Sticky-probe active microrheology: Part 2. The influence of attractions on non-Newtonian flow

We derive a theoretical framework for the non-Newtonian viscosity of a sticky, attractive colloidal dispersion via active microrheology by modeling detailed microscopic attractive and Brownian forces between particles. Actively forcing a probe distorts the surrounding arrangement of particles from equilibrium; the degree of this distortion is characterized by the Péclet number, Pe≡F^ext/(2kT/a), where kT is the thermal energy and a the probe size. Similarly, the strength of attractive interactions relative to Brownian motion is captured by the second virial coefficient, B₂. We formulate a Smoluchowski equation governing the pair configuration as it evolves with external and attractive forces. The microviscosity is then computed via non-equilibrium statistical mechanics. For active probe forcing, the familiar hard-sphere boundary-layer and wake structures emerge as Pe grows strong, but attractions alter its shape: changes in relative probe motion arising from its attraction to the bath particles can lead to a high-Pe, strong-attraction flipping of the microstructure, where an upstream depletion boundary layer forms, along with a downstream accumulation wake. This highly distorted structure is analyzed at the micro-mechanical level, where changes in the time spent upstream or downstream from a bath particle lead to hypo- and hyper-viscosity. When attractions are strong, separating the interparticle microviscosity into contributions from attractions and repulsions reveals an attractive undershoot and a repulsive overshoot, as advection grows strong enough to break interparticle bonds downstream and drain the wake. In contrast to linear-response rheology that is predictable entirely by B₂ for short-ranged attractions, here the non-Newtonian viscosity is not, owing to the additional length scale introduced by the boundary layer. The ratio of external to attractive forces eventually supersedes B₂ as the relevant predictor of structure and rheology. This behavior may provide interesting connections to active motion in biological systems where attractive forces are present.