Brownian Dynamics of a Sphere Between Parallel Walls

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Original Entry by Tom Dimiduk, AP225 Fall 2010

Brownian Dynamics of a Sphere Between Parallel Walls E. R. Dufresne, D. Altman and D. G. Grier

Soft matter Keywords

Brownian motion, Surface

Summary

Video microscope setup used to make these measurements. B) shows that their sample was a 2cm x 1 cm cover slip on a glass slide. It was anchored with an 8 micron gap above the slide using norland optical adhesive. Glass capillaries at either end allow flow through the cell. A) Shows the slide as imaged in an inverted microscope.
z)</math>, fit to eq. (1) for <math>D_i(h)</math>."
Fig 3: "Height dependence of the in-plane diffusion coefficient for a Brownian sphere in a slit pore. The solid curve results from the stokeslet approximation and is indistinguishable from the linear superposition approximation’s prediction. The dashed curves show the range of predicted values due to the uncertainty in the sphere’s radius. The squares show Faxen's predictions from eq. (6)."

To the right is a diagram showing my best interpretation of the apparatus the authors used to make high resolution measurements of the brownian motion of a sphere confined along one dimension by two glass planes 8 microns apart. They positioned a 1 micron sphere in a precise position using an optical trap, then released it to study motion. They imaged with a video microscope at 60 Hz and used particle centroiding to obtain a resolution of 20 nm in x and y. They obtained z information by repeating the experiment releasing the particle from the same location with the microscope focused at differing heights. They measure distance from the wall by running a steady poiseuille flow and observing its effect on the particle velocity.

Their results are shown in figures 1 and 3. Figure 1 is nothing remarkable, simply showing that (a) the distribution of locations widens over time, (b) the centers drift linearly in the applied flow, and (c) the squared deviation increases linearly.

Figure 3 is their real result, showing that the velocity falls off towards each wall. They present several equations that predict this behavior

The simplest, which models the flow well is linear superposition of a basic sphere near a wall equation

<math> \gamma_2(h) \approx \gamma_0 + [\gamma_1(h) - \gamma_0] + [\gamma_1(H-h) - \gamma_0] </math>

where <math> \frac{\gamma_0}{\gamma_1(h)} \approx 1 - \frac{9}{16} \frac{a}{h} </math>

They also work out a more complicated solution using greens functions and image flow, which exactly satisfies the boundary values, but for their experiment, it gives identical predictions to the superposition solution.

Conclusions

The influence of boundaries on diffusion of particles is of significant importance to a number of industrial processes and much of biology. Quantitative treatment for the interaction of particles with multiple boundaries or multiple particles near a boundary has been difficult. This paper presents both an experimental and theoretical approach for attacking these kinds of situations.