The Freedericksz Transition

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

Problem 10.1(b) (Worked Example) Calculate the critical magnetic field required to induce the Freedericksz transition described in part (a), where d is the gap between the plates. 

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.

Case H homeotropic alignment, Ca < 0 era < 0. Above the Freedericksz transition where no standard EC is predicted, convection (ns-EC) builds up with properties similar to those listed for case G. The patterns are disordered (see Fig. 14) as expected for an initial homeotropic alignment. 

In this paper we have reviewed the structures appearing at onset of electro-convection in nematic liquid crystals. The influence of the relevant material parameters (ca and ao) and the role of the initial director alignment were explored. Our calculations using a linear stability analysis of the standard model of electroconvection (performed for zero frequency) revealed that four different scenarios characterized by different ranges of the wavenumber q can be identified (1) the Qf= 0 mode (a homogeneous deformation known as the Freedericksz transition) predicted and observed in cases C, D, E and H, which is... 

The subcritical nature of the Freedericksz transition can be explained as follows. When the director settles to the precession state, light becomes el-liptically polarized inside the nematic. On the other hand, it is known that the Freedericksz transition for elliptically polarized light depends on ellipticity and... 

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. ... 

Experimental observations of the flexodomains have been made in planar geometry for various nematics either with < 0 (where the Freedericksz transition is excluded) or with > 0.1142,34,39 pj-e ise compar-... 

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. 

We consider the dynamics of the Freedericksz transition in the splay geometry upon the removal of the applied field [24-27]. Initially the liquid crystal director is aligned vertically by the applied field, as shown in Figure 5.17(a). When the applied field is removed, the liquid crystal relaxes back to the homogeneous state. The rotation of the molecules induces a macroscopic translational motion known as the backflow effect. The velocity of the flow is... 

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. 

Now we consider the Freedericksz transition in a polymer-stabilized nematic liquid crystal in the splay geometry. The liquid crystal director is given by... 

A suitable order parameter for monitoring the Freedericksz transition (involving a significant number of nematic particles) is (P ) = 5(3(ufy) —1), where the average (...) is performed over all particles and MC cycles, and y represents a unit vector along the field direction. If, on the other hand, the average (...) is taken over particles in the first layer next to the fiber only (of thickness o), one obtains (Pj ) sensitive... 

The Freedericksz transition is also the basis of operation for most liquid crystal displays. In a twisted-nematic display cell, the surface anchoring at the two substrates is incommensurate, inducing a twist to the director field that rotates light polarization. Application of an electric field sufficiently strong to align the director... 

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. 

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