Week 2 : Hydrodynamic Coupling of Two Brownian Spheres to a Planar Surface, E. R. Dufresne, T. M. Squires, M. P. Brenner and D. G. Grier, Phys. Rev. Lett,85, 3317 (2000).

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The authors present a theoretical analysis of brownian motion for a two particle system in the presence of boundary using the method of images, approximating the flow around the sphere as a stokeslet. They also present experimental evidence to validate their results.


The authors used a single laser to position two colloidal spheres in a plane perpendicular to the optical axis simulatnesouly at a set horizontal distance with respect to one antoher. Two spheres with simultaneously trappes by oscialltion the x-y position of the beam using a mirror, and the particules were allowed to freely diffuse after setting the initial positions by diverting the beam. A salt solution was used to reduce the debye length to 7 nm, ensuring minimization of surface forces interfering with measurements. The authors did not report the mean fluctuation from the focal plane, but commented that it did not interefere with measurements. They made measurements for several different distances from the surface between 1 micron and 30 microns (using 0.5 micron particles) and particle-particle distances of 2 and 10 microns. It would be interesting to see near field interactions on the order of nm. It is unclea rto the reader what the distance threshold is for their theory to hold (though we do know how the error scales).

The authors computed cooperative motion <math>{\vec{p}}</math>, and relative motion <math>{\vec{r}}</math>, for particle positions of <math>{\vec{r_1}}</math> and <math>{\vec{r_2}}</math>.

<math>{\vec{p} = \vec{r_1} + \vec{r_2}}</math>

<math>{\vec{r} = \vec{r_1} - \vec{r_2}}</math>

For different time steps, the authors computed the variance in the cooperative motion and found the diffusion (observeed, not nominal due to addition/subtraction of potisions) coefficient by fitting a linear model to the variance of the ensemble of p and r values vs time steps. An example of this analysis is shown below in Figure 1 for a set separation and height from the surface.


The authors employ the method of images where the symmetry plane is taken as the coverslip surface. They consider the particle to be coupled to itself, it's image, the neigboring particle, and the neighboring particle's image. They approximate the flow around the sphere as a stokeslet, and arrive at the following diffusion coefficients for cooperative (C) and relative (R) motion paralell and perpendicular to the surface: EQ1.png


<math>{\xi = 4(h^2/a^2)}</math>


The observed diffusion coefficients along with the nominal values from the theory is shown below in Figure 2. D_|| refers to diffusion in the direction paralell to the surface. Dashed lines result from linear superposition of diffusion coefficients, which is a naive initial analysis. Bold curves are the theoretical values resulting from the theory proposed above.


Figure 1 - Experimental and theoretical results. Dashed curves result from linear superposition of diffusion coefficients, while bold curves result from the above theory. a,b are for one h = 1.4 microns, c,d for h= 25.5 microns for 0.45 micron beads.

Their theoretical results match the data well.

The mean square deivations averaged over all r values is shown below in Figure 2.


Figure 2: "Mean-squared deviations between measured and predicted diffusion coefficients for relative perpendicular motion, averaged over initial separations r. Dashed lines are guides to the eye and emphasize the a�h leading-order error in the linear superposition model’s predictions."