Carr-Helfrich Model

The current carriers in the nematic phase are ions whose mobility is greater along the preferred axis of the molecules than perpendicular to it. Because of conductivity anisotropy, space charges will accumulate with signs as shown in Fig. 6 at the distortion maxima and minima. The applied field acts on the charges to give rise to material flow in alternating directions, which in turn exerts a torque on the molecules. This is reinforced 

Earlier it was thought that Williams domains could only be observed in materials 

In calculating the threshold voltage, Hel-frich assumed that the spatial periodicity of the fluid deformation was proportional to the thickness of the cell. Penz and Ford [19-21] solved the boundary value problem associated with the electrohydrodynamic flow process. They reproduced Helfrich s results and showed several other possible solutions that may account for the higher order instabilities causing turbulent fluid flow. 

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Taking into account flexoelectricity, it is possible to explain the appearance of a certain angle a, which the Kapustin-Williams domains form in some cases with the y-axis (the usual domain strips are parallel to the 2/-axis, Fig. 5.5). This oblique roll motion was observed in [90] and cannot be explained within the framework of the usual three-dimensional Carr-Helfrich model with strong anchoring at the boundaries [91]. The angle of the domain pattern a was shown [88, 89] to depend on the flexoelectric moduli eii, 633, the dielectric Ae, and the conductive Aa anisotropy. In certain intervals of the en — 633 and 611/633 values the angle A = 0 (the usual Kapustin-Williams domains) or A = 7t/2 (the longitudinal domains, also seen in experiment near the nematic-smectic A transition [91]). 

At high frequencies, with a reduction in temperature, the threshold field of the normal prechevron domains increases smoothly, not displaying any peculiarity when the temperature is T, where the anisotropy of the electrical conductivity disappears. Without doubt, the high-frequency electrohydrodynamic mode is caused by the isotropic mechanism of destabilization, since when cr /crx = 1, the Carr-Helfrich model does not hold. Analysis shows [123] that the new low-frequency mode (longitudinal domains) is also caused by the isotropic mechanism. 

The mathematical treatment of the Carr-Helfrich model leads to the following expression for the threshold voltage for Williams domains [64] ... 

The basic mechanism for electric field induced instabilities is now very well understood in terms of the Carr-Helfrich model based on field induced space charges due to conductivity and dielectric anisotropies [16, 31], Helfrich [16] made derivations for only DC fields, which were further extended to AC fields by Dubois-Violette and co-... 

Electrically driven convection in nematic liquid crystals [6,7,16] represents an alternative system with particular features listed in the Introduction. At onset, EC represents typically a regular array of convection rolls associated with a spatially periodic modulation of the director and the space charge distribution. Depending on the experimental conditions, the nature of the roll patterns changes, which is particularly reflected in the wide range of possible wavelengths A found. In many cases A scales with the thickness d of the nematic layer, and therefore, it is convenient to introduce a dimensionless wavenumber as q = that will be used throughout the paper. Most of the patterns can be understood in terms of the Carr-Helfrich (CH) mechanism [17, 18] to be discussed below, from which the standard model (SM) has been derived... 

A distinguishing feature of this model is the absence of a positive feedback between the magnitude of the space charge and the orientation of the director, which is required for the Carr-Helfrich mechanism. These physical parameters are independent of each other here. 

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