Single particle motion in colloidal dispersions: a simple model for active and nonlinear microrheology

Abstract

The motion of a single Brownian probe particle subjected to a constant external body force and immersed in a dispersion of colloidal particles is studied with a view to providing a simple model for particle tracking microrheology experiments in the active and nonlinear regime. The non-equilibrium configuration of particles induced by the motion of the probe is calculated to first order in the volume fraction of colloidal particles over the entire range of Pe, accounting for hydrodynamic and excluded volume interactions between the probe and dispersion particles. Here, Pe is the dimensionless external force on the probe, or Péclet number, and is a characteristic measure of the degree to which the equilibrium microstructure of the dispersion is distorted. For small Pe, the microstructure (in a reference frame moving with the probe) is primarily dictated by Brownian diffusion and is approximately fore–aft symmetric about the direction of the external force. In the large Pe limit, advection is dominant except in a thin boundary layer in the compressive region of the flow where it is balanced by Brownian diffusion, leading to a highly non-equilibrium microstructure. The computed microstructure is employed to calculate the average translational velocity of the probe, from which the ‘microviscosity’ of the dispersion may be inferred via application of Stokes drag law. For small departures from equilibrium (Pe), the microviscosity ‘force-thins’ proportional to $\hbox{\it Pe}^2$ from its Newtonian low-force plateau. For particles with long-range excluded volume interactions, force-thinning persists until a terminal Newtonian plateau is reached in the limit $\hbox{\it Pe}\,{\rightarrow}\,\infty$. In the case of particles with very short-range excluded volume interactions, the force-thinning ceases at $\hbox{\it Pe}\,{\sim}\, O(1)$, at which point the microviscosity attains a minimum value. Beyond $\hbox{\it Pe}\,{\sim}\, O(1)$, the microstructural boundary layer coincides with the lubrication range of hydrodynamic interactions causing the microviscosity to enter a continuous ‘force-thickening’ regime. The qualitative picture of the microviscosity variation with Pe is in good agreement with theoretical and computational investigations on the ‘macroviscosity’ of sheared colloidal dispersions, and, after appropriate scaling, we are able to make a direct quantitative comparison. This suggests that active tracking microrheology is a valuable tool with which to explore the rich nonlinear rheology of complex fluids.