Threshold Freedericksz transition

If we compare with figure C2.2.I I, we can see that this defonnation involves bend and splay of the director field. This field-induced transition in director orientation is called a Freedericksz transition [9, 106, 1071. We can also define Freedericksz transitions when the director and field are both parallel to the surface, but mutually orthogonal or when the director is nonnal to the surface and the field is parallel to it. It turns out there is a threshold voltage for attaining orientation in the middle of the liquid crystal cell, i.e. a deviation of the angle of the director [9, 107]. For all tliree possible geometries, the threshold voltage takes the fonn [9, 107]... 

Note 1 The Freedericksz transition occurs when the strength of the applied field exceeds a certain threshold value (see Definition 5.12). 

BaTiC>3 particles are another very attractive and intensively studied type of nanoparticles in nematic liquid crystals. Cook et al. reported on an asymmetric Freedericksz transition, where doping nematic TL205 with single domain ferroelectric BaTiC>3 nanoparticles (9 nm in diameter) reduced or increased the threshold voltage by 0.8 V depending on the polarity of the applied voltage [149]. 

Blach and co-workers also observed a lower Freedericksz transition threshold voltage (V ii,) for 5CB doped with rather large BaTi03 particles (150 nm in diameter) [316], which is surprising considering an earlier report by West and Reznikov et al., who found no such reduction of Vth using smaller, chemically similar nanoparticles [317]. 

The Optical-Induced Freedericksz Transition of Nematic Liquid Crystal (5CB) doped with l%(w/w) of 5, 10, 15, 20-tetraphenylporphyrinatozinc( II) (ZnTPP) were studied. Excited by Ti Sappire laser with the 82MHz repetition rate and lOOfs pulse duration, the optical Freedericksz threshold of a 23.6pm-thickness planar alignment sample occurred at an intensity level of0.35mW/mm in contrast to the normally observed 83mW/mm value for pure 5CB. The coordination-bonding interaction between 5CB and ZnTPP were discussed by UV-vis and fluorescence spectra. We attribute the reduction of the optical Freedericksz threshold to the coordination-bonding interaction. 

Due to their large optical anisotropies, liquid crystals (LCs) have a large optical nonlinearity which is the result of molecular reorientation (Freedericksz transition) in an external field which exceeds the critical field [1], The high external field inhibits the application of LCs, and decreasing the threshold as low as possible is a difficult task [2], LCs doped with a small amount of absorbing dyes that could decrease the needed optical field intensity have been reported [3]. The basic assumption is that the anomalous reorientation of the director results from the interaction between the excited dye molecules and the host. However, this sample would easily degrade under the influence of laser radiation. 

Figure 2. Threshold voltage Uth/Uo and the critical wavenumber qc versus the dimensionless dielectric anisotropy eajev calculated from Eqs. (7) and (8). a b Planar alignment with a a > 0, c d homeotropic ahgnment with a a < 0. Dashed lines correspond to the Freedericksz transition, solid lines to the direct EC transition.

Figure 7. Threshold voltages Uth/Co and the critical wavenumber qc versus the relative dielectric anisotropy eajei. calculated from Eq. 8. Homeotropic alignment with <7a > 0. The upper (a b ) and lower (c d ) plots differ only in the axis scales. Dashed lines correspond to the Freedericksz transition, solid lines correspond to the direct transition to an ("a-induced") EC patterned state, dotted lines represent a secondary transition to EC.

The Freedericksz transition in the nematic and smectic C phases of 3- -heptyl-6-(4- -hexyloxyphenyl)-l,2,4,5-tetrazine has been studied. In both phases, the threshold voltage and the switching times were measured. In the smectic C phase, two thresholds have been observed which can be explained by an asymmetric chevron structure <1996MI131>. 

The Freedericksz transition discussed in 3.4.1 may be called a homogeneous transition since the distortion occurring above the threshold is uniform in the plane of the sample. In low-molecular-weight nematics, which as a rule have relatively small elastic anisotropy k i kjj 2 22), it is the homogeneous transition that is generally observed. Some polymer nematics, however, are known to exhibit high elastic anisotropy - an example is a racemic mixture of poly-y-benzyl-glutamate (PEG) which has k Jk =11.4 and k /k = 13.0 - and in such cases more complex types of field-induced deformations are possible. ... 

Much more exciting is the possibility of qualitatively new phenomena, which are generically related to flexopolarization. A prominent example is provided by the so-called flexodomains. They appear as the result of an equilibrium transition from the basic planar state if the applied electric field strength exceeds a certain threshold, Er. Flexodomains are stripe patterns parallel to the imposed preferred direction no x, i.e. with a wave vector qc T In contrast to the standard Freedericksz transition, the sign of... 

The liquid crystal parameters K2, K3/Kx> na and cell design parameters 3g OCg, d and the orientation of polarizers were varied one at a time. The ratio K3/KX was chosen as a parameter in order to keep the Freedericksz transition threshold voltage constant while varying the elastic constants. Variations in the threshold voltage due to the natural cholesteric pitch were not compensated for. Several calculations were also conducted for the case of infinite cholesteric pitch (no cholesteric compound in the liquid crystal mixture). 

The dependence of the merit figure on the dielectric anisotropy is a rather sensitive function of the choice of the threshold voltage. Had we selected the Freedericksz transition (capacitance) threshold voltage Instead of an optical threshold V9o(40 ), we would observe a decrease in figure of merit with increasing such as reported elsewhere.15 However, for an optical reponse, the latter seems more relevant. 

It is worthwhile to point out two characteristics of the flexoelectric effect. First, there is no threshold for the applied field, which is different from Freedericksz transition, where there is a threshold below which no deformation occurs. Deformation of the director configuration occurs under any field. Second, the direction of the bend depends on the polarity of the applied field, which is also different from Freedericksz transition where the deformation is independent of the polarity of the applied field. 

In Freedericksz transition, when the applied field is slightly above the threshold E, the tilt is proportional to /E-Ec and the retardation is proportional to E- Ec) (see Chapter 5 for details). 

Strictly speaking, when the mbbing angle <1> is not equal to zero the Freedericksz transition threshold is smeared. However, in a normally black IPS mode, the transmittance is proportional to the phase retardation 8 = Ind rdX of the LC cell asT sin 8/2). In the small voltage regime, the phase retardation is small and transmittance exhibits a threshold-like transition. 

The spectral response of a holographic PDLC to applied electric fields is shown in Figure 11.27, where white incident hght is used [37]. At 0 V, due to the periodic refractive index, the cell has a high narrow reflection peak. When the applied voltage is increased, the liquid crystal is ahgned toward the layer normal direction. The amplitude of the oscillation of the refractive index decreases and the reflection of the cell also decreases. The drive voltage is approximately equal to the product of the field threshold of the Freedericksz transition of the liquid crystal layer and the cell thickness. 

Note that the threshold field here is smaller than the threshold field of the Freedericksz transition in the regular liquid crystal cell consisting of two parallel substrates with the same cell gap, for the following reason. In the regular cell the hquid crystal is anchored by the two-dimensional surface of the substrate, while in the polymer-stabilized hquid crystal cell here, the liquid crystal is anchored by the one-dimensional polymer fiber. 

The simplest display application of a nematic polymer liquid crystal, using polarized light, involves the splay Freedericksz transition shown in Fig. 5(a) for a material with positive Ac and planar surface alignment. The device is constructed using two polarizers crossed with each other, placed above and below the cell, with their polarization direction at 45° to the director of the planar texture. Above a threshold voltage given by... 

To understand how liqtrid crystals respond to external fields, it must be realised that in most cases interactiorrs between the liqrrid crystal molecules and botmdaries have a large effect. These boundaries can be with a solid material such as glass, but the air-liqtrid crystal boundary is also both important and interesting. In marty cases, the influence of the boundary opposes the resportse to the electric field, and the resrrlt is a threshold phenomenon called the Freedericksz transition. 

The optical reorientation processes discussed up to now were qualitatively similar to the corresponding low-frequency field effects. As mentioned earlier this is not always the case. A breakdown of the analogy with static fields was first reported by Zolotko et al. who observed in a homeotropic layer a drastic increase of the Freedericksz threshold power for an o-ray as the angle of incidence was increased. Durbin et al. mentioned that in a planar cell Freedericksz transition cannot be induced by a light beam polarized perpendicularly to the director. From a simple analogy one would expect for these cases a threshold not deviating significantly... 

To observe the optical-field-induced Freedericksz transition, the angle of incidence of the pump beam was set at 0°. The observed birefringence versus pump intensity at Tffj - T = 9.2 K is shown in Fig. 1. It exhibits a threshold intensity at 155 W/cm. According to Eq. (5),... 

As is shown in Fig. 5, the experimentally observed threshold field dependence on wo/d also follows the theoretical prediction. For E46, a 100-Axm sample has a Freedericksz transition field intensity of 200 W/cm (using the values = 0.3, K lO" , d = 0.01 cm, and n no 1.5). For large values of Wo/d (for example, wo/d > 5), the observed threshold field approaches this value. However, as the incident laser beam size decreases to a value comparable with the thickness d or less, the threshold optical intensities increase dramatically. At wq d, the threshold intensity increases by almost an order of magnitude. In general, the experimentally observed relative increase of the threshold field is slightly larger than the theoretical value, probably because of a systematic difference between the experimental observation of the onset of reorientation (by the appearance of the self-focusing effect on the exit Ar+ laser beam) and also... 

FIG. 1. Ratio of threshold power P,hr(Freedericksz transition in the held of an ordinary light wave obliquely incident on a crystal at an angle a to the threshold power (0) at normal incidence 1— experimental plot 2—theoretical plot in the plane-wave approximation 3—theoretical plot calculated by a variational method /, hr(0) = 50 mW. 

It is experimentally demonstrated that a circularly polarized laser beam normally incident on a homeotropically aligned nematic film can induce a collective precession of the molecules in the film if the laser intensity is above the threshold for the Freedericksz transition. The effect is shown to result from a transfer of angular momentum from the laser beam to the medium. 

With the input intensity changing slowly, the hysteresis loop is well reproducible. The upward transition BC in Fig. 2 is much faster than the downward transition DA The switching times are up— 1 s and down—15 s, respectively. The observed threshold power for up transition at point B was Pth 110 mW. With a beam cross section of 5x 10 cm at the sample, the corresponding threshold intensity was 7th 2.2 kW/cm. This transition should result from the field-induced molecular reorientation usually known as the Freedericksz transition. We found that the observed threshold for a circularly polarized input beam was twice that for a linearly polarized input beam, as expected from theory." However, no polarization rotation of the output was observed in the latter case. 

We now consider some quantitative features of the problem. The threshold Ah for the Freedericksz transition with circularly polarized light is given by... 

An experimental investigation was made of aberration self-focusing due to orientational deformation of a nematic with a hybrid orientation subjected to the held of an incident ordinary optical wave. When the radiation was incident on a sample from the homeotropic substrate side, the reorientation occurred at intensities higher than the threshold for a photoinduced Freedericksz transition in a homeotropic cell of the same thickness, and it depended strongly on the angle of incidence of the ordinary wave on the sample. The effect was not observed when light was incident from the planar substrate side. 

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