Dense Colloidal Suspensions under Time-Dependent Shear

We consider the nonlinear rheology of dense colloidal suspensions under a time-dependent simple shear flow. Starting from the Smoluchowski equation for interacting Brownian particles advected by shearing (ignoring fluctuations in fluid velocity), we develop a formalism which enables the calculation of time-dependent, far-from-equilibrium averages. Taking shear stress as an example, we derive exactly a generalized Green-Kubo relation and an equation of motion for the transient density correlator, involving a three-time memory function. Mode coupling approximations give a closed constitutive equation yielding the time-dependent stress for arbitrary shear rate history. We solve this equation numerically for the special case of a hard sphere glass subject to step strain.

Figure 1

The long-time stress $\sigma (\infty )$ as a function of strain $\gamma$ for a hard sphere glass just above the MCT glass transition ($\varphi =0.52$) following a step strain. The dashed line is the linear response result, and the solid line is $\sigma (\infty )$ from the isotropic approximation [23]. The inset shows the decaying stress as a function of time following the step.