Orientation functions

Noncrystalline domains in fibers are not stmctureless, but the stmctural organization of the polymer chains or chain segments is difficult to evaluate, just as it is difficult to evaluate the stmcture of Hquids. No direct methods are available, but various combinations of physicochemical methods such as x-ray diffraction, birefringence, density, mechanical response, and thermal behavior, have been used to deduce physical quantities that can be used to describe the stmcture of the noncrystalline domains. Among these quantities are the amorphous orientation function and the amorphous density, which can be related to some of the important physical properties of fibers. 

Me(A) increases between entanglement points due to the retraction process at constant X. A more detailed treatment of disentanglement would account for the orientation function of the entanglements and lateral contraction, as discussed elsewhere [1]. Eq. 8.4 becomes... 

Plugging, corrosion, physical damage, correct bolt torque, contaminants, correct arrester orientation, functioning instrumentation... 

Table 5 Crystallites Orientation Function (fc) of Differently Drawn PET Fibers...

Draw ratio Density of the amorphous material da) (g/cm-" ) Amorphous orientation function fa) Crystallite length Oc) (nm) Long period (L) (nm) Degree of crystallinity (X=>) Substructure parameter (A) Axial elastic modulus ... 

Fibers of a diversified draw ratio in the range 2.0-5.2 X were considered, determining the following parameters of their fine structure the crystalline and amorphous orientation functions,and/a, degree of crystallinity, X, and critical dissolution time (CDS) in seconds. The results obtained are listed in Table 11. 

Annealing temperature rc) Annealing time (min) Birefringence (An) Anid Volume crystallinity (%) TTM fraction Critical dissolve time (s) Amorphous orientation function (/ )... 

The low electrical conductivity of PET fibers depends essentially on their chemical constituency, but also to the same extent on the fiber s fine structure. In one study [58], an attempt was made to elucidate the influence of some basic fine structure parameters on the electrical resistivity of PET fibers. The influence of crystallinity (jc) the average lateral crystallite size (A), the mean long period (L), and the overall orientation function (fo) have been considered. The results obtained are presented in the form of plots in Figs. 9-12. 

Information on molecular orientation can be useful in two primary ways. First, it is possible to use the orientation functions or averages to gain an understanding of the mechanisms of plastic deformation. Secondly the orientation averages can provide a basis for understanding the influence of molecular orientation on physical properties, especially mechanical properties. 

In this review the definition of orientation and orientation functions or orientation averages will be considered in detail. This will be followed by a comprehensive account of the information which can be obtained by three spectroscopic techniques, infra-red and Raman spectroscopy and broad line nuclear magnetic resonance. The use of polarized fluorescence will not be discussed here, but is the subject of a contemporary review article by the author and J. H. Nobbs 1. The present review will be completed by consideration of the information which has been obtained on the development of molecular orientation in polyethylene terephthalate and poly(tetramethylene terephthalate) where there are also clearly defined changes in the conformation of the molecule. In this paper, particular attention will be given to the characterization of biaxially oriented films. Previous reviews of this subject have been given by the author and his colleagues, but have been concerned with discussion of results for uniaxially oriented systems only2,3). 

Fig. 10. Comparison of the measured refractive indices for PET film with values calculated from the orientation functions determined from n.m.r. assuming transversely isotropic structural units. I, Experimental points predicted values. Reproduced from Polymer by permission of the publishers, Butterworth Co (Publishers) Ltd. (C)...

Additional software has been developed to merge data from various data collection steps and to model the data using suitable statistical distribution functions. We are working on software to perform corrections for absorption, specimen shape, and misalignment. Library routines for 2-diraensional data smoothing and integration are being adapted to the calculation of orientation functions and other moments of the probability distributions. 

Most interpretations of orientation texture data are based on the assumption that the specimen is a sphere. A spherical specimen shape is not easily achieved for the kinds of materials we are dealing with and departures from sphericity interfere with the calculation of orientation functions (8). We expect to address this problem by calculating empirical correction functions that can be applied to specimen shapes that closely simulate the ones we use. This problem must be attacked using numerical integration in 3 dimensions, which is a computer-intensive activity. 

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

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...

In the deduction Ruland determines, which contribution to the observed intensity, /, is added by each reflection ring of the likewise fiber symmetrical function, Iopt. Then he adds up all the rings weighted by the orientation function g 3). In this way Eq. (9.11) is simplified. A general solution is obtainable by multipole expansion. 

Pathway to the Solution. All orientation functions are defined on the orientation sphere. At the best, their period is n. Thus the mapping of an analytical function h (x) on the orientation sphere is equivalent to... 

Figure 9.9. Infinite functions in a periodic world. Using a function (black line) as an orientation function means to wrap it around the orientation sphere. Only one branch of a LORENTZ distribution at the right side of the equator is sketched. Shape change occurs... 

Fig. 12 The meridianal orientation function p(0) for a Gaussian distribution, the distribution function p(0)sin0,the shear strain energy function tan20sin0p(0) and the function tan20...

The polymer fibre is considered to be composed of a parallel array of identical fibrils. A single fibril is a series arrangement of domains with a varying angle to the fibre axis described by the orientation function p(Q) in the unloaded state. The strength of a domain is supposed to be given by... 

A trademark of amyloid fibrils is their cross-/ structure. This structure is the basis of the repetitive hydrogen-bonding extension of the fibril (Makin et al., 2005). Cross-/ structures are observed in the silk fibers of some insects (Geddes et al., 1968 Hepburn et al., 1979), although none are observed in spiders or lepidoptera (Craig, 1997). This absence has been explained by the possibility that cross-/ silks or a-silks may be converted into collinear /1-silks by stretching the fiber and an increased orientation-function correlated to the speed at which silk is formed (Riekel et al., 2000). 

Figure 11.16 PTT WAXD pattern and indices of the reflections [45]. From Polym. Bull., Crystal orientation function of poly(trimethylene terephthalate) by wide-angle X-ray diffraction, Chuah, H. H. and Chang, B. T. A., 46, p. 310, Figure 2, Copyright Springer-Verlag (2001). Reproduced by permission of Springer-Verlag GmbH Co. KG...

In the above, the variable R is the radius between center to center fiber spacing, while r is the fiber radius. The shear modulus (Gm) can be approximated as Em/3. The matrix modulus is effected by the level of crystallinity and it is important that the samples are fully crystallized to ensure reproducibility. The value of (> for 30wt% glass-fiber-reinforced PET has been calculated as 3.15 x 104. Using the mathematical analysis shown above, the orientation function of the glass fiber... 

The orientation functions (F-values) of as-spun fibers were calculated according to the following equation ... 

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