Small-angle scattering and scale-dependent heterogeneity

Abstract

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 scaledependent 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.

Structural Analysis of Molecular Materials Using the Pair Distribution Function

“…SAS data are commonly assessed by various forms of real-space distribution functions that come with different requirements for the validity of their use and varying benefits for assessing different physical properties of structural heterogeneities. ,− The most common forms are the generalized 3D and 1D correlation functions. These functions represent the longer-range fluctuation in the density with respect to the average and are related to the probability that a rod of length r will have both ends in the same phase.…”

Scattering intensity limit value at very small angles

“…Thus, ρ = 0 and, consequently, I V (0 + ) is equal to zero for simple cubic crystals. We refer to Gommes (2016) for further geometries characterized by vanishing I V (0 + ) values.…”

Unwarping GISAXS data

“…Fourthly, a range of sophisticated data-analysis methods could be more easily applied to GISAXS data if they were remapped to an undistorted reciprocal space. For example, modern developments in correlation methods such as angular correlation analysis (Wochner et al, 2009;Altarelli et al, 2010;Lehmkü hler et al, 2014Lehmkü hler et al, , 2018Lhermitte et al, 2017), fluctuation scattering (Chen et al, 2012;Malmerberg et al, 2015;Martin, 2017), or variance scattering (Yager & Majewski, 2014;Gommes, 2016) are not currently used in a GISAXS context. Finally, we note that healing data can be a useful pre-processing step (Liu et al, 2017), allowing such data to be used with existing data-analysis pipelines, or input into modern machine-learning methods (Kiapour et al, 2014;Wang et al, 2016Wang et al, , 2017Meister et al, 2017).…”