Orientation distribution characterising

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

The presented derivations of the load rate and the lifetime relationships applying the shear failure criterion are based on a single orientation angle for the characterisation of the orientation distribution. Therefore these relations give only an approximation of the lifetime of polymer fibres. Yet, they demonstrate quite accurately the effect of the intrinsic structural parameters on the time and the temperature dependence of the fibre strength. 

Nematic phases are characterised by a uniaxial symmetry of the molecular orientation distribution function f(6), describing the probability density of finding a rod with its orientation between 6 and 6 + d0 around a preferred direction, called the director n (see Fig. 15.49). An important characteristic of the nematic phase is the order parameter (P2), also called the Hermans orientation function (see also the discussion of oriented fibres in Sect. 13.6) ... 

The microstructure of all mouldings was characterised by determining fibre content (weight and volume fraction), average fibre dimensions (diameter and length), fibre length distribution and fibre orientation distribution. 

The 3D orientation distributions of structural features within damage zones are important for modelling of fluid flow as the non-parallel members of the arrays induce an intersection network which will control the connectivity of barriers. It is insufficient to characterise the average fault orientations or to identify average fault trends or families. A more detailed statistical analysis of fault orientations is needed in order to evaluate the 3D distribution of flow paths and barriers. An analysis of fault dips from North Sea wells (-20 km of total core studied) yields an average dip of 59°. However the standard deviations of these data sets, which will control the density and pattern of intersections, is typically between 15° and 26°. Fig. 15 illustrates one example which com... 

Because the angle 0 refers to the absorption/emission axis of the fluorescent molecule, the method does not characterise the orientation of the polymer molecules directly when probe molecules are used rather than fluorescence of the polymer itself. Some relationship between the orientations of the probes and the polymer chains must then be assumed if information about the orientation distribution of the chains is to be deduced. 

Although/is a convenient index to characterise orientation, one must note that it is only an average value—specifically the second moment of the whole orientation distribution for the specific set of (hkl) planes utilised. Hence, two different orientation distributions could exist having... 

To see the nature of the orientational order parameters needed to characterise the biaxial nematic we start with the singlet orientational distribution function. Since the molecules are taken to be rigid the distribution is a function of the three Euler angles, a 3y, which we denote by Q connecting the director... 

For smectic phases the defining characteristic is their layer structure with its one dimensional translational order parallel to the layer normal. At the single molecule level this order is completely defined by the singlet translational distribution function, p(z), which gives the probability of finding a molecule with its centre of mass at a distance, z, from the centre of one of the layers irrespective of its orientation [19]. Just as we have seen for the orientational order it is more convenient to characterise the translational order in terms of translational order parameters t which are the averages of the Chebychev polynomials, T (cos 2nzld)-, for example... 

The two singlet distribution functions are not in themselves sufficient to characterise the order in a smectic A phase because there is, in general, a correlation between the position of a molecule in a smectic layer and its orientation. We need, therefore, the mixed singlet distribution function P(z,cos ) which gives the probability of finding a particle at position z and at an orientation P with respect to the director [18,19]. At the level of description provided by the order parameters it is necessary to introduce the mixed order parameter... 

We turn now to the orientational correlations which are of particular relevance for liquid crystals that is involving the orientations of the molecules with each other, with the vector joining them and with the director [17, 28]. In principal they can be characterised by a pair distribution function but in view of the large number of orientational coordinates the representation of the multi-dimensional distribution can be rather difficult. An alternative is to use distance dependent orientational correlation coefficients which are related to the coefficients in an expansion of the distribution function in an appropriate basis set [17, 28]. 

In a system of nanoparticles, thermal fluctuations of their magnetic moments severely reduce the anisotropy of the resonance magnetic field, resulting in superparamagnetic spectra narrowing. This reduction is the more pronounced the smaller is the particle size. Therefore, the SPR spectra of macroscopically isotropic nanoparticle systems characterised by a distribution in size usually maintain a distinct shape asymmetry characteristic of powder patterns of randomly oriented anisotropic particles. From an inspection of such spectra, one can conclude that the angular dependence of the resonance magnetic field of individual particles is not completely reduced. 

In the model, the uniform contribution (and thus, the doublet splitting) is proportional to the overall average orientation . The interaction parameter u characterises the strength of orientational interactions between segments (0 < u < 1). Thus, for a given deformation ratio X, the spectrum contains one constant splitting and a distribution of additional shifts, which is clearly seen in Figure 15.4. 

X-ray diffraction (in crystalline polymers) Unoriented crystalline polymers show X-ray diffraction patterns, which resemble powder diagrams of low-molecular crystals, characterised by diffraction rings rather than by spots. As a result of orientation the rings contract into arcs and spots. From the azimuthal distribution of the intensity in the arcs the degree of orientation of the crystalline regions can be calculated (Kratky, 1941). 

Compared to crystalline materials, the production and handling of amorphous substances are subject to serious complexities. Whereas the formation of crystalline materials can be described in terms of the phase rule, and solid-solid transformations (polymorphism) are well characterised in terms of pressure and temperature, this is not the case for glassy preparations that, in terms of phase behaviour, are classified as unstable . Their apparent stability derives from their very slow relaxations towards equilibrium states. Furthermore, where crystal structures are described by atomic or ionic coordinates in space, that which is not possible for amorphous materials, by definition, lack long-range order. Structurally, therefore, positions and orientations of molecules in a glass can only be described in terms of atomic or molecular distribution functions, which change over time the rates of such changes are defined by time correlation functions (relaxation times). 

The discussion of the most general form of biaxial orientation involves fairly complicated mathematics. Further discussion in this section and in the sections on the characterisation of orientation is therefore largely restricted to a special simple type of distribution of orientations of the structural units. This distribution is the simplest type of uniaxial orientation, for which the following conditions apply. 

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. 

The second type of explanation for finding values of R less than 3 involves the assumption that the emission and absorption axes of the fluorescent molecule are not coincident. Kimura et al. have considered a model in which the absorption and emission axes each have, independently, a cylindrically symmetric distribution of orientations around a third unique axis in the molecule, and Nobbs et al. have considered a model which includes both this and the possibility that there is a fixed angle between the emission and absorption axes which are otherwise uniformly distributed around a third unique axis. In the more general model at least three parameters are required to specify the relationships between the directions of the emission and absorption axes and that of the unique axis of a fluorescent molecule and these are not generally known. For orthotropic symmetry, five v, are required to characterise the distribution of orientations of the unique axes and if the constant NqIo is included, there is a total of at least nine unknown quantities. No attempt has so far been made to evaluate these from intensity measurements on an orthotropic sample. For a uniaxial sample only two parameters, cos O and cos O, are required for characterising the distribution of orientations and by making various approximations the total number of unknown quantities can be reduced to six. Their evaluation then becomes a practical possibility. 

The choice of fluorescent probe depends on a variety of factors. It has already been pointed out that what is determined directly is information which characterises the distribution of orientations of the fluorescent molecules. The ideal experiment would be one in which the polymer molecules themselves contained fluorescent groups. Stein has considered the theory of the fluorescence method specifically for a uniaxially oriented fluorescent rubber but no experiments to study orientation have been reported for such a system. Nishijima et al have, however, made some qualitative observations on the polarisation of the fluorescent light from polyvinylchloride films which had been first stretched and then irradiated with light of wavelength 185 nm to produce fluorescent polyene segments. 

Polar plots of /, Ix and p were made and were compared with those predicted on the basis of simple models for possible distributions of orientations in an attempt to characterise the distribution in terms of these models. Figure 3 shows some polar plots of In for polyvinyl alcohol. 

If values of cos 0p and cos 0p are required, the relationship between the two distributions of orientations niust be known. This can only be established initially by comparison of the fluorescence results with those of other methods which give direct information about cos 0p and cos 0p, such as infra-red and Raman spectroscopy, and studies along these lines are in progress (see Section 5.3.2). If the relationship can be satisfactorily established the fluorescence method could become a much simpler method of quantitatively characterising molecular orientation in amorphous polymers than any of the other methods that offer information about both cos 0p and cos 0p. 

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