Effects of shear and walls on the diffusion of colloids in microchannels

Colloidal suspensions flowing through microchannels were studied for the effects of both the shear flow and the proximity of walls on the particles' self-diffusion. Use of hydrostatic pressure to pump micron-sized silica spheres dispersed in water-glycerol mixture through poly(dimethylsiloxane) channels with a cross section of $30\times 24\mu {\mathrm{m}}^2$, allowed variation in the local Peclet number (Pe) from 0.01 to 50. To obtain the diffusion coefficients, image-time series from a confocal scanning laser microscope were analyzed with a method that, after finding particle trajectories, subtracts the instantaneous advective displacements and subsequently measures the slopes of the mean squared displacement in the flow $\left(x\right)$ and shear $\left(y\right)$ directions. For dilute suspensions, the thus obtained diffusion coefficients $(D_x$ and $D_y)$ are close to the free diffusion coefficient at all shear rates. In concentrated suspensions, a clear increase with the Peclet number (for Pe > 10) is found, that is stronger for $D_x$ than for $D_y$. This effect of shear-induced collisions is counteracted by the contribution of walls, which cause a strong local reduction in $D_x$ and $D_y$.

Figure 1

(a) Schematic of the experimental setup. Typical (X,Y,Z) channel dimensions are 2 cm, 30, and $24\mu \text{m}$, and CSLM represents a confocal scanning laser microscope. (b) Confocal image taken at a height of $12\mu \text{m}$ from the bottom of a suspension at volume fraction 0.3, flowing through the channel from left to right with a maximum velocity of $5.4\mu \text{m}/\text{s}$. Scale bar: $10\mu \text{m}$.

Figure 2

Particle velocity profiles across the channel for different pressure drops. Panels (a) and (b) show the measurements from the videos. Panels (c) and (d) show the comparison of normalized velocity profiles (black points) with theory (red lines). Particle volume fractions are 0.03 for (a) and (c) and 0.3 for (b) and (d).

Figure 3

MSDs for a suspension at $\Phi =0.3$ for the experiment where $v_{\max }=5.4\mu \text{m}/\text{s}$. Different datasets in the same panel correspond to different locations: From bottom to top $y=4.5$, 5.6, 7.5, 9.7, and 10.5 $\mu \text{m}$ from the center line. This order corresponds to an increasing local shear rate $\left(\dot{\gamma }\right)$. Panel (a) MSDs in flow direction (after correction for advection) and panel (b) MSDs in the velocity gradient direction. Solid lines are linear fits to the experimental data.

Figure 4

Experimental results for a concentrated suspension $\left(\Phi =0.3\right)$ (a) nearly at rest and (b) in a strong flow. Upper graph: (black square) diffusion coefficients in the flow $\left(x\right)$ and (red circle) shear $\left(y\right)$ directions as a function of the lateral $\left(y\right)$ position in the channel. Lower graph: local velocity and shear rate in the same channel. The walls are located at $y=\pm 15\mu \mathrm{m}$. Symbols represent experimental data whereas lines are to guide the eye.

Figure 5

Diffusion coefficients in the flow $\left(x\right)$ and shear $\left(y\right)$ directions as a function of lateral $\left(y\right)$ position for a suspension at $\Phi =0.3$ studied at different flow rates. At $y=0$, fluid elements are advected without any shear (irrespective of flow rates), and all the measured $D_x$ and $D_y$ coincide. Maximum flow velocities $\left(v\max \right)$: black squares:$1.44\mu \text{m}/\text{s}$; red circles:$2.24\mu \text{m}/\text{s}$; violet up-triangles:$3.05\mu \text{m}/\text{s}$; blue down-triangles: $3.89\mu \text{m}/\text{s}$; and green diamonds: $4.75\mu \text{m}/\text{s}$. Symbols represent experimental data whereas lines are to guide the eye.

Figure 6

Short-time self-diffusion coefficients in the flow $\left(x\right)$ and velocity gradient $\left(y\right)$ directions as a function of Pe. (a) At Φ = 0.03, diffusion is isotropic and independent of Pe number. (b) At $\Phi =0.3$, the diffusion coefficient is isotropic up to $\mathrm{Pe}\approx 5$, after which it increases with Pe. Beyond Pe = 10, diffusion becomes strongly anisotropic. Red points (circle) are for $D_y$, and black points (square) are for $D_x$. Blue dashed lines indicate the diffusion coefficient at Pe = 0.

Figure 7

Change in normalized diffusion coefficients as a function of normalized distance $\left(\Delta y/d\right)$ from the wall. $d$ $\left(=2a\right)$ is the particle diameter. Symbols show the experimental data, and solid and dotted lines are the analytical solutions [Eqs. (3) and (4)] whereas dashed lines indicate the free diffusion coefficient. Left panels (a) and (c) show the data for Φ = 0.03. Maximum flow speed $\left(v\max \right)$: black square: $0.16\mu \text{m}/\text{s}$; red circle: $5.4\mu \text{m}/\text{s}$; and green down-triangles: $10.6\mu \text{m}/\text{s}$. Right panels (b) and (d) show the data for $\Phi =0.3$. Here the dotted lines are plotted just to visualize the difference with the experimental data for Φ = 0.03. $v\max$: black square: $0.13\mu \text{m}/\text{s}$; red circle: $0.80\mu \text{m}/\text{s}$; and green down-triangles $4.0\mu \text{m}/\text{s}$.

Figure 8

Pair potential normalized by kT (blue circles) of silica spheres $(d=1\mu \text{m}$; diameter) in water-glycerol solvent. $r$ is the distance between two particle centers. $\Phi =0.03$.