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Next: Hydrodynamic coupling to a Up: Pair Diffusion Previous: Mobility and diffusivity

Normal mode diffusivity

In the particular case of two identical spheres, diagonalizing the diffusivity tensor yields diffusion coefficients for collective (C) motion of the center-of-mass coordinate $ \vec R = \vec r_1 + \vec r_2$ and relative (R) motion $ \vec r = \vec r_1 - \vec r_2$, with one set of normal modes directed perpendicular ($ \perp$) to the initial separation, and the other parallel ($ \Vert$):

$\displaystyle D^{C,R}_\perp (r)$ $\displaystyle = \frac{D_0}{2} \, \left[ 1 \pm \frac{3}{2} \, \frac{a}{r} + \ensuremath{{\cal O} \left(\frac{a^3}{r^3}\right)}\right]$ (33)
$\displaystyle D^{C,R}_\Vert (r)$ $\displaystyle = \frac{D_0}{2} \, \left[ 1 \pm \frac{3}{4} \, \frac{a}{r} + \ensuremath{{\cal O} \left(\frac{a^3}{r^3}\right)}\right].$ (34)

Positive corrections apply to collective modes and negative to relative. The collective diffusion coefficients $ D^C_\perp$ and $ D^C_\Vert$ are enhanced by hydrodynamic coupling because fluid displaced by one sphere entrains the other. Relative diffusion is suppressed by the need to transport fluid into and out of the space between the spheres.

Batchelor [60] obtained a series solution of the Stokes equation for this system in 1976, obtaining additional terms up to $ {\cal O} \left(a^6/r^6\right)$. This observation highlights both the weakness and the strength of stokeslet analysis. The stokeslet $ \ensuremath{\mathsf{G}}\xspace ^S$ depends only on a sphere's position and not on its radius. Consequently, Eq. (28) contains information only to linear order in $ a$. On the other hand, the same approach can be applied readily to other configurations, and the associated diffusivity tensor diagonalized to obtain the independent dynamical modes.

Deriving Eqs. (33) and (34) required enough approximations that the reader might be concerned about the results' accuracy. Figure 5 compares their predictions with measured diffusion coefficients for silica spheres of radius $ a = 0.495 \pm 0.025~\ensuremath{\mathrm{\mu m}}\xspace $ (Lot 21024, Duke Scientific, Palo Alto, CA) suspended in water. These measurements were obtained with optical tweezers using techniques similar to those in Sec. 2 except that electrostatic interactions were minimized with added salt.

Two spheres were captured with optical tweezers, raised to $ h = 25~\ensuremath{\mathrm{\mu m}}\xspace $ above the nearest surface and released, their motions being tracked through digital video microscopy for 5/30 sec thereafter. Repeatedly positioning and releasing the spheres over a range of initial separation yielded statistically large samples of the spheres' hydrodynamically coupled motions in the plane. These were binned in $ r$ and analyzed according to Eq. (32) to extract the plotted diffusivities. In the absence of hydrodynamic coupling, and thus at large separations, the spheres' pair diffusivities should be $ 2D_0$, with the free self-diffusion coefficient expected to be $ D_0 = 0.550 \pm 0.028~\ensuremath{\mathrm{\mu m}}\xspace ^2/\mathrm{sec}$ at the experimental temperature $ T = 29.00 \pm 0.05\ensuremath{^\circ\mathrm{C}}\xspace $. This limiting behavior appears as a dashed line in Fig. 5 and agrees with the experimental data at large separations. Eqs. (33) and (34) yield the dashed curves in Fig. 5 and agree well with the measured diffusivities, with no adjustable parameters.

Figure 9: Measured pair diffusion for 1  $ \mathrm{\mu m}$-diameter silica spheres in water at $ T = 29\ensuremath{^\circ\mathrm{C}}\xspace $. Dashed curves result from stokeslet analysis (Eqs. (33) and (34)). Solid curves include corrections for proximity to a wall (Sec. 3.4).

Crocker [61] previously reported a larger data set for two polystyrene microspheres somewhat closer to the nearest bounding wall whose dynamics agreed less well with Eqs. (33) and (34). Crocker surmised that the spheres' hydrodynamic coupling to the nearer wall reduced their diffusivities, and that this affected collective diffusion more than relative. However, a quantitative analysis of a wall's influence was not available and the correction was treated semi-empirically. Stokeslet analysis provides the necessary accounting for confinement's influence.

next up previous
Next: Hydrodynamic coupling to a Up: Pair Diffusion Previous: Mobility and diffusivity
David G. Grier 2001-01-16