Over the past decade, microrheology has burst onto the scene as a technique to ascertain the viscoelastic properties of complex fluids and biological materials at the micrometer scale. While advances in experimental microrheology methods continue unabated, theoretical developments, in comparison, struggle to keep pace. To address this imbalance, we study the motion of a Brownian 'probe' particle under the action of an external force and immersed in a colloidal dispersion, as a simple theoretical model for particle-tracking microrheology experiments in the active, or nonequilibrium, regime.

The forced probe drives the microstructure of the dispersion out of equilibrium; counteracting this is the Brownian diffusion of the probe and colloidal 'bath' particles. The degree of microstructural distortion is governed by the ratio of external to Brownian forces, or Peclet number, Pe. The limit Pe → 0 is the realm of passive, or linear, microrheology; increasing Pe drives the system into the active, or nonlinear, regime. We calculate the nonequilibrium microstructure over the entire range of Pe, accounting for hydrodynamic and excluded-volume interactions between the probe and bath particles. With the microstructure in hand, the average velocity of the probe is computed, from which one can define a 'microviscosity' of the dispersion, via application of Stokes drag law.

Depending on the amplitude and time-dependence of the external force on the probe, the linear or nonlinear response of the dispersion may be inferred. In this study, we consider two cases: (i) a steady force of arbitrary magnitude — from which the (nonlinear) steady-force microviscosity is calculated; and (ii) a small-amplitude oscillatory force — from which the (complex) frequency-dependent microviscosity is obtained. In both cases, after appropriate scaling, our results are in qualitative agreement with traditional (macro-) rheological studies. This suggests active microrheology to be a valuable tool for studying the rich nonlinear rheology of colloidal dispersions, and perhaps other complex materials.