Work

X-ray to visible light scattering by fractal agglomerates: A model comparison applied to soot

Year: 2026

Type: article

Abstract: Characterizing nanoparticles in terms of size and morphology is essential to understand their physical and optical properties. Small or Ultra Small-Angle X-ray Scattering (SAXS/USAXS) is a powerful in situ technique that allows the analysis of such nanostructures exhibiting size and morphology through the Guinier and Porod regimes, which can be repeated at different scales (e.g., primary spheres and Agglomerates for mass fractal agglomerates, as focused on in this study). This is the principle of the widely used Beaucage’s unified model, built in reciprocal q -space (the momentum transfer or scattering wave vector) in order to interpret experimental results. An alternative method based on pair correlation modeling can be used. Indeed, by definition, the scattered signal corresponds to the Fourier transform of the particles’ pair correlation function. The present study compares the two approaches for fractal agglomerates of point-contact spheres. It is shown that the Beaucage’s model is limited to agglomerates consisting of a large number of primary spheres and that certain destructive interferences, not taken into account in this model, can affect data inversion, potentially affecting the extraction of the primary sphere size distribution. In this work, both models are compared in the direct way applied on synthetic data, and used to invert experimental SAXS data from soot particles in a laminar ethylene diffusion flame measured at the European Synchrotron Radiation Facility (ESRF) and processed via Abel inversion. This study puts in evidence the fundamental differences between both models and highlight their respective limitations. Nevertheless, both models provide similar results except for the mass fractal dimension and primary sphere geometric standard deviation. The present study demonstrates thus the potential of the pair correlation modelling and encourages its further development. • A critical analysis and comparison is carried out between unified scaling approaches, commonly used in interpreting electromagnetic wave scattering signals in visible or X-ray ranges, and a more rigorous method based on modeling the pair correlation function. • The results highlight a significant shortcomings in classical modeling, particularly its neglect of destructive interference effects that crucially influence the amplitude and slope of modeled signals. • A new expression for the polydispersity index (PDI) tailored for lognormal distributions is introduce. • The application of Abel inversion to scattering spectra in laminar diffusion flames has a strong impact on inverted parameters. • Applying both models to experimental data obtained by small-angle X-ray scattering (SAXS) reveals significant impacts on most of the measured parameters when the new expression for PDI is taken into account.

Source: Journal of Quantitative Spectroscopy and Radiative Transfer

DOI: 10.1016/j.jqsrt.2026.109860