Small-angle x-ray scattering functions in the vicinity of zero scattering angle with an application to polymer blends

The small-angle X-ray scattering (SAXS) of isotropic scatterers is studied in the region near zero scattering angles, where void scattering may contribute to the scattering power Q. Since Q is evaluated by an integration, the corresponding integrands I-1(s) (for point collimation) and (I) over tilde(1)(s) (for slit focus) as well as the relations between them are studied. The properties of both their moments and their Taylor series expansions are discussed. For a lamellar two-phase system, which is frequently observed in semicrystalline or microphase-separated polymers, analytical equations are presented. They can help (1) to extrapolate I-1(s) and (I) over tilde(1)(s) toward zero scattering angle, (2) to detect void scattering, and (3) to improve the modeling of the paracrystalline stack even in the case of considerable roughness of the layer surfaces. An example for a set of semicrystalline HDPE samples illustrates how to quantify void scattering and the effects of interfacial statistical structure.