Patterns driven by combined ac and dc electric fields in nematic liquid crystals

The effect of superimposed ac and dc electric fields on the formation of electroconvection and flexoelectric patterns in nematic liquid crystals was studied. For selected ac frequencies, an extended standard model of the electrohydrodynamic instabilities was used to characterize the onset of pattern formation in the two-dimensional parameter space of the magnitudes of the ac and dc electric field components. Numerical as well as approximate analytical calculations demonstrate that depending on the type of patterns and on the ac frequency, the combined action of ac and dc fields may either enhance or suppress the formation of patterns. The theoretical predictions are qualitatively confirmed by experiments in most cases. Some discrepancies, however, seem to indicate the need to extend the theoretical description.

Figure 1

Phase diagram for EC patterns in the $U_{\mathrm{dc}}\text{−}{U}_{\mathrm{ac}}$ plane under combined dc and ac voltages with $\omega {\tau }_q=0.3$ ($f=10$ Hz) in the conductive regime: Boundary curve enclosing the pattern-free region (a); the critical wave number $|{\mathit{q}}_c|$ (b); angle $\alpha$ between the critical wave vector ${\mathit{q}}_c$ and the $x$ axis (c) along the boundary curve. Material parameters of Phase 5, thickness $d=10$ $\mu$m, flexocoefficients $e_1=e_3=0$.

Figure 2

Phase diagram for EC patterns in analogy to Fig. 1 except for the ac-voltage component with $\omega {\tau }_q=6$ ($f=200$ Hz) in the dielectric regime. The crossover between the conductive EC patterns and the dielectric ones is marked by the filled circle in (a).

Figure 3

Phase diagram for flexodomains and conductive EC rolls for the ac-voltage component with $\omega {\tau }_q=0.3$ ($f=10$ Hz) in analogy to Fig. 1 except for finite flexocoefficients $e_1=12$ pC/m, $e_3=0$. The crossover between flexodomains and conductive EC rolls is marked by the filled circle in (a).

Figure 4

Phase diagram for flexodomains and dielectric EC rolls in analogy to Fig. 3, but for the much higher ac frequency $\omega {\tau }_q=6$ ($f=200$ Hz). The crossover between flexodomains and dielectric EC rolls is marked by the filled circle in (a).

Figure 5

Experimental boundary curve of the pattern-free region in the $U_{\mathrm{dc}}\text{−}{U}_{\mathrm{ac}}$ plane in the presence of combined dc and ac voltages for $f=10$ Hz, in the conductive regime for Phase 5. Cell thickness $d=11.4$ $\mu$m. Stars indicate the locations where the snapshots of Fig. 6 were taken.

Figure 6

Snapshots of EC patterns along the boundary curve at the three locations indicated in Fig. 5: (a) pure dc driving; (b) superimposed dc and ac voltages; and (c) pure ac driving. The wave number $q_c$ and the obliqueness angle $\alpha$ of the observed oblique conductive rolls are (a) $q_c\approx 1.42\pi /d$, $\alpha \approx \pm {37}^{\circ }$; (b) $q_c\approx 1.12\pi /d$, $\alpha \approx \pm {26}^{\circ }$; (c) $q_c\approx 1.30\pi /d$, $\alpha \approx \pm {22}^{\circ }$. The arrow bars of length 20 $\mu$m are parallel to the initial planar director orientation $\mathit{n}\parallel \widehat{\mathbf{x}}$.

Figure 7

Experimental phase diagram of EC patterns for Phase 5 in the $U_{\mathrm{dc}}\text{−}{U}_{\mathrm{ac}}$ plane for an ac frequency $f=200$ Hz, in the dielectric regime: Boundary curve enclosing the pattern-free region (a); the critical wave number $|{\mathit{q}}_c|$ along the boundary curve (b). Cell thickness $d=10.8$ $\mu$m. Stars indicate the locations where the snapshots of Fig. 8 were taken.

Figure 8

Snapshots of EC patterns along the boundary curve at the four locations indicated in Fig. 7: (a) oblique rolls at pure dc driving; (b) oblique rolls at superposed dc and ac voltages; (c) dielectric rolls at pure ac driving; and (d) dielectric rolls at superposed dc and ac driving. The arrow bars [of length 50 $\mu$m for (a) and (b), and 20 $\mu$m for (c) and (d)] are parallel to the initial planar director orientation $\mathit{n}\parallel \widehat{\mathbf{x}}$.