Microrheology of colloidal dispersions: Shape matters

Abstract

We consider a “probe” particle translating at constant velocity through an otherwise quiescent dispersion of colloidal “bath” particles, as a model for particle-tracking microrheology experiments in the active (nonlinear) regime. The probe is a body of revolution with major and minor semiaxes a and b, respectively, and the bath particles are spheres of radii b. The probe's shape is such that when its major or minor axis is the axis of revolution the excluded-volume, or contact, surface between the probe and a bath particle is a prolate or oblate spheroid, respectively. The moving probe drives the microstructure of the dispersion out of equilibrium; counteracting this is the Brownian diffusion of the bath particles. For a prolate or oblate probe translating along its symmetry axis, we calculate the nonequilibrium microstructure to first order in the volume fraction of bath particles and over the entire range of the Péclet number (Pe), neglecting hydrodynamic interactions. Here, Pe is defined as the non-dimensional velocity of the probe. The microstructure is employed to calculate the average external force on the probe, from which one can infer a “microviscosity” of the dispersion via Stokes drag law. The microviscosity is computed as a function of the aspect ratio of the probe, â=a/b, thereby delineating the role of the probe's shape. For a prolate probe, regardless of the value of â, the microviscosity monotonically decreases, or “velocity thins,” from a Newtonian plateau at small Pe until a second Newtonian plateau is reached as Pe-->[infinity]. After appropriate scaling, we demonstrate this behavior to be in agreement with microrheology studies using spherical probes [Squires and Brady, “A simple paradigm for active and nonlinear microrheology,” Phys. Fluids 17(7), 073101 (2005)] and conventional (macro-)rheological investigations [Bergenholtz et al., “The non-Newtonian rheology of dilute colloidal suspensions,” J. Fluid. Mech. 456, 239–275 (2002)]. For an oblate probe, the microviscosity again transitions between two Newtonian plateaus: for â<3.52 (to two decimal places) the microviscosity at small Pe is greater than at large Pe (again, velocity thinning); however, for â>3.52 the microviscosity at small Pe is less than at large Pe, which suggests it “velocity thickens” as Pe is increased. This anomalous velocity thickening—due entirely to the probe shape—highlights the care needed when designing microrheology experiments with non-spherical probes.