Carr-Helfrich Mode

The physical mechanism of the instability is related to several coupled phenomena discussed by Helfrich [42]. His elegant calculation of the instability threshold is reproduced here for the simplest steady state one-dimensional model shown in Fig. 11.33a. A homogeneously aligned nematic liquid crystal layer of thickness d is stabilised by the rubbed surfaces of the limiting glasses. The dielectric torque is considered negligible (s = 0). At first, a small director fluctuation 9(x) with a period dh postulated  

With the field applied, this fluctuation causes a slight periodic deflection of the electric current lines along the director proportional to the anisotropy of conductivity (ja = J — a L 0. This creates the x-component of the current that, in turn, results in the accumulation of a space charge Q(x) close to the points where angle 0 = 0. Therefore, the x-component of the field (E ) emerges. The electric current density is Jt = OtjEj where the tensor of the electric conductivity has a standard form  

11 Optics and Electric Field Effects in Nematic and Smectic A Liquid Crystals 

Combining (11.85-11.87), we get the periodic space charge distribution over x  

Due to the space charge and corresponding force —Q(x)E the nematic liquid begins to move with a velocity determined by reduced form of the Navier-Stokes equation (7.16)  

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We now explicitly include in the discussion the anisotropy of the electrical conductivity (Xj of a liquid crystal. This anisotropy itself turns out to be a reason for electrohydrodynamic destabilization. First, we discuss the Carr-Helfrich mode of the instability [236, 237], which arises in a homogeneously oriented liquid crystal layer in a sandwich cell between transparent electrodes. 

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. 

Thus, to date we have three theoretically predicted modes for a high-frequency instability caused by the Carr-Helfrich anisotropic mechanism. They are the conductance and dielectric regimes and the inertia mode. Two of these (the conducting regime and the inertia mode) correspond to the steady-state motion of the liquid and the stationary deviations of the... 

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 other two instabilities shown in Fig. 28 may be observed only in liquid crystals (nematic, cholesteric, and smectic C). The first is the Carr-Helfrich instability, which is caused by a low-frequency electric field and occurs in the form of elongated vortices with their axis perpendicular to the original director alignment. The vortices cause a distortion of the director orientation, which is observed optically as a one-dimensional periodic pattern (Kapustin-Williams domains). The other anisotropic mode is observed only in highly conductive liquid crystals. For its interpretation the inertial term dvidt for the fluid velocity must be taken into account, which is why this mode may be called inertial mode. 

Carr-Helfrich one in nematics and may be studied quantitatively theoretically [280]. At high frequencies, an instability is observed with a characteristic frequency dependence of the threshold field Elayer thickness (fundamental domains). This has been regarded as an analog of the dielectric regime [282], but it can also be interpreted as the electrolytic mode [283] with some specific features. In some special cases a new domain mode is observed [284], which has been referred to the inertial (anisotropic) mode (discussed in Section 9.4.1.3 of this Chapter). 

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