Fluctuations director axis

Fig. 5 Schematic depiction of the optical field propagating as an o-wave in a planar nematic liquid crystal cell, ba is the orientational fluctuation of the director axis. 

Solving for the director axis fluctuation in in the steady state, and equations (10) and (11) for an input e-wave (E ) gives a steady state... 

The order parameter, defined by Equation (2.2) and its variants such as Equations (2.4) and (2.8), is an average over the whole system and therefore provides a measure of the long-range orientation order. The smaller the fluctuation of the molecular axis from the director axis orientation direction, the closer the magnitude of S is to unity. In a perfectly ahgned hquid crystal, as in other cry stalhne materials, = 1 and S= 1 on the other hand, in a perfectly random system, such as ordinary Uquids or the isotropic phase of hquid crystals, [Pg.25]

Scattering of light in a medium is caused by fluctuations of the optical dielectric constants 5s(f, t). In isotropic liquids 58(f, t) are mainly due to density fluctuations caused by fluctuations in the temperature. For liquid crystals in their ordered phases, an additional and important contribution to 5 (f, t) arises from director axis fluctuations. 

SCATTERING FROM DIRECTOR AXIS FLUCTUATIONS IN NEMATIC LIQUID CRYSTALS... 

As shown in Figure 5.2, the director axis fluctuation 8n (which is normal to h since h = 1) is decomposed into two orthogonal components 8ni and 8n2, along the unit vectors e and 2 respectively. Note that one of them, 8nj is in the plane defined by q and h (taken as f ), and the other, 802, is perpendicular to the q-z plane. 

From Equation (5.1) the change in e p associated with the director axis fluctuation comes from the second term ... 

If an external field is present (e.g., a magnetic field applied in the z direction), the director axis fluctuations may be reduced. Quantitatively, this may be estimated by including in the free-eneigy equation [Eq. (5.12)] the magnetic interaction term... 

In the isotropic phase director axis orientations are random. The optical dielectric constant, a thermal average, is therefore a scalar parameter. The fluctuations in this case are due mainly to fluctuations in the density of the liquid caused by temperature fluctuations. 

Just as the scattered light is distributed over a spectmm of wave vectors ky for a given incident wave vector k the frequency of the scattered light is distributed over a spec-tram of frequencies (Hf, for a given incident light frequency cd,. This spread in frequency Ao)= ay-co, is inherently related to the fact that the director axis fluctuations are characterized by finite relaxation time constants. 

These influences of the intermolecular and the elastic torques in cw-laser-induced nonlinear diffraction effects in nematic films are reported in the work by Khoo. There it is also noted that the optical nonlinearity associated with nematic director axis reorientation is proportional to the factor AcVAT [see Eq. (8.44)] typical of orientational fluctuations induced by light scattering processes (see Chapter 5). Although both Ae and K are strongly dependent on the temperature, the combination AsV is not. This is because Ae is proportional to the order parameter S, whereas K is proportional to S. ... 

Figure 11.18. (a) Scattering of a polarized laser by director axis fluctuations inaNLC (b) Optical polar-... 

From many perspectives, liquid crystals are particularly suited for applications in this regime. First, they are nonabsorptive and possess large dielectric anisotropy (A8=8g-8o l) over the entire near-UV to infrared spectrum and beyond (400 mn-20 pm cf. Chapter 3). Secondly, the scattering loss a in this regime is an order of magnitude smaller than in the visible region, since the principal mechanism for scattering loss are director axis fluctuations [a 1/A," ( >2)]. 

Such order can be described in terms of the preferential alignment of the director, a unit vector that describes the orientation of molecules in a nematic phase. Because the molecules are still subject to random fluctuations, only an average orientation can be described, usually by an ordering matrix S, which can be expressed in terms of any Cartesian coordinate system fixed in the molecule. S is symmetric and traceless and hence has five independent elements, but a suitable choice of the molecular axes may reduce the number. In principle, it is always possible to diagonalize S, and in such a principal axis coordinate system there are only two nonzero elements (as there would be, for example, in a quadrupole coupling tensor). In the absence of symmetry in the molecule, there is no way of specifying the orientation of the principal axes of S, but considerable simplification is obtained for symmetric molecules. If a molecule has a threefold or higher axis of symmetry, its selection as one of the axes of the Cartesian coordinate system leaves only one independent order parameter, with the now familiar form ... 

The local order in a cholesteric may be expected to be very weakly biaxial. The director fluctuations in a plane containing the helical axis are necessarily different from those in an orthogonal plane and result in a phase biaxiality . Further, there will be a contribution due to the molecular biaxiality as well. It turns out that the phase biaxiality plays a significant role in determining the temperature dependence of the pitch. Goossens has developed a general model taking this into account. The theory now involves four order parameters the pitch depends on all four of them and is temperature dependent. However, a comparison of the theory with experiment is possible only if the order parameters can be measured. 

To construct a continuum theory of S, we have to take into account firstly, the orientational fluctuations of the director about the layer normal (z axis), and secondly, as in smectic A, the distortions of the layers themselves. Expressions for the former were given by Saupe, but the complete theory including the latter contributions and the coupling between the two was derived by the Orsay group. We chose a cartesian... 

The mean square amplitude of the orientational fluctuations of the director and the corresponding intensity of light scattering may be worked out in a similar fashion. For example, when the incident and scattered beams are both polarized in the plane perpendicular to the optic axis (or column axis) the scattered intensity is given by... 

In a uniaxial nematic liquid crystal, the spatial orientation of the optical axis is determined by the orientation of the director. Due to thermally excited orientational director fluctuations, the spatial direction of the optical axis is not constant in time. As a result, any light illuminating the sample is... 

In confined geometries, however, this is not the case. In a planar sample, for example, the magnitude of the wave vector component parallel to the boundaries is arbitrary, whereas the fluctuation wave vector component perpendicular to the boundaries can only have certain values. In this case, the allowed wave vector components are determined by the sample thickness, viscoelastic properties of the liquid crystal, and also by the interaction of a liquid crystal with the aligning surface. If the director is strongly bound to the aligning substrate, it cannot deviate from the induced direction (easy axis) and the fluctuation amphtude at the boundary is zero. On the other hand, if the surface only weakly anchors the director orientation, the director can fluctuate to a certain degree around the easy axis. 

In the following, and / q denote the order parameter fluctuations with respect to the nematic director parallel to the x and axis, respectively. The other three fluctuation modes are uncoupled and represent either director fluctuations and low j3 2 modes) or biaxial fluctuations, high /3 2 inodes. 

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