Scattering from `Babinet' particles (or not…): spherical particles made up of spheres and spherical particles with spherical voids

The small-angle scattering form factors of two classes of composite particles with contrasting internal architectures have been studied: one consisting of inclusions of smaller spheres embedded within a larger sphere, and the other comprising a solid sphere with randomly distributed spherical voids. These systems serve as material- and void-based analogues, providing a model framework for examining how internal material distribution in porous particles influences scattering signatures. Monte Carlo simulations were used to generate scattering curves across a range of volume fractions and polydispersities, which were then employed to benchmark analytically derived form factor expressions. Steric repulsion between, respectively, spheres and voids was taken as hard-sphere interactions. The results reveal that internal structural asymmetries, especially in spatial correlations and contrast topology, significantly affect scattering patterns, despite the particles having similar overall structures and volume fractions. In particular, spheres-of-spheres structures exhibit features in the scattering signal from internal modulations, while void-based particles display smoother shell-like scattering features. The analytical models show excellent agreement with the simulated data, capturing both the global shape and fine structural characteristics. These findings demonstrate that relatively simple analytical approaches, validated against numerical simulations, can reliably describe complex heterogeneous particles. This methodology provides a robust basis for interpreting scattering data from porous and composite materials across a wide range of applications.