View source for 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).

Wiki Entry by Robin Kirkpatrick, AP 225, Fall 2011
== Introduction ==
The authors present a theoretical analysis of brownian motion for a two particle system in the presence of a boundary using the method of images, approximating the flow around the sphere as a stokeslet. They also present experimental evidence to validate their results.

== Methodology ==
The authors used a single laser to position two colloidal spheres in a  plane perpendicular to the optical axis simultanesouly at a set horizontal distance with respect to one another. Two spheres were simultaneously trapped by oscillation the x-y position of the beam using a mirror, and the particles 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 interfere 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. Note that the results are linear, validating the assumption that convenction terms can be neglected for the time scales used. 

[[Image:RRLKFig1.png]]

Figure 1 - Example analysis of computing the diffusion coefficient for collective motion. This is not to be confused with the diffusion coefficient for a single particle.

== Theory ==
The no-slip condition at the particle surfaces and and the coverslip surface results in coupled motion from creating local velocity fields. 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:

[[Image:EQ1.png]]


[[Image:EQ2.png]]

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

It should be noted that the results only depend on a dimensionless distance parameter. Therefore, near wall hindrance and hydrodynamic coupling occurs on the scales of the particles being studied.

== Results ==
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 that fails near the surface. Bold curves are the theoretical values resulting from the theory proposed above.

[[Image:RLKFig1.png]]

Figure 2- 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 are on the same order of magnitude of the data and their theoretical results are an improvement over simply considering the linear superposition of diffusion coefficient as seem in Figure 2. The mean square deivations averaged over all r values is shown below in Figure 2.

[[Image:Fig4bR.png]]

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."

The authors also show the theoretical eigenvectors of the diffusivity tensor which the authors also derived using the method of images, which reveals the independent directions of the brownian motion at different distances from the surfaces. As seen below in Figure 3, collective and relative motion become coupled when h is on the same order as r. The scale over which coupling is significant can occur over 5 particle diameters.

[[Image:RLKFig3.png]]

Fig 3: "Cross-sectional view of the diffusive modes for two
spheres near a wall. Collective motion normal to the wall becomes
increasingly coupled with relative motion parallel to the
wall as h approaches r. Collective normal modes at large r cross
over continuously to relative parallel modes as r decreases. The
dashed line at x � 0 indicates the symmetry plane."


== Keywords ==
[[Method of Images]], [[Brownian Motion]], [[Hydrodynamic Coupling]]