Small-angle scattering and scale-dependent heterogeneity

Although small-angle scattering is often discussed qualitatively in terms of material heterogeneity, when it comes to quantitative data analysis this notion becomes somehow hidden behind the concept of correlation function. In the present contribution, a quantitative measure of heterogeneity is defined, it is shown how it can be calculated from scattering data, and its structural significance for the purpose of material characterization is discussed. Conceptually, the procedure consists of using a finite probe volume to define a local average density at any point of the material; the heterogeneity is then quantitatively defined as the fluctuations of the local average density when the probe volume is moved systematically through the sample. Experimentally, it is shown that the so-defined heterogeneity can be estimated by projecting the small-angle scattering intensity onto the form factor of the chosen probe volume. Choosing probe volumes of various sizes and shapes enables one to comprehensively characterize the heterogeneity of a material over all its relevant length scales. General results are derived for asymptotically small and large probes in relation to the material surface area and integral range. It is also shown that the correlation function is equivalent to a heterogeneity calculated with a probe volume consisting of two points only. The interest of scale-dependent heterogeneity for practical data analysis is illustrated with experimental small-angle X-ray scattering patterns measured on a micro- and mesoporous material, on a gel, and on a semi-crystalline polyethylene sample. Using different types of probes to analyse a given scattering pattern enables one to focus on different structural characteristics of the material, which is particularly useful in the case of hierarchical structures.