  Soft particulate gels include materials we can eat, squeeze, or 3D print.
From foods to bio-inks to cement hydrates, these gels are composed of a small
amount of particulate matter (proteins, polymers, colloidal particles, or
agglomerates of various origins) embedded in a continuous fluid phase. The
solid components assemble to form a porous matrix, providing rigidity and
control of the mechanical response, despite being the minority constituent. The
rheological response and gel elasticity are direct functions of the particle
volume fraction $\phi$: however, the diverse range of different functional
dependencies reported experimentally has, to date, challenged efforts to
identify general scaling laws. Here we reveal a hidden hierarchical
organization of fractal elements that controls the viscoelastic spectrum, and
which is associated with the spatial heterogeneity of the solid matrix
topology. The fractal elements form the foundations of a viscoelastic master
curve, which we construct using large-scale 3D microscopic simulations of model
gels, and can be described by a recursive rheological ladder model over a range
of particle volume fractions and gelation rates. The hierarchy of the fractal
elements provides the missing general framework required to predict the gel
elasticity and the viscoelastic response of these ubiquitous complex materials.
