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Pair diffusion near a wall

Figure 7: Measured in-plane pair diffusion coefficients for two 1  $ \mathrm{\mu m}$-diameter silica spheres in water at $ T = 29\ensuremath{^\circ\mathrm{C}}\xspace $ at height $ h = 1.6~\ensuremath{\mathrm{\mu m}}\xspace $ above a glass wall. Dashed curves result from linear superposition of drag coefficients, while solid curves result from stokeslet analysis.

In his 1927 treatise, Oseen suggested that Faxén's results might be applied to more complicated systems even if his methods could not [63]. Oseen pointed out that the drag on a sphere moving in direction $ \alpha$ near a wall can be factored into the sphere's drag in an unbounded system and an additional contribution $ \gamma^W_\alpha(h)$ due to the wall:

$\displaystyle \gamma_\alpha(h) = \gamma_0 + \gamma^W_\alpha(h).$ (40)

He proposed that the drag coefficient for motion in a given direction might be approximated by linearly superposing individual contributions $ \gamma_{\alpha i}$ from all bounding surfaces and neighboring particles:

$\displaystyle \gamma_\alpha(\vec r) \approx \gamma_0 + \sum_{i=1}^N \gamma_{\alpha i}(\vec r - \vec r_i).$ (41)

Oseen emphasized that this cannot be rigorously correct because it violates boundary conditions on all surfaces. Even so, if the surfaces are well-separated, the errors may be acceptably small. Given this hope, Oseen's linear superposition approximation has been widely adopted.

The data in Fig. 7 show measured in-plane pair diffusion coefficients for 1  $ \mathrm{\mu m}$-diameter spheres positioned by optical tweezers at $ h = 1.55 \pm 0.66~\ensuremath{\mathrm{\mu m}}\xspace $ above a glass wall. Naively adding the drag coefficients [58] due to sphere-sphere and sphere-wall interactions yields $ D_\psi^{-1}(r,h) = D_\psi^{-1}(r) + [D_\Vert^{-1}(h) - D_0^{-1}]/2$, whose predictions appear as dashed curves in Fig. 7 and agree poorly with measured diffusivities.

Figure 8: Left: Hydrodynamic interactions for two spheres near a planar boundary. Right: Normal modes of motion obtained from stokeslet analysis of this system [64].
\includegraphics[width=2in]{figures/conf_pair} \includegraphics[width=2.5in]{figures/diffmodes}

A more complete treatment not only resolves these quantitative discrepancies but also reveals an additional influence of the bounding surface: the highly symmetric and experimentally accessible modes parallel to the wall are no longer independent. As shown in Fig. 8, each sphere interacts with its own image, its neighbor, and its neighbor's image. These influences contribute $ \ensuremath{\mathsf{b}}\xspace ^e_{i\alpha,j\beta} = (1 - \delta_{ij}) \,
...vec r_j)
+ \ensuremath{\mathsf{G}}\xspace ^W_{\alpha\beta}(\vec r_i - \vec R_j)$ to the mobility of sphere $ i$ in the $ \alpha$ direction. Eigenvectors of the corresponding diffusivity tensor appear in Fig. 8. The independent modes of motion are rotated with respect to the bounding wall by an amount which depends strongly on both $ r$ and $ h$. Even though the experimentally measured in-plane motions are not independent, they still satisfy Eq. (32) with pair-diffusion coefficients $ D^{C,R}_\alpha(r,h) =
D_{1\alpha,1\alpha}(r,h) \pm D_{1\alpha,2\alpha}(r,h)$, where the positive sign corresponds to collective motion, the negative to relative motion, and $ \alpha$ indicates directions either perpendicular or parallel to the line connecting the spheres' centers. Explicitly, we obtain [64]

$\displaystyle \frac{D^{C,R}_\perp(r,h)}{2D_0}$ $\displaystyle =$ $\displaystyle 1 - \frac{9}{16} \, \frac{a}{h} \pm
\frac{3}{4} \, \frac{a}{r} \left[
1 - \frac{1 + \frac{3}{2} \, \xi}{(1 + \xi)^{3/2}}
\right]$   and (42)
$\displaystyle \frac{D^{C,R}_\Vert(r,h)}{2D_0}$ $\displaystyle =$ $\displaystyle 1 - \frac{9}{16} \, \frac{a}{h} \pm
\frac{3}{2} \, \frac{a}{r} \left[
1 - \frac{1 + \xi + \frac{3}{2} \, \xi^2}{(1 + \xi)^{5/2}}
\right]$ (43)

up to $ {\cal O} \left(a^3/r^3\right)$ and $ {\cal O} \left(a^3/h^3\right)$, where $ \xi = 4h^2/r^2$. These results appear as solid curves in Figs. 6 and 7. Not only does the stokeslet approximation perform better than linear superposition for spheres close to a wall, it performs equally well at all separations we have examined [64].

next up previous
Next: Electrohydrodynamic Coupling Up: Hydrodynamic Interactions Previous: Hydrodynamic coupling to a
David G. Grier 2001-01-16