Soluto-inertial phenomena: Designing long-range, long-lasting, surface-specific interactions in suspensions

Significance

Liquid suspensions of micron-scale particles and drops play a ubiquitous role in a broad spectrum of materials of central importance to modern life. A suite of interactions has long been known and exploited to formulate such suspensions; however, all such interactions act over less than a micron in water—and often much less. Here we present a concept to design and engineer nonequilibrium interactions in suspensions, which are particle surface-dependent, may last for hundreds of seconds, and extend hundreds of times farther than is currently possible. The conceptual versatility of the results presented here suggests new capabilities for manipulating suspensions, sorting particles, and synthesizing novel materials and particles.

Abstract

Equilibrium interactions between particles in aqueous suspensions are limited to distances less than 1 μm. Here, we describe a versatile concept to design and engineer nonequilibrium interactions whose magnitude and direction depends on the surface chemistry of the suspended particles, and whose range may extend over hundreds of microns and last thousands of seconds. The mechanism described here relies on diffusiophoresis, in which suspended particles migrate in response to gradients in solution. Three ingredients are involved: a soluto-inertial “beacon” designed to emit a steady flux of solute over long time scales; suspended particles that migrate in response to the solute flux; and the solute itself, which mediates the interaction. We demonstrate soluto-inertial interactions that extend for nearly half a millimeter and last for tens of minutes, and which are attractive or repulsive, depending on the surface chemistry of the suspended particles. Experiments agree quantitatively with scaling arguments and numerical computations, confirming the basic phenomenon, revealing design strategies, and suggesting a broad set of new possibilities for the manipulation and control of suspended particles.

Colloidal suspensions and emulsions of 10-nm to 10-μm particles play a central role in a wide variety of industrial, technological, biological, and everyday processes. Everyday goods, including shampoos, inks, vaccines, paints, and foodstuffs as well as industrial products such as drilling muds, ceramics, and pesticides, rely fundamentally on stably suspended microparticles for their creation and/or operation. This incredible versatility derives from the extensive variety of properties (e.g., mechanical, optical, and chemical) attainable in suspension through a generic set of physicochemical strategies (1–4). A proper understanding of the stability and dynamics of suspensions in general thus underpins both fundamental science and technological applications.

The properties and performance of suspensions depend preeminently on the effective interactions between particles. The celebrated Derjaguin–Landau–Verwey–Overbeek (DLVO) theory (5–7) balances electrostatic interactions (typically repulsive) between charged colloids—as screened by ions in the surrounding electrolyte—against van der Waals attractions, and successfully predicts the stability, phase behavior, and response of electrostatically stabilized suspensions. Additional (non-DLVO) forces can also be used to stabilize or destabilize colloidal suspensions. Grafted or adsorbed macromolecules provide short-range steric repulsions that stabilize suspended particles against van der Waals-induced flocculation (8–11). By contrast, nonadsorbed macromolecules that remain dispersed in solution introduce entropic depletion attractions whose strength and range is set by the size and concentration of depletants (12, 13). Such depletion interactions scale with thermal energy ($k_{B}T$), and thus enable tunable and reversible attractions (14, 15). Clever design of shaped or patterned colloids yields “lock-and-key” colloidal interactions (16, 17) and so-called “colloidal molecules” (18, 19). Grafting ligand-functionalized molecules to colloidal surfaces enables molecular sensing (20, 21), and sophisticated design of colloidal self-assembly (22, 23).

Despite the past century’s advances in the understanding, control, and engineering of colloidal forces, the range of colloidal interactions remains fundamentally limited. van der Waals interactions extend for tens of nanometers (24–26). Steric repulsions (24, 27) and depletion attractions (12, 24, 28) are limited by the size of the adsorbed, grafted, or suspended macromolecules, typically in the ∼10- to 100-nm range. Of colloidal interactions, electrostatics have the longest range, yet they are fundamentally limited by the (Debye) length scale over which the surface charge is screened by electrolyte ions (10, 24, 27): The largest possible Debye screening length in room temperature aqueous suspensions is ≤1 μm, and more typically between 1 nm and 100 nm. Longer screening lengths are possible in nonpolar solvents, but control and stabilization of nonaqueous charges and suspensions remains challenging (29–32). Magnetic and hydrodynamic interactions are unscreened and can extend to longer ranges (10, 27) but are essentially indiscriminate and less easily controlled, designed, or tuned. Lastly, defects introduced by particles in liquid crystals give rise to interparticle forces over particle length scales (33, 34), yet they are limited to liquid crystalline materials.

In what follows, we demonstrate a versatile strategy to design and establish nonequilibrium interactions that range over hundreds of microns, persist for tens of minutes, and can be designed to attract or repel suspended colloids, depending on the colloids’ surface chemistry. These interactions exploit diffusiophoresis (DP)—the migration of suspended particles and droplets in response to gradients or fluxes of solute (35–38) or solvent (“solvophoresis”) (39, 40). Such gradients arise spontaneously around membranes, reactive surfaces, electrodes, dissolving solids, evaporating liquids, or generally in the vicinity of equilibrating surfaces. Recent years have seen a burst of interest in DP in areas ranging from membrane fouling (41), to transport into dead-end pores (42, 43), to self-propelling particles (44–47), to active matter (48–50).

Soluto-Inertial Beacon

Fig. 1 illustrates the mechanism we propose for this interaction, which requires three key ingredients. First, a particle or structure must act as a beacon that generates a long-lasting solute flux. Second are the suspended objects (e.g., colloidal particles, polymers, or emulsion drops), which may be attracted or repelled from the beacon. Third is the solute whose flux mediates the interaction by driving suspended objects into diffusiophoretic migration. Careful choice of these three ingredients (beacon, solute, and suspended particle) enables the duration, direction, and range of the beacon–suspension interactions to be designed and engineered.

Fig. 1.

Long-range SI interactions. An SI beacon (gray), initially loaded with a high solute concentration, is placed in a solute-free suspension. A solute outflux is established during equilibration, driving nearby suspended particles into diffusiophoretic migration. The magnitude and direction of migration depends on interactions between the particle surface and the solute, depicted here by particles of different surface chemistries (orange and green) that migrate either up or down the solute gradient. (Inset) Schematic radial profile of solute concentration inside and outside of the beacon.

Whether the beacon attracts or repels colloids in suspension depends on how those colloids migrate under the chosen solute flux. Theories for diffusiophoretic mobilities involve the relative excess (or depletion) of the solute near the particle surface, which is forced into motion by solute gradients in the bulk solution. Such theories have been developed for gradients of electrolytes (35, 37) and nonelectrolytes (51), yet difficulties in establishing sufficiently strong, stable gradients have prevented systematic experimental measurements (as is routine for electrophoresis). Recent developments in microfluidics, however, have enabled more direct studies of DP (40, 52–56). In particular, we have recently developed a microfluidic device (56) that enables gradients to be directly imposed, and diffusiophoretic migration to be visualized and measured under various solute and solvent gradients (40).

Fig. 2 highlights the surface specificity of DP under a given solute flux. High- and low-concentration SDS solutions flow through the outer “reservoir” channels of a three-channel device, to establish and maintain an SDS gradient of controllable strength across the central sample channel. Even the direction of migration depends on the specific solute/colloid pair. Fluorescent, sulfonated polystyrene (PS) colloids (Materials and Methods) move diffusiophoretically down SDS concentration gradients (Fig. 2 A and B) (Movie S1), consistent with electro-DP (35, 37), treating the (ionic) surfactant SDS as an electrolyte. By contrast, decane drops (Materials and Methods) migrate up SDS gradients (Fig. 2 C and D) (Movie S2), consistent with nonuniform SDS adsorption onto the droplet surface that creates either surface tension gradients, and thus “soluto-capillary” migration (57), or DP under strong adsorption (58). Irrespective of the detailed mechanism at play, a “soluto-inertial” (SI) beacon designed to emit SDS should attract decane droplets but repel PS particles.

Fig. 2.

Particle surface specificity of DP. (A and B) PS colloids, initially uniformly distributed (A), migrate diffusiophoretially down SDS concentration gradients, as seen after 100 s (B). (C and D) By contrast, the DP of fluorescently-dyed decane drops is directed up SDS gradients. In A and C, t = 0 s; in B and D, t = 100 s.

Having identified suitable combinations of solute and suspended particles, we now turn to the beacon itself. The beacon must be designed to emit a long-lived solute flux, because the longer the beacon takes to equilibrate with its surrounding solution, the longer the diffusiophoretic interaction lasts. We achieve this long-lived flux by developing the solute analog of “thermal inertia,” wherein materials with high volumetric heat capacity resist changes in temperature and thus maintain long-lasting heat flux. SI beacons can be made from materials that strongly partition the solute, so that solute within the beacon ($C_B$) equilibrates at a concentration that exceeds the concentration in the neighboring solution ($C_S$) by a large partition coefficient K, giving $C_B=K{C}_S$, where $K\gg 1$ (59). With such a choice, a beacon loaded with solute that is placed in a solute-free suspension equilibrates over long time scales, as demonstrated and described in Proof of Principle, ensuring a long-lived solute outflux (and thus SI interaction). Therefore, the key physicochemical property required of the SI beacon is that it strongly partitions whatever solute has been selected to attract or repel the colloids of interest.

Proof of Principle

Fig. 3 shows proof-of-principle demonstrations of the SI interaction described above, and specifically confirms the range, duration, and particle surface-specificity. Fig. 2 suggests that an SI beacon that emits SDS will repel PS colloids, and attract decane drops. SDS is known to associate with polyethylene glycol (PEG) (60, 61), suggesting that PEG hydrogels will strongly partition SDS, and thus function as SI beacons. Fig. 3A shows a cylindrical SI beacon of radius R ≈ 200 μm created by photopolymerizing a PEG-diacrylate (PEG-DA) precursor solution in situ within a microfluidic device by exposure to masked, ultraviolet light (56, 62, 63), then flushing unreacted precursor from the device.

Fig. 3.

Experimental demonstration of the range, duration, and particle surface specificity of SI interactions. (A) Microfluidic device showing beacon structure (hydrogel post) in the center. (B) Loading beacon with solute. (C) Flushing loading solution. (D) Slow equilibration of stored solute into particle suspension. Insets show radial concentration profiles throughout the experiment. (E–H) PS particles respond to SDS gradient by migrating over hundreds of microns for tens of minutes: (E) t = 0 s, (F) t = 150 s, (G) t = 300 s, and (H) t = 1,000 s. Beacon location is indicated by red circle. (I) Time-stamped streak lines of decane droplet migration in first 350 s of experiment, directed towards the SI beacon. Notably, PS colloids migrate down SDS gradients whereas decane droplets migrate up the SDS gradients.

The experimental procedure is shown schematically in Fig. 3 B–D. An SI beacon is initially loaded by immersion in a 5-mM SDS solution (Fig. 3B). The SDS loading solution is then flushed by flowing in a suspension of colloids and/or drops (Fig. 3C). The SDS that had partitioned into the SI beacon is no longer in equilibrium with the surrounding solution, and therefore diffuses out of the SI beacon to equilibrate. The resulting SDS concentration gradient persists as long as SDS outfluxes from the SI beacon (Fig. 3D), within which suspended colloids and/or drops migrate diffusiophoretically.

Fig. 3 E–I reveals this system to behave as predicted: PEG-DA beacons partition SDS, and therefore establish a long-lived, long-ranged SDS flux that effectively repels PS colloids but attracts decane drops. In particular, PS particles are repelled from the SI beacon (down the SDS gradient), forming a 300- to 400-μm-thick particle-free region around the beacon (Movie S3). PS particle migration is evident for at least 1,000 s, by which time most of the particles have migrated out of the microscope field of view. The particle surface specificity of SI interactions is verified by following the same procedure but introducing decane droplets instead of PS colloids. As expected, decane droplets experience a long-range, long-lasting SI attraction toward the beacon, moving up the imposed SDS gradient as shown by the streak lines in Fig. 3I and Movie S4. The range and duration in this case is comparable to that observed with PS particles. Notably, the SI interaction range (here 100–1,000 μm) is 10³–10⁴ times larger than the Debye screening length that limits the electrostatic interaction.

SI Model

Having shown that the SI effect can be exploited to generate and direct diffusiophoretic motion in suspensions, we now develop a model of this behavior, focusing initially on spherical SI beacons to avoid the mathematical subtleties of 2D diffusion. A spherical SI beacon is initially loaded by immersion in a loading solution of concentration $C_S^0$, and thus equilibrates with some beacon concentration $C_B^0=K{C}_S^0$. When suspended in a solute-free environment ($C_S=0$), the solute in the beacon $C_B\left(t\right)$ diffuses out into the surrounding solution. We assume that the solute concentration within the beacon evolves rapidly enough that intrabeacon concentration gradients can be neglected [$C_B\left(r,t\right)\mathrm{\approx }{C}_B\left(t\right)$], and that the concentration within the beacon $C_B\left(t\right)$ changes on time scales much slower than are required for $C_S\left(r,t\right)$ to evolve. Under these quasi-steady assumptions, the concentration field around a spherical, SI beacon of radius R that partitions solute with a partition coefficient K obeys

$$C_S\left(r,t\right)=\frac{C_B\left(t\right)}{K}\frac{R}{r}.$$ [1]

This concentration field gives rise to a diffusion-limited solute flux $J=4\pi DR{C}_B\left(t\right)/K$ out of the beacon, where D is the diffusion coefficient of the solute. This outflux must equal the rate at which solute molecules are lost from the beacon,

$$J=\frac{4\pi DR{C}_B\left(t\right)}{K}=-\frac{d}{dt}\left[\frac{4\pi R^3}{3}{C}_B\left(t\right)\right],$$ [2]

which can be solved to give the beacon concentration

$$C_B\left(t\right)=C_B^0{\text{exp}}^{-\frac{3D}{K{R}^2}t}.$$ [3]

Eq. 3 reveals a natural SI time scale,

$${\tau }_{SI}=K\frac{R^2}{3D},$$ [4]

over which SI beacons emit solute, which exceeds the diffusion time scale ${\tau }_D=R^2/D$ by the partition coefficient K, and may thus be many orders of magnitude longer when $K\gg 1$.

We now investigate the range over which colloidal particles migrate diffusiophoretically under the concentration gradient set by the SI beacon. Diffusiophoretic migration velocities under electrolyte gradients are predicted (35, 51) to obey

$$u_{DP}=D_{DP}\mathrm{\nabla }\ln C_S,$$ [5]

where $D_{DP}$ is the diffusiophoretic mobility of the particle, whose magnitude and sign both depend on the surface chemistry of the particle and solute. Using Eq. 5 for the quasi-steady concentration field around a spherical SI beacon (Eq. 1) reveals colloids to migrate under a quasi-steady SI outflux with velocity

$$u_{DP}\left(r\right)=-\frac{D_{DP}}{r},$$ [6]

decaying with distance like $r^{-1}$. Several features are notable: The SI migration velocity (i) is independent of $C_B\left(t\right)$, the instantaneous concentration of solute in the beacon, (ii) decays slowly with distance from the SI beacon, and (iii) is particle surface-specific, as determined by $D_{DP}$.

Quantitative Measurements of Migration Velocity

Trajectories of individual particles can be extracted from micrograph series, allowing particle velocities to be measured directly in space and time. Fig. 4 shows raw (Inset) and scaled velocity profiles at different times, around cylindrical SI beacons of two different radii (R_P = 130 μm and 200 μm).

Fig. 4.

Radial SI velocity profiles for PS colloids migrating around cylindrical SI beacons of radii R_P = 130 μm (open red symbols) and 200 μm (filled blue symbols), at different times. Points represent data measured in experiment, with unscaled data shown in Inset. Scaling distance by post radii $R_P$, velocities by the maximum velocity $u_{\text{max}}$ measured at any place and time in each experiment, and time by the radial diffusion time $R_P^2/D_{SDS}$ collapses measured data for both posts onto the profiles computed from the quasi-steady mass transport model. Measured and computed velocity profiles at different (scaled) times are represented with different colors, with corresponding $t/\tau$ values indicated in the key.

No steady-state concentration profile exists for 2D structures like the SI beacons shown in Fig. 3. We therefore solve the transient mass transport problem analytically and numerically using COMSOL Multiphysics, under the same quasi-steady assumptions described in SI Model: Concentration fields $C_S\left(r,t\right)$ in bulk solution evolve much more rapidly than in the SI beacon (Supporting Information and Fig. S1), and so we impose a quasi-steady boundary condition $C_B\left(t\right)$. Guided by the scaling arguments for spherical SI beacons, we scale distance by the beacon radius $R_P$, time by the diffusion time $R_P^2/D_{SDS}$ of dissolved SDS, and concentrations by a concentration scale $C_0$. In cylindrical coordinates, the nondimensionalized diffusive mass transport equation in the radial direction is given by

$$\frac{\partial \tilde{C}}{\partial \tilde{t}}=\frac{1}{\tilde{r}}\frac{\partial }{\partial \tilde{r}}\left(\tilde{r}\frac{\partial \tilde{C}}{\partial \tilde{r}}\right),$$ [7]

where $C_S=C_0\tilde{C}$, $r=R_P\tilde{r}$, and $t=\left(R_P^2/D_{SDS}\right)\tilde{t}$. The diffusion coefficient $D_{SDS}$ of aqueous SDS below the critical micelle concentration is taken as 780 μm²/s (64). The concentration field $\tilde{\mathit{C}}\left(\tilde{r},\tilde{t}\right)$ is then computed by enforcing $\tilde{C}\left(\tilde{r}=1,\tilde{t}\right)=1$ and $d\tilde{C}/d\tilde{r}\left(\tilde{r}\to \mathrm{\infty },\tilde{t}\right)\mathrm{\to }0$. The analytical solution (Supporting Information and Fig. S2) is in good agreement with the numerical model. The DP migration velocities of suspended particles can then be computed at any position and time from $\tilde{C}\left(\tilde{r},\tilde{t}\right)$ using Eq. 5, and are simply proportional to $\nabla \ln \tilde{\mathit{C}}\left(\tilde{r},\tilde{t}\right)$.

Fig. S1.

Time evolution of the concentration profile of SDS in solution, corresponding to different boundary conditions at the beacon–solution interface. (A) Boundary concentration calculated by solving coupled mass transport equations inside and outside the beacon, with D_B = 30 μm²/s and $K=30$. (B) Concentration at boundary assumed to decay exponentially with decay constant = $D_{SDS}/K{R}_P^2$. (C) Fixed boundary condition imposed at the interface. (D–F) Velocity profiles corresponding to the concentration profiles depicted in A–C, respectively.

Fig. S2.

Comparison between the numerical and analytical solutions corresponding to R_P = 200 μm and $R_{\infty }=\mathrm{2,000}\hspace{0.17em}\mathrm{\mu }\text{m}$. The smooth curves represent the analytical solution found by computing the first 100 terms of the Fourier series given by Eq. S25, and the points represent the numerical solution. The comparison is demonstrated for five time steps, represented by different colors, as indicated in the key.

The diffusiophoretic mobility $D_{DP}$ of PS particles under SDS gradients is not known a priori. To compare the measured and calculated SI migration velocities, we normalize all measured or calculated velocities by a single value, corresponding to the maximum value that was measured or calculated,

$$\tilde{u}=\frac{u}{u_{\text{max}}}=\frac{\nabla \ln C_S}{\text{max}|\nabla \ln C_S|}.$$ [8]

So long as $D_{DP}$ is constant, normalizing velocities in this way enables direct comparisons between measured and calculated velocity profiles at different times, irrespective of the single (unknown) parameter $D_{DP}$.

Fig. 4 shows the nondimensional radial velocity profiles of PS colloids measured in experiments with SI beacons of radii R_P = 200 μm (filled blue points) and R_P = 130 μm (empty red points). Each differently colored set of data corresponds to a finite time window, and depict the time evolution of the measured velocity profile around the beacon. Appropriate nondimensionalization allows the experimental data to be compared with the SI model, the predictions from which are plotted as solid lines corresponding to each of the experimental time intervals shown. When properly scaled according to the SI arguments presented above, measured velocity profiles show excellent agreement, in space and time, not only between the two different-sized SI beacons but also between the experimental measurements and the SI model predictions. This collapse justifies our choice of the length, time, and velocity scales as well as confirms that the mass transport model coupled with the quasi-steady state assumption captures the observed SI migration phenomenon quantitatively. It should be noted that, although the velocity profile appears increasingly flat as time progresses, it remains nonzero in both the SI model and the experiment. The success of the scalings for the SI time scale (Eq. 4) and distance scale R_P, along with the quantitative agreement between theory and experiment in Fig. 4, underscores the quantitative capability to design a long-range, long-lasting suspension interaction.

Discussion and Conclusion

The general SI strategy described here shares many features in common with previous observations involving DP, specifically involving reacting or dissolving interfaces. Derjaguin et al. (35) elucidated the existence and influence of DP on latex film formation onto salt-soaked surfaces. Prieve (36) and coworkers noted an analogy with chemically reacting systems, e.g., as steel dissolution drives the diffusiophoretically accelerated deposition of latex particles. More recently, McDermott et al. (65) showed that calcium carbonate particles dissolving in unsaturated aqueous solutions act as diffusioosmotic micropumps, driving flows along neighboring surfaces. Zheng and Pollack (66) reported long-range exclusion near hydrogel boundaries, and Florea et al. (67) revealed ion exchange reactions to form a colloidal exclusion zone near membrane surfaces.

We have established a conceptual framework for the design and engineering of long-range, nonequilibrium interactions in suspension whose magnitude and direction depends on the surface chemistry of the suspended particles. Our results highlight the versatility and generality enabled by combining the slow, SI release of solute with the diffusiophoretic migration of suspended particles. The direction and speed of suspended particle migration can be controlled by appropriate choice of solute, and the range and duration of the SI interaction can be tuned by choosing size and material of the SI beacon to maximize the partition coefficient. With the specific PEG-DA (beacon)–SDS (solute) system, we have revealed that SI interactions last for tens of minutes and extend over hundreds of microns. Moreover, fairly simple scaling arguments and numerical computations capture the quantitative and qualitative characteristics of SI interactions. Although specific experiments here used relatively large SI beacons held fixed in place, analogous physics and scaling arguments should also hold for freely suspended beacons, although beacon sedimentation introduces additional complexity. The generality of the SI concept naturally suggests a variety of new directions and applications, including the in situ separation and collection of particular suspended colloids, accelerated or triggered flocculation of emulsions and suspensions, layer-by-layer deposition, and other novel synthesis strategies.

Materials and Methods

Device Fabrication.

A single inlet/outlet microfluidic device is used, with a large central circular chamber (Fig. 3). A computer-controlled laser cutter (Trotec Speedy 100) cuts the channel into 60-μm-thick scotch tape. The cut tape is then stuck to a Petri dish, which is used as a master for making a polydimethylsiloxane (PDMS) replica of the design. The PDMS master is used to fabricate the device in “microfluidic stickers” (NOA – 81; Norland Adhesive) (68). The central chamber has a radius of 2 mm, and the inlet and outlet channels are 500 μm wide. A glass cover slide is used to seal the device, with holes drilled to provide access for inlet and outlet tubing. A PDMS inlet is ozone-bonded to the cover slide to provide support for inlet and outlet pins and tubings. The device is then baked at 80 °C for at least 4 h to strengthen bonding.

Sample Preparations.

SDS solutions are prepared by diluting a 10-mM SDS (Sigma Aldrich) stock solution in deionized water. PEG-DA precursor solution is prepared by mixing 33% (vol/vol) PEG(700)-DA (Sigma-Aldrich) with 4% (vol/vol) photoinitiator (2-hydroxy-2-methylpropiophenone; Sigma-Aldrich) in deionized water; 0.25% vol/vol fluorescent PS beads, 1 μm in diameter (FS03F; Bangs Laboratories), are suspended in clear deionized (DI) water to form the PS suspension. The decane emulsion is prepared by first adding 0.5% vol/vol fluorescent yellow 131SC dye (Keystone) to decane (Sigma Aldrich); 1% vol/vol of the dyed decane is vortexed for 30 s with a 1-mM SDS solution (in DI water) and then sonicated for 15 s to create 1- to 2-μm decane droplets in water.

Experimental Setup.

PEG-DA gels are used as SI structures and fabricated using the microscope projection lithography technique (62, 69). A UV lamp is set to 30 mW/cm² (measured at an empty objective slot). A 1,000-μm-diameter circular photomask is inserted into the microscope and aligned as described previously (56). PEG-DA precursor solution is injected until the channel is filled. The syringe is disconnected, and 2 min are allowed for flow to relax. Then a 500-ms UV exposure is used with a 10× objective to photopolymerize the gel. The precursor solution is flushed from the device by flowing DI water for 30 min. This results in hydrogel posts of diameter 375–425 μm (Fig. 3). Different sizes of posts are obtained by changing the size of the photomask, the exposure time, and the objective magnification.

Experiments are performed using an inverted microscope (Nikon TE2000U). The hydrogel SI structure is initially loaded with a fixed concentration (5 mM) of SDS, by maintaining flow in the channel for 20 min. SDS solution is then flushed out by displacing with the suspension of PS particles or decane drops. The inlet is pressurized to 500 mbar, and the channel is flushed for 5 s before reducing the pressure to 20 mbar. The focus is adjusted to the center of the channel using a 10× objective. Video recording is started (Andor iXon 885 fluorescence camera) and flow in the channel is stopped using the technique described in ref. 40. In each experiment, images are recorded for 1,000 s at 1 frame per second, with 0.1-s exposure times.

Data Analysis.

The 2D particle trajectories are extracted from the fluorescence micrograph series using algorithms adapted from those of Crocker and Grier (70) and implemented in the R programming language, previously used for analysis of bright-field micrographs (71). For each image series, a background image was calculated by finding the time-averaged brightness for each pixel. This background was subtracted from each image in the series. Images were further processed with a spatial band-pass filter and a local background subtraction to eliminate pixel noise and long wavelength brightness fluctuations. This processing has the added benefit of removing out-of-focus particle images, allowing the analysis to focus on particles in the microscope’s focal plane. Local brightness maxima are identified as candidate particle positions, and a brightness-weighted centroiding over the particle diameter is performed to obtain particle coordinates with subpixel precision. Finally, trajectories are obtained by linking particle positions between frames in the acquired videos.

The PS particles explore all three dimensions on the timescale of the experiments. However, due to the axisymmetry of the concentration gradient, 2D tracking in the x–y plane is sufficient to observe and explore the diffusiophoretic particle motion. Instantaneous particle velocities in the radial direction are obtained from frame-to-frame displacements, and velocity profiles are calculated by averaging the velocities within annular regions of 50 pixels width with the origin fixed at the center of the beacon. The concentration profile evolves with time and, therefore, so does the velocity profile, and, as such, velocity profiles are calculated independently over 4.5-s (for R_P = 130 μm) and 10-s (for R_P = 200 μm) intervals throughout the experiment. Splitting the experiment into chunks in this way provides more samples in each annular bin, improving the statistics of averaging and suppressing noise in the velocity profiles.

Validity of Quasi-Steady-State Assumption

The quasi-steady-state assumption stated in the SI model is applicable only when the rate of mass transport via diffusion inside and outside the chemical beacon is much faster than the rate of change of solute concentration inside the beacon, due to the high partitioning of solute by the beacon. Mathematically, this would imply that the SI time scale (${\tau }_{SI}$) is much greater than the characteristic diffusion time scales both inside and outside the beacon.

$${\tau }_{SI}\gg {\tau }_B,{\tau }_S,$$ [S1]

where ${\tau }_B$ and ${\tau }_S$ are the diffusion time scales in the beacon and solution, respectively. This implies

$$\frac{K{R}^2}{D}\gg \frac{R^2}{D}\Rightarrow K\gg 1,$$ [S2]

$$\frac{K{R}^2}{D}\gg \frac{R^2}{D_B}\Rightarrow K\gg \frac{D}{D_B},$$ [S3]

where $D_B$ and D are the diffusion coefficients of the solute in the beacon and solution, respectively. Thus, we conclude from this analysis of the quasi-steady-state model for a spherical SI beacon that the partition coefficient should be greater than a certain critical number, given by Eqs. S2 and S3 for the quasi-steady-state assumption to hold.

Boundary Condition at Beacon Surface

The particle tracking velocity measurements of the PS beads are compared with the quasi-steady mass transport model, assuming a fixed concentration of SDS at the beacon–solution interface. Again, this assumption is only valid under the conditions stated in Validity of Quasi-Steady-State Assumption, specifically when K is large. Here we examine whether this assumption captures the observed SI migration of colloidal particles in our experimental system. Fig. S1 shows a comparison between the concentration profiles generated in the solution by imposing different boundary conditions at the beacon surface. The profile in Fig. S1A is evaluated by solving the complete mass transport equations, both inside and outside the beacon, such that the boundary condition at $\tilde{r}=1$ is not specified, but rather emerges from the calculation. For the purpose of demonstration, we have assumed D_B = 30 μm²/s and $K=30$. In Fig. S1B, we assume that the concentration inside the beacon [$C_B\left(t\right)$] and consequently at the beacon surface decays quasi-steadily, with an SI time scale given by ${\tau }_{SI}=K{R}_P^2/D_{SDS}$. Finally, Fig. S1C shows the time evolution of the concentration profile in the solution, when the concentration at the beacon boundary is kept fixed; $\tilde{C}\left(\tilde{r}=1,t\right)=1$. Fig. S1 D–F shows the nondimensional diffusiophoretic velocity profiles ($\tilde{u}=\tilde{\nabla }\ln \tilde{C}$) under the concentration gradient in the three cases (Fig. S1 A–C).

We conclude that, for a large value of K, the concentration at the boundary of the beacon (and within) changes very slowly, as is evident from the complete solution to the coupled mass transport equations in Fig. S1A, which allows us to make simple quasi-steady approximations for the boundary concentration, similar to the profiles depicted in Fig. S1B. One may note that, because the decay constant in this case is inversely proportional to K, the rate of change of the boundary concentration diminishes for a very high value of K. Further, because the values of $D_B$ and K are not known a priori for our system, and because PEG-DA strongly partitions SDS (K is large), we simply impose a fixed boundary condition at the beacon surface (Fig. S1C) to describe the SI migration of colloidal particles in the bulk solution. The validity of this simplification is further justified by the almost identical trends in the velocity profiles obtained under the three different scenarios (Fig. S1 D–F).

Analytical Solution: Fixed Cylindrical SI Beacon

The experimental geometry is 2D axisymmetric, and thus we solve the radial component of the mass transport equation in a cylindrical coordinate system, given by

$$\frac{\partial C_S}{\partial t}=D_{SDS}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial C_S}{\partial r}\right).$$ [S4]

We assume that $C_i\left(\mathrm{\sim }0\right)$ is the initial solute concentration in the channel. The appropriate boundary conditions for our system are as follows: (i) Quasi-steady boundary condition at the edge of the beacon (fixed concentration), and (ii) no-flux boundary condition at the channel edge.

$$C_S\left(r,0\right)=C_i,$$ [S5]

$${C_S\left(r,t\right)|}_{r=R_P}=C_0,$$ [S6]

$${\frac{\partial C_S\left(r,t\right)}{\partial r}|}_{r=R_{\infty }}=0,$$ [S7]

where $R_P$ is the radius of the beacon and $R_{\infty }$ is the channel radius. We solve these equations via separation of variables. The nonhomogeneous Dirichlet boundary conditions motivate decomposing the concentration field into steady-state ($C_{ss}$) and transient ($\widehat{C}$) components,

$$C_S=C_{ss}\left(r\right)+\widehat{C}\left(r,t\right).$$ [S8]

At steady state, the concentration everywhere in the channel is equal to the concentration at the beacon boundary,

$$C_{ss}\left(r,t\to \infty \right)=C_0.$$ [S9]

We now solve for the transient concentration profile $\widehat{C}\left(r,t\right)$, with the following initial and boundary conditions:

$$\widehat{C}\left(r,0\right)=C_i-C_0,$$ [S10]

$${\widehat{C}\left(r,t\right)|}_{r=R_P}=0,$$ [S11]

$${\frac{\partial \widehat{C}\left(r,t\right)}{\partial r}|}_{r=R_{\infty }}=0.$$ [S12]

We attempt a variables-separable solution to Eq. S4,

$$\widehat{C}\left(r,t\right)=\theta \left(r\right)T\left(t\right),$$ [S13]

which is solved by

$$T\left(t\right)\mathrm{=}\text{A}{\exp }^{\mathrm{-}{\mathit{\lambda }}^2{D}_{SDS}t},$$ [S14]

$$\theta \left(r\right)=\text{B}{J}_0\left(\lambda r\right)+\text{C}{Y}_0\left(\lambda r\right),$$ [S15]

where $J_0$ and $Y_0$ are zeroth-order Bessel functions of the first and the second kind, respectively, and λ, $\text{A}$, $\text{B}$, and $\text{C}$ are constants. The transient concentration field thus has the form

$$\widehat{C}\left(r,t\right)=\left(\alpha J_0\left(\lambda r\right)+\beta Y_0\left(\lambda r\right)\right){\text{exp}}^{-{\lambda }^2{D}_{SDS}t},$$ [S16]

where $\alpha =\text{AB}$ and $\beta =\text{AC}$. Eigenvalues λ are determined by imposing boundary conditions [S11] and [S12] on $\widehat{C}\left(r,t\right)$, giving a transcendental equation

$$Y_0\left(\lambda R_P\right)J_1\left(\lambda R_{\infty }\right)-J_0\left(\lambda R_P\right)Y_1\left(\lambda R_{\infty }\right)=0,$$ [S17]

where $J_1$ and $Y_1$ are first-order Bessel functions of the first and the second kind, respectively. The values of ${\lambda }_n$ that satisfy this equation are the eigenvalues for the corresponding (basis) eigenfunctions

$$\widehat{C_n}\left(r,t\right)=\left({\alpha }_n{J}_0\left({\lambda }_{n}r\right)+{\beta }_n{Y}_0\left({\lambda }_{n}r\right)\right){\exp }^{-{\lambda }_n^2{D}_{SDS}t}.$$ [S18]

The most general solution for the transient concentration profile can thus be expressed as a superposition of these basis functions,

$$\widehat{C}\left(r,t\right)={\displaystyle \sum _n\left({\alpha }_n{J}_0\left({\lambda }_{n}r\right)+{\beta }_n{Y}_0\left({\lambda }_{n}r\right)\right){\text{exp}}^{-{\lambda }_n^2{D}_{SDS}t}},$$ [S19]

where ${\alpha }_n$ and ${\beta }_n$ are related via

$${\alpha }_n=-{\beta }_n\frac{Y_0\left(\lambda R_P\right)}{J_0\left(\lambda R_P\right)},$$ [S20]

to satisfy the boundary condition [S11]. Coefficients ${\beta }_n$ (and consequently ${\alpha }_n$) are determined from initial condition [S10],

$$\widehat{C}\left(r,0\right)={\displaystyle \sum _n{\alpha }_n{J}_0\left({\lambda }_{n}r\right)+{\beta }_n{Y}_0\left({\lambda }_{n}r\right)}=C_i-C_0.$$ [S21]

We define the quantity inside the summation as a new function $f_n\left(r\right)$ and mutiply both sides of Eq. S21 by ${\displaystyle {\int }_{R_P}^{R_{\infty }}r{f}_m\left(r\right)dr}$ to get

$${\displaystyle {\int }_{R_P}^{R_{\infty }}r{f}_m\left(r\right)\widehat{C}\left(r,0\right)dr}={\displaystyle \sum _n{\displaystyle {\int }_{R_P}^{R_{\infty }}r{f}_m\left(r\right)f_n\left(r\right)dr}}.$$ [S22]

Finally, we use the orthogonality of the basis functions to show

$${\beta }_n=\frac{{\displaystyle {\int }_{R_P}^{R_{\infty }}r{S}_0\left({\lambda }_{n}r\right)\left(C_i-C_0\right)dr}}{{\displaystyle {\int }_{R_P}^{R_{\infty }}r{S}_0\left({\lambda }_{n}r\right)S_0\left({\lambda }_{n}r\right)dr}},$$ [S23]

where

$$S_0\left({\lambda }_{n}r\right)=Y_0\left({\lambda }_{n}r\right)-\frac{Y_0\left({\lambda }_n{R}_P\right)}{J_0\left({\lambda }_n{R}_P\right)}{J}_0\left({\lambda }_{n}r\right).$$ [S24]

With ${\beta }_n$ given by Eq. S23, the concentration profile of the solute in the channel is given by

$$\begin{array}{c}{C}_S\left(r,t\right)=C_0+{\displaystyle \sum _n{\beta }_n(-\frac{Y_0\left({\lambda }_n{R}_P\right)}{J_0\left({\lambda }_n{R}_P\right)}{J}_0\left({\lambda }_{n}r\right)}\\ +Y_0\left({\lambda }_{n}r\right))\hspace{0.17em}{\text{exp}}^{-{\lambda }_n^2{D}_{SDS}t},\end{array}$$ [S25]

We compute the first 100 terms of the Fourier series to get an analytical solution for the concentration profile generated by a 200-μm-radius beacon in a 2,000-μm-radius channel and compare that to the numerical solution obtained from COMSOL Multiphysics. Fig. S2 shows the agreement between the analytical solution (Eq. S25), represented by the solid lines and the numerical solution, represented by points at different time steps (as indicated by different colors). The “spikes” in the numerical data result from the finite mesh size in the COMSOL solution.