Active microrheology: Fixed-velocity versus fixed-force

Abstract

In active microrheology, a probe particle is driven by an external force through a complex medium and its motion studied in order to infer properties of the embedding material. It is conducted in two limiting forms: either the probe is propelled by a fixed force, as with magnetic tweezers, or it is driven at a fixed velocity, as with optical tweezers. Recent work has shown that the mean probe motion can be interpreted as an effective material viscosity, but that this viscosity depends on whether the fixed-force or fixed-velocity mode is employed. We compute the effective viscosity probed by fixed-velocity active microrheology of a dilute colloidal dispersion. A comparison is made between this new result and the effective viscosity probed in the fixed-force mode. In the absence of hydrodynamic interactions, the particle-phase contributions to the effective viscosity for the two modes differ by exactly a factor of two. A simple scaling argument has been previously advanced to rationalize this difference: in the fixed-force mode, the probe is free to diffuse, and thus the relaxation time scale is set by the relative diffusivity between probe and bath. However, in the fixed-velocity mode, thermal motion of the probe particle is “frozen out” because the probe cannot diffuse; the relaxation rate is thus halved. The ratio of the two rates is independent of how quickly the probe particle is driven through the suspension—the extent and shape of microstructural deformation is the same for the two cases. In contrast, when the suspended particles interact hydrodynamically, the distortions to the suspension microstructure in the fixed-velocity versus fixed-force modes differ. We show that, depending on the strength of the hydrodynamic interactions, the ratio of the fixed-velocity to the fixed-force microstructural contributions to the effective viscosity may be as small as 1.3, and only approaches 2.0 when hydrodynamic interactions among the particles are negligibly weak. While this ratio varies both as a function of the strength of the deformation imposed and of the strength of hydrodynamic interactions, the fixed-velocity effective viscosity agrees qualitatively with that already measured for the fixed-force mode: the colloidal dispersion thins in the limit of weak hydrodynamic interactions; and it first thins and then thickens in the limit of strong hydrodynamic interactions, as the strength of deformation increases, recovering characteristics of shear-(force-) thinning and thickening well known in colloidal dispersions. The agreement between the two, and with traditional macrorheological approaches, shows that both fixed-force and fixed-velocity provide a useful tool for the interrogation of complex fluids.