Electroconvection

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

Electrodecantation or electroconvection is one of several operations in which one mobile component (or several) is to be separated ont from less mobile or immobile ones. The mixture is introduced between two vertical semipermeable membranes for separating cations, anion membranes are used, and vice versa. When an electric field is apphed, the charged component migrates to one or another of the membranes bnt since it cannot penetrate the membrane, it accu-mnlates at the snrface to form a dense concentrated layer of particles which will sink toward the bottom of the apparatus. Near the top of the apparatus immobile components will be relatively pure. Murphy [/. Electrochem. Soc., 97(11), 405 (1950)] has used silver-silver chloride electrodes in place of membranes. Frilette [ J. Phys. Chem., 61, 168 (1957)], using anion membranes, partially separated and Na K and Li, and K and Na. ... 

In a hydrodynamically free system the flow of solution may be induced by the boundary conditions, as for example when a solution is fed forcibly into an electrodialysis (ED) cell. This type of flow is known as forced convection. The flow may also result from the action of the volume force entering the right-hand side of (1.6a). This is the so-called natural convection, either gravitational, if it results from the component defined by (1.6c), or electroconvection, if it results from the action of the electric force defined by (1.6d). In most practical situations the dimensionless Peclet number Pe, defined by (1.11b), is large. Accordingly, we distinguish between the bulk of the fluid where the solute transport is entirely dominated by convection, and the boundary diffusion layer, where the transport is electro-diffusion-dominated. Sometimes, as a crude qualitative model, the diffusion layer is replaced by a motionless unstirred layer (the Nemst film) with electrodiffusion assumed to be the only transport mechanism in it. The thickness of the unstirred layer is evaluated as the Peclet number-dependent thickness of the diffusion boundary layer. 

The top level of the electro-diffusion hierarchy is formed by the electroconvection phenomena, of which electro-osmosis is in several respects the simplest one. Certain aspects of electro-osmosis will be treated in Chapter 6. The higher we climb the hierarchy outlined the less rigorous our mathematics will become and the more vague heuristic statements will appear. 

In this section we shall consider the simplest model problem for the locally electro-neutral stationary concentration polarization at an ideally permselective uniform interface. The main features of CP will be traced through this example, including the breakdown of the local electro-neutrality approximation. Furthermore, we shall apply the scheme of 4.2 to investigate the effect of CP upon the counterion selectivity of an ion-exchange membrane in a way that is typical of many membrane studies. Finally, at the end of this section we shall consider briefly CP at an electrically inhomogeneous interface (the case relevant for many synthetic membranes). It will be shown that the concentration and the electric potential fields, developing in the course of CP at such an interface, are incompatible with mechanical equilibrium in the liquid electrolyte, that is, a convection (electroconvection) is bound to arise. 

The electroviscous effects are observed as variations of viscosity upon application of outer electric fields, and as build-up of potential gradients upon flow of such fluids. See also -> electroconvection, electrorheological... 

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]. 

The overview is organized as follows. In section 1, we outline briefly the relevant properties of liquid crystals and sketch the theoretical description. In section 2, we discuss electroconvection for different material parameter sets and geometries focusing mainly on the onset of convection. A summary concludes the paper. 

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 paper we have reviewed the structures appearing at onset of electro-convection in nematic liquid crystals. The influence of the relevant material parameters (ca and ao) and the role of the initial director alignment were explored. Our calculations using a linear stability analysis of the standard model of electroconvection (performed for zero frequency) revealed that four different scenarios characterized by different ranges of the wavenumber q can be identified (1) the Qf= 0 mode (a homogeneous deformation known as the Freedericksz transition) predicted and observed in cases C, D, E and H, which is... 

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-... 

For the parameter combination o < 0 and <7a < 0, which can be found in some nematic compounds, electroconvection is definitely excluded within the standard model. Nevertheless, EC has surprisingly been observed in this case (for recent examples see, e.g. Kochowska et alP and Toth-Katona et al ). The theoretical analysis has proved that flexoelectricity is crucial for understanding this non-standard EC because in Eq. (4.7) the contribution V Pfi to Pel is dominant. It is interesting that the flexotorque on the director is determined by the difference (ei — 63) of the flexocoefScients while the sum (ei - - 63) governs the flexocharge and thus its contribution to the viscous torque. Further details will be sketched in Section 4.3.2. 

This section deals with the influence of flexoelectricity on electroconvection with planar geometry and the most studied material parameter combination a < 0 and Oa > 0. The analysis makes use of the common nemato-electrohydrodynamic equations, 4s jjj addition the flexopolarization... 

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