Freedericksz threshold

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.

Case C homeotropic alignment, Ca < 0 0. In this combination of the material parameters, the linear stability analysis of the basic state does not predict a direct transition to EC since the resulting expression for U q) in Eq. (8) is negative for all q 0 (except for e in the immediate vicinity of zero, see below). The reason is that the two terms in the denominator act differently compared to the case B ca > 0, aa < 0) described in the previous subsection. The Carr-Helfrich torque is now stabilizing while the dielectric torque (oc is destabilizing. At q = qp = 0, this term dominates and describes, at the threshold Ups (see Eig. 7a), the continuous bifurcation to the Freedericksz distorted state of homogeneous (along the x direction) bend (see Fig. 8a). 

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.

Upon increasing the voltage the first transition is usually to the homogeneous (q = qp = 0) splay Freedericksz state with a frequency-independent threshold voltage Upi (Fig. 10a), i.e the absolute minimum of the neutral curve is at = 0. However, very near to e = 0 the absolute minimum appears at a finite q. Thus we have the planar counterpart of the a-induced FC described in case C. The Freedericksz threshold smoothly transforms into an FC threshold Ua at eo/ej = 0.0057 (see Fig. lOc-d). Below this, a direct transition to FC is predicted with q growing continuously from zero and remaining extremely small. This transition seems to be a better candidate for experimental observation than its homeotropic counterpart, because both Ua and qa are substantially lower than in case C. 

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>. 

In other words, deformation occurs only above a certain critical field This is referred to as the Freedericksz effect. The threshold condition can be used for a direct determination of the splay modulus n. 

Fig. 3.4.5. Theoretical curves illustrating the relaxation of the Freedericksz threshold as the magnetic field is tilted away from the normal to the initial orientation of the director. The calculations have been made for PAA in the twist geometry for a sample of thickness 12.7 y m. (a) Field normal to the director, q> = 90°, (b) q> = 89°, (c) f = 85° and (d)

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. ... 

Fig. 3.4.11. Threshold plots of R (the ratio of the periodic distortion threshold and the normal (homogeneous) Freedericksz threshold) and of (the dimensionless wavevector of periodicity at threshold) versus Periodic...

Besides the elastic and the electric torques the so-called flexoelectric (or flexo) torques on the director play an important role as well. Their effect on pattern-forming instabilities in nematics is the main issue of this chapter. Flexotorques originate from the fact that typically (in some loose analogy to piezoelectricity) any director distortion is accompanied by an electric flexopolarization Pa (characterized by the two ffexocoefScients ei, 63). From a microscopic point of view, finite ei and 03 naturally arise when the nematic molecules have a permanent dipole moment. But also for molecules with a quadrupolar moment, finite ei and 63 are possible (see also Chapter 1 in this book ). Flexopolarization has to be incorporated into the free energy P n) for finite E. It is not surprising that this leads to quantitative modifications of phenomena, which exist also for ci = 63 = 0. Though, for example, the Freedericksz threshold field Ep is not modified, the presence of flexoelectricity leads to considerable modifications of the Freedericksz distorted state for E > Ep- ... 

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. 

From Figure 8.19, an excellent dark state is observed at normal incidence. As the applied voltage exceeds the Freedericksz threshold voltage (V,ft-2.1 V s), LC directors are reoriented by the applied electric field, resulting in light transmission fi om the crossed analyzer. At -6 Vrms, the normalized transmittance reaches 100% for the green light (X = 550 nm). 

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... 

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