Fiber patterns

It is a 2D slice of the fiber pattern. Only fiber symmetry makes that it is completely represented by a curve as a function of a transversal (cf. Bonart [16]) coordinate... 

Spinning a crystal during measurement of WAXS patterns is an old method that turns any scattering pattern into a fiber pattern. The rotational axis becomes the principal axis. Thereafter isotropization of the scattering data is simplified because the mathematical treatment can resort to fiber symmetry of the measured data. In the literature the method is addressed as the rotating-crystal method or oscillating-crystal method. 

Isotropization in the Case of Fiber Symmetry. If methods for the analysis of isotropic data shall be applied to scattering patterns with uniaxial orientation, the corresponding isotropic intensity must be computed. By carrying out this integration (the solid-angle average in reciprocal space) the information content of the fiber pattern is reduced. One should consider to apply an analysis of the longitudinal and the transversal structure (cf. Sect. 8.4.3). 

Particularly useful in materials science is a special ID projection the projection of a fiber pattern on the fiber axis, S3... 

In order to demonstrate completeness of a SAXS fiber pattern in the 3D reciprocal space, it is visualized in Fig. 8.16. The sketch shows a recorded 2D SAXS fiber pattern and how it, in fact, fills the reciprocal space by rotation about the fiber axis. V3. Let us demonstrate the projection of Eq. (8.56) in the sketch. It is equivalent to, first, integrating horizontal planes in Fig. 8.16 and, second, plotting the computed number at the point where each plane intersects the S3-axis. 

Figure 9.6. Fanning-out of an equatorial streak in a fiber pattern caused from misorienta-tion. Dashed arcs indicate azimuthal scans that are performed in practical measurements. The recorded scattering curves are used to separate the effects of misorientation and extension of the structural entities... 

If the intended evaluation can be carried out on isotropic material, and thus the observed anisotropy is rather an obstacle than an advantage, the fiber pattern can be isotropized (cf. Sect. 8.4.2). This may, in particular, be helpful if lamellar structures are analyzed. If the focus of the study is on the anisotropic structure, the multidimensional CDF (cf. Sect. 8.5.5) may be a suitable tool for analysis. Several studies have demonstrated the power of the CDF method for the study of structure evolution during straining [174,177,181-183],... 

Figure 10.2. Plot of the positions (x,y) of peak maxima extracted on radial rays in a moderately oriented SAXS fiber pattern according to BRANDT RULAND [265], A microscopical draw ratio Xj = 1.41 of the structural entities is determined from the slope... 

Compression molded (150°C for 3 minutes press chilled with cold water immediately thereafter) samples of poly(trans-l,4-hexadiene) (14) and poly(5-methyl-l,4-hexadiene) were examined with a General Electric (XRD-3) X-ray unit. Transmission Laue X-ray photographs were taken using nickel filtered copper X-radiation. Samples were stretched to four times of their original lengths to obtain oriented fibers. The fiber patterns were obtained in a flat plate film holder with the specimen to film distance standardized at 5 centimeters. X-ray diffraction patterns were similarly obtained for the hydrogenated sample of poly(5-methyl-l,4-hexadiene). 

To clarify the tacticity problem, trans-l,4-hexadiene and 5-methyl-l,4-hexadiene polymers were examined by X-ray diffraction. Fiber diagrams were obtained from samples stretched to four times their original lengths. Eight reflections from the poly(trans-1,4-hexadiene) fiber pattern may be interpreted on the0basis of a pseudo-orthorhombic unit cell with a = 20.81 + 0.05 A b =... 

The poly(5-fnethyl-l, 4-hexadiene) fiber pattern (Figure 6) gave an identity period of 6.3 A, indicating a 3 isotactic helix structure. The X-ray diffraction pattern was not very sharp, which may be due to the difficulty of the side chain with a double bond to fit in a crystalline lattice. The crystallinity was determined to be 15% using the Hermans and Weidinger method (27). A Chloroform-soluble fraction free from catalyst residues showed no improvement in the sharpness of the X-ray diffraction pattern. These data show that the configuration of the 1,2-polymerization units in the homopolymer of 5-methyl-1,4-hexadiene is isotactic. 

Although it has been found that the separated amylose component can be readily orientated to yield fiber patterns, amylopectin usually gives poor or amorphous patterns. In the granule, however, amylopectin does exhibit crystallinity, since waxy maize starch gives a diffraction pattern and other waxy starches behave similarly.193 -195 (This suggests that the branch points in the amylopectin molecule may be in the amorphous part of the granule.)... 

While a polar-zipper model was initially proposed for Asp2Glni3Lys2 (Perutz et ah, 1994), more recently a water-filled nanotube was proposed in which the homopolymer in the / -conformation forms a helical array having 20 residues per turn (Perutz et ah, 2002a,b). (In the earlier work, the diffraction pattern had been interpreted as a fiber pattern—that is, with the 4.8-A reflection in the fibril direction.) Rather than being attributed to intersheet stacking, the 8.4-A reflection was not accounted for. Further,... 

Therefore, uniaxially oriented samples should be prepared for this purpose, which give so-called fiber pattern in X-ray diffraction. The diffraction intensities from the PPX specimen of P-form, which had been elongated 6 times at 285°C, were measured by an ordinary photographic method. The reflections were indexed on the basis of the lattice constants a=ft=2.052nm, c(chain axis)=0.655nm, a=P=90°, and y=120°. Inseparable reflections were used in the lump in the computation by the least square method. 

Later on we succeeded in partially aligning microcrystals within the sample they were placed in X-ray capillaries and pulled down by gently sucking their mother liquor, whereupon the needle-shaped crystals oriented along the capillary axis. Such samples sometimes, but only under perfect measuring conditions, produced pseudo fiber patterns which consisted of oriented arcs with average length of 60° (Fig. lb). In many cases these arcs were comj sed of distinct spots that could be clearly resolved by eye. Since such patterns may arise from partial orientation of the nucleic acid component within fairly well packed particles, these patterns indicated reasonable internal order. 

For a fiber pattern, lines are marked through the center of the pattern parallel and perpendicular to the machine direction [3]. The line parallel to the machine direction is the meridian and the other line is the equator. The spots fall on layer lines that are roughly parallel to the equator. The equator is designated the = 0 layer line whereas the layer line on the other side is = 1 [3]. 

Fig. 57a. A typical multilayered cloth made cf potassium tartaric amide 31b at pH 5. Its observed physical shape is fortuitous. (Negative stain, uranyl acetate 1%). b Freeze-etching erf a similar multilayer made of the sodium salt 31a. At higher magnification (below), the bilayer profiles become visible (Pt/C shadowed), c Fiber pattern of 31a. (Negative stain, uranyl acetate 1%). d Digitized area of the fiber bundle taken from (c). e Fourier transform of the input image from (d), as obtained by calculating the reciprocal space frequencies (x-y exchanged). Two intense spots yield a periodical pattern of 38.78 A [376]...

The structural anisotropy in crystalline or structurally ordered BPDA-PFMB films was studied in this laboratory with wide-angle X-ray diffraction (WAXD) methods. In brief, WAXD experiments were designed to examine both the reflection and transmission modes of thin-fihn samples. In addition, uniaxially oriented polyimide fiber WAXD patterns were obtained to aid in the identification of the film structure. The film WAXD pattern obtained from the reflection mode corresponded well to the fiber pattern scanned along the equatorial direction (Figure 16.3), " which indicates that the reflection mode pattern represents the (hkQ) diffractions. On the other hand, as shown in Figure 16.4, the (001) diffractions were predominant in the film WAXD pattern obtained via the transmission mode. This pattern corresponded to the fiber pattern scanned along the meridian direction. These experimental observations clearly indicate that the c-axes of the crystals are preferentially oriented parallel to the film surface however, within the film, they are randomly oriented. 4.2 5 j( should be pointed out that the WAXD experiments are only sensitive to crystalline or ordered structures in polyimide films. They do not provide any information on the amorphous regions. 

Figure 16.3. Comparison of BPDA-PFMB WAXD film patterns obtained with a reflection mode and fiber patterns scanned along the equatorial direction.

Figure 16.6 BPDA-PFMB WAXD fiber patterns for fibers with different draw ratios (reduced 15% for reproduction).

To calculate the WAXS fiber pattern of PTFE phase I, we assumed that the CF2 units are equally distributed around the chain axis due to the high mobility of twin reversals in the high-temperature phase. In other words, the electron density of the PTFE chains is cylindrically symmetric in space average. Thus we are able to calculate the mean molecular structure amplitude according to Eq. (26). 

Fig. 24. WAXS fiber pattern of a hexagonal chain arrangement consisting of 10 x 12 PTFE stems of length 40 A, assuming perfect orientation. The sharp spots are the consequence of finite model size...

The chain conformation it was treated mathematically in terms of atoms regularly spaced along helices, an approach equivalent to defining the contents of the unit cell in terms of cylindrical coordinates. Packing of the chains was characterized by defining the distances between molecules as determined from the equatorial reflections on an x-ray fiber pattern. j ... 

The determination of crystal structure in synthetic polymers is often made difficult by the lack of resolution in the diffraction data. The diffuseness of the reflections observed in most x-ray fiber patterns results from the small size and imperfect lattice nature of the polymer crystallites. Resolution of individual reflections is also made difficult from misorientation of the crystallites about the fiber axis. This lack of resolution leads to poor accuracy in measurement of peak positions. In particular, this lack of accuracy makes determination of layer line heights difficult with a corresponding loss of significant figures in evaluation of the repeat distance for the molecular conformation. In the case of helical conformations, the repeat distance may be of considerable length or, as we shall show, indeterminate and, in effect, nonperiodic. This evaluation requires high accuracy in measurements of layer line heights. 

In this paper we examine electron diffraction fiber patterns of the homopolymer polytetrafluoroethylene (-CF2 CF2-)n PTFE, in which the resolution is sufficient to yield much more accurate values of layer line heights than were available from the previous x-ray diffraction experiments (1) on the crystal structure of Phase II, the phase below the 19°C transition (2). On the basis of x-ray data, the molecule was assigned the conformation 13/6 or thirteen CF2 motifs regularly spaced along six turns of the helix. This is equivalent to a 132 screw axis. The relationship between the molecular conformation and the helical symmetry has been studied by Clark and Muus (3) and is illustrated in Figure 1. The electron diffraction data of high resolution enabled us to determine if this unusual 13-fold symmetry was exact or an approximation of the true symmetry. We have also... 

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