A theoretical bridge between linear and nonlinear microrheology

Abstract

Passive 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 fluctuation-dissipation 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 (ω→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 ω→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.