Colloidal suspensions and the relation between their rheology and their microstructure is investigated. The literature showed great evidence of the relation between rheological quantities and particle volume fraction, ignoring the influence of the cluster. We propose to extend previous models using a new double fractal structure which allows, first, to recover the well-known models on the case of percolated system and, second, to capture the influence of the cluster size. This new model emphasises the necessity of such structure to account for recent experimental results. Then, the model is compared with data coming from the literature and shows close agreement.

Colloidal gels are out of equilibrium soft solids composed of attractive Brownian particles that form a space-spanning network at low volume fractions. The elastic properties of these systems result from...

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.