  We show that Hertzian particle contacts are the underlying cause of the
as-yet-unexplained noninteger power laws in weakly nonlinear rheology. In the
medium amplitude oscillatory shear (MAOS) region, the cubic scaling of the
leading order nonlinear shear stress ($\sigma_\mathrm{3} \sim
\gamma_\mathrm{0}^{m_\mathrm{3}}$, $m_\mathrm{3}=3$) is the standard
expectation. Expanding on the work by Natalia et al. [J. Rheol. 64 625-635
(2020)], we report an extensive data set of noncubical, noninteger power law
scalings $m_\mathrm{3}$ for particle suspensions in two immiscible fluids with
a capillary attractive interaction, known as capillary suspensions. Here, we
show that distinct power law exponents are found for the storage and loss
moduli and these noninteger scalings occur at every secondary fluid
concentration for two different contact angles. These compelling results
indicate that the noninteger scalings are related to the underlying
microstructure of capillary suspensions. We show that the magnitude of the
third harmonic elastic stress scaling $m_\mathrm{3,elastic}$ originates from
Hertzian-like contacts in combination with the attractive capillary force. The
related third harmonic viscous stress scaling $m_\mathrm{3,viscous}$ is, found
to be associated with adhesive-controlled friction. These observations,
conducted for a wide range of compositions, can help explain previous reports
of noninteger scaling for materials involving particle contacts and offers a
new opportunity using the variable power law exponent of MAOS rheology to
reveal the physics of particle bonds and friction in the rheological response
under low deformation instead of at very high shear rates.
