  Nanocomposites formed by mixing nanoparticles and polymers offer a limitless
creative space for the design of functional advanced materials with a broad
range of applications in materials and biological sciences. Here we focus on
aqueous dispersions of hydrophobic colloidal soot particles, namely carbon
black (CB) dispersed with a sodium salt of carboxymethylcellulose (CMC), a food
additive known as cellulose gum that bears hydrophobic groups, which are liable
to bind physically to CB particles. Varying the relative content of CB
nanoparticles and cellulose gum allows us to explore a rich phase diagram that
includes a gel phase. We investigate this hydrogel using rheometry and
electrochemical impedance spectroscopy. CB-CMC hydrogels display two radically
different types of mechanical behaviors that are separated by a critical
CMC-to-CB mass ratio $r_c$. For $r<r_c$, i.e., for low CMC concentration, the
gel is electrically conductive and shows a glassy-like viscoelastic spectrum,
pointing to a microstructure composed of a percolated network of CB
nanoparticles decorated by CMC. In contrast, gels with CMC concentration larger
than $r_c$ are non-conductive, indicating that the CB nanoparticles are
dispersed in the cellulose gum matrix as isolated clusters, and act as physical
crosslinkers of the CMC network, hence providing mechanical rigidity to the
composite. Moreover, in the concentration range, $r>r_c$ CB-CMC gels display a
power-law viscoelastic spectrum that depends strongly on the CMC concentration.
These relaxation spectra can be rescaled onto a master curve that exhibits a
power-law scaling in the high-frequency limit, with an exponent that follows
Zimm theory, showing that CMC plays a key role in the gel viscoelastic
properties for $r>r_c$. Our results offer a characterization of CB-CMC
dispersions that will be useful for designing nanocomposites based on
hydrophobic interactions.
