  In medium amplitude oscillatory shear (MAOS), 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 typical
expectation. Expanding on the work by Natalia et al. [J. Rheol., 64(3): 625
(2020)], we report non-cubical, non-integer 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 non-integer
scalings occur at every secondary fluid concentration for two different contact
angles. These compelling results indicate that the non-integer 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, therefore, associated with adhesive
controlled friction. These observations, conducted for a wide range of
compositions, can help explain previous reports of non-integer scaling for
materials involving particle contacts and offers a new opportunity to study the
importance of the particle bonds and friction in the rheological response under
low deformation instead of at very high shear rates.
