Kapustin domains

Note 1 In DSM the Williams Kapustin) domains become distorted and mobile, and macroscopic director alignment is completely disturbed. 

In nematic liquid crystals, subjected to an external electric field at a certain critical voltage, a periodic distribution of the space charge Q and the electric potential appears, resulting in the corresponding periodic variations of the initial director orientation L and the hydrodynamic fiow with the velocity v. This effect, known as the electrohydrodynamic instability (EHDI), could be visualized optically as a periodic pattern of domains. Fig. 5.5. In a screen, domains become visible as black and white stripes perpendicular to the distortion plane, where periodic director deformation and vortex liquid crystal movement is observed. These stripes are caused by the periodicity of the change in the refractive index for an extraordinary ray due to variations in the director. Fig. 5.6. These spatially periodic variations of the refractive index (domains) were first detected by Zvereva and Kapustin [32]. Then Williams [33] investigated transverse domains in detail, and it is current practice to call this type of instability Williams or Kapustin-Williams [34] domains. 

If a monochromatic beam, for example, from a laser, is transmitted through a cell showing domain structure, a different pattern appears on a screen placed behind the cell. The diflEraction pattern takes the form of a chain of reflections arranged in the plane perpendicular to the domains [40, 41]. The angular distribution of the maxima and minima is described by the usual equation for diffraction from a grating with period w w is the period of the Kapustin-Williams domains) ... 

The threshold voltages for the Kapustin-Williams domains, calculated from (5.29) and (5.30) for doped MBBA and their dependence on the dielectric anisotropy, are shown in Fig. 5.7 (curve 2). As will be seen below, (5.29) and (5.30) significantly underestimate the value of Uth- This is a consequence of the unidimensionality of the model or, in other words, a consequence of not allowing for the boundary conditions. 

The threshold voltage Uth and the f)eriod Wth of the Kapustin-Williams domains are found from the linearized system of equations of nematody-namics in an electric field, as a condition of nontriviality of the fluctuations amplitudes 0 , v , t , where... 

The threshold of the instability in a nematic, with Ae > 0 when there is a planar initial orientation, is calculated in [54, 55]. In [31] it was also shown that the threshold voltage C/th of the Kapustin-Williams domains in homogeneous orientation is proportional to the following nematic viscoelastic parameters ... 

The Kapustin-Williams domains have not been directly observed experimentally with homeotropic orientation. For Ae < 0 they are only observed for a voltage exceeding the Frederiks threshold, i.e., essentially with a quasi-planar orientation. In the region Ae 0, when the threshold of reorientation is high, a different, very specific, instability is observed, namely, a lattice with a small period (wave vector qx y 57r/d) [31], as shown in Fig. 5.6(b). 

FIGURE 5.8. Experimental dependences on Ae of (1) the threshold for the (bend) Prederiks effect Ub (4) the threshold of an electrohydrodynamic instabihty with a homeotropic initial orientation (3) the threshold of the Kapustin-Williams domains with planar initial orientation. The calculation of Uthr using the Helfrich s one-dimensional model for a homeotropic orientation (5.30) is shown in curve 2 [31]. 

Finally, we remember that the Kapustin-Williams domains take place due to the effect of the positive conductive anisotropy of the nematic liquid crystal cr /a and disappear in the region or /a < 0 (Fig. 5.7(a)). The considerable decrease in the threshold voltage of the Kapustin-Williams domains for the large conductive anisotropy c7 /(jj proved to be a useful tool for developing liquid crystal mixtures for a dynamic scattering display with low controlling voltages [56]. 

References [59] investigate the static and dynamic behavior of the electrohydrodynamic instability in fireely suspended layers of nematic liquid crystals. The existence of a domain mode was shown, which consists of adjacent elongated domains with a spatial period proportional to the thickness of the layer. This mode occurs only if the thickness of the layer exceeds a critical value 7 /x), and can be understood in terms of the same anisotropic mechanism as the Carr-Helfrich-type, as in the case of the Kapustin-WiUiams modulated structure. 

The two-dimensional model [66] of this domain structure shows that its threshold considerably depends on the value of the Leslie viscosity coefficient as and the dielectric anisotropy Ae. Unlike the Kapustin-Williams domains, this instability could also be observed for negative conductivity anisotropy. There remains only one specific point where the instability ceases to exist, namely, the conductivity isotropy point. Act = 0. 

The dependences of the threshold of the Kapustin-Williams domains [68] and the critical frequency [79] on physical parameters are in good agreement with the theoretical estimations (5.43). Only a certain correction of (5.43) is needed to explain the variation of critical frequency for different substances [79]. However, the anisotropic dielectric regime of the electrohydrodynamic instability in homogeneously oriented nematic hquid crystals seems not to have been observed in experiment yet. 

The Kapustin-Williams domains were also investigated for an electric field applied parallel to the substrates (perpendicular to the light beam). 

Domain patterns arise in sufficiently strong fields with both a homogeneous [80, 81] and a homeotropic [82] initial orientation of the nematic liquid crystal. The period of these domains is always of the order of the thickness of the layer (but not of the separation between electrodes). The Kapustin-Williams domains also occur in twist cells in this case, the strips are oriented at an angle of 45° to the direction of rubbing of the electrodes [87]. 

A magnetic field applied to the cell also effects the formation of an anisotropic domain pattern. When Ae < 0 a magnetic field, oriented at right angles to the electric field, normally increases the threshold for the Kapustin-Williams domains by compensating for the destabilizing role of A(7, whereas H E lowers it. 

Calculations have been made [49] of the magnetic field dependences of the threshold voltage and the period of the Kapustin-Williams domains. The increase in the threshold voltage and the decrease in the period of the domains with an increasing external field [84] confirms the theoretical predictions. 

The threshold conditions and optical appearance of the anisotropic modulated structure strongly depend on the cell thickness. Experiment shows that in thin cells (approximately 10 /xm or less) and, in particular, with well-purified nematic liquid crystals (electrical conductivity ca. 10 ohm cm ), the Kapustin-Williams domains (and the dynamic scattering of light) do not actually appear. Either longitudinal Vistin domains [5, 8,... 

As the thickness decreases the threshold voltage increases very sharply. This can be seen in Fig. 5.14, where the calculated values for Uth are given as a function of the electrical conductivity for typical values of s, = 4.7 and ) = 10 cm s , and the cell thickness is a parameter. Figure 5.14 also explains why the Kapustin-Williams domains and the dynamic scattering of light are not observed in relatively thick samples with a small electrical conductivity of 10 -10 ohm cm . ... 

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]). 

Let us now briefly describe electrohydrodynamic instabilities in polymer nematics. The first observation of the Kapustin-Williams domains in nematic polymers were reported in [117, 118]. The qualitative picture of the phenomenon is, in fact, the same as that for the conventional nematics (domains perpendicular to the initial director orientation in a planar cell, typical divergence of the threshold voltage at a certain, critical frequency, etc.). The only principal difference is a very slow dynamics of the process of the domain formation (hours for high-molecular mass compounds [117]). The same authors have observed longitudinal domains in very thin samples which may be referred to as the flexoelectric domains [5-14] discussed in Section 5.1.1. 

FIGURE 5.23. Threshold of Kapustin-Williams domains in polymer nematics versus frequency. 

Both the Kapustin-Williams domains and the light scattering textures can be frozen on cooling the polymeric samples in the electric field. Thus, light diffraction gratings can be prepared. Such memory effect has been reported [120]. 

In the pretransition temperature region, where the nematic phase has a certain degree of short-range smectic A order, it is easy to detect both the motion of the liquid and the formation of the domain patterns [122, 123, 126]. The latter have marked differences from the domain patterns arising in the pure nematic phases. For instance, in the temperature region where the ratio of the electrical conductivity cr /crx becomes less than 1, the Kapustin-Williams domains are not observed with an initial planar orientation. Instead, different domains are formed at low frequencies (a < ujc) of the applied field. They are arranged parallel to the initial orientation of the director and have a well-defined field strength threshold, Uth oc d [123]. 

The threshold voltage for the Kapustin-Williams domains has a discontinuity at a temperature T corresponding to the condition = 1. The... 

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