The Freedericksz effect

4 Distortions due to magnetic and electric fields static theory 3.4.1 The Freedericksz effect 

The simplest method of measuring the three elastic constants of a nematic liquid crystal is by studying the deformations due to an external magnetic field (Freedericksz and Tsvetkov, Zocher ). The geometry has to be so chosen that the orienting effect of the field conflicts with the orientations imposed by the surfaces with which the liquid crystal is in contact. To develop a static theory of such deformations we apply the equation of 

Let us consider first the case of a nematic film in which the initial undisturbed orientation of the director is throughout parallel to the glass plates. The magnetic field H is now applied perpendicular to the director and to the plates (fig. 3.4.1(a)). For this geometry, n = (cos, 0,sin, H = (0,0,7/) and G = (O,O, fa sin0). The free energy of elastic deformation (3.3.6) reduces in this case to 

By straightforward substitution in (3.3.17) and simplification, we obtain the equilibrium condition 

As is to be expected, the deformation involves the splay and bend moduli, fcii and fcjj respectively, and not the twist modulus k. Because of the orienting influence of the glass surfaces, 0 = 0 at z = 0 and d, where rfis the thickness of the film. Therefore 9 attains a maximum value at z = djl and from symmetry considerations 9 z) = 0(d—z). Since d0/dz = 0 at z = djl, we get 

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The many different kinds of LCDs " based on electrically controlled birefringence (ECB) are variants of the Freedericksz effect first reported in the 1930s. These related display types based on the same electro-optical effect, but a slightly different display configuration are denominated by a multitude of names. However, current versions of this type of display are now often referred... 

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. 3.4.3. (a) The usual experimental configuration for the optical observation of the Freedericksz effect. Li t is incident normal to the film. However, for reasons discussed in the text, this arrangement is unsuitable for observing a twist deformation, b) Oblique configuration which enables the optical detection of a twist deformation. The magnetic field is perjjendicular to the plane of the paper... 

The Freedericksz effect in UgUy anisotropic nenuttics periodic... 

We shall now extend the theory of the Freedericksz effect to study the dynamical behaviour when the magnetic field is switched on or off suddenly. The analysis is particularly simple for a twist deformation (fig. 3.4.1 b)) because the torsion exerted on the director does not result in a translational motion of the centres of gravity of the molecules. Neglecting director inertia in (3.3.2) we obtain the following equation of motion for this geometry ... 

Dynamics of the Freedericksz effect Neglecting inertial effects, (3.3.1) reduces in the present case to... 

Reorientation of a nematic liquid crystal above a certain threshold value of the electric held of an optical wave, known as the photoinduced Freedericksz transition (PFT)—recently discovered and investigated both experimentally and theoretically -—exhibits a certain number of special features which are absent in the case of the rf analog of the Freedericksz effect. 

FIGURE 6 A scheme of epical observation of the Freedericksz effect N - nonatic film between glass plates, L - light source, P -polariser, A - analyso-, and D photodetection system. 

Homeotropic to planar transition backflow and kickback effects The other two geometries used in the Freedericksz experiment are more interesting as they result in a new effect, namely, hydrodynamic flow induced by orientational deformation. This is the inverse of the more familiar property of flow alignment that has been discussed at length in previous sections. 

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- ... 

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. 

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... 

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... 

We present a detailed theoretical calculation, with experimental verification, of the nonlocal molecular reorientation of the nematic-liquid-crystal director axis induced by a cw Gaussian laser beam. The natures of the torque balance equations and the solutions are significantly different for normally and nonnormally incident laser beams. The nonlocal effects resulting from molecular correlation effects are particularly important for laser spot sizes that are different (smaller or larger) from the sample thickness. Experimental measurements for the transverse dependence of the molecules and the dependence of the Freedericksz threshold as a function of the laser beam sizes are in excellent agreement with theoretical results. We also comment on the effect of these nonlocal effects on transverse optical bistability. 

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. 

In 1990, Janossy showed that a small amount of dye added to a nematic liquid crystal dramatically reduces the threshold intensity of the optical Freedericksz transition [68]. Subsequently, it was demonstrated that the underlying process is an optically driven Brownian ratchet mechanism [69-71]. Here, energy, but not momentum, from the radiation field causes unidirectional continuous rotation of dye molecules in the nematic, exerting a torque on the director that exceeds the direct optical torque by orders of magnitude. Similar mechanisms could, in principle, be realized in LCEs. Whether such processes are viable in overcoming the orienting effect of the network is not clear the viability of such Brownian motor processes in LCEs is an intriguing open problem. 

From the point of view of physics, LCs are partially oriented fluids that exhibit anisotropic optical, dielectric, magnetic, and mechanical properties. The most important property of LCs is the reorganization of their supramolecular structures on external stimuli such as electric and magnetic fields, temperatnre, and mechanical stress, which lead to changes in their optical properties. In particular, electric tiled-induced control of optical properties of LCs (electro-optical effects based on the Freedericksz transition ) is at the heart of the multi-billion dollar liquid crystal display (LCD) industry. Most current LCD technologies rely on nematic " and to a lesser extent on ferroelectric LCs, while the recently discovered bent-core and orthoconic LCs still require significant investment into fundamental research and development. These and other applications and technologies continne to drive the search for new liquid crystal materials, and provide impetus to continue fundamental studies on new, often exotic, classes of compounds. 

Karapinar R, Neill MO, Hird M (2002) Polymer dispersed ferroelectric liquid crystal films with high electro-optic quality. J Phys D Appl Phys 35(9) 900-903 Kitzerow HS, Molsen H, Heppke G (1992) Linear electro-optic effects in polymer-dispersed ferroelectric liquid crystals. Appl Phys Lett 60 3093 Kossyrev PA, Qi J, Priezjev NV, Pelcovits RA, Crawford GP (2002) Virtual surface, director domain and the Freedericksz transition in polymer-stabilized liquid crystals. Appl Phys Lett 81 2986... 

Sugimura, A., and Ou-Yang Zhong-can. 1992. Anomalous photocurrent transients in nematic liquid crystals The nonlinear optical PockeTs effect induced by the Freedericksz transition. Phys. Rev. A. 45(4) 2439-2448. 

Figure 12.7 shows a numerical simulation for an input beam power of 3.9 mW and a beam waist of 3 pm (the laser used is the 514.5 nm fine of an Ar laser). In the absence of the external bias (0q=O), the linearly polarized electric field of tire input light is perpendicrrlar to the director axis. Since its intensity is below the optical Freedericksz threshold value (cf. Chapter 8), it cannot create director axis reorientation that is, there is no self-action effect. The focused beam thus diffracts freely in X and as it propagates along z (Fig. 12.7a). When the external voltage above the Freedericksz threshold is applied, the director axis is reoriented (for the numerical simulation, the angle is assumed to be 45°). In this above Freedericksz condition, the ophcal field will reorient the director axis without threshold, and initiate the self-guiding process leading to the formation of a spatial soliton—a beam that maintains its beam waist over marty Rayleigh lengths (see Fig. 12.7b).

These values have been taken from [21], and wa e measured using ftie magnetic Freedericksz effect. They are lower ftian the oftier values quoted, whidi may reflect some uncertainty ovCT the magnetic susceptibility anisotropy. 

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