Director frame

Figure 1. Schematic of the mesomorphous structure in a lamellar phase. The different coordinate systems used in the text are outlined laboratory frame (L), director frame (D), and molecular frame (M). 0LD and 0DM are angles between the z axis in laboratory-director systems and director-molecular systems, respectively (13).

Figure 22 A schematic illustration of the various coordinate frames considered within the two-step model, for the case of a specifically deuterium-labeled methylene segment in the surfactant hydrocarbon chain. The laboratory frame (L) is set by the direction of the external magnetic field, where Zjl is the field direction. In this frame the nuclear quadrupolar moment tensor is diagonal. The director frame ( >) is associated with the micellar aggregate where Z/> specifies the micellar surface normal. It is assumed that the fast local dynamics occur with an essentially cylindrical symmetry around Z/). The molecular frame (Af) corresponds to the principal axis of the electric field gradient tensor. For the case of a methylene segment, Zm specifies the direction maximum component of the field gradient tensor, which is furthermore cylindrically symmetrical around Zm-...

Before discussing the different parameters used in the diffusion operator SLE formalism, it is necessary to define a number of different coordinate systems, depicted in Fig. 6a and b. The first frame of interest is the director frame (xp, yjj, zjj), which is fixed relative to the structure of the polymer to which the label is attached. The Zp axis is used to define the energy potential that imposes orientational order on the... 

The orientation of molecules in a mesophase can be specified by a singlet distribution function /(fi), where Q, denotes the Eulerian angles (0,0,-0) that transform between the molecular frame and the director frame. The average of any single-molecule property X(n) over the orientations of all molecules is defined by... 

FIGURE 3.2. The coordinate system required to describe the interaction between two asymmetric molecules, (a) The upper diagram illustrates the ri2 frame in which the intermolecular vector ri2 is the mutual polar axis, while (b) the lower diagram illustrates the director frame in which the director is the mutual polar axis. 

Assuming the validity of Fick s laws for liquid crystals, various theories [7-15] have been developed to describe the diffusion constants D and in particular their anisotropy ratio a, in terms of the mesophase order and other adequate macroscopic and microscopic material parameters. However, in view of the unsatisfactory agreement with experimental data, so far this has only achieved limited success. By transforming the diffusion tensor from a local (cluster) to the laboratory director frame and by taking the orientational ensemble average, Blinc et al. [7] showed that the two tensor components of D for thermotropic N, S, and Sc phases should be coupled to the order parameter S, the average diffusion constant (D), and the limiting values D, of a perfectly ordered cluster (S = 1) in the... 

The above spectral densities can be modified for the occurence of chain flexibility, and for the director being oriented at dLD w.r.t. the external BQ field in the L frame. For CD bonds located in the flexible chain, the effect of DF is reduced due to an additional averaging of the time dependent factor (/f g) by conformational transitions in the chain. Consequently, the spectral densities given in Eqs. (60)-(62) are modified by replacing Soc%0(Pm,q) by the segmental order parameter YCD of the C-D bond at a particular carbon site on the chain.146,147 As observed experimentally,148,149 the spectral densities in a flexible chain show a SqD dependence when DF dominate the relaxation rates. The general expression of Jm(co 0LD) due to DF in uniaxial nematic phases is given by... 

Another rotational diffusion model known as the anisotropic viscosity model156,157 is very similar to the above model, and its main feature is to diagonalize the rotational diffusion tensor in the L frame defined by the director. A similar (but not the same) expression as Eq. (71) is J R(r)co)... 

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Anisotropic fluids, of which nematic liquid crystals are the most representative and simplest example, are characterized by an anisotropic dielectric permittivity. The nematic phase has D,yuh symmetry, and in a laboratory frame with the Z axis parallel to the C , symmetry axis (the director) the permittivity tensor has the form ... 

Liquid crystals can display different degrees of long-range order, dependent on temperature, chemical composition, and the presence or absence of electric fields. In the nematic phase, the molecular axes point in a common direction, denoted by the director n but the molecular centers are otherwise arranged randomly. Because of the low degree of long-range order, nematic LCs have viscosities typical of ordinary liquids, and displays based on nematic LCs can operate at television frame rates. The most popular nematic-based display, the twisted nematic (TN), will be discussed in more detail below. 

Figure 4.6-3 Eulerian angles ), p, and the molecular frame z (long axis), y, and x (short axis) forms with the laboratory frame z, y, x of which r agrees with the director and the optical axis of the macroscopic sample also given are the angles a and 0 specifying within the molecular frame the orientation of a transition moment M and its projection m onto the x, y -plane.

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