# Hydrodynamic Coupling of Two Brownian Spheres to a Planar Surface

## Overview

Reference:

- [1] Dufresne, E. R., Squires, T. M. & Brenner, M. P.
*Phys. Rev. Lett.***85**, 3317-3320 (2000).

Keywords: Brownian Motion, Diffusion, Colloid, Digital Video Microscopy

## Summary

Dufresne, Squires, Brenner, and Grier studied the relative and cooperative diffusion of two micron-sized silica spheres in electrolyte solution. The authors found that the proximity of a flat plate of glass affected the diffusion of the colloidal spheres and were able to characterize the effect.

In the figure below (Fig. 1 from [1]), the inset diagram shows the two spheres, the position vector between them, the height above the glass plate, and the axes defined to study the motion of the spheres parallel to the position vector, perpendicular to the position vector, and along the z-axis perpendicular to the glass plate.

The particles were rapidly trapped and released by optical tweezers. This method allowed the experimenters to gather many diffusion measurements for each separation r and each height h they were interested in. Letting the particles diffuse for only a short time allowed the researchers to minimize diffusion in the z-direction. The particle tracking would not be as accurate if the particles had moved substantially in the z-direction. The figure above (Fig. 1 from [1]) shows the relationship between the cooperative motion (<math>\vec{r}_1+\vec{r}_2</math>) in the perpendicular direction of the two particles over time for the starting conditions: <math> h = 25.5 \pm 0.7 \mu m </math> and <math> r=7.00 \pm .25 \mu m </math>. Such a procedure was repeated for r between 2 and 10 microns and h between 1.55 and 25.5 microns, and the resulting diffusion was split up into parallel and perpendicular components

For spheres far above the plate, the following set of equations which simply ignore the plate should describe the particles' diffusion quite well:

(1) <math>\frac{D_\perp^{C,R}(r)}{2D_{0}} = 1\pm\frac{3a }{4r}+O\frac{a^3}{r^3}</math>

(2) <math>\frac{D_\parallel^{C,R}(r)}{2D_{0}} = 1\pm\frac{3a }{2r}+O\frac{a^3}{r^3}</math>

When the authors include the plate in their calculation, they use the "method of images," and derive the following equations:

(3) <math>\frac{D_\perp^{C,R}(r,h)}{2D_{0}} = 1-\frac{9a}{16h}\pm\frac{3a}{4r}\left(1-\frac{1+\frac{3}{2}\xi}{(1+\xi)^\frac{3}{2}}\right)</math>

(4) <math>\frac{D_\parallel^{C,R}(r,h)}{2D_{0}} = 1-\frac{9a}{16h}\pm\frac{3a}{2r}\left(1-\frac{1+\xi+\frac{3}{2}\xi^2}{(1+\xi)^\frac{5}{2}}\right)</math>

<math> \xi=\frac{4h^2}{r^2}</math>

These last two equations depend on the spheres' height above the plane, h, in addition to r. When the spheres are close to the wall, the interaction with the wall dominates over the spheres' interactions with each other.

## Soft Matter Details

**Interactions on the scale of kT:**

The experiments were carried out in an electrolyte solution to minimize electrostatic interactions and isolate the spheres' diffusive behavior. Under the experimental conditions, one lone sphere's diffusion coefficient was expected to be given by: <math>{D_0} = \frac{k_bT}{6\pi\eta a}</math> where <math>\eta</math> is viscosity and a is radius This diffusion coefficient scales with <math>k_bT</math> as do many quantities in soft matter physics. The relationship here is not surprising because the diffusion is driven by brownian motion.

**Determining behavior near the critical point between two different regimes:**

The authors of this letter present two sets of equations on their way to developing equations (3) and (4). The first set (1) and (2) is relevant far away from the glass plate, and the second (not in this summary) applies when the spheres are very near the glass plate. The authors carefully state that neither set of equations adequately describes the diffusion in between these two regimes. The field of Soft Matter involves many instances of transitions from one phase (or regime) to another phase (or regime). Oftentimes, the extremes are well understood, while the area right around the transition is very complicated and poorly understood. For example, it is simpler to understand a Hookean solid and a viscous liquid, than it is to understand the viscoelastic transition between these two regimes. In the case of Dufresne *et. al.*, the authors were able to develop equations (3) and (4) which fit all their experimental data well.

**Experimental Methods:**

To study the diffusion of two colloidal particles near a wall, the scientists used optical tweezers and digital video microscopy.