Colloid-colloid hydrodynamic interaction around a bend in a quasi-one-dimensional channel

We report a study of how a bend in a quasi-one-dimensional (q1D) channel containing a colloid suspension at equilibrium that exhibits single-file particle motion affects the hydrodynamic coupling between colloid particles. We observe both structural and dynamical responses as the bend angle becomes more acute. The structural response is an increasing depletion of particles in the vicinity of the bend and an increase in the nearest-neighbor separation in the pair correlation function for particles on opposite sides of the bend. The dynamical response monitored by the change in the self-diffusion $[D_{11}\left(x\right)]$ and coupling $[D_{12}\left(x\right)]$ terms of the pair diffusion tensor reveals that the pair separation dependence of $D_{12}$ mimics that of the pair correlation function just as in a straight q1D channel. We show that the observed behavior is a consequence of the boundary conditions imposed on the q1D channel: both the single-file motion and the hydrodynamic flow must follow the channel around the bend.

Figure 1

Snapshots of (a) a $3\mu \mathrm{m}$ q1D channel with a ${60}^{\circ }$ sharp bend and (b) a $3\mu \mathrm{m}$ q1D channel with a ${60}^{\circ }$ smooth bend. The bend apices are marked with a red “A” and the centerline coordinate $x$ is marked in red. These geometries are approximated by the geometry shown in (c) and (d) for smooth and sharp bends, respectively.

Figure 2

The measured pair potentials between $1.58\mu \mathrm{m}$ colloids in $3.0\mu \mathrm{m}$ channels. This figure is reproduced with permission from Fig. 8 of [10].

Figure 3

Linear particle density as a function of distance from the apex of the bend in q1D channels. The density values have been reduced by dividing by the mean density ($\eta =0.6$).

Figure 4

Linear density as a function of distance from the apex of the bend in a ${60}^{\circ }$ q1D channel. The density values have been reduced by dividing by the mean density.

Figure 5

Pair distribution functions for particles on the same side (${180}^{\circ }$) and on opposite sides of acute bends in smooth (a) and sharp (b) $3\mu \mathrm{m}$ q1D channels.

Figure 6

Separation of nearest-neighbor particles on opposite sides of the bend in a $3\mu \mathrm{m}$ q1D channel as a function of the bend angle for a suspension with $\eta =0.6$. The solid line is a plot of Eq. (7).

Figure 7

Position of the first peak of the pair correlation function for particles on opposite sides of the bend in ${60}^{\circ }$ and ${120}^{\circ }$ q1D channels as a function of packing fraction.

Figure 8

Pair distribution functions obtained from Monte Carlo simulations for suspensions in q1D channels with bend angles ${60}^{\circ }$, ${75}^{\circ }$, ${150}^{\circ }$, and ${180}^{\circ }$.

Figure 9

Dependence of $D_{12}\left(x\right)$ on the pair separation for particles on opposite sides of the bend in channels with (a) sharp and (b) smooth bends.

Figure 10

Dependence of $D_{12}\left(x\right)$ on $x$ in the ${60}^{\circ }$ channel, for several packing fractions.

Figure 11

Comparison of the locations of the first neighbor separation obtained from $g_2\left(x\right)$, the peak in $D_{12}\left(x\right)$, and Eq. (7) (solid line).

Figure 12

Diagrams demonstrating the sharp and smooth bend geometries.

Figure 13

Packing fraction around a particle located at the apex of the bend in a smooth channel with width $\sigma$ and centerline radius of curvature $R\sigma$.