Orientation uniaxial distribution

In order to evaluate quantitatively the orientation of vibrational modes from the dichroic ratio in molecular films, we assume a uniaxial distribution of transition dipole moments in respect to the surface normal, (z-axis in Figure 1). This assumption is reasonable for a crystalline-like, regularly ordered monolayer assembly. An alternative, although more complex model is to assume uniaxial symmetry of transition dipole moments about the molecular axis, which itself is tilted (and uniaxially symmetric) with respect to the z-axis. As monolayers become more liquid-like, this may become a progressively more valid model (8,9). We define < > as the angle between the transition dipole moment M and the surface normal (note that 0° electric field of the evenescent wave (2,10), in the ATR experiment are given by equations 3-5 (8). 

Fig. 17. (a) 2D CSA-DECODER spectrum of polyethyleneterephthalate fibres oriented perpendicular to the rotor axis with a mixing time of 9.6 ms equivalent to a rotation of 125° (b) Simulation of the experimental data using a uniaxial distribution with FWHM = 25°. (Adapted from Lewis et al.260 with permission.)... 

Uniaxial drawing of polymer samples introduces a uniaxial distribution of molecular orientations which often can be approximated by a Gaussian. The NMR spectrum then depends on the angle between the drawing direction and the magnetic field. For example, the spectrum of unoriented FIFE is well approximated by a symmetric powder pattern (Fig. 10.3.1(a), cf. Fig. 3.1.3), while the spectrum of a uniaxially oriented sample exhibits two peaks when the direction of molecular order is parallel to the magnetic field (Fig. 10.3.1(b)) [Hepl]. 

Fig. 4a and b. Fig. 4a and b represent hypothetical pole figure plots of polyethylene as discussed in the text Numbers refer to relative diffracted intensities, (a) This shows uniform orientation distribution of a axes in the N-TD plane and a preference for perpendicular alignment to MD. (b) This shows a preference for alignment of the a axes along MD with a uniform (uniaxial) distribution about this same axis... 

For the simplest type of uniaxial orientation the distribution function reduces to N(6), where N(6)do) is the fraction of units for which OX3 lies within any small solid angle dco at angle 6 to 0X3. By characterising the distribution is meant finding out as much as possible about N 6) for the various types of structural unit that may be present in the polymer. 

A hypothetical uniaxial distribution of orientations has 20% of the axes of the units oriented randomly, 50% oriented on a cone of semi-angle a = 30° and the remainder parallel to the axis of the cone. Calculate (cos 0), (cos ), ( 2( 08 )) and (P4(cos0)), where 6 is the angle between the axis of a unit and the cone axis. 

Simplifying assumptions a highly oriented polymer has the simplest form of uniaxial distribution and is fully oriented, there is no correction for reflection at the surface of the sample, no internal field corrections and no scattering of radiation. 

The plot is as shown, with for set A and O for set B. The orientation distribution of set A corresponds to uniaxial distribution of chain axes towards the OXj axis with no specific preference for chains to lie in the plane of the film. The orientation distribution of set B corresponds to uniaxial orientation away from the OX2... 

In polarized fluorescence measurements, the sample is Uluminated with polarized light, and the emitted intensity is measured for polarization parallel and perpendicular to the excitation radiation. The experiment yields both f2((j)) and the 4 Legendre polynomial,/4( ) = (35cos - 30cos + 3)/8. The latter is a more strongly decreasing function of (j) than is/2( ), so that it weights more the more oriented end of the orientation distribution function. For uniaxial distributions, these two functions alone are adequate to describe the orientation [202]. 

The principal components of the real part of the refractive index, n, can also be expressed in terms of the order parameters. General formulas for the ordinary and extraordinary refractive indices have been developed that depend upon the molecular orientation in the film [585]. For an ensemble of uniaxially distributed hydrocarbon chains, the principal components are... 

It was therefore with some confidence in the infra-red and Raman spectroscopic methods that a much more complex investigation was carried out on the molecular orientation in one-way drawn PET films which show uniplanar axial orientation 5). In such films the condition of fibre symmetry is removed in two ways (1). There is no longer uniaxial symmetry of the distribution of chain axes. 

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]... 

Polarized Raman and fluorescence spectroscopies, NMR and X-ray diffraction allow the determination of at least (P2) and (P4) for uniaxial systems. This is a great advantage since the shape of the orientation distribution can then be estimated [7], even if not all the coefficients of the ODF s expansion are known. While P2 has fixed boundary limits, those of (P4) depend on the (P2) value such as... 

The multidimensional chord distribution function (CDF) for oriented materials (in particular useful for the study of materials with uniaxial orientation, i.e., fibers) (Sect. 8.5.5)... 

Of highest practical relevance is the case in which the scattering pattern, the structural entities, and even the orientation distribution g(uniaxial symmetry (F3-materials). If the structure is ruled by polydispersity and the material is uniaxially oriented, F3 is most frequently fulfilled. In this case the mathematical relations are considerably simplified. Suitably the orientation distribution is normal-ized... 

Sampling of x-ray diffraction does not occur with uniaxially oriented (but non-crystalline) specimens which only produce continuous intensity distribution on layer lines. Information on intermolecular association cannot be inferred in these cases. 

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