Three length-scales colloidal gels: The clusters of clusters versus the interpenetrating clusters approach

Typically, in quiescent conditions, attractive colloids at low volume fractions form fractal gels structured into two length scales: the colloidal and the fractal cluster scales. However, when flow interferes with gelation colloidal fractal gels, it may display three distinct length scales [Dagès et al., Soft Matter 18, 6645–6659 (2022)]. Following those recent experimental investigations, we derive two models that account for the structure and the rheological properties of such atypical colloidal gels. The gel elasticity is inferred from scaling arguments, and the structure is translated into scattering intensities following the global scattering functions approach proposed by Beaucage and, typically, measured in small-angle x-ray scattering (SAXS). In both models, we consider that the colloids condensate into fractal clusters. In the clusters of the clusters model, the clusters form superagregates that then build the gel network. In the interpenetrating clusters model, the clusters interpenetrate one another to form the gel network. Those two models are then used to analyze rheo-SAXS experiments carried out on carbon black gels formed through flow cessation.