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We show that Hertzian particle contacts are the underlying cause of theas-yet-unexplained noninteger power laws in weakly nonlinear rheology. In themedium amplitude oscillatory shear (MAOS) region, the cubic scaling of theleading order nonlinear shear stress ($\sigma_\mathrm{3} \sim\gamma_\mathrm{0}^{m_\mathrm{3}}$, $m_\mathrm{3}=3$) is the standardexpectation. Expanding on the work by Natalia et al. [J. Rheol. 64 625-635(2020)], we report an extensive data set of noncubical, noninteger power lawscalings $m_\mathrm{3}$ for particle suspensions in two immiscible fluids witha capillary attractive interaction, known as capillary suspensions. Here, weshow that distinct power law exponents are found for the storage and lossmoduli and these noninteger scalings occur at every secondary fluidconcentration for two different contact angles. These compelling resultsindicate that the noninteger scalings are related to the underlyingmicrostructure of capillary suspensions. We show that the magnitude of thethird harmonic elastic stress scaling $m_\mathrm{3,elastic}$ originates fromHertzian-like contacts in combination with the attractive capillary force. Therelated third harmonic viscous stress scaling $m_\mathrm{3,viscous}$ is, foundto be associated with adhesive-controlled friction. These observations,conducted for a wide range of compositions, can help explain previous reportsof noninteger scaling for materials involving particle contacts and offers anew opportunity using the variable power law exponent of MAOS rheology toreveal the physics of particle bonds and friction in the rheological responseunder low deformation instead of at very high shear rates.

Particle contact dynamics as the origin for non-integer power expansion rheology in attractive suspension networks