Spatiotemporal complexity of electroconvection patterns in nematic liquid crystals

We investigate a number of complex patterns driven by the electroconvection instability in a planarly aligned layer of a nematic liquid crystal. They are traced back to various secondary instabilities of the ideal roll patterns bifurcating at onset of convection, whereby the basic nematohydrodynamic equations are solved by common Galerkin expansion methods. Alternatively these equations are systematically approximated by a set of coupled amplitude equations. They describe slow modulations of the convection roll amplitudes, which are coupled to a flow field component with finite vorticity perpendicular to the layer and to a quasihomogeneous in-plane rotation of the director. It is demonstrated that the Galerkin stability diagram of the convection rolls is well reproduced by the corresponding one based on the amplitude equations. The main purpose of the paper is, however, to demonstrate that their direct numerical simulations match surprisingly well new experiments, which serves as a convincing test of our theoretical approach.

Figure 1

Critical rms voltage $U_c$ and critical wave number $q_c$ for MBBA as function of ${\omega }^{\prime }=\omega {\tau }_q$ in the conductive regime from theory (solid lines) in comparison with experimental data (circles).

Figure 2

Stability diagram for normal rolls in MBBA in the ${\omega }^{\prime }-\varepsilon$ plane for the same parameters as in Fig. 1.

Figure 3

Stability diagram for normal rolls in MBBA in the ${\omega }^{\prime }-\varepsilon$ plane in the framework of the CAE.

Figure 4

Transient zigzag pattern clearly visible in a fs-$A$ picture ($\text{Re}[A(x,y)exp\left(i{q}_{c}x\right)]$) (a) and the modulus of the roll amplitude $\left|A\right(X,Y\left)\right|$ (b) for ${\omega }^{\prime }=1$ and $\varepsilon =0.1$.

Figure 5

Quasistationary fs-$A$ zigzag pattern (a) and the corresponding twist field $\phi$ (b) for ${\omega }^{\prime }=1$ and $\varepsilon =0.12$.

Figure 6

Experimental snapshots for ${\omega }^{\prime }=1$ and $\varepsilon =0.12$, above threshold of zigzag instability: optical geometry O1 (a) and optical geometry O2 (b). The image size is 800 $\mu \mathrm{m}$ $\times 800 \mu \mathrm{m}$.

Figure 7

Abnormal fs-$A$ roll pattern (a) and the corresponding twist field $\phi$ (b) for ${\omega }^{\prime }=1.6$ and $\varepsilon =0.1$.

Figure 8

Experimental snapshots for ${\omega }^{\prime }=1.6$ and $\varepsilon =0.11$, above threshold of abnormal rolls: optical geometry O1 (a) and optical geometry O2 (b). The image size is 800 $\mu \mathrm{m}$ $\times 800 \mu \mathrm{m}$.

Figure 9

Defect chaotic fs-$A$ patterns (a), the corresponding twist field $\phi$ (b), and the modulus of the roll amplitude $\left|A\right|$ (c) for ${\omega }^{\prime }=1.6$ and $\varepsilon =0.19$.

Figure 10

Experimental snapshots of chaotic defect patterns for ${\omega }^{\prime }=1.6$ and $\varepsilon =0.2$: optical geometry O1 (a) and optical geometry O2 (b). The image size is 800 $\mu \mathrm{m}$ $\times 800 \mu \mathrm{m}$.

Figure 11

Formation of vertical domain walls for ${\omega }^{\prime }=1.9$ and $\varepsilon =0.11$: fs-$A$ picture (a), the corresponding twist field $\phi$ (b), and modulus of rolls amplitude $\left|A\right|$ (c) after integration time $T={10}^4$. The corresponding pictures, fs-$A$ (d) and $\phi$ (e), after continuing the simulations by $T={10}^4$.

Figure 12

Experimental snapshots for ${\omega }^{\prime }=1.9$ and $\varepsilon =0.1$: optical geometry O1 (a) and optical geometry O2 (b). The image size is 500 $\mu \mathrm{m}$ $\times 500 \mu \mathrm{m}$.

Figure 13

Wall dynamics viewed via twist field $\phi$ during the generation of new walls for ${\omega }^{\prime }=1.9$ and $\varepsilon =0.13$. The single pictures are shown at times $T=351$ (b), 367 (c), and 950 (d) when starting with picture (a).

Figure 14

Time sequence of the defect and wall dynamics in $\left|A\right|$ during the generation of a pair of new walls for the parameters of Fig. 13. The pictures are shown for the times $T=340$ (1), 351 (2), 358 (3), 367 (4), 382 (5), 389 (6), and 700 (7) after the first one (0).

Figure 15

Defect lattice in the roll pattern fs-$A$ (a) and the corresponding twist field $\phi$ (b) for ${\omega }^{\prime }=1.9$ and $\varepsilon =0.19$.

Figure 16

Experimental snapshots for ${\omega }^{\prime }=1.9$ and $\varepsilon =0.18$: optical geometry O1 (a) and optical geometry O2 (b). The image size is 500 $\mu \mathrm{m}$ $\times 500 \mu \mathrm{m}$.