Data analysis, SAXS

When coupled with revolutionary advances in data analysis, whereby a low resolution three-dimensional electron density map may be recovered from the one-dimensional X-ray scattering profile, SAXS has now become a routine technique for characterizing conformational changes in biomolecules (Lipfert and Doniach, 2007 Petoukhov and Svergun, 2007 Putnam et al., 2007). To date such methods have been used to study proteins in solution. Only recently have these methods been applied to the study of RNA molecules in solution (Lipfert et al., 2007a). 

Application of SVD analysis to the time-resolved SAXS data of Russell et al. (2002) clearly indicated two significant singular values, and therefore enabled data analysis by projection of each independent scattering profile onto two states. The folded and unfolded states were selected because of... 

Bimodal or not bimodal—critical comments Is the bimodal distribution as observed by beam based positron lifetime analysis (BPALS) real or a systematic effect of the data analysis To date the answer cannot be given with certainty. Arguments could be made why such a distribution is not observed by SAXS and is observed by BPALS in data shown here [62], and in work by Gidley et al.[46] Positrons are implanted at specific depths and only after measuring at different mean depth can one... 

The objective of the following part of this section is to assess the effect of different parameters on the craze microstructure. In these studies the SAXS curve of each sample has been subjected to the data analysis outlined above. 

A more detailed analysis of data from SAXS can be carried out by means of the interface distribution function gi(r), introduced by Ruland [30]. This function is the second derivative of the one-dimensional correlation function and... 

There are a number of works on rubber-based blends which exploited a SAXS data analysis based on approximations such as the Guinier, Porod or Debye-Bueche. These approaches are very interesting because they offer valuable information on the size of dispersed domains within the matrix of a blend, without the need of intensive calculation and without having to develop complex theoretical models for the fitting of SAXS patterns. 

Wide- and small-angle scattering are fully developed X-ray techniques for studying structural features. The theory is complete, the experimental devices are well developed, laboratory systems are commercially available, and dedicated X-ray facilities for special applications exist in several international research centers. Data analysis techniques have improved substantially in the last two decades through computers use of numerical methods. By nature, SAXS is used in the study of relatively large-scale structures, while wide-angle X-ray diffraction (WAXD) deals mainly... 

If we consider a two-phase system consisting of two different substances, a and b, respectively, each substance has a constant electron density and is filling the volume, Vj and Vb, respectively, from the total volume V. If the phases are homogenous and have a distinct, sharp interface (small particles embedded in a polymer matrix), the SAXS data analysis consists of resolving the equation ... 

In the present multiscale approach, SAXS data analysis reveals that already at low deformations (lower than e(start)) the lamellar crystals oriented with their normal parallel to the tensile force, in order to alleviate compressive transversal stress, undergo buckling instability, with consequent formation of undulated chevron-like super-structures (Fig. 11.12b ). However, at deformations close to e(end) up to the breaking the effect of lamellar stacking and parallel orientation of the lamellar normal to the stretching direction on the SAXS intensity becomes buried by the large diffuse scattering localized on the equator due to cavitation (Fig. 11.12c"). 

Small-angle X-ray scattering (SAXS) 1 bulk, > 1 mm Electron density — Dedicated data analysis necessary... 

Stribeck N. SAXS data analysis of a lamellar two-phase system. Layer statistics and compansion. Colloid Polym Sci 1993 271(11) 1007-1023. 

Stribeck N. Complete SAXS data analysis and synthesis of lamellar two-phase systems. Deduction of a simple model for the layer statistics. J Phys IV 1993 3(C8) 507-510. 

Figure 5.14 Time dependence of the various structural parameters estimated in the isothermal crystallization process of isotactic polypropylene. The FTIR data show the growth of regular helical segments. The Rg, and L are the radius of gyroid of the higher-density domains, the correlation distance between the neighboring domains, and the long period of the stacked lamellae, respectively, revealed by the SAXS data analysis. The Q is the invariant and is approximately proportional to the degree of crystallinity Xc, which was evaluated also using the WAXD data [67].

Published papers indicate the rapid progress of radiation sources, but also a considerable backlog as far as the advance of data analysis in the field of polymer nanostructure studies by small-angle X-ray scattering (SAXS) is concerned. In several fields of materials science, the importance of method developments has been early recognized and supported. For instance, the great success of protein crystallography is mainly based on the development and standardization of methods for the evaluation of synchrotron radiation data [10], which evidenced an unparalleled precision in crystal structure studies. A similar awareness is still insufficient in the field of polymer materials science. 

SAXS data analysis gives an insight in the nanostructure in the physical space (r), but the SAXS intensity is gathered in the reciprocal space. Because both (r) and /(s) are present in the Fourier relation (Eq. (1)), projections should play the major role in SAXS analysis [23], despite that numerous examples are found in the literature, where nanostructure parameters are determined from scattering intensity curves cut from the SAXS pattern along a straight line. From the mathematical point of view, this cut is a section, and the informational content of sections and projections is readily established by means of mathematical reasoning. 

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