The medium amplitude oscillatory shear of semi-dilute colloidal dispersions. Part I: Linear response and normal stress differences

We study the medium amplitude oscillatory shear (MAOS) of a semi-dilute, hard-sphere colloidal dispersion. Through solution of the Smoluchowski equation governing the statistical distribution of suspended particles in the semi-dilute limit, we calculate the linear viscoelasticity and the normal stress differences that arise from oscillatory shear as an expansion in terms of small rates of deformation. The effects of hydrodynamic versus conservative interactions are studied via the excluded-annulus model for the hard-sphere interactions. This treats the hard-sphere interactions as a steric barrier that resides beyond the hydrodynamic radius of the particles and allows for continuous variation of the strength of hydrodynamic interactions. We see remarkable distinctions between suspensions that are freely draining and those for which hydrodynamic lubrication among the particles dominate their motion. In oscillatory shear flow of a freely draining suspension at high-frequency, the normal stress differences are generated by hard-sphere stresses. In contrast, for a hydrodynamically interacting suspension at high-frequency, the first normal stress difference derives from Brownian stresses while the second normal stress difference is dominated by hydrodynamic stresses. When the first normal stress difference is plotted parametrically against the oscillatory rate of strain, the resulting Lissajous-Bowditch curves for freely draining suspensions and those with full hydrodynamic interactions take on distinct and disparate shapes. Additionally, we use an asymptotic analysis to predict that the third harmonic of the suspension stress for hard spheres is dominated by the hydrodynamic lubrication in the limit of high frequency oscillation. Our calculations demonstrate that under MAOS, hydrodynamic interactions play a central and qualitative role in the stress response of a semi-dilute colloidal dispersion.