Orientation parameters

3 Nielsen Modification of Halpin-Tsai Equations with Respect to the Maximum Packing Fraction  

Flexural modulus (GPa) for short glass fibers filled PP 

max fiber volume ratio range Average data various suppliers 

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Comparing the alteration in a values with the established changes in fine structure parameters presented in Table 12, it can be seen that the alteration in a values does not correlate with variations in any particular structure parameter. This leads to the conclusion that the ascertained alternation in a must be evoked by a complex change in different substructure, microstructure, and orientation parameters of the fiber. 

The deflection tool orientation parameters can be obtained using the vectorial method of D. Ragland, the Ouija Board , or the three-dimensional mathematical deflecting model. 

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]. 

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...

If the principal axes of the structural entities are all oriented in a plane perpendicular to the principal axis of the material (film orientation), the orientation parameter returns for = —0.5. 

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... 

In practice, either a pole figure has been measured in a texture-goniometer setup, or a 2D SAXS pattern with fiber symmetry has been recorded. In the first case we take the measured intensity g (pole figure. In the second case we can choose a reflection that is smeared on spherical arcs and project in radial direction over the range of the reflection. From the measured or extracted intensities I (numerical integration and normalization... 

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],... 

If the observed reflections are not on spherical arcs, the computation of an orientation parameter becomes an arbitrary operation that is not exclusively related to misorientation of structure. Most probably the topology of the structural entities is coupled to their orientation6, and Chap. 10 applies. 

Figures 26 and 27 present the modulus and strength as a function of the orientation parameter sin2 for the PpPTA, PBO and PIPD-HT fibres, assuming a fibre with a single orientation angle. As the precise value of g for the PBO fibre is not known, we have taken the same value as for PpPTA. This enables us to demonstrate the effect of the chain modulus on the modulus and the strength of the fibres, in particular at medium values of the orientation parameter and for highly oriented fibres. For example, at an orientation parameter value of sin2 =0.028 the modulus and strength for a PpPTA fibre are =84 and crb=3.9 GPa, for PBO =104 and ab=5.2 GPa, and for PIPD-HT =241 and crb=8.6 GPa.

Fig. 26 The modulus as a function of the initial orientation parameter for PIPD-HT, PBO and PpPTA fibres...

For crb 0 the lifetime fb °°,so Eq. 115 presents a non-linear relation between log( b) and the creep stress crb, which is different from the Coleman relation. According to Eq. 115, at constant load the lifetime of a fibre decreases with increasing orientation parameter. Figure 61 compares the observed data for a PpPTA fibre by Wu et al. with the calculated lifetime curve using the parameter values /J=0.08, tan =0.1483, g= 1.6 GPa, j O.032 (GPa)-1, which implies a fibre with a sonic modulus of 91.8 GPa [30]. As shown by Wu et al., fibres that were tested at high stresses had shorter lifetimes than those calculated from the experimental lifetime relation. 

According to the simple Eq. 115 and the full Eq. 140, the lifetime of a fibre measured at a constant load decreases with increasing orientation parameter. The dependence of the slope of the curve, log( b) vs ob, on the initial orientation distribution has been calculated for PpPTA fibres using Eq. 140. Figure 70 shows that at constant load for increasing orientation angle the lifetime curves become steeper, while at the same time the lifetime decreases. This effect has been observed for nylon 66 yarns as shown in Fig. 71, where the lifetime data... 

Note 3 For mesophases eomposed of cylindrically symmetric molecules there is a precise relationship between the magnetic anisotropy, A, and the second-rank orientational parameter P ). 

The IA phase coexistence equations could not be solved in a low E, region for the aqueous schizophyllan system, where the degree of orientation parameter a2 of the lower molecular weight polymer component 2 becomes very small. This is mainly due to poor approximations in the asymptotic expansions of p , and a,(N) at small ot2 values. The broken binodals in Panel b of Fig. 9 are not calculated results but interpolation curves. 

The order parameter S is the orientational average of the second-order Legendre polynomial P2(a n) (n = the director), and if the orientational distribution function is approximated by the Onsager trial function, it can be related to the degree of orientation parameter ot by... 

The fibrillar structure of crystalline polymers is determined by molecular characteristics, the initial morphology and orientation conditions. Recently, a complex investigation of the effect of molecular parameters (MW, MWD and degree of branching) and orientation parameters (temperature and draw ratio) on the morphology of PE and its thermomechanical behaviour has been reported 181 185). 

Figure 41. Distance and orientational parameters associated with the efficiency of y-hydrogen abstraction by a carbonyl oxygen atom. (Copied with permission [282a].)...

Fig. 17a and b. Temperature dependence of the orientational parameters of spacer Qsp (1) and mesogenic groups Qmes (2) for polymers IX (a) and X (6). The arrows indicate the transition temperatures between the different smectic phases... 

In the case of the ethane (45) and butane (46) monolayers adsorbed on graphite, it has been possible to analyze the neutron diffraction patterns using all three Euler angles of the molecule as orientational parameters. Here we limit discussion to the butane monolayer which we have taken as a model system and whose vibrational spectrum was discussed in Sec. II. 

Adjustments in other factors must also be made. These may include the lattice parameters, a 20-zero correction parameter, and a preferred orientation parameter. Principal among the non-structural parameters to be adjusted are (1) any adjustable parameters in the expression used for y and (2) the parameters in G(04-0h). G(0.,--0h) is itself a convolution of the instrumental... 

The whole-pattern-fitting structure-refinement method is not limited to specimens containing randomly oriented crystallites. The method does permit an orientation parameter to be refined, though it was not done in this work. Crystallites in the molded polypropylene specimen probably were oriented to some small extent nevertheless, the refinement method worked well. The sensitivity of the method to crystallite orientation and the applicability of the method to polymers with oriented crystallites such as in drawn films or fibers are yet to be determined. 

The isolated 5 helices were free of any objectionable contacts. Since there is no special relationship between two chains in the 5-fold double helix, the relative orientation parameters, Ay and AW, were obtained from the contacts map for each conformation. For both 5, and 5, double helices, conformations without fatal short contacts could be constructed. These double helices were subjected to a packing analysis in the... 

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