Drop-in additives for suspension manipulation: Colloidal motion induced by sedimenting soluto-inertial beacons

Soluto-inertial (SI) suspension interactions involve nonequilibrium colloidal migration over millimeters, driven by “beacons” that establish long-lasting chemical gradients. Previous demonstrations of the SI concept were restricted to fixed, cylindrical beacons that would sustain solute fluxes in controlled two-dimensional geometries. Here we examine soluto-inertial interactions established by a spherical beacon that sediments through a suspension. The sedimenting beacon leaves a trailing solute wake, the flux from which drives the motion of neighboring suspended objects. Experiments reveal wakes that can attract or repel particles, depending on the solute-colloid pair, creating local regions in the suspension that are enriched in or devoid of particles. Theoretical descriptions accurately capture experimental observations and provide predictive ability to gauge the strength of these interactions in terms of the design parameters. We anticipate that long-range, bulk suspension interactions driven by “drop-in” beacon additives will enable on-demand flocculation of dilute suspensions and separation of colloidal mixtures.

Figure 1

Freely falling SI beacons induce diffusiophoretic colloidal migration. (a) A cylindrical beacon post fixed within a microfluidic chamber establishes a radial concentration profile of fluorescein. (b) The solute flux driven by a fixed cylindrical beacon drives suspended colloids to migrate diffusiophoretically in the radial direction. (c) Experimental visualization and (d) computed profile of solute emitted by a sedimenting beacon, which forms a convection-diffusion boundary layer along the beacon and a solute wake behind. The experiment (c) comprises a water drop unloading Oil Red O dye as it sediments through an oil bath. (e) As the solute in the wake diffuses radially outward, its concentration gradients drive suspended particles to migrate toward (or away) from the wake, creating regions in the suspension that are enriched or depleted in particles. (f) SI attraction of fluorescent colloids by a solute wake trailing a sedimenting beacon, illuminated from the side by a laser sheet and visualized from below. Color-coded streak lines show diffusiophoretic migration directed toward the center of the dark circle, which arises because the sedimenting beacon physically blocks the camera's view.

Figure 2

The experimental setup used to record the transmitted light intensity through a suspension of negatively charged polystyrene colloids. A uniform, diffuse white light source is held behind the sample cell containing the suspension. The sample cell itself is 40 mm tall, 6 mm wide, and 3 mm thick, with acrylic side walls. The front, back, and top of the cell are enclosed with glass slides. A hole drilled through the top slide provides access into the cell. A charge-coupled device (CCD) camera attached to a zoom lens is used to record images of the experimental setup at 10 frames per second.

Figure 3

The falling velocity of ionogel beacons of different sizes. Points correspond to velocities and radii measured experimentally, while the solid line represents the prediction of Eq. (1). Error bars show standard deviations in the measured velocities and radii.

Figure 4

Falling beacons loaded with the ionic liquid $\left[{\mathrm{C}}_4\mathrm{mim}\right]\left[\mathrm{I}\right]$ [(a)–(g)] and the surfactant SDS [(h)–(n)] create wakes that attract and repel particles, respectively. Images show snapshots in time for each of the two beacons. (a) The $\left[{\mathrm{C}}_4\mathrm{mim}\right]\left[\mathrm{I}\right]$ beacon enters the frame at $t=0$ s and (b) hits the bottom of the container after 0.6 s; the wake diffuses out and collects particles, as seen after (c) 5 s, (d) 15 s, (e) 30 s, and (f) 60 s. (g) A pixel thick slice (at $z=11.7$ mm), when stacked together, shows the temporal evolution of the particle concentration in the wake. The axis on the right shows the time elapsed after the beacon crosses the dashed red line in (f) and the axis on the left shows the distance the beacon would have fallen had the fluid been unbounded. The dark band between 12 and 13 s is an artifact of the illumination. (h) The SDS beacon enters at $t=0$ s and (i) hits the bottom after 1.7 s. Unlike the ionic liquid, the SDS wake repels particles to create a depletion zone, as seen after (j) 10 s, (k) 20 s, (l) 40 s, and (m) 60 s. (n) A pixel thick slice (at $z=8.6$ mm), when stacked together, shows the temporal evolution of the particle concentration in the wake. The axes on the left and right have the same interpretation as (g).

Figure 5

Soluto-inertial attraction of suspended particles by the solute wake of a falling ionogel beacon leads to colloidal accumulation within the wake. (a) Projected concentration $C_P^{\mathrm{proj}}(x,z_0)$ at height $z_0$ in the sample cell, normalized by the background concentration $C_{P0}^{\mathrm{proj}}(x\to \infty )$ as a function of time. Particles migrate from the bulk suspension ($\tilde{x}>{\tilde{x}}_i)$ to accumulate near the center of the solute wake ($\tilde{x}<{\tilde{x}}_i)$, with intensity that increases over time. Lengths $x$ are normalized by the beacon radius $R_B$, here $225\mu \mathrm{m}$. Circle represent measured data points, and lines reflect predictions described in Sec. 3. (b) The number $\mathrm{\Delta }N\left(t\right)$ of particles collected by the solute wake increases with time, with a collection rate that decreases with beacon radius $R_B$. Theoretical predictions [Eqs. (23) and (9)] capture measured data.

Figure 6

The dimensionless total mass of solute collected in the trailing wake of a sedimenting beacon, per unit length of the wake. Points show numerically computed values of dimensionless mass per length vs Pe [Eq. (20)], while the solid line represents the ${\mathrm{Pe}}^{-2/3}$ scaling derived in Eq. (17). Computations are performed in the Stokes flow regime with $\mathrm{Re}=0$.

Figure 7

Particle response to the solute wake established by a falling beacon that attracts colloids. (a) Evolution of particle concentration in suspension after the formation of the wake. Dotted teal lines show the initial increase in the particle concentration, until it reaches a maximum value at $t_{\max }$. This maximum concentration profile is shown by the solid yellow line. The concentration relaxes as the DP velocity diminishes and the particles diffuse out, which is shown by the dashed orange lines. The concentration profiles are calculated with ${\tilde{D}}_{\mathrm{DP}}={10}^{-1}$, the order of magnitude appropriate for the experiments, but with an unrealistically large particle diffusivity ${\tilde{D}}_P={10}^{-1}$ to match the strengths of the particle diffusive flux and the DP flux and therefore show their competition. (b) Excess particles ($\mathrm{\Delta }N=N_{\mathrm{SI}}-N_0$) collected by the wake of sedimenting beacons vs Pe. Computations for the excess particles are performed with more realistic particle diffusivity ${\tilde{D}}_P={10}^{-4}$ and ${\tilde{D}}_{\mathrm{DP}}={10}^{-1}$, as appropriate for these experiments.

Figure 8

Effect of particle diffusivity (${\tilde{D}}_P$) and DP mobility (${\tilde{D}}_{\mathrm{DP}}$) on colloidal collection in the beacon wake. (a) Excess particles collected in the wake ($\mathrm{\Delta }N=N_{\mathrm{SI}}-N_0$), due to their DP migration toward higher solute concentration vs nondimensional time. The different curves collapse when scaled by the dimensionless DP mobility (${\tilde{D}}_{\mathrm{DP}}$). Inset shows the raw, unscaled results. (b) Maintaining a constant mass of solute yet changing the width of the initial solute pulse does not impact the number of particles collected, shown by the dashed red and dotted green lines. Moreover, the increase in particle count follows a logarithmic scaling in time (black curve). Time is nondimensionalized by solute diffusion timescale, $R_B^2/D_S$.

Figure 9

Dimensionless total solute mass in the wake per unit length ($\tilde{\lambda }$), as a function of Pe, at different values of Re. Computations (points) reveal $\tilde{\lambda }$ to be independent of Re, scaling as ${\mathrm{Pe}}^{-2/3}$ (line), irrespective of Re.

Figure 10

2D axisymmetric geometry, representative of the COMSOL computational domain used to solve Eqs. (A1)–(A10).

Figure 11

A calibration curve is prepared to calculate the concentration of colloids in the suspension, as a function of the transmitted light intensity.

Figure 12

Comparison between the strength of the diffusiophoretic particle flux driven by the solute wake (${\tilde{D}}_{\mathrm{DP}}{\tilde{R}}_i\tilde{\nabla }ln{\tilde{C}}_S{|}_{{\tilde{R}}_i}$) and the diffusive particle flux (${\tilde{D}}_P{\tilde{R}}_i\tilde{\nabla }{\tilde{C}}_P{|}_{{\tilde{R}}_i}$), at different combinations of the scaled DP mobility ${\tilde{D}}_{\mathrm{DP}}$ and particle diffusivity ${\tilde{D}}_P$.

Figure 13

Temporal evolution of the location ${\tilde{R}}_i$ beyond which, the excess particles in the center of the wake is balanced by a net depletion of particles in the suspension, at different combinations of the scaled DP mobility ${\tilde{D}}_{\mathrm{DP}}$ and particle diffusivity ${\tilde{D}}_P$.