A theoretical bridge between linear and nonlinear microrheology

Abstract

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.

Contributions to membrane-embedded-protein diffusion beyond hydrodynamic theories

“…However, we suspect that our model of Section I A will also differ from the linear coupling model of Démery and Dean in terms of the nonlinear response. In [16], the distortion in φ caused by the object-field interaction vanishes as 1/V p at large velocities, but because of the boundary condition we have chosen, there will always be a non-vanishing distortion in φ, though it may occur in a boundary layer near the protein, as in calculations of the nonlinear microrheological drag in colloidal model systems [53,59].…”

“…The integral dℓ (σ + Π) ·n may also be evaluated simply by using an identity derived from the reciprocal theorem [52] of low-Reynolds number fluid mechanics (see [53,54] and references within). This trick lets us determine the total drag force on an object in a fluid flow v in terms of a simpler "reference" flowṽ in the same geometry.…”

“…We can do this in two major routes: 1) directly via Eq. 24, or 2) applying an identity derived from the reciprocal theorem [52][53][54]. Each route has separate advantages: if we use Eq.…”

Enhanced dispersion by elastic turbulence in porous media

“…To account for these differences we determined the shear-dependent viscosity TJ( . :Y) of the solution by active microrheology in geometries similar to our microfluidic device [23][24][25][26] . This was achieved by dragging a silica bead ( a = 3p.m) through an unpatterned sample cell of h = 7 p.m height using a scanning optical tweezer.…”

Single particle motion in a sheared colloidal dispersion

“…Brady (2010, 2012) demonstrated that fluctuations in the motion of a fixed-force probe can interpreted as a force-induced "microdiffusivity," which encodes information on the normal stress differences in the dispersion. The aforementioned studies have focused on the microstructure between the probe and a bath particle; Squires (2008) and DePuit et al (2011) suggest that it is the (unsteady) nonequilibrium microstructure between pairs of bath particles advected past a moving probe that could be more indicative of the macrorheology of the dispersion, particularly when the probe is much larger than the bath particles. It should also be mentioned that the active microrheology of concentrated dispersions has been addressed via dynamic simulations [Carpen and Brady (2005); Winter et al (2012)] and mode-coupling theories [Gazuz and Fuchs (2013); Voigtmann and Fuchs (2013)].…”