The medium amplitude oscillatory shear of semidilute colloidal dispersions. Part II: Third harmonic stress contribution

Abstract

In Paper I [J. W. Swan et al., J. Rheol. 58, 307–338 (2014)], we derived an exact theoretical description of medium amplitude oscillatory shear for a semidilute colloidal dispersion. Through solution of the Smoluchowski equation governing the spatial distribution of suspended particles in the semidilute limit, we calculated the stresses that arise from an oscillatory linear flow as an expansion in powers of the rate of deformation. Here, this is extended to calculation of the first departures from linearity in the first and third harmonics of the suspension stress driven by oscillatory deformation. The role of hydrodynamic interactions is investigated via the excluded-annulus model in which particles are given an impenetrable core with a radius larger than their hydrodynamic radius. The ratio of these length scales controls the strength of hydrodynamic interactions. The third harmonic of the suspension stress is predicted to be dominated by hydrodynamic stresses at high frequency, a result that is shown to be valid experimentally for the oscillatory shear response of concentrated near hard-sphere dispersions. The calculations anticipate recent experimental observations on model near hard-sphere colloidal dispersions, and quantitative agreement is demonstrated when the predictions are scaled appropriately to account for volume fraction effects. The first departures from linearity in harmonics of the suspension stress are separated into several material functions that are independent of the flow geometry. These functions are generated from detailed numerical solutions, while asymptotic analysis is shown to predict the values of these functions at high frequency. These exact calculations provide a basis for understanding the onset of nonlinear rheological behavior of colloidal suspensions under dynamic oscillatory flow.