Phase function anisotropic

Liquid crystal is a term that is now commonly used to describe materials that exhibit partially ordered fluid phases that are intermediate between the three dimensionally ordered crystalline state and the disordered or isotropic fluid state. Phases with positional and/or orientational long-range order in one or two dimensions are termed mesophases. As a consequence of the molecular order, liquid crystal phases are anisotropic, i.e., their properties are a function of direction. 

Isotropic scattering indicates that the radiant energy incident on a volume element is uniformly distributed to all directions. For an isotropically scattering medium, all a, coefficients of the phase function are zero, except a0. If only a0 and the first coefficient a, are considered, then one obtains the linearly anisotropic phase function, which means that the phase function is a linear function of cos 0 (or, in the case of an azimuthally symmetric medium, a linear function of p = cos 9). 

If the phase function is approximated as a linearly anisotropic one (there is no justification for the use of a more elaborate one since the method itself will not pick the resolution), such as ... 

S2 and S4 Models for Cylindrical Geometries. For the solution of the RTE in cylindrical media, the formulations for the S2 and S4 discrete ordinates (DO) approximations (based on Ref. 69) will be presented here. Note that, in a more recent study, Jendoubi et al. [75] used a similar DO approximation in cylindrical geometry and evaluated the effect of anisotropic phase function on the accuracy of the model. 

Here, let us assume that the phase function is a linearly-anisotropic one, expressed as ... 

It is important to ask the following questions before a specific model is chosen. (1) Is the medium geometry simple (2) Are there steep temperature and species concentration distributions in the medium (3) Are there anisotropically scattering particles (4) If there are, what kind of scattering phase function approximations can be used for them Having some approximate answers to these questions will help to expedite the selection process. 

Because of the orientational order of the molecules, liquid crystal phases are anisotropic. This is reflected in the anisotropic response of liquid crystals to fields, specifically electric, magnetic or mechanical fields. All of these are important to the functioning of devices based on liquid crystals. 

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

The range of systems that have been studied by force field methods is extremely varied. Some force fields liave been developed to study just one atomic or molecular sp>ecies under a wider range of conditions. For example, the chlorine model of Rodger, Stone and TUdesley [Rodger et al 1988] can be used to study the solid, liquid and gaseous phases. This is an anisotropic site model, in which the interaction between a pair of sites on two molecules dep>ends not only upon the separation between the sites (as in an isotropic model such as the Lennard-Jones model) but also upon the orientation of the site-site vector with resp>ect to the bond vectors of the two molecules. The model includes an electrostatic component which contciins dipwle-dipole, dipole-quadrupole and quadrupole-quadrupole terms, and the van der Waals contribution is modelled using a Buckingham-like function. 

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

Note 4 The extent of the positional correlations for the molecules in a nematic phase is comparable to that of an isotropic phase although the distribution function is necessarily anisotropic. 

One of the most popular refinement programs is the state-of-the-art package Refmac (Murshudov et ah, 1997). Refmac uses atomic parameters (xyz, B, occ) but also offers optimization of TLS and anisotropic displacement parameters. The objective function is a maximum likelihood derived residual that is available for structure factor amplitudes but can also include experimental phase information. Refmac boasts a sparse-matrix approximation to the normal matrix and also full matrix calculation. The program is extremely fast, very robust, and is capable of delivering excellent results over a wide range of resolutions. 

It bears repeating that the values are effective diffusivities and that, in fact, diffusivity is a function of surfactant concentration, as shown by interferometry for the Li phase of Ci2(EO)5 [9]. For the anisotropic phases diffusivity is also a function of orientation, and Dej depends on the number and orientation of domains of the phase as formed during dissolution. Thus, the value shown in Table 1 for DgHi of Ci2(EO)6 is intermediate between the diffusivities parallel and perpendicular to the rodlike micelles measured in fully oriented samples of the hexagonal phase Hi for this surfactant [26]. 

In isotropic materials, and along symmetry directions in anisotropic materials, all the traction components in the third equation vanish, as does the component of A associated with the SH mode. There are then three equations for three unknowns. For the general anisotropic case the four equations can be solved to give the amplitude A4 of the reflected wave. The amplitude is a complex quantity, because there may be a change in phase upon reflection. For an incident wave at an angle d to the normal and 0 to some direction lying in the surface (usually the lowest index direction available) the reflectance function may be written... 

Fourier transform infrared spectroscopy (FTIR) was also used to study the anisotropic structure of polyimide films. This work was based on the fact that there are characteristic absorptions associated with in-plane and out-of-plane vibrations of some functional groups, such as the carbonyl doublet absorption bands at 1700-1800 cm . The origin of this doublet has been attributed to the in-phase (symmetrical stretching) and out-of-phase (asymmetrical stretching) coupled... 

Nematic phases are characterized by an unordered statistical distribution of the centers of gravity of molecules and the long range orientational order of the anisotropically shaped molecules. This orientational order can be described by the Hermans orientation function 44>, introduced for l.c. s as order parameter S by Maier and Saupe 12),... 

The 2,2,6,6-tetramethylpiperidinoxyl radical (TEMPO) was first prepared in 1960 by Lebedev and Kazarnovskii by oxidation of its piperidine precursor.18 The steric hindrance of the NO bond in TEMPO makes it a highly stable radical species, resistant to air and moisture. Paramagnetic TEMPO radicals can be employed as powerful spin probes for elucidating the structure and dynamics of both synthetic and biopolymers (e.g., proteins and DNA) by ESR spectroscopy.19 Unlike solid-phase 1H-NMR where magic angle spinning is required in order to reduce the anisotropic effects in the solid-phase environment, solid-phase ESR spectroscopy can be conducted without specialized equipment. Thus, we conducted comparative ESR studies of various polymers with persistent radical labels, and we also determined rotational correlation times as a function of... 

---