Application of SAXS alone and in concert with other techniques, such as WAXS, rheology, FTIR, and others, has provided powerful insights into polyurethane formation and structure [67-72]. The broad concepts related to SAXS are quite similar to those of WAXS [73, 74]. The detector is moved quite a bit farther from the sample than in a WAXS experiment, on the order of 0.5 m to more than 3 m when employing synchrotron radiation sources . Wide-angle scattering signal is not detected or is faintly registered by the detector. Instead, only the scattering at low angular dispersion from the main beam intensity is recorded. Low-angle scattering detection results in the need to have good removal of the main beam (and other background sources as will be discussed later) to obtain the most useful and interesting information. A means of removing this direct beam intensity is shown in Figure 5.16 where a physical beam stop is employed. The passed scattered intensity will fall upon the detector representing the amplitude of the Fourier transform of the shape in the beam. Polydispersity of the size and shape of randomly oriented scattering objects, as usually exist in polyurethanes, results in a superposition of scattering patterns leading to a smooth function such as shown in Figure 5.17. Alignment of nonspherical scattering objects can lead to anisotropic scattering patterns. Interpretation of scattering using classical analyses involves fitting the measured scattering to models based on polydisperse known shapes such as spheres . The modeling and interpretation become more challenging as complex shapes must be accounted for. From the fit of data to models, insightful information on the spatial distribution of particles in the sample, their potential agglomeration, their interfacial characteristics, and their shapes can be obtained .
The center of the scattering peak is indicated in Figure 5.17, and the calculated d-spacing showed based on the Bragg equation given by Equation 5.6. The d-space quantity is of interest by itself in polyurethane science, since it is a quantitative measure of the interhard segment spacing that can then be correlated to various formulation and processing variables. It should be pointed out that d-spacing based on picking out the center of the first peak can sometimes be
Figure 5.16 Relationship of sample to detector for SAXS analysis. Compare to Figure 5.13 for WAXS analysis.
Figure 5.17 SAXS data for a polyurethane foam. Inset is the 2-D data.
fairly easy as in the pattern of Figure 5.17. Other times, as will be shown in the following text, picking the peak intensity will be imprecise enough that a d-space calculation will be best characterized as an estimate. Data analysis techniques can be employed to improve the accuracy and precision of the d-space calculation. One such correction is the Lorentz correction for removal of vestigial main beam intensity. This technique is appropriate for lamellar scattering structures, which many polyurethane hard segments are, but certainly not all. Systematic control of variables and careful analysis of data can then allow for improved understanding and prediction of polyurethane properties. Furthermore, it has been demonstrated that simultaneous measurement of the growth of SAXS with rheological, spectroscopic, and thermal properties can provide new insights into polyurethane structure and property development .
The fact that scattering is a function of geometry and that ideal shapes scatter X-rays in well-understood ways has provided additional opportunity to understand polyurethanes for the frequent instances where ideal response models do not perfectly fit. The variance with expectations can be modeled and the closeness of fit to data used to provide additional structural insights that can be tied to polymer properties. Among the most useful models is based on the theoretical analysis by Porod on deviations from ideal scattering behavior due to the real interfacial thickness of phase separated systems [77-79]. These interfacial thicknesses are also referred to as boundary widths or diffuse boundary thicknesses.
It is vital to note that the Porod analysis requires that for a 2-phase system with a perfectly sharp interface, the long q values must decrease according to an inverse 4th power:
where K is a constant defined by Porod as
In Equation 5.9, / is the Porod inhomogeneity length and Q is a term described variously as the "invariant" or the "Porod invariant" defined explicitly by
Kp can also be calculated as the limiting value reached by the product of I(q)q4. In Equation 5.10, 0 1 and 0 2 are the volume fractions of the respective two phases and p1 and p2 are the respective electron densities.
It is in the variation of data from expected ideal values that interpretation of polyurethane structure can be understood and properties systematically refined. In the cases where I(q)q4 does not reach a limiting value, the inferences can be particularly illuminating. In polyurethane research, the usual case is for a graph of I(q)q4 versus q2 to have a negative slope. This result can be interpreted as a result of diffuse phase boundaries (as opposed to a very sharp phase boundary) and phase mixing . The result of the phase mixing is a depletion of scattering intensity at high scattering angles that are more closely associated with the region closest to each scattering center and less with the regions between the scattering centers [81, 82].
As seen in Figures 5.17 and 5.18a, the scattering intensity at high values of q is small and declines rapidly (at or near the 4th power) as predicted by the Porod analysis. Thus, the means of subtraction for background intensity, beam collimation, and even vestigial wide-angle scattering is critical to obtaining unbiased reliable results. In a critical review of correction techniques, it has been shown by Koberstein et al.  that representation of the background correction as a constant will always lead to overestimation of interfacial mixing. This is due to overestimation of the decline in background intensity that leads to an exaggerated negative deviation from ideal expectations based on the Porod analysis.
Obtaining the correlation function of scattering requires calculation of the Fourier transform of the one-dimensional SAXS curve [83, 84]. This yields a curve of y1 termed the correlation function that is the spatial correlation of the electron density correlations. A useful mental image is to imagine the function y1(r) as the probability of scattering from a rod of length r with equal electron densities at either end. A peak in the resulting correlation function will then occur when there is a frequently occurring spacing within the structure. An example of SAXS and the resulting correlation function is given in Figure 5.18. The first peak in the correlation function represents the d-space distance that would normally be estimated from the peak in the scattering intensity. In the provided example, picking out a maximum may not be a particular challenge, but there are many examples where defining a d-space could be somewhat more challenging or arbitrary as shown in Figure 5.19. Features of the correlation function can lead to direct calculation of various useful quantities related
Figure 5.18 SAXS data for a molded polyurethane foam and the derived correlation function. (a) SAXS data for a molded polyurethane foam and (b) the derived correlation function. Reprinted with permission from Refs.  and . © Elsevier Pub.
Figure 5.19 SAXS data for a TDI-flexible slab foam. The d-spacing determination requires more judgment that required for data such as Figure 5.17.
TABLE 5.6 Data derived from a porod analysis of Figure 5.17
to the 2-phase structure, the values of which can be used to determine the relevant structure-processing-property relationships of polyurethanes. An example of such data is provided in Table 5.6, which gives available phase information from the scattering pattern and resulting correlation function of Figures 5.18.