Particle Motion in Colloidal Dispersions: Applications to Microrheology and Nonequilibrium Depletion Interactions

Abstract

Over the past decade, microrheology has burst onto the scene as a technique to interrogate and manipulate complex fluids and biological materials at the micro- and nano-meter scale. At the heart of microrheology is the use of colloidal 'probe' particles embedded in the material of interest; by tracking the motion of a probe one can ascertain rheological properties of the material. In this study, we propose and investigate a paradigmatic model for microrheology: an externally driven probe traveling through an otherwise quiescent colloidal dispersion. From the probe's motion one can infer a 'microviscosity' of the dispersion via application of Stokes drag law. Depending on the amplitude and time-dependence of the probe's movement, the linear or nonlinear (micro-)rheological response of the dispersion may be inferred: from steady, arbitrary-amplitude motion we compute a nonlinear microviscosity, while small-amplitude oscillatory motion yields a frequency-dependent (complex) microviscosity. These two microviscosities are shown, after appropriate scaling, to be in good agreement with their (macro)-rheological counterparts. Furthermore, we investigate the role played by the probe's shape --- sphere, rod, or disc --- in microrheological experiments. Lastly, on a related theme, we consider two spherical probes translating in-line with equal velocities through a colloidal dispersion, as a model for depletion interactions out of equilibrium. The probes disturb the tranquility of the dispersion; in retaliation, the dispersion exerts a entropic (depletion) force on each probe, which depends on the velocity of the probes and their separation. When moving 'slowly' we recover the well-known equilibrium depletion attraction between probes. For 'rapid' motion, there is a large accumulation of particles in a thin boundary layer on the upstream side of the leading probe, whereas the trailing probe moves in a tunnel, or wake, of particle-free solvent created by the leading probe. Consequently, the entropic force on the trailing probe vanishes, while the force on the leading probe approaches a limiting value, equal to that for a single translating probe.