Pattern electroconvection

P. Tadapatri, U.S. Hiremath, C.V. Yelamaggad and K.S. Krishnamurthy, Patterned electroconvective states in a bent-core nematic liquid crystal, J. Phys. Chem. B 114(1), 10-21, (2010). doi 10.1021/jp9058802... 

Abstract A systematic overview of various electric-field induced pattern forming instabilities in nematic liquid crystals is given. Particular emphasis is laid on the characterization of the threshold voltage and the critical wavenumber of the resulting patterns. The standard hydrodynamic description of nematics predicts the occurrence of striped patterns (rolls) in five different wavenumber ranges, which depend on the anisotropies of the dielectric permittivity and of the electrical conductivity as well as on the initial director orientation (planar or homeotropic). Experiments have revealed two additional pattern types which are not captured by the standard model of electroconvection and which still need a theoretical explanation. 

Keywords Pattern formation, instabilities, liquid crystals, electroconvection... 

In the second group we find pattern forming phenomena based on new instability mechanisms arising from the specific features of liquid crystals, which have no counterpart in isotropic fluids or at least are difficult to assess. Some examples are shear (linear, elliptic, oscillatory, etc.) induced instabilities, transient patterns in electrically or magnetically driven Freedericksz transitions, structures formed in inhomogeneous and/or rotating electric or magnetic fields, electroconvection (EC), etc. [5-7]. 

Figure 1. Schematic morphological phase diagram in the f/ — / plane. Solid lines correspond to the threshold voltage of standard electroconvection, the dashed line denotes the threshold of the prewavy patterns or wide domains (see later). For details see [16].

Figure 3 exhibits the director and charge distribution as well as the velocity field in the rr — z plane at onset of electroconvection, where the x direction is parallel to the initial (planar) director ahgnment and A is the pattern wavelength. 

The dashed line in Fig. 1 is the experimental threshold curve for prewavy patterns or wide domains (A 4 — lOd) that also represent electroconvecting structures though not captured by the standard model (see later in Section 2.2.). 

Figure 8. Schematic director profile in case C. a Freedericksz distorted state, b with superposed electroconvection pattern.

Figure 9. Snapshots of electroconvection patterns superposed on the Freedericksz state in case C. a oblique rolls, b normal rolls.

Case G planar alignment, 6 < 0 o-q < 0. Standard EC (based on the CH mechanism) cannot occur for the material parameter combination e < 0, da < 0 [2] except the a induced" pattern type. Nevertheless, convection associated with roll formation has been observed in ac electric field in the homologous series of N-(p-n-alkoxybenzylidene)-n-alkylanilines, di-n-4-4 -alkyloxyazoxybenzenes and 4-n-alkyloxy-phenyl-4-n alkyloxy-benzoates [52-54]. The characteristics of the patterns the orientation of the rolls, contrast, frequency dependence of the wavevector and the threshold, director variation in space and time etc. - are substantially different from those observed in the standard EC. Since this roll formation process falls outside of the frame of the standard model, it has been called nonstandard electroconvection (ns-EC). 

Figure 13. Snapshots of nonstandard electroconvection pattern in case G taken with crossed polarizers, a Oblique rolls, b parallel rolls. Contrast was enhanced hy digital processing. The initial director orientation is horizontal. The depicted image is 0.225 x 0.225mm, d = ll/rm.

In this chapter the influence of flexoelectricity on pattern formation induced by an electric Held in nematics will be summarized. Two types of patterns will be discussed in the linear regime, the equilibrium structure of flexoelectric domains and the dissipative electroconvection (EC) rolls. In a separate section, recent experimental and theoretical results on the competition and crossover between the flexoelectric domains and EC patterns will be described. 

More frequently, instead of the equilibrium pattern sketched so far, one observes electroconvection (EC) patterns in nematics, which present dissipative structmes characterized by director distortions, space charges and material flow. A necessary requirement for their existence is the presence of charge carriers in the nematic. In a distorted nematic, where n is neither parallel nor perpendicular to E, the generation of a non-zero space charge, pei, by charge separation is then inevitable. The resulting Coulomb force in the flow equations (generalized Navier-Stokes equations) drives a... 

Patterns in nematics are easily observed by optical means where the anisotropy of the refractive index is exploited. In this way the stripe patterns in electroconvection in the planar geometry are easily discriminated from flexodomains the angle a between the wave vector q of the EC stripes and the preferred direction no a is small (normal or oblique rolls) in contrast to a = 90° (longitudinal stripes) in flexodomains. 

Electroconvection in nematics is certainly a prominent paradigm for nonequilibrium pattern-forming instabilities in anisotropic systems. As mentioned in the introduction, the viscous torques induced by a flow field are decisive. The flow field is caused by an induced charge density p i when the director varies in space. The electric properties of nematics with their quite low electric conductivity 10 (fl m) ] are well described within the electric quasi-static approximation, i.e. by charge conservation and Pois-... 

As already stated, electroconvection cannot be explained by the standard model for < 0 and Ca < 0. Surprisingly, EC has been observed also for this parameter combination in certain calamitic nematics.These EC patterns differ clearly from the standard EC patterns the rolls are dominantly parallel to the initial director alignment (see Fig. 4.6a). They are not observable using the common shadowgraph technique (single polarizer) but are by using crossed polarizers (plus sometimes an additional... 

It is very interesting to see that a crossover between the equilibrium flexodomains and the dissipative EC patterns can be observed in the same experiment (planar geometry) by merely increasing the AC frequency co. Inspection of Fig. 4.5 reveals that at very small u) and for suitable material parameter combinations of the nematics, the critical voltages of the equilibrium flexodomains and those of the dissipative electroconvection patterns might approach each other. In fact, very recent theoretical and experimental studies on the calamitic nematic Phase 4, have demonstrated the... 

V.A. Delev, A.P. Krekhov and L. Kramer, Crossover between flexoelectric stripe patterns and electroconvection in hybrid ahgned nematics. Mol. Cryst. Liq. Cryst. 366(1), 849-856, (2001). doi 10.1080/10587250108024026... 

F. Hoemer, 1. Rehberg Using thermal noise to measure the correlation lengths of electroconvection, in S. Kai (ed.) Pattern formation in complex dissipitative systems and Global Dynamics, World scientific, p. 429 (1992)... 

S. Kai, Y. Adachi, S. Nasimo Stability diagram, defect turbulence, and new patterns in electroconvection in nematics, in RE. Cladis, P. Palffy-Muhoray (eds.) Spatio-Temporal Patterns in Nonequilibrium Complex Systems, SFI Studies in the Sciences of Complexity, Addison-Wesley, (1994). 

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