eventually merge with the preexisting defects at the corners. For both transition pathways stochastic resonance can be observed, but the corresponding phase transition dynamics show a dramatic difference in their responses to the boundary anchoring strength. In small systems we observe a sticky-boundary effect for a certain range of anchoring strength in which the phase transition gets stuck and stochastic resonance becomes deactivated. Our work demonstrates the dynamical interplay among defects, noises, and boundary conditions in confined liquid crystals.

Figure 1

Metastable states obtained from simulation without external potential. The system size is $25\times 25$ excluding particles at the boundary. Boundary particles are shown in black. The energy of each particle, normalized to $[-1,0]$, is represented by color (blue means low energy and red means high energy).

Figure 2

The $\mathsf{S}$ path from configuration A to configuration B driven by an external potential. The system size is $25\times 25$ excluding particles at the boundary. Boundary particles are colored in black and internal particles are colored according to the normalized energy (blue for low energy and red for high energy). The parameters used are $A_0=0.5$, $W=20$, and $D=0.01$.

Figure 3

The $\mathsf{X}$ path from configuration A to configuration B driven by an external potential. The system size is $50\times 50$ excluding particles at the boundary. Boundary particles are colored in black and internal particles are colored according to the normalized energy (blue for low energy and red for high energy). The parameters used are $A_0=0.5$, $W=20$, and $D=0.01$.

Figure 4

Time evolution of the elastic energy and critical points. (a) Elastic energy of the $25\times 25$ system (black curve). Black vertical dashed lines correspond to the states shown in Fig. 2. The red vertical dashed line corresponds to the configuration with the highest elastic energy. The gray dashed curve indicates the strength of the external potential. (b) Critical point corresponding to the red vertical line in (a). (c) Elastic energy of the $50\times 50$ system (solid black curve). Black vertical dashed lines correspond to the states shown in Fig. 3. The red vertical dashed line corresponds to the configuration with the highest elastic energy. The gray dashed curve indicates the strength of the external potential. (d) Critical point corresponding to the red vertical line in (c). The parameters used are $A_0=0.5$, $W=20$, and $D=0.01$.

Figure 5

Threshold of external potential $A_0^{*}$ separating the $\mathsf{S}$ path and $\mathsf{X}$ path in systems with different sizes. The solid line corresponds to $D=0.01$, while dashed line corresponds to $D=0.1$. In addition, $W=20$.

Figure 6

Three-dimensional heatmap of the particle synchronization level in response to the oscillating potential. Both the altitude and the color represent the value of ${\gamma }_i$ at different particles in the square. The $25\times 25$ system is shown for (a) $A_0=0.05$, (b) $A_0=0.2$, and (c) $A_0=0.5$. The $50\times 50$ system is shown for (d) $A_0=0.05$, (e) $A_0=0.2$, and (f) $A_0=0.5$. In addition, $W=20$ and $D=0.01$ for both systems.

Figure 7

Plot of (a) $\mathrm{\Gamma }$ vs $D$ and (b) $S$ (defined as $s_1+s_2+s_3+s_4$) vs time $t$. Here I, II, and III correspond to the three points marked in (a). The black line represents the oscillating external potential. The system size is $25\times 25$, $W=20$, and $A_0=0.35$.

Figure 8

Plot of $\mathrm{\Gamma }$ for different parameters. Each panel corresponds to $\mathrm{\Gamma }$ obtained for different $D$ and $A_0$, with the value of $\mathrm{\Gamma }$ shown in color, for (a), (c), and (e) the $25\times 25$ system and (b), (d), and (f) the $50\times 50$ system, with (a) and (b) $W=20$, (c) and (d) $W=1.5$, and (e) and (f) $W=0.5$. For each set of parameters, the simulation time is $t=10000$, which contains $n=20$ oscillating periods.

Figure 9

Plot of $\mathrm{\Gamma }$ vs $W$. The system size is $25\times 25$, with $A=0.7$ and $D=0.05$. The simulation time is $t=10000$, which contains $n=20$ oscillating periods.