“Microviscoelasticity” of colloidal dispersions

Abstract

We study the motion of a Brownian probe particle subjected to a small amplitude oscillatory external force and immersed in a colloidal dispersion, as a theoretical benchmark for particle-tracking microrheology experiments. The application of an external force on the probe drives the microstructure of the dispersion out of equilibrium; opposing this is the Brownian diffusion of the probe and colloidal “bath” particles. The degree to which the microstructure is driven away from equilibrium is given by the nondimensional external force on the probe, or Péclet number, Pe. The nonequilibrium microstructure of the dispersion is calculated for small departures from equilibrium, i.e., to first order in Pe, and to leading order in the bath particle volume fraction, accounting for excluded volume and hydrodynamic interactions between particles. The nonequilibrium microstructure is used to calculate the average velocity of the probe, from which one may infer the complex microviscosity (or modulus) of the dispersion via application of Stokes drag law. The microviscosity is computed over the entire range of oscillation frequencies, thereby determining the linear viscoelastic response of the dispersion. After appropriate scaling, our results are in qualitative agreement with traditional (macro-) rheology studies, suggesting that oscillatory-probe microrheology can be a useful tool to examine the viscoelasticity of hard-sphere colloidal dispersions and perhaps other complex fluids.