Freedericksz transition field

In this paper we present several new aspects of this problem. They include an experimental confirmation of the theoretical results for the oblique-incidence case as well as the theory for the reorientation of normally incident lasers (i.e., when the laser polarization is orthogonal to the director axis of the nematic film) and experimental results for the observed Freedericksz transition field and broadened (or narrowed) radial dependence. 

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... 

In the following two Sections we will discuss and compare theoretical and experimental studies on RBC and EHC. We often use non-dimensionalized units. Thus we write wave numbers as qi = q/n/d, where the prime is sometimes omitted, and magnetic fields as Hi = hxHf with the splay Freedericksz transition field H/ = (n/d)[ku/ fioXa)] - Other quantities have been introduced before. Note that for the Cartesian components of the wave-vector we use two symbols, q = (qx, qy) = q,p)-... 

The preceding discussion and results apply to the case where an extraordinary wave laser is obliquely incident on the (homeotropic) sample (i.e., (3 0). For the case where a laser is perpendicularly incident on the sample (i.e., its optical electric field is normal to the director axis), there will be a critical optical field >, the so-called Freedericksz transition field [see Eq. (8.54)], below which molecular reorientation will not take place. Second, the tum-on time of the molecular reorientation depends on the field strength above Ep (i.e., on op-. ). For small E op the tum-on time can approach many minutes Studies with nanosecond and picosecond lasers" have shown that under this perpendicularly incident (i.e., (3=0) geometry, it is very difficult to induce molecular reorientation through the mechanism discussed previously. 

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). 

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. 

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. 

Problem 10.1 (c) Compute the critical magnetic field for a Freedericksz transition in which the director is initially oriented in the z direction, perpendicular to the plates, and strongly... 

Wall defects are also very important in nematic phases, especially in electric or magnetic fields. This will be considered further in section C2.2.4.1. which discusses Freedericksz transitions in a nematic in an electric or magnetic field. 

Here nd are elastic constants. The first, is associated with a splay deformation, K2 is associated with a twist deformation and with bend (figure C2.2.11). These three elastic constants are termed the Frank elastic constants of a nematic phase. Since they control the variation of the director orientation, they influence the scattering of light by a nematic and so can be determined from light-scattering experiments. Other techniques exploit electric or magnetic field-induced transitions in well-defined geometries (Freedericksz transitions, see section (C2.2.4.1I [20, M]. 

In the second group we find pattern forming phenomena based on new instability mechanisms arising from the specific features of liquid crystals, which have no counterpart in isotropic fluids or at least are difficult to assess. Some examples are shear (linear, elliptic, oscillatory, etc.) induced instabilities, transient patterns in electrically or magnetically driven Freedericksz transitions, structures formed in inhomogeneous and/or rotating electric or magnetic fields, electroconvection (EC), etc. [5-7]. 

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. ... 

A. J. Karn, S. M. Arakelian, Y. R. Shen and H. L. Ong, "Observation of Magnetic-field-induced First-order Optical Freedericksz Transition in... 

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... 

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). 

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... 

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. 

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... 

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