A simple paradigm for active and nonlinear microrheology

Abstract

In microrheology, elastic and viscous moduli are obtained from measurements of the fluctuating thermal motion of embedded colloidal probes. In such experiments, the probe motion is passive and reflects the near-equilibrium ͑linear response͒ properties of the surrounding medium. By actively pulling the probe through the material, further information about material properties can be obtained, analogous to large-amplitude measurements in ͑macro-͒ rheology. We consider a simple model of such systems: a colloidal probe pulled through a suspension of neutrally buoyant bath colloids. We choose a system with hard-sphere interactions but neglect hydrodynamic interactions, which is simple enough to permit analytic solutions, but nontrivial enough to raise issues important for the interpretation of experiments in active and nonlinear microrheology. We calculate the microstructural deformation for arbitrary probe size and pulling rate ͑expressed as a dimensionless Péclet number Pe͒. From this, we determine the average retarding effect on the probe due to the microstructure, as well as fluctuations about this average. The high-Pe limit is singular, giving a finite Brownian contribution even in the limit of negligible diffusion. Significantly, different results are obtained for probes driven at constant velocity and constant force. Furthermore, we demonstrate that a probe pulled with an optical tweezer ͑roughly a harmonic well͒ can behave as fixed-force, fixed-velocity, or as a mixture of those modes, depending on the strength of the trap and on the pulling speed. More generally, we discuss how these results relate to previous work on the rheology of colloidal suspensions. Not surprisingly, the present theory ͑which ignores hydrodynamic interactions͒ gives shear thinning but no shear thickening; we expect that the incorporation of hydrodynamics would result in shear thickening as well. The effective micro-and macro-viscosities, when appropriately scaled, are in semi-quantitative agreement. This seems remarkable, given the rather significant difference in the two methods of measurement. However, for more complicated or unknown materials, where such scaling relations may not be known in advance, the comparison between micro-and macro may not be so favorable, which raises important questions about the relation between micro-and macrorheology. Finally, by analogy with previous work on macrorheology, we propose methods to scale up the present ͑dilute͒ theory to account for more concentrated suspensions, and suggest new active microrheological experiments to probe different aspects of suspension behavior.

The force on a boundary in active matter

“…And note that this is true regardless of the relative magnitudes of k B T and k s T s (including the singular athermal limit k B T = 0). This same independence of k B T occurs in the analogous hard-sphere rheology problem at high Péclet numbers (Brady & Morris 1997;Squires & Brady 2005). Also, the ratio ( /δ) 2 = 6D swim /D T = U 0 /D T = Pe is a Péclet number based on the run length that measures the relative importance of swimming to Brownian diffusion.…”

“…With F P = 0, the force the active colloidal particles exert on the body is given exactly by (Brady 1993;Squires & Brady 2005) F = −k B T S B P(x, q, t)n dS, where n is the outer normal to the body surface as shown in figure 3. The force averaged over the orientation of the active particles is…”

Active microrheology in active matter systems: Mobility, intermittency, and avalanches

“…In active microrheology, the properties of a medium are probed using the mobility and velocity fluctuations of an externally driven probe particle that is roughly the same size as the particles that comprise the medium [1][2][3][4] . For example, the nonlinear mobility of a probe particle dragged through a colloidal system changes across the glass transition [5][6][7][8] .…”

Shear Thinning and Local Melting of Colloidal Crystals

“…Driving the probe particle by optical tweezing can lead to three different modes: constant force, constant velocity or a mixed mode [9,14]. At constant force the probe can move around obstacles while at constant velocity the particle will largely resist lateral motions thus leading to significantly larger deformations of the material.…”