Nonlinear microrheology: Bulk stresses versus direct interactions

In passive microrheology, the linearviscoelastic properties of complex fluids are inferred from the Brownian motion of colloidal tracer particles. Active (but gentle) forcing may also be used to obtain such linear-response information. More significant forcing may drive the material significantly out of equilibrium, thus potentially providing a window into the nonlinear response properties of the material. In leaving the linear-response regime, however, the theoretical underpinning for passive microrheology is lost, and a variety of issues arise. Most generally, what exactly can be measured, and how can such measurements be interpreted? Here we motivate and discuss a variety of theoretical issues facing the interpretation of active microrheology. First., in the continuum limit, the inhomogeneous velocity field around the probe gives rise to rheological inhomogeneities, whereupon an assumed generalized Stokes drag yields a weighted average of the viscosities around the probe rather than the (homogeneous) viscosity measured macroscopically. We then explicitly treat the material microstructure using a model system (a large colloidal probe pulled through a dilute suspension of small bath particles). We examine the different sources of stress upon the probe particle (e.g., direct probe-bath collisions as well as microstructural deformations within the bulk suspension) and discuss their analog (or lack thereof) in the corresponding macrorheological system. We discuss several crucial issues for the interpretation of nonlinear microrheology: (1) how to interpret the inhomogeneous and nonviscometric nature of the deformation field around the probe, (2) the distinction between direct and bulk stresses and their deconvolution, and (3) the (Lagrangian) time-dependent nature of the stress histories experienced by material elements as they advect past the probe. Having identified these issues, we briefly discuss adaptations of the basic technique to recover bulk rheology more faithfully. Whereas we specifically discuss a model colloidal suspension, we ultimately envision a technique capable of measuring the nonlinear rheology of general materials.