Spatiotemporal chaos in electroconvection of a homeotropically aligned nematic liquid crystal.

We present patterns of electroconvection (EC) for the homeotropically aligned nematic liquid crystal MBBA. A voltage $V=\sqrt{2}{V}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}(2\ensuremath{\pi}ft)$ was applied. With increasing ${V}_{0}$, the bend Freedericksz transition at ${V}_{F}$ was followed by the onset of EC at ${V}_{c}g{V}_{F}$. We found four distinct pattern types. First, a primary supercritical Hopf bifurcation to traveling waves (TW's) of convection rolls occurred. The structure factor $S(\mathbf{k})$ of this state reflected the azimuthal anisotropy of the underlying Freedericksz state. For $fl{f}_{L}\ensuremath{\simeq}75\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$ there was a superposition of two oblique-roll modes (pattern I). These patterns were chaotic in space and time. For larger $f$ the patterns consisted of chaotic TW normal rolls (pattern II). Here the chaos was attributable to the motion of dislocations and domain walls between left- and right-traveling waves. A secondary bifurcation yielded pattern III; it had no dominant TW frequency but had broadband chaotic dynamics dominated by the motion of dislocations. This pattern type had been referred to by others as a ``chevron pattern;'' its structure factor still revealed azimuthal anisotropy. Finally, at somewhat larger $ϵ\ensuremath{\equiv}{V}^{2}∕{V}_{c}^{2}\ensuremath{-}1$ a highly disordered pattern IV with defect dynamics was found. This state had been studied before by Kai and co-workers and was referred to by them as ``phase turbulence.'' It had a structure factor that was (within our resolution) invariant under rotation. For patterns I, II, and III, $S(\mathbf{k})$ contained crescent-shaped peaks. The peak shape was qualitatively different from the case of planar EC where the structure factor has an elliptical cross section. We present measurements of the widths $1∕{\ensuremath{\xi}}_{k}$ and $1∕{\ensuremath{\xi}}_{\ensuremath{\theta}}$ in the radial $(k)$ and the azimuthal $(\ensuremath{\theta})$ directions. For small $ϵ$ (patterns I and II) we found that ${\ensuremath{\xi}}_{k}$ was consistent with the usual Ginzburg-Landau scaling ${\ensuremath{\xi}}_{k}\ensuremath{\sim}{ϵ}^{\ensuremath{-}{\ensuremath{\nu}}_{k}}$ with ${\ensuremath{\nu}}_{k}\ensuremath{\simeq}1∕2$. However, for ${\ensuremath{\xi}}_{\ensuremath{\theta}}$ we found ${\ensuremath{\xi}}_{\ensuremath{\theta}}\ensuremath{\sim}{ϵ}^{\ensuremath{-}{\ensuremath{\nu}}_{\ensuremath{\theta}}}$ with ${\ensuremath{\nu}}_{\ensuremath{\theta}}\ensuremath{\simeq}3∕4$. Presumably this anomalous scaling of ${\ensuremath{\xi}}_{\ensuremath{\theta}}$ is associated with the Goldstone mode of homeotropic EC. We also show data for the height ${S}_{0}$ of the structure factor that are consistent with ${S}_{0}\ensuremath{\sim}{ϵ}^{\ensuremath{\beta}}$ with $\ensuremath{\beta}\ensuremath{\simeq}\ensuremath{-}0.5$, implying that ${S}_{0}$ diverges at onset. This differs from the case of domain chaos in rotating Rayleigh-B\'enard convection where experiment is consistent with $\ensuremath{\beta}=1∕2$ and thus with a vanishing ${S}_{0}$. The difference between the shape of the structure-factor cross section and between the exponents, for the present case, for planar EC, and for domain chaos suggests that there are different universality classes for spatiotemporal chaos.