^{1,a)}and Markus Rauscher

^{1}

### Abstract

We investigate the nonequilibrium fluid structure mediated forces between two colloids driven through a suspension of mutually noninteracting Brownian particles as well as between a colloid and a wall in stationary situations. We solve the Smoluchowski equation in bispherical coordinates as well as with a method of reflections, both in linear approximation for small velocities and numerically for intermediate velocities, and we compare the results to a superposition approximation considered previously. In particular, we find an enhancement of the friction (compared to the friction on an isolated particle) for two colloids driven side by side as well as for a colloid traveling along a wall. The friction on tailgating colloids is reduced. Colloids traveling side-by-side experience a solute induced repulsion while tailgating colloids are attracted to each other.

The authors thank S. Dietrich for support and fruitful discussions. One of the authors (M.R.) acknowledges the funding by the Deutsche Forschungsgemeinschaft within the priority program SPP 1164 “Micro- and Nanofluidics” under Grant No. RA 1061/2-1.

I. INTRODUCTION

II. TRANSPORT EQUATIONS

A. Bispherical coordinates

1. Numerical solution

2. Expansion in

B. Method of reflections

III. COLLOIDS DRIVEN SIDE BY SIDE

A. Expansion in

1. Bispherical coordinates

2. Method of reflections

B. Discussion and numerical solution

IV. COLLOIDS DRIVEN BEHIND EACH OTHER

A. Expansion in

1. Bispherical coordinates

2. Method of reflections

B. Discussion and numerical solution

V. COLLOID NEAR A WALL

VI. DISCUSSION

### Key Topics

- Colloidal systems
- 113.0
- Friction
- 16.0
- Numerical solutions
- 12.0
- Boundary value problems
- 11.0
- Laplace equations
- 9.0

## Figures

The two-colloid system (a) and the colloid-wall system (b). The colloids are driven with velocity . The gray areas and are forbidden for the center of the solute particles, and the dashed lines represent their surfaces .

The two-colloid system (a) and the colloid-wall system (b). The colloids are driven with velocity . The gray areas and are forbidden for the center of the solute particles, and the dashed lines represent their surfaces .

Contour plots of the stationary density of solute particles near two driven colloids for and . Bright areas correspond to high densities and dark areas to low densities. In (a) the colloids are driven from left to right, and the density is enhanced between the colloids which leads to a repelling force. In (b) they are driven from bottom to top, the bow wave effect in front of the rear colloid and the depletion behind the one in front are reduced, so the friction forces on the two colloids are reduced as well.

Contour plots of the stationary density of solute particles near two driven colloids for and . Bright areas correspond to high densities and dark areas to low densities. In (a) the colloids are driven from left to right, and the density is enhanced between the colloids which leads to a repelling force. In (b) they are driven from bottom to top, the bow wave effect in front of the rear colloid and the depletion behind the one in front are reduced, so the friction forces on the two colloids are reduced as well.

Friction force of ideal solute particles on one of the two colloids driven side by side to first order in . The horizontal line at is the limit for . The same force applies to a colloid which is driven parallel to a planar wall at distance .

Friction force of ideal solute particles on one of the two colloids driven side by side to first order in . The horizontal line at is the limit for . The same force applies to a colloid which is driven parallel to a planar wall at distance .

Repelling force of ideal solute particles between two colloids driven side by side with velocity . The same force acts on a colloid which is driven parallel to a planar wall at distance .

Repelling force of ideal solute particles between two colloids driven side by side with velocity . The same force acts on a colloid which is driven parallel to a planar wall at distance .

Repelling force of ideal solute particles between two colloids driven side by side at distance . The same force acts on a colloid which is driven parallel to a planar wall at distance .

Repelling force of ideal solute particles between two colloids driven side by side at distance . The same force acts on a colloid which is driven parallel to a planar wall at distance .

Friction force induced by the solute particles on one of two colloids driven behind each other to first order in . The horizontal line is the limit for , i.e., the friction force for a single colloid.

Friction force induced by the solute particles on one of two colloids driven behind each other to first order in . The horizontal line is the limit for , i.e., the friction force for a single colloid.

Friction forces induced by solute particles on the two colloids driven behind each other for . The limit of (i.e., the force on a single colloid) is given by .

Friction forces induced by solute particles on the two colloids driven behind each other for . The limit of (i.e., the force on a single colloid) is given by .

Contour plots of the stationary density of ideal solute particles for a colloid driven from left to right parallel to a wall at distance (a) and (b) . (a) is exactly the upper half of Fig. 2(a).

Contour plots of the stationary density of ideal solute particles for a colloid driven from left to right parallel to a wall at distance (a) and (b) . (a) is exactly the upper half of Fig. 2(a).

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