SAXS patterns

The interpretation of this function is straightforward, since the CDF has been defined [16] by the Laplacian of Vonk s multidimensional correlation function [19]. For this reason, it presents the autocorrelation in space of the surfaces from the domains in a similar manner as Ruland s interface distribution function does [20-22] for one-dimensional (ID) structures as a function of distance. For samples with uniaxial symmetry, the CDF z ri2,rz) is a function of two coordinates only (transverse direction and draw direction r ). Therefore, it can be displayed by means of contours or density plots in a plane. Positive peaks found in the vicinity of the origin are size distributions of the primary domains. In this way, their size, shape, and orientation in space are depicted. The farther negative peaks exhibit long periods , i.e., the distance between two adjacent domains. The next positive peaks describe the size and orientation of super domains i.e., clusters made from two adjacent domains), and correlations among more distant domains are manifested by consecutive peaks at even longer distances. 

The contributions to the value of the CDF are arising from correlations between domain surfaces. For instance, a cylindrical domain is characterized by two sharp peaks on the meridian, their distance from the origin denoting the cylinder height. In the equatorial direction, the peak is not sharp, falls off almost linearly, and becomes zero at the diameter of the cylinder. 

The CDF interpretation results in a detailed, yet only qualitative description of the complex nanostructure at each stage of the straining experiment. Several features are frequently superimposed, and the quantitative analysis requires a complex three-dimensional (3D) adapted model to be fitted to the CDF in order to retrieve precise data concerning the nanostructure evolution. For this reason, it may be reasonable to quantitatively study only a partial aspect of the nanostructure topology [23]. 

Considering TPEs under uniaxial load, Bonart [23] has proposed the study of two aspects of the nanostructure, called longitudinal and transverse structure. They can readily be extracted from the scattering pattern by projections [24]. In both cases, the result is a curve and, obviously, curves are analyzed with less computational effort than 2D scattering patterns. 

The longitudinal structure evaluates the (chord) lengths of soft and hard domains, only along lines running parallel to the draw direction. Thus a quantitative representation is obtained, describing both the average extensions 

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The SAXS intensity distribution was measured with a rotating anode x-ray generator (Rigaku Denki, Rotaflex, RTP 300 RC) operated at 40 kV and 100 mA. The x-ray source was monochrolmatized to CuK (A = 0.154 nm) radiation. The SAXS patterns were taken with a fine-focused x-ray source using a flat plate camera (Rigaku Denki, RU-lOO). In the measurement of the solution sample, we used a glass capillary (< = 2.0 mm Mark-Rohrchen Ltd.) as a holder vessel. 

N2 adsorption-desorption isotherms revealed that MCs had hi surface area (>1200 m /g) and large pore volume (>1.0 cm /g). From SAXS patterns of the prepared materials, it was confirmed that pores of SBA-15 and CMK-3 retained highly ordered 2-dimensional hexagonal type arrangement [5], while MCM-48 had 3-dimensional cubic type pore structure. It should be noted that a new scattering peak of (110) appeared in the CMK-1 after the removal of MCM-48 template. Furthermore, the pore size of CMK-1 and the wall thickness of MCM-48 were found to be 2.4 nm and 1.3 nm, respectively. This result demonstrates that a systematic transformation of pore structure occurred during the replication process from MCM-48 to CMK-1 [6]. 

Limits of Time-Resolved and Simultaneous Measurements. Structure evolution studies are based on the ability to carry out time-resolved scattering experiments. The power of this scattering technique is a function of the minimum cycle time during which a scattering pattern with sufficient signal-to-noise ratio can be recorded. As cycle times for anisotropic 2D SAXS patterns have fallen below a value... 

Peaks in SAXS patterns rest on a rapidly decaying background. Figure 8.7 shows an example for a typical isotropic bulk semicrystalline polymeric material. The long period of such data should never be determined from the peak maximum found in the... 

The classical treatment of diffuse SAXS (analysis and elimination) is restricted to isotropic scattering. Separation of its components is frequently impossible or resting on additional assumptions. Anyway, curves have to be manipulated one-by-one in a cumbersome procedure. Discussion of diffuse background can sometimes be avoided if investigations are resorting to time-resolved measurements and subsequent discussion of observed variations of SAXS pattern features. A background elimination procedure that does not require user intervention is based on spatial frequency filtering (cf. p. 140). 

Any ID projection of a SAXS pattern contains specific one-dimensional information on the nanostructure70 of the material. Therefore /, (s,) is called the one-dimensional scattering intensity in the direction of s,. 

CDFs are computed from scattering data which are anisotropic and complete in reciprocal space. Thus the minimum requirement is a 2D SAXS pattern of a material with fiber symmetry taken in normal transmission geometry (cf. p. 37, Fig. 4.1). Required pre-evaluation of the image is described in Chap. 7. 

Figure 8.31 shows central sections of two original SAXS patterns of PEE 1000/4394 in strained and relaxed state. In the strained state (Fig. 8.31a) a 6-point-diagram is detected. During relaxation (Fig. 8.31b) a well-separated 4-point-diagram is observed. Interpretation of the patterns is restricted to description and speculation. 

A programming package PRIMUS for the evaluation of isotropic SAXS patterns is offered by Svergun [196], Although the focus is on biopolymers, it can also evaluate general particle scattering. 

Figure 8.36. SAXS pattern of a thermoplastic elastomer during straining. The thin horizontal line in the center is called an equatorial streak. In this case it is well-separated from the long-period peaks above and below... 

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 (orientation parameter by numerical integration and normalization... 

Figure 10.3. Shape of the maximum peak intensity, J((p ), extracted from radial sections of a moderately anisotropic (Xj = 1.41) SAXS pattern with fiber symmetry as a function of the sectioning angle

fiber axis. Dots experimental values. Solid line Theoretical shape according to RULAND [265]... 

A comparison of the calculated relative peak heights using the very simplified electron density profiles of Figs. 23b and 23d with the SAXS pattern of the sample, Fig. 24, unequivocally indicates the presence of the non-centrosysmmetric lamellae structure depicted in Fig. 23a. In agreement with SCMF calculations [90], the centrosymmetric two-way arrangement (Fig. 23c) of A-B-C-A-C-B pattern was not observed. 

Fig. 18. Time-resolved small-angle X-ray scattering patterns from polypropylene sheet under quick stretch in the horizontal direction. A speed of stretch was 233 mm/min (367 % stietch/min). An exposure time for each pattern was 0.1 s. Intervals between exposures were 0.2 s. An X-ray wavelength was 0.155 nm. A slight deformation of the symmetric SAXS pattern was already observed in the second patterns, suggesting some degree of orientation was brought about in quite an early stage. The SAXS patterns changed abruptly and drastically in the sixth pattern just when the sample began to yield (when the tension began to decrease).

Fig. 4 Effect of nanoclay loading on neat SEBS a Lorentz -corrected SAXS profiles (vertically shifted for better clarity) showing effect of nanoclay arrows indicate peak positions, b Lengths corresponding to first- and second- order scattering vector positions along with the 2D SAXS patterns for each sample of clay-loaded nanocomposites...

Figure 4. Thermal evolution of the SAXS patterns of the system C12PO4/ water (30/70 w/w). The lamellar lyotropic phase is indexed doot - 9.83 nm (20°C).

Fig. Z13 SAXS pattern from an PI2PS Y-shaped polymer (details as Fig. 2.12) (Pochan et at. 1996/)). The pattern shows four-fold symmetry from an approximate [100] zone axis of a BCC structure.

Fig. 2.16 Results obtained from an oscillatory shear experiment on an/ps = 0.103 PS-PEP diblock (Okamoto et al. 1994a). (a) SAXS patterns obtained at four representative strain phases as shown in (b) (c) a model showing (110) and (110) planes that give rise to the four diffraction peaks in (a). The pattern at each phase was obtained by integrating over [ A + 0 A + 0,-], where 0 = 0, id2, ittH and A = 0.194tt. Each pattern represents an average over 80 strain cycles.

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