Preferred orientation uniaxial

The mathematical expression of N(6, q>, i//) is complex but, fortunately, it can be simplified for systems displaying some symmetry. Two levels of symmetry have to be considered. The first is relative to the statistical distribution of structural units orientation. For example, if the distribution is centrosymmetric, all the D(mn coefficients are equal to 0 for odd ( values. Since this is almost always the case, only u(mn coefficients with even t will be considered herein. In addition, if the (X, Y), (Y, Z), and (X, Z) planes are all statistical symmetry elements, m should also be even otherwise = 0 [1]. In this chapter, biaxial and uniaxial statistical symmetries are more specifically considered. The second type of symmetry is inherent to the structural unit itself. For example, the structural units may have an orthorhombic symmetry (point group symmetry D2) which requires that n is even otherwise <>tmn = 0 [1], In this theoretical section, we will detail the equations of orientation for structural units that exhibit a cylindrical symmetry (cigar-like or rod-like), i.e., with no preferred orientation around the Oz-axis. In this case, the ODF is independent of t/z, leading to n — 0. More complex cases have been treated elsewhere [1,4]... 

The uniaxial orientation parameter is the most simple way to characterize preferred orientation. It is simple, because it is only a number - in fact, for = a is the first non-trivial expansion coefficient in a multipole expansion of the normalized... 

Uniaxially oriented films generally have high strength in only one direction, whereas biaxially oriented films are relatively strong in both machine direction (MD) and transverse direction (TD). Although intuitively one might expect biaxially oriented films to be preferred over uniaxially oriented films. 

The absence of preferred orientation statistically averages out the parallel and perpendicular conduction effects in Sr2Ru04. The bidimensional metallic conductivity, which is clearly observed in a single crystal, is completely inhibited in free powders composed of millions or billions of randomly oriented crystallites. This is shown in Figure 2.12c. By contrast, it can be expected that the application of several thousand pounds of uniaxial... 

Most HIPS fail in the uniaxial loading of a cantilever impact (Izod) test only if the test specimens have been notched beforehand. In the notched impact test, the stress direction in injection molded specimens is the same as the preferred orientation direction. This increases the measured impact toughness and the flexural impact test and, therefore, serves primarily for comparing the toughness of different products. 

When powder particles have thin platelet-like shapes, they will tend to agglomerate, aligning their flat surfaces nearly parallel to one another Figure 3.20, left). As a result, the orientations of platelets are randomized via rotations about a common axis normal to their largest faces, and such samples are expected to have a uniaxial preferred orientation (or texture). 

Nematic liquids can be homogeneously ordered in thick layers by a magnetic field of only a few thousand gauss. Homogeneously ordered samples behave optically like uniaxial crystals. The optical axis coincides with the preferred orientation of the long molecular axis. It turns parallel to the magnetic field direction. 

If there is no preferred orientation around Oxy, as is the case in the simplest form of uniaxial orientation, U31) = ( 2)- Equation (A.5) then shows that... 

Orientation of thin polymer films may be accomplished by stretching, which results in uniaxial orientation where a crystallographic axis, usually the long chain axis, tends to line up parallel to the stretching direction but no preferred orientation of crystallites about this axis takes place. Another way to orient a polymer is to roll the film between rollers. This can also be accomplished by placing the polymer film between silver chloride plates (Elliott et ai, 19486) and rolling, removing the silver chloride with a sodium thiosulfate solution if necessary. Orientation in the direction... 

Since only three independent intensities were measured, only two unknown quantities could be determined, and in order to reduce the number of unknowns various approximations and assumptions were made. It was first assumed that the polymer chains had no preferred orientation around their long axes. This assumption, together with the fact that the specimens had uniaxial symmetry, reduced the number of unknown orientation parameters to two, which were M200 and M400 or, equivalently, cos and cos 6, where 9 is the angle between a typical chain axis and the draw direction. Secondly, it was assumed that i s a2 = a,. The ratio r = a,/as could then be determined from measurements on a random sample. Substitution of cos 0 =, cos 0 = 5 into eqns. (13) and rearranging shows that... 

Two special types of orientation are of particular importance, uniaxial and biaxial orientation. In the former case, there is preferred orientation in one direction, in the latter case in two directions. The diagram in Fig. 54 may convey an idea of the two cases The structural units are represented by oblong lamellae. The length. 

Films can be divided into three classes—isonopic, uniaxial, and biaxial—depending on the form of their permittivity tensor s (1.1.2°). For isotropic films, the three principal values of the permittivity tensor are equal, i = 62 = s-, and there is no preferred orientation within the film. The permittivity tensor of a uniaxial film has two different principal values, one that describes the propagation of radiation in the film plane and another that is perpendicular to the film plane, so that = Sy s. For a biaxial film all of the principal values of the permittivity tensor are different, so that Ex By For crystalline films with triclinic... 

In general, the orientation of cracks will depend on the processing conditions of the specimens or components furthermore, their distribution may be random, or there may be some preferred orientation. Although, until now, it has been assumed that cracks are perpendicularly oriented to a uniaxial tensile stress field, this assumption is not generally necessary and has only been made to simplify the above arguments. 

In terms of elastic or electromagnetic properties, if two of the three directions in a material are equivalent, the material is said to be uniaxial. The nematic and smectic A phases of liquid crystals are uniaxial, since all directions perpendicular to the director are equivalent and different from the direction of preferred orientational order. Solids with hexagonal, tetragonal, and trigonal symmetry are also uniaxial. If all three directions in a material are inequivalent, then the material is biaxial. The liquid crystalline smectic C phase is biaxial because one direction perpendictrlar to the director is in the plane of the layers while the other direction perpendictrlar to the director makes an angle equal to the tilt angle with the layers. Solids of orthorhombic, monoclinic, and triclinic symmetry are also biaxial. 

A fibre or a uniaxially drawn film will usually show no preferred orientation in the plane normal to the fibre axis or the draw direction (see Figure 7.2). The compliance matrix then reduces to a form with only five independent constants ... 

Locally oblate lyotropic elastomers with lamellar phase structure (L -phase) can be oriented by uniaxial compression, as outlined above for thermotropic smectic-A elastomers. Fischer et al. synthesized crosslinked polysiloxane elastomers carrying non-ionic amphiphilic side-groups attached with their hydrophobic end to the polymer backbone. They were able to compress elastomer samples between Teflon half-cylinders to about half of their original thickness. The orientation of the phase structure - except for some unoriented domains - was demonstrated by means of H-NMR spectroscopy on the directly deuterated samples as well as by X-ray scattering. The preferred orientation of the director, and hence the amphiphilic side chains, was found to be parallel to the axis of compression with the amphiphilic bilayers aligned perpendicularly [98, 99]. 

Similar to fibers, films are often oriented. The simplest case of orientation is called uniaxial orientation, when the macromolecular segments are oriented in one preferred orientation. But a film may also be oriented in two directions (called biaxial orientation) having no fiber analog. Isotropic, unoriented film would correspond to the as-spun fiber (both having no preferred molecular orientation). Thus, DSC experiments carried out on unoriented films give results similar to those obtained from experiments performed on chips, powders, or as-spun fibers, but with minimum external thermal resistance. 

Figure 13. EISF of a particle undergoing uniaxial rotational diffusion on a circle of radius 2 A (0.2 nm) in a powder sample. The lower line is for no preferred orientation, so that (cos 2 )=0. The upper line is for two sites separated by Jt, strongly preferred, so that (cos2fl))=l, and the intermediate line is for (cos 2 )=0.5, which occurs when the restricting potential is roughly comparable with k T.

Interpretation of Anisotropic 2D Scattering Pattern Polymer crystallization under external fields (e.g., shear or extensional flows, stretching of polymer solids) usually exhibits preferred orientation, which results in anisotropic scattering patterns. For example, a four-point SAXS pattern was seen during solid-state uniaxial deformation of an ethylene-propylene copolymer (both experimental data and fitted 3D plot are shown in Fig. 1.17) [108]. 

We develop the approach adopted by Frank [91]. Let n be a unit vector representing the preferred orientation of a uniaxial liquid crystal at any point x and assume that it varies slowly with position. The sign of n has no physical significance in most cases. However, for molecules with permanent dipole moments this may not be the case and then the sign of n becomes important, but this will not be considered here. We introduce a local system of Cartesian coordinates x, z with z parallel to... 

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