Threshold voltage frequency dependence

From this equation, Ey (T)/E- -(0) increases monotonically from unity at T=0K to 1,33 at T=Tc. This behaviour differs from one observed in the CDW materials, where Ey exhibits a divergence at T=Tc and a minimum slightly below Tq, which results from an increase in Ey at low temperatures, due to phase fluctuations. Hence, one should expect to observe similar properties of the SDW current-carrying state to ones of the CDW nonlinear current-voltage characteristics, accompanied by broad and narrow band noise, with sharp threshold fields, frequency-dependent conductivity, interference effects between the ac voltage generated in the sample, and an external rf field, hysteresis and memory effects etc. 

Figure 33 (a) Frequency-dependent conductivity in the absence of pinning (A) and with pinning and damping (B) (after Ref. 107) (b) nonlinear conductivity in the SDW phase cf (TMTSF)2AsF6 (c) current-voltage characteristics, (d) The zEx = const, relation is verified in the series of alloys (TMTSF)2AsF6(1 x) SbF when the threshold field ET is determined by the amount of lattice defects

Measurements have shown that the prewavy pattern appears in a forward bifurcation [50]. Its threshold voltage Upw has a weak, nearly linear frequency dependence. It usually occurs at higher frequencies (see Fig. 1). Conductive normal rolls, dielectric rolls and the prewavy pattern may follow each other with increasing / (dielectric rolls may be skipped in compounds with higher conductivity). Near the crossover frequency /c, the conductive (or dielectric) rolls can coexist with the prewavy pattern resulting in the defect-free chevron structure [51]. 

The threshold voltage is usually a few volts and is practically independent of the sample thickness. It is, however, strongly dependent on the frequency > (fig. 3.10.4). There is a cut-off frequency above which the domains do not appear, the value of co increasing with the conductivity of... 

Fig. 3.10.6. Frequency dependence of the threshold voltage in />,/> -di-n-hept-oxyazobenzene, a nematic of positive dielectric anisotropy. (After Gruler and...

We have so far discussed only materials of negative dielectric anisotropy. Electrohydrodynamic distortions are observed even in weakly positive materials,but only when the initial orientation of the director is perpendicular to the applied field. Striations appear above a threshold voltage but vanish at still higher voltages and there is no dynamic scattering. The frequency dependence of the threshold voltage is shown in fig. 3.10.6. 

FIGURE 4.41. Frequency dependence of the threshold voltage for the Frederiks transition in a nematic polymer with sign inversion of the dielectric anisotropy... 

FIGURE 5.3. Experimental frequency dependence (x) of the threshold voltage for the Prederiks modulated structure appearing from the initial homeotropic director orientation [17] (o) denotes the corresponding threshold for the uniform Frederiks transition C/p = Tr AnKzz/ i. -... 

The inertia mode has been observed in experiment [44, 93, 94]. Figure 5.16 presents the frequency dependences of the threshold for the appearance of a stationary domain pattern in strongly conducting nematic liquid crystals with Ae < 0 [93]. The domains here have a width exceeding the thickness of the cell wide domains), and are oriented at right angles to the initial orientation of the director. The threshold voltage is practically independent of the thickness of the layer and is proportional to The frequency dependence of the threshold is different for diflFerent samples, which may be caused by the firequency dependence of the electrical conductivity which also includes the dielectric losses. These results correspond to the theoretical predictions for an inertia mode of an electrohydrodynamic instability [92]. 

The thresholds for the formation of vortices in silicone oil, carbon tetrachloride, and acetone were measured [76] and it was shown that Uth oc and that it decreases with decreasing viscosity. The frequency dependence of the threshold voltage for the appearance of vortices in the isotropic phase of MBBA was investigated by Kirsanov [110]. It was shown that this dependence could also be represented in the form Uth oc... 

More detailed measurements of the dependences Uth f) in pure and doped MBBA at various temperatures (for sandwich cells) were performed [76, 109]. The results of these measurements are represented in Fig. 5.20. The threshold of the vortical motion was taken as the onset of the circular tion of the solid impurity particles in the electrode plane. The shape of the curves in Fig. 5.20 depends on the electrical conductivity. With a high electrical conductivity the curves have a plateau in the low-frequency region and a characteristic dependence I7th oc at frequencies above the critical frequency. At the transition point to the nematic phase the threshold voltage of the instability does not change. It is shown in [109] that the height of the low-frequency plateau is proportional to and at frequencies of u > 47r(j/e the threshold field does not depend on <7. Moreover, it does not depend on the thickness of the sample, i.e., on the separation between the electrodes. 

Figure 13. Frequency dependence of the threshold voltage f/,1, and domain period for Frederiks domains near the dielectric anisotropy sign inversion frequency [83, 84], (—) Calculated values. Experimental values (+) o- j =+4.75 (O) o-ej,=+0.35 (A) o- i=+0.05. ois for...

The solution of set (Eqs (120) and (121)) represents a threshold voltage that is independent of frequency in the range gxVTq conductance regime) and increases critically when co co IXq. Such a dependence is shown qualitatively in Fig. 28. 

The optical pattern of the perturbed state in the high frequency (/>/,) region is characterized by periodic parallel striations of a much shorter period (a few micrometers) than the Williams domains [23]. Above the threshold, these striations move, and bend and give rise to what has been called a chevron pattern (Fig. 5) [8-12,17,23-26]. In this regime, the threshold is determined by a critical field rather than a critical voltage. The threshold field strength increases with the square root of the frequency. The spatial periodicity of the chevron pattern is also frequency dependent it is... 

In this chapter we will extend the threshold calculations to show how the instability arises and to combine both frequency dependence and two-dimensional features. We will compare the results with measured values of threshold voltage and domain spacing as function of ac frequency. Good quantitative agreement will be demonstrated. 

Eq. [32] shows that instability can also take place for Ae > 0, as discussed by Dubois-Violette and Penz. In that case, the threshold voltage becomes independent of frequency at high frequency, but this is not a conduction regime because Vth is then only dependent on k and 6. Furthermore, this threshold is higher than that of the dielectric Freedericksz transition for the case of the entire sample deforming uniformly (S = 0, Vth = Tx/kn/Ae), and so will not occur. In fact. 

Orsay Liquid Crystal Group solved the electrohydrodynamic problem for a variable frequency, sinusoidal voltage source. The fluid instability occurs at the frequency-dependent threshold voltage... 

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