# Hydrodynamic Coupling of Two Brownian Spheres

Original entry by Hyerim Hwang, AP 225, Fall 2011.

## Reference

Eric R. Dufresne, Todd M. Squires, Michael P. Brenner, and David G. Grier, "Hydrodynamic Coupling of Two Brownian Spheres to a Planar Surface", *Physics Review Letters* **2000** *85(15)*, 3217-3320

## Keywords

Diffusion, Particles, Sedimentation, Motion, Flow

## Introduction

Hydrodynamic properties of colloidal systems have been researched but still remain controversial or unexplained. This study describes an experimental and theoretical investigation of diffusion of colloidal spheres near a flat plate. The geometry they chose is in the range of nonadditive hydrodynamic coupling in confined colloidal suspensions.

## Results

They combined optical tweezer manipulation and digital video microscopy to measure four components of the pair diffusion tensor for two colloidal spheres as a function of their center-to-center separation r and of their height h above a flat glass surface. (Radius of silica spheres : 0.495 um, Height of a layer of water : 140 um, Temperature of sample volume : 29'C)

The diffusion coefficients associated with each mode of motion were obtained from the equation when the angle brackets indicate an ensemble average. Figure 1 shows for one mode of motion at one height and starting separation. Diffusion coefficients extracted from least squares fits to equation appears in Figure 2 as functions of r for the smallest and largest accessible values of h. In the absence of their interactions, the observed trends reflect hydrodynamic coupling between the spheres and bounding surface. Dashed curves in Figure 2 result from linear superposition of drag coefficients and solid lines result from the theory described with no adjustable parameters. Horizontal lines indicate the asymtotic self-diffusion coefficient.

Each sphere interacts with its own iamge, its neighbor, and its neighbor's image. These influences contribute to the mobility of sphere i in the a direction. Eigenvectors of the corresponding diffusivity tensor are shown in Figure 3. The independent modes of motion are rotated with respect to the bounding wall by an amount which depends strongly on both r and h.

## Discussion

Confining surface influences colloidal dynamics at large separations, and three-surface coupling can be descibed by a leading-order stokeslet approximation. Wall-induced hydrodynamic interactions can have influenced nonequilibrium optical tweezer measurements of confined colloidal interactions and contributed to the observed attractions between same-charged particles.