Uniaxial orientation parameter

Uniaxial orientation parameter (Hermans orientation function) Scattering data evaluation program by A. Hammersley (ESRF) Free Electron Laser Hamburg Full width at half-maximum... 

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... 

Figure 9.4. The orientation of structural entities (rods) in space with respect to the (vertical) principal axis and the values of for, the uniaxial orientation parameter (Hermans orientation function) for (a) fiber orientation, (b) isotropy, (c) film orientation...

As are the other multipole-expansion coefficients, the uniaxial orientation parameter is computed from Eq. (9.6). For materials with fiber symmetry the relation simplifies4 and... 

The uniaxial orientation parameter related to the orientation of polymer chains gains particular importance, because it can also be determined by measurement of birefringence [250,252],... 

Normal incidence transmission IRLD measurements are used to study thin films (typically 100 pm thickness and less, depending on the molar extinction coefficient of the bands) with in-plane uniaxial orientation. Two spectra are recorded sequentially with the radiation polarized parallel (p) and perpendicular (s) to the principal (machine) direction of the sample. The order parameter of the transition moment of the studied vibration is calculated from either the dichroic ratio (R — Ap/As) or the dichroic difference (AA = Ap—As) as ... 

Specular reflection IR spectroscopy has been used by Cole and coworkers to study the orientation and structure in PET films [36,37]. It has allowed characterizing directly very highly absorbing bands in thick samples, in particular the carbonyl band that can show saturation in transmission spectra for thickness as low as 2 pm. The orientation of different conformers could be determined independently. Specular reflection is normally limited to uniaxial samples because the near-normal incident light does not allow measuring Ay. However, it was shown that the orientation parameter along the ND can be indirectly determined for PET by using the ratio of specifically selected bands [38]. This approach was applied to the study of biaxially oriented PET bottles [39]. 

An approach simpler than the "general" method has been used by Bower and coworkers [52,53] and has proved to be useful to determine the order parameters of uniaxially oriented polymers. This approach, called "cylindrical" method, requires a smaller number of spectra that are easily measurable, but it assumes that the Raman tensor associated with the vibrational mode under study is cylindrical. In such case, at — a2 — a so that the depolarization ratio R so of an... 

The orientation dependent parameter p defined by Eq. (11) becomes unity in the isotropic state, and decreases as the polymers are uniaxially oriented. Therefore, it follows from Eqs. (9) and (10) that the wormlike hard spherocylinder system has a smaller translational entropy loss from the ideal solution in the liquid crystal state than in the isotropic state. This difference drives the system to form a liquid crystal phase. However, in order to determine the equilibrium orientation of the system, the orientation dependence of Sor has to be formulated, and this is done in Sect. 2.3. 

Hereafter, taking into account the axial symmetry of the problem that implies m = —m [see Eq. (4.333)], we will use only one upper index in the notation of the coefficient (4.334) Qfv. Further on, however, when expanding Q with respect to the probing field amplitude we will have to indicate the order of the approximation by an additional upper index. To avoid confusion, this index will be kept inside curled braces [e.g. <2 y2 ]- In a reduced form, namely, at l = m = m = 0, the parameters Q render customary uniaxial orientational order parameters of the particle axes in a suspension (magnetic fluid). In this case their meaning is illustrated by... 

Ro is given by 2 cot a, where a is the angle between the dipole moment vector and the chain axis. (P2 cos0)) is the one-parameter measure of orientation normally given in the literature for uniaxially oriented samples, a is a function of the vibration in question and is known for a number of vibrations useful in studying thermoplastic elastomers (Myers and Cooper, 1994). 

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... 

The molecular orientation of the epoxy networks was investigated using rheooptical FTIR spectroscopy and uniaxial deformation was carried out above and below the Tg. The effects of diol chain length and molar ratio on the mechanical properties and orientation parameter were discussed (79). 

At that stage a uniaxial, orientational order parameter S is introduced in terms of the mean square projections of molecular vector a ... 

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