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Active and nonlinear microrheology experiments involve a colloidal probe that is forced to move within a material, with the goal of recovering the nonlinear rheological response properties of the material. Various mechanisms cause discrepancies between the nonlinear rheology measured microrheologically and macroscopically, including direct probe-bath collisions, the Lagrangian unsteadiness experienced by the material elements, and the spatially inhomogeneous and rheologically mixed strain field set up around the probe. Here, we perform computational nonlinear microrheology experiments, in which a colloidal probe translates through a dilute suspension of Brownian ellipsoids, whose results we compare against analogous computational experiments on the macroscopic shear rheology of the same model material. The quantitative impact of each of the mechanisms for micro-macro-discrepancy can thus be computed directly, with additional computational experiments performed where the processes in question are 'turned off'. We show that all three discrepancy mechanisms impact the microrheological measurement quantitatively, and that none can be neglected. This motivates a search for microrheological probes whose geometry or forcing is optimized to minimize these impacts, which we present in a companion article.

Stress development, relaxation, and memory in colloidal dispersions: Transient nonlinear microrheology J. Rheol. 57, 457 (2013); 10.1122/1.4775349Rheo-SANS investigation of acicular-precipitated calcium carbonate colloidal suspensions through the shear thickening transitionPassive microrheology exploits the fluctuation-dissipation theorem to relate thermal fluctuations of a colloidal probe to the near-equilibrium linear response behavior of the material through an assumed generalized Stokes Einstein relation (GSER). Active and nonlinear microrheology, on the other hand, measures the nonlinear response of a strongly driven probe, for which fluctuationdissipation does not hold. This leaves no clear method for recovering the macroscopic rheological properties from such measurements. Although the two techniques share much in common, there has been little attempt to relate the understanding of one to the other. In passive microrheology, the GSER is generally assumed to hold, without the need for explicit calculation of the microstructural deformation and stress, whereas in nonlinear microrheology, the microstructure must be explicitly determined to obtain the drag force. Here we seek to bridge the gap in understanding between these two techniques, by using a single model system to explicitly explore the gentle-forcing limit, where passive (x ! 0) and active (U ! 0) microrheology are identical. Specifically, we explicitly calculate the microstructural deformations and stresses as a microrheological probe moves within a dilute colloidal suspension. In the gentle-forcing limit, we find the microstructural stresses in the bulk material to be directly proportional to the local strain tensor, independent of the detailed flow, with a prefactor related to the effective shear modulus. A direct consequence is that the probe resistance due to the bulk stresses in passive (linear response) microrheology quantitatively recovers the results of macroscopic oscillatory shear rheology. Direct probe-bath interactions, however, lead to quantitative discrepancies that are unrelated to macroscopic shear rheology. We then examine the microstructural equations for nonlinear microrheology, whose U ! 0 limit reduces to the x ! 0 limit in passive microrheology. Guided by the results from passive microrheology, we show that direct probe-material interactions are unrelated to the macroscopic shear rheology. Moreover, we show that the bulk microstructural deformations (which quantitatively recover macroscopic shear rheology in the linear limit) now obey a governing equation that differs qualitatively from macroscopic rheology, due to the spatially dependent, Lagrangian unsteady mixture of shear and extensional flows. This inherently complicates any quantitative interpretation of nonlinear microrheology.

In this second article devoted to 'computational experiments' of nonlinear microrheology, we examine the effect that changing the probe shape or motion has upon the three sources of discrepancy that we previously examined for spheres. In particular, prolate ellipsoidal probes have relatively long regions of relatively constant strain rate, giving predominantly shear and relative Lagrangian steadiness. The micro-macro discrepancy is shown not to arise from Lagrangian unsteadiness, but largely from the non-viscometric nature of the flows. Second, an oblate ellipsoidal probe exacerbates the extensional regions in front of and behind the probe. However, the relatively low extensional rates around such 'disks' would require them to be pulled at much higher rates through the fluid in order to excite the extensional deformations. Because our model material thickens under uniaxial extension, but thins under biaxial extension, the contribution of each to the total drag is partially negated by the other. Finally, we examine a rotating spherical probe, which is Lagrangian steady and pure shear. We show that the apparent viscosity thus recovered is close to the true shear viscosity, and furthermore that the true shear viscosity can be extracted quantitatively from the apparent microviscosity.

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