Frederiks threshold

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). 

Frank theory, nematics 60 Frank-Oseen energy 27 Frederiks threshold... 

Three kinds of Frederiks transitions can be used to measure the three elastic constants of nematic polymers. The magnetic threshold fields Hci are functions of Ku(i = 1,2,3) respectively,... 

The oscillations of I (U) are well seen in the experimental plot. Fig. 11.21. The measurements were made at 27°C on 55 nm thick cell filled with a mixture having ta = 22. From the I (U) curve, the field dependence of the phase retardation 8(17) and the Frederiks transition threshold Uc were obtained. In mm, from Ec = UJd and Fq. (11.56) the splay elastic constant Ku was found. The bend modulus "33 was calculated from the derivative dbldU. The same material parameters may be found for the whole temperature range of the nematic phase. 

Fig. 11.21 The oscillating experimental curve I(U) right axis) is voltage dependent intensity of the light transmitted by the 50 pm thick planar nematic cell placed between crossed polarizers (the logarithmic voltage scale for /([/ j is the bottom axis). The pointed curve is the voltage dependence of phase retardation 5 calculated from curve I(U) with a Frederiks transition threshold at Uc (the scale for 5(1/) is on the top axis and its argument i.e. voltage is on the left axis)...

The dependencies 6 ocd and 8 ocE agree well with experiment [30]. Therefore, in principle, we can find from the measured value of the cell retardation because usually 33 is known from the Frederiks transition threshold. However, in a real experiment it is almost impossible to have zero anchoring energy. For the finite anchoring energy, we can only find ratio ej,/W and the accuracy of determination... 

In Chapter 11 we have found that, for the Frederiks transition in nematics, the threshold field coherence length is determined by the cell thickness, = din, see Eq.(11.53). Now we shall briefly discuss another type of instability with a threshold determined by the geometrical average of the two parameters mentioned [17],... 

From Eq. (13.29) is seen that, at (p = 0, there is no electric torque exerted on the director. Thus, there should be a threshold for the distortion as in the case of the Frederiks transition in nematics. We can find the threshold field Ec, considering a small distortion (p 0. The equation... 

For instance, for T(p = 10 dyn,ii = 2 x 10 " cm,Fs = 300 statC/cm (1 mC/m ), the threshold field is 0.1 statV/cm, i.e. 3 kV/m. Due to a high value of the Frederiks type distortion in SmC can be observed at extremely low voltage across the cell Uc = dE 30 mV for 10 pm thick cell). However, independently of the field magnitude, after switching the field off, the distortion relaxes to the initial uniform structure, cp(x) = 0. The relaxation time of the distorted structure is owed to pure elastic, nematic-like torque and for small distortion only fundamental Fourier harmonic is important. 

If we are only interested in the Frederiks-type threshold we should add the surface term VF cp to Eq. (13.32). Then, a finite anchoring only increases the apparent cell thickness by two extrapolation lengths b = d + where... 

Here, the first term describes the nematic-like elastic energy in raie crmstant approximation (K in 9). This term allows a discussion of distortions below the AF-F threshold (a kind of the Frederiks transition as in nematics in a sample of a finite size). In fact, the most important specific properties of the antiferroelectric are taken into account by the interaction potential W between molecules in neighbour layers the second term in the equation corresponds to interaction of only the nearest layers (/) and (/ + 1). Let count layers from the top of our sketch (a) then for odd layers i, i + 2, etc. the director azimuth is 0, and for even layers / + 1, / + 3, etc. the director azimuth is n. The third term describes interactimi of the external field with the layer polarization Pq of the layer / as in the case of ferroelectric cells. Although for substances with high Pq the dielectric anisotropy can be neglected, the quadratic-in-field effects are implicitly accounted for by the highest order terms proportiOTial to P. ... 

Carrying out the experiment under the microscope, we observe that nothing happens for very small values of H. The visual field remains dark when viewed with crossed polarisers. Then at a threshold value Hq, the field of view is suddenly lit up. The sample has been deformed. This is the Frederiks transition., named after the Russian physicist who first observed the phenomenon in the 1930s. The experiment is easy to carry out in the laboratory, with quite rudimentary equipment. For a nematic slab of thickness 25 gm. He is about 0.2 Tesla and Eq about 400 V/cm. 

Fig. 9.7. Electrical Frederiks transition for a twisted nematic, (a) Anchoring of molecules on the substrate creates a constant twist throughout the sample, (b) Electric field E is applied parallel to the twist axis. From a threshold value Ec, molecules stand up, and for E Ec they lie parallel to E throughout most of the sample...

If the light wave is incident normally on a homeotropi-cally oriented cell, the director will become reoriented if the light intensity exceeds a certain threshold (this is called a light-induced Frederiks transition, or LFT). There may also be a threshold intensity for director reorientation when an o-wave is incident on a cell with a uniform planar orientation, but only if the adiabatic condition is violated. This leads naturally to the question of what happeits when an o-waye is normally incident on a hybrid cell (Fig. 1). 

Elastic moduli could be measured from the electro- (and magneto)-optical characteristics of the Frederiks transition, such as the threshold voltage C/p oi the threshold magnetic field ifp, according to the relationships... 

When the applied field is much higher than the threshold of the Frederiks transition the field coherence length becomes comparable with the surface extrapolation length itself H,E b. This condition corresponds to the second threshold of the complete reorientation of a liquid crystal including surface layers, curve 3 in Fig. 3.11(a). Thus, the second threshold field allows 6 (and W ) to be calculated. 

In this case the threshold of the Frederiks transition disappears, i.e., the deformation of the director alignment begins at infinitely small voltages [7, 8]. Figure 4.2 shows that for small values of 6q the corresponding response of the nematic cell assumes a quasi-threshold character [8]. 

Dynamics of fiuctuations, arising below the threshold of the Frederiks transition, were studied in [42]. The fiuctuations are modulated and amplified just above the critical (threshold) strength of the field. 

It is shown in [26] that the (a) and (c) deformations in Fig. 4.30 do not have a threshold, i.e., they can occur with any small qxternal voltage, and the stabilizing dielectric torque (because of Ae < 0) partially depresses the flexoelectric deformation. For (b) and (d) the deformation has a threshold and, with the conditions Wsi Ws2, the threshold depends on the polarity of the field. When the flexoelectric and dielectric torques are simultaneously destabilizing, the flexoelectric effect reduces the threshold of the Frederiks transition. In the opposite case, the dielectric torque raises the threshold of the flexoelectric effect. 

The field-induced reorientation of nematic polymers caused by the coupling of the electric field with their dielectric anisotropy was studied in a variety of papers [229, 231-238]. Unfortunately, only in a few papers (e.g., [237]) is a certain preliminary orientation of a polymer specified and we can speak of the true Frederiks transition with a well-defined threshold voltage. Nevertheless, the general opinion is that the Prank elastic moduli of both comb-like [232-235, 238] and linear-chain [236, 237] nematic polymers are of the same order of magnitude as of their low-molecular mass counterparts. 

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

The moduli were calculated from the threshold of the Frederiks transition ((4.9) induced by a magnetic (Ax > 0) and electric (Ae < 0)) field in homeotropically oriented liquid crystal layers. The same order of magnitude (10 -10 dyn), which is typical of conventional nematics, has been found for elastic moduli Kn and for other nematic polymers [233, 234]. Unwinding of the helical structure of chiral nematic polymers allowed the elastic constant K22 to be calculated K22 10" dyn for an arylic comb-like copolymer with cholesterol and cyanobiphenyl side-chair mesogens [229]). 

The firequency region of the modulated Frederiks structure is very narrow near a = a . For u) < uji the uniform Frederiks transition becomes more favorable, while for u > Ui the threshold, calculated according to (5.17), rapidly diverges (Fig. 5.2). 

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. -... 

Experimental data confirm the main conclusion of the theory that near the sign reversal frequency of Ae there occurs the Frederiks transition. The observed modulated structure is static, and its threshold depends on neither the value of conductivity cr) nor its anisotropy experimentally measured threshold conditions... 

A theoretical investigation of the stability of nematic liquid crystals with homeotropic orientation requires a three-dimensional approach. Helfrich s one-dimensional theory predicts the dependence of the threshold of the instability on the magnitude of Ae, as shown by curve 2 in Fig. 5.8, according to which the electrohydrodynamic instability should be observed when either Ae < 0 (and consequently the bend Frederiks effect reorientation will not take place), or when small Ae > 0. In Helfirich s model the destabilizing torque as dvzjdx is responsible for this instability, which replaces the destabilizing torque a dvzjdx in the equation for the director rotations (5.27). Although the torque is small ( a3 -C o 2 ) it is not compensated for (e.g., when Ae = 0) by anything else apart from the elastic torque. 

FIGURE 6.16. Bistable switching in long-pitch cholesterics with a tilt of the director o- (a) Tilted states with n/2 turns in zero field, = 55 . (b) Free energies g as functions of thickness to pitch ratio d/Po at zero field, dlPoY = 0.89 is the operating point, (c) g(d/PoY in an electric field versus reduced volteige U/Ufi Uf is the Frederiks transition threshold, Ae > 0. 

The threshold voltage observed experimentally for a transition from a planar to a homeotropic texture depends on the layer thickness according to Uq oc (for the Frederiks transition the threshold voltage would be independent of thickness). The model accounting for the experimental data has been developed by Parodi [109]. 

When the field is oriented along the director, a Frederiks transition is possible when 2 > 3- The director should rotate around the z-axls not changing its angle ft, so that in its final position, i.e., in the strong field limit, its projection onto the smectic planes, coincides with the y-axis. The corresponding threshold field (Eo)3 is proportional to ( 2 A field... 

Figure 10. Three typical geometries for observing the Frederiks transition the splay (a), bend (b), and twist (c) distortions induced by a magnetic field H. (Top) Initially, the director n is along the L axis, but when the field exceeds a certain threshold (bottom) distortion occurs.

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