^{1,a)}and John F. Brady

^{1}

### Abstract

We consider a “probe” particle translating at constant velocity through an otherwise quiescent dispersion of colloidal “bath” particles, as a model for particle-tracking microrheology experiments in the active (nonlinear) regime. The probe is a body of revolution with major and minor semiaxes and , respectively, and the bath particles are spheres of radii . The probe’s shape is such that when its major or minor axis is the axis of revolution the excluded-volume, or contact, surface between the probe and a bath particle is a prolate or oblate spheroid, respectively. The moving probe drives the microstructure of the dispersion out of equilibrium; counteracting this is the Brownian diffusion of the bath particles. For a prolate or oblate probe translating along its symmetry axis, we calculate the nonequilibrium microstructure to first order in the volume fraction of bath particles and over the entire range of the Péclet number , neglecting hydrodynamic interactions. Here, is defined as the non-dimensional velocity of the probe. The microstructure is employed to calculate the average external force on the probe, from which one can infer a “microviscosity” of the dispersion via Stokes drag law. The microviscosity is computed as a function of the aspect ratio of the probe, , thereby delineating the role of the probe’s shape. For a prolate probe, regardless of the value of , the microviscosity monotonically decreases, or “velocity thins,” from a Newtonian plateau at small until a second Newtonian plateau is reached as . After appropriate scaling, we demonstrate this behavior to be in agreement with microrheology studies using spherical probes [Squires and Brady, “A simple paradigm for active and nonlinear microrheology,” Phys. Fluids17(7), 073101 (2005)] and conventional (macro-)rheological investigations [Bergenholtz *et al.*, “The non-Newtonian rheology of dilute colloidal suspensions,” J. Fluid. Mech.456, 239–275 (2002)]. For an oblate probe, the microviscosity again transitions between two Newtonian plateaus: for (to two decimal places) the microviscosity at small is greater than at large (again, velocity thinning); however, for the microviscosity at small is less than at large , which suggests it “velocity thickens” as is increased. This anomalous velocity thickening—due entirely to the probe shape—highlights the care needed when designing microrheology experiments with non-spherical probes.

This research was supported in part by Grant No. CTS -0500070 from the National Science Foundation.

I. INTRODUCTION

II. NONEQUILIBRIUM MICROSTRUCTURE

A. Prolate probe

B. Oblate probe

III. MICROVISCOSITY

IV. ANALYTICAL RESULTS

A. Near equilibrium

B. Far from equilibrium

1. Oblate probe

2. Prolate probe

V. NUMERICAL METHODS

A. Legendre polynomial expansion

B. Finite differences

VI. RESULTS

VII. DISCUSSION

### Key Topics

- Colloidal systems
- 14.0
- Torque
- 12.0
- Materials properties
- 7.0
- Diffusion
- 6.0
- Boundary value problems
- 4.0

## Figures

Definition sketch for the prolate probe. Here, the aspect ratio . The circle of radius unity represents a bath particle, which is contacting the probe. The probe moves at constant velocity along its axis of revolution, the axis. The excluded-volume surface (broken curve) is formed by “rolling” the bath particle over the probe’s surface.

Definition sketch for the prolate probe. Here, the aspect ratio . The circle of radius unity represents a bath particle, which is contacting the probe. The probe moves at constant velocity along its axis of revolution, the axis. The excluded-volume surface (broken curve) is formed by “rolling” the bath particle over the probe’s surface.

Definition sketch for the oblate probe. Here, the aspect ratio , and the legend is the same as in Fig. 1. Again, the probe moves along its axis of revolution (the axis) at constant velocity.

Definition sketch for the oblate probe. Here, the aspect ratio , and the legend is the same as in Fig. 1. Again, the probe moves along its axis of revolution (the axis) at constant velocity.

Plot of the mobility factors versus probe aspect ratio . The broken line is the oblate factor and the solid line is the prolate factor .

Plot of the mobility factors versus probe aspect ratio . The broken line is the oblate factor and the solid line is the prolate factor .

Microviscosity increments at small as a function of probe’s aspect ratio . The broken line is the oblate increment and the solid line is the prolate increment .

Microviscosity increments at small as a function of probe’s aspect ratio . The broken line is the oblate increment and the solid line is the prolate increment .

Sketch of the microstructure around an oblate probe at large .

Sketch of the microstructure around an oblate probe at large .

Sketch of the microstructure around an prolate probe at large .

Sketch of the microstructure around an prolate probe at large .

Microviscosity increments at large as a function of probe’s aspect ratio . The legend is the same as in Fig. 4.

Microviscosity increments at large as a function of probe’s aspect ratio . The legend is the same as in Fig. 4.

Difference in microviscosity increments at small and large as a function of probe aspect ratio . The broken(solid) line is the difference in the oblate(prolate) increment

Difference in microviscosity increments at small and large as a function of probe aspect ratio . The broken(solid) line is the difference in the oblate(prolate) increment

Sample finite difference grid for a prolate probe. Here, ; there are grid points; and . Note the high density of mesh points near the excluded-volume surface.

Sample finite difference grid for a prolate probe. Here, ; there are grid points; and . Note the high density of mesh points near the excluded-volume surface.

Microstructural deformation, , in the symmetry plane of the prolate probe as a function of . Here, and the probe moves from left to right. The excluded-volume surface is shown with zero deformation, ; darker regions imply accumulation, ; and lighter regions represent depletion, . The closed curve inside the excluded-volume surface is the probe itself.

Microstructural deformation, , in the symmetry plane of the prolate probe as a function of . Here, and the probe moves from left to right. The excluded-volume surface is shown with zero deformation, ; darker regions imply accumulation, ; and lighter regions represent depletion, . The closed curve inside the excluded-volume surface is the probe itself.

Microstructural deformation, , in the symmetry plane of the oblate probe as a function of . Here, and the probe moves from left to right. The shading scheme is the same as in Fig. 10.

Microstructural deformation, , in the symmetry plane of the oblate probe as a function of . Here, and the probe moves from left to right. The shading scheme is the same as in Fig. 10.

Microviscosity increment for a prolate probe, , as a function of for different : circles, ; squares, ; and triangles, . The filling pattern indicates the method of numerical solution: filled, Legendre expansion; and empty, finite differences. The solid line is the microviscosity increment for a spherical probe, [Squires and Brady (2005)].

Microviscosity increment for a prolate probe, , as a function of for different : circles, ; squares, ; and triangles, . The filling pattern indicates the method of numerical solution: filled, Legendre expansion; and empty, finite differences. The solid line is the microviscosity increment for a spherical probe, [Squires and Brady (2005)].

Microviscosity increment for an oblate probe, , as a function of for different . The legend is the same as in Fig. 12.

Microviscosity increment for an oblate probe, , as a function of for different . The legend is the same as in Fig. 12.

Comparison of microviscosity increments from prolate and oblate probes with the macroviscosity [Bergenholtz *et al.* (2002)] and the microviscosity from a spherical probe [Squires and Brady (2005)]. Symbol legend: diamonds, oblate ; squares, oblate ; triangles, prolate ; circles, prolate . A sketch of the probe for each case is also shown. The solid line is the macroviscosity and the broken line is the microviscosity for a spherical probe, . For the microviscosity , and for the macroviscosity (with the shear rate).

Comparison of microviscosity increments from prolate and oblate probes with the macroviscosity [Bergenholtz *et al.* (2002)] and the microviscosity from a spherical probe [Squires and Brady (2005)]. Symbol legend: diamonds, oblate ; squares, oblate ; triangles, prolate ; circles, prolate . A sketch of the probe for each case is also shown. The solid line is the macroviscosity and the broken line is the microviscosity for a spherical probe, . For the microviscosity , and for the macroviscosity (with the shear rate).

Sketch of a prolate probe translating at angle to its symmetry axis: (a) near-longwise motion, ; and (b) near-broadside motion, . Note, the axis is directed out of the page.

Sketch of a prolate probe translating at angle to its symmetry axis: (a) near-longwise motion, ; and (b) near-broadside motion, . Note, the axis is directed out of the page.

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