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In microrheology, elastic and viscous moduli are obtained from measurements of the fluctuating thermal motion of embedded colloidal probes. In such experiments, the probe motion is passive and reflects the near-equilibrium (linear response) properties of the surrounding medium. By actively pulling the probe through the material, further information about material properties can be obtained, analogous to large-amplitude measurements in (macro-) rheology. We consider a simple model of such systems: a colloidal probe pulled through a suspension of neutrally buoyant bath colloids. We choose a system with hard-sphere interactions but neglect hydrodynamic interactions, which is simple enough to permit analytic solutions, but nontrivial enough to raise issues important for the interpretation of experiments in active and nonlinear microrheology. We calculate the microstructural deformation for arbitrary probe size and pulling rate (expressed as a dimensionless Peclet number Pe). From this, we determine the average retarding effect on the probe due to the microstructure, as well as fluctuations about this average. The high-Pe limit is singular, giving a finite Brownian contribution even in the limit of negligible diffusion. Significantly, different results are obtained for probes driven at constant velocity and constant force. Furthermore, we demonstrate that a probe pulled with an optical tweezer (roughly a harmonic well) can behave as fixed-force, fixed-velocity, or as a mixture of those modes, depending on the strength of the trap and on the pulling speed. More generally, we discuss how these results relate to previous work on the rheology of colloidal suspensions. Not surprisingly, the present theory (which ignores hydrodynamic interactions) gives shear thinning but no shear thickening; we expect that the incorporation of hydrodynamics would result in shear thickening as well. The effective micro- and macro-viscosities, when appropriately scaled, are in semi-quantitative agreement. This seems remarkable, given the rather significant difference in the two methods of measurement. However, for more complicated or unknown materials, where such scaling relations may not be known in advance, the comparison between micro- and macro may not be so favorable, which raises important questions about the relation between micro- and macrorheology. Finally, by analogy with previous work on macrorheology, we propose methods to scale up the present (dilute) theory to account for more concentrated suspensions, and suggest new active microrheological experiments to probe different aspects of suspension behavior.

A simple paradigm for active and nonlinear microrheology