Size-dependent control of colloid transport via solutegradients in dead-end channelsSangwoo Shina,1, Eujin Uma, Benedikt Sabassa, Jesse T. Aulta, Mohammad Rahimib, Patrick B. Warrenc,and Howard A. Stonea

aDepartment of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544; bLewis-Sigler Institute for Integrative Genomics,Princeton University, Princeton, NJ 08544; and cUnilever R&D Port Sunlight, Bebington, Wirral CH63 3JW, United Kingdom

Edited by Herbert Levine, Rice University, Houston, TX, and approved November 17, 2015 (received for review June 11, 2015)

Transport of colloids in dead-end channels is involved in wide-spread applications including drug delivery and underground oiland gas recovery. In such geometries, Brownian motion may beconsidered as the sole mechanism that enables transport of colloidalparticles into or out of the channels, but it is, unfortunately, anextremely inefficient transport mechanism for microscale particles.Here, we explore the possibility of diffusiophoresis as a means tocontrol the colloid transport in dead-end channels by introducing asolute gradient. We demonstrate that the transport of colloidalparticles into the dead-end channels can be either enhanced orcompletely prevented via diffusiophoresis. In addition, we showthat size-dependent diffusiophoretic transport of particles can beachieved by considering a finite Debye layer thickness effect,which is commonly ignored. A combination of diffusiophoresis andBrownian motion leads to a strong size-dependent focusing effectsuch that the larger particles tend to concentrate more and residedeeper in the channel. Our findings have implications for all mannersof controlled release processes, especially for site-specific deliverysystems where localized targeting of particles with minimal disper-sion to the nontarget area is essential.

colloid | diffusiophoresis | solute gradient | size effect | dead-end channel

The ability of a particle to migrate along a local solute con-centration gradient, which is referred to as diffusiophoresis,

has been exploited to direct transport in a variety of systems, e.g.,artificial swimmers (1, 2) and collective behaviors of active col-loids (3, 4). One physical mechanism for diffusiophoresis origi-nates from surface–solute interactions, where the solute gradientsets up an osmotic pressure gradient within a narrow interactionregion. This gradient leads to fluid flow along the surface of aparticle, in which case propulsion occurs in the opposite di-rection and is referred to as chemiphoresis (5, 6). In addition,differences in diffusivities between anions and cations lead tospontaneous electrophoresis of a particle, giving an additionalpropulsion mechanism. A particular feature of diffusiophoresis isthat the diffusiophoretic mobility, or the phoretic velocity, of aparticle is independent of its size, as long as the thickness of theinteraction region, e.g., the Debye screening layer when the in-teraction is electrostatic, is much thinner than the size of theparticle (6). This feature allows the utilization of diffusiophoresisfor enhancing transport of microscale particles, leading to ordersof magnitude higher transport rates compared with pure diffu-sion (7). However, this size independence could also be a sourceof frustration because it precludes useful effects such as sortingor controlling transport by particle size.We anticipate that size-independent particle mobility breaks

down when the thickness of the surface–solute interaction regionbecomes comparable to the size of the particle. Already morethan a century ago this feature has been well appreciated in thefield of electrokinetics as the Hückel limit (8) where the elec-trophoretic mobility of a particle is 2/3 of the Smoluchowskimobility if the thickness of Debye screening layer κ−1 is larger thanthe particle radius a, i.e., κa< 1. Likewise, there are a number ofinvestigations, mostly theoretical, on the effect of finite Debye layer

thickness on the diffusiophoretic mobility, which have shown thatthe influence of finite κa is much stronger than for electrophoresisdue to the presence of chemiphoresis (9–11). Because a finiteDebye layer effect can lead to size-dependent particle mobility, werevisit the influence of finiteness of κa on the diffusiophoretic mo-bility, and exploit it in a useful way.One application of size-dependent diffusiophoresis is particle

transport into dead-end geometries, which is important to manyindustrial applications including drug delivery and disinfection.When characterizing transport processes, transferring fluidand/or particles into dead-end geometries or pores is a signifi-cant challenge as such geometries do not allow any net fluid flowwithin the system. Recently, it has been shown that diffusiophoresiscan be used to pump colloidal particles and oil emulsions in and outof dead-end capillaries, which have significant implications for oilrecovery systems (12).Here, we use size-dependent diffusiophoresis to control col-

loid transport in a dead-end channel. The observed size de-pendence is attributed to the abovementioned finite Debye layereffect. This key insight can be exploited pragmatically, to achieveuseful end points such as size-dependent particle sorting andseparation. We show that theory accounting for finite κa is in agood agreement with the experimental observations.Further, we demonstrate that a judicious choice of ions for

multicomponent solutes can also be used to gain exquisite con-trol of the phenomenon, importantly at constant osmolarity,which is critical for in vivo applications, either preventing colloidparticles from entering a dead-end channel, or promoting entry

Significance

Dead-end geometries are commonly found in many poroussystems. Particle transport into such dead-end pores is oftenimportant, but is difficult to achieve owing to the confinement.It is natural to expect that Brownian motion is the sole mech-anism to deliver the particles into the pores, but that mecha-nism is, unfortunately, slow and inefficient. Here, we introducesolute gradient to control the transport of particles in dead-endchannels. We demonstrate a size effect such that larger parti-cles tend to focus more and reside deeper in the channels. Ourfindings suggest a potential pathway to many useful applica-tions that are difficult to achieve in dead-end geometries suchas particle sorting and sample preconcentration, which areimportant in pharmaceuticals and healthcare industries.

Author contributions: S.S., P.B.W., and H.A.S. designed research; S.S., E.U., B.S., J.T.A., M.R., P.B.W.,and H.A.S. performed research; S.S., E.U., B.S., J.T.A., P.B.W., and H.A.S. analyzed data;and S.S., P.B.W., and H.A.S. wrote the paper.

Conflict of interest statement: P.B.W. discloses a substantive (>$10,000) stock holding inUnilever PLC.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: sangwoos@princeton.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1511484112/-/DCSupplemental.

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as for a single binary ionic solute such as NaCl. These observa-tions suggest future applications in drug delivery (13, 14).To impose a solute gradient along a dead-end channel, we

have used a microfluidic approach (Fig. 1A), where the entranceof the dead-end channel is connected to the main flow channelso that the solute concentration (co) can be regulated at the en-trance. Additionally, the height of the dead-end channel is muchthinner (≈10 μm) than the main channel (≈100 μm), such thatthe disturbance from the main channel can be minimized (15).Moreover, unlike electrophoresis or thermophoresis, where

the field gradient is generally constant, diffusion in a dead-endchannel is time dependent. Therefore, to observe the transientdynamics, especially at the early stages of the solute and colloidtransport, a steplike initial concentration profile is desired. Toachieve this condition, we inserted in the main channel an im-miscible fluid such as an oil droplet or an air bubble to separatethe leading solution, which fills in the dead-end channel first,with the trailing solution that has different solute conditions andcontains the colloidal particles (Fig. 1B). This sequential ap-proach works as an effective gate leading to a repeatable con-stant inlet concentration at the start of the experiment. In theabsence of this flow design, the trailing fluid that contains thecolloidal particles will be mixed with the leading fluid duringthe delivery to the inlet of the dead-end channel. This results in>10 min of dead time before the steady-state inlet condition isreached (Fig. 1C); such a delay would obscure and compromisethe results we report here.Using this setup, we study the colloid transport in a dead-end

channel induced by a solute gradient. Because the colloidalparticles migrate toward higher solute concentrations with our

current choice of solutes, we keep the initial inner solute con-centration (ci) high and the outer solute concentration (co) low,unless otherwise noted. Typical transport dynamics of colloids inthe presence of a solute gradient are shown in Fig. 1 D and E andMovie S1. Owing to the solute (NaCl) gradient, the colloidtransport into the dead-end channel is accelerated by nearly twoorders of magnitude compared with the pure diffusion case (i.e.,no solute gradient) as shown in Fig. 1F.The enhanced colloid transport is not surprising as it has al-

ready been reported in a number of studies (7, 16, 17). However,we have also observed that the density profile of the leadingcolloids is nonuniform in the lateral direction, and the colloidalparticles tend to concentrate as they transport along the channel(Fig. 1 D and E). These features stem from the intrinsic geo-metrical confinement of the dead-end channel. In the presenceof a solute gradient along a channel, a diffusioosmotic flow isinduced, which has a pluglike flow profile. Because a dead-endgeometry does not allow any net flux, a pressure gradient isestablished that opposes the osmotic flow. In consequence, as thepressure-driven flow has an essentially parabolic profile, summingup these two components results in a circulating flow where thevelocity at the center of the channel is opposite to the wall slipvelocity. This flow is analogous to an electrokinetic pump wherean electroosmotic flow that is induced by an external electric fieldis balanced by a Poiseuille flow in a dead-end channel and leads toa circulating flow developed nearly instantaneously within theentire channel (18).In our system, because the solute concentration is diffusing out

over time (ci > co), the circulating flow slowly propagates towardthe closed end of the channel (Supporting Information, Movie S2).

A B C

D E F

Fig. 1. Colloid transport in a dead-end channel induced by a solute gradient. (A) Setup for enabling transport experiments in a dead-end channel withminimum disturbance from a continuous flow. (B) Steplike initial solute and colloid concentrations were realized by inserting an oil droplet or an air bubble inbetween two liquids with different solute concentrations. (C) Otherwise, a gradual concentration was observed due to mixing. Insets are the fluorescentintensity distributions of colloidal particles (polystyrene latex beads, diameter 0.19 μm) along the dead-end channel. (D) Sequential images and (E) intensitydistributions of colloids (particle diameter 0.19 μm) migrating along a dead-end channel in the presence of a solute gradient (NaCl: ci = 2 mM, co = 0.02 mM).Channelwise direction, x, is normalized by the length of the dead-end channel, L (=400 μm). (F) Time taken for the colloidal particles to reach the middle ofthe dead-end channel with 50% of the inlet fluorescent intensity (0.5Ii) under different solute gradients. (Scale bars: 50 μm.)

258 | www.pnas.org/cgi/doi/10.1073/pnas.1511484112 Shin et al.

Along with this flow, particles experience phoretic motion,and thus the net motion of the particles is the sum of the fluidflow and the phoretic movement, which leads to the lateral cur-vature of the density profile, as also observed in a recent study byKar et al. (12).The transient nature of solute diffusion in a dead-end channel

also leads to another significant effect: particle focusing. Becausethe gradient of the solute concentration is always decreasingalong the channel due to diffusion, the particle phoretic velocity,which is proportional to ∇lnc, decreases as the particles movedeeper into the channel (Fig. S1). Thus, particles tend to accu-mulate near the leading edge of the migrating colloidal front (17,19), which creates a pluglike colloidal “wave” that can be quanti-tatively identified from the time-dependent fluorescence intensitydistribution along the channel (Fig. 1 D and E).This colloidal wave leads to preconcentration, separation, and

sorting of particles, which could be useful to many applications.The quantitative factors that define the colloidal wave are thelocation of the peak, the amplitude, and the width of the wave. Intheory, these factors are set by the transport properties of theparticle and the solute, namely the diffusiophoretic mobility ofthe particle (Γp) and diffusivity of the particle (Dp) and the solute(Ds). In detail, the transient distribution of colloids (concentra-tion n) in the presence of a solute gradient can be described byan advection–diffusion equation that is coupled to the solutediffusion, which drives the advection of colloids. The velocity ofthe particles is defined as v= vp + vf where vp =Γp∇lnc is theparticle velocity driven by diffusiophoresis and vf is the flow ve-locity driven by solute gradient (see Supporting Information for fullderivation of vf). Using the conservation law ∂n=∂t  +  ∇  ·   j  =   0,where j  =   −Dp∇n  +   vn is the particle flux, the dimensionlessequation for the colloid density N can be expressed as

∂N∂τ

=Dp

Ds∇2N −

Γp

Ds∇ðN∇lnCÞ − ∇ · ðVNÞ, [1]

where C is the dimensionless solute concentration, τ is the di-mensionless time (tDs=L2, where L is a characteristic lengthscale), and V is the dimensionless fluid velocity, which is definedas V = vfL=Ds. The first term on the right-hand side of Eq. 1represents the diffusion of colloids, whereas the latter terms rep-resent the colloid advection driven by a solute gradient.In general, Γp is commonly regarded as a size-independent

value in the thin Debye layer approximation (κa→∞). When thethickness of the Debye layer becomes comparable to the size ofthe particle, the diffusiophoretic mobility becomes size dependent,ΓpðκaÞ. Following the pioneering work of Prieve and coworkers (9,10), the size-dependent diffusiophoretic mobility of the particles isexpressed as

Γp =e

2η

�kBTZe

�2 u01− u1=ðu0   κaÞ , [2]

where e is the permittivity of the medium, η is the viscosity of themedium, kB is the Boltzmann constant, T is the temperature, Z isthe valence of the solute, and e is the elementary charge. Also, u0and u1 are functions of the zeta potential. The lowest order contri-bution for very large particles or vanishing thickness of the Debyelayer (κa→∞) is

u0 = 2βZeζpkBT

+ 8 ln  cosh�Zeζp4kBT

�. [3]

Here, ζp is the zeta potential of the particles andβ= ðD+ −D−Þ=ðD+ +D−Þ, where D+ and D− are the diffusivitiesof cations and anions, respectively. The first and second terms onthe right-hand side of Eq. 3 represent electrophoresis and

chemiphoresis, respectively. For u1, lengthy expressions can befound in (9) (see Supporting Information for further discussion).A remarkable fact is that even when κa≈ 100, unlike electro-

phoretic mobility (20), the predicted diffusiophoretic mobilitydeviates noticeably from Γpðκa→∞Þ (10). When κa≈ 10, which is ageneral condition for common diffusiophoresis experiments (7, 12,16, 17), the diffusiophoretic mobility is predicted to be almost anorder of magnitude lower, depending on ζp (10). Our experimentalconditions are in a similar range, κa≈ 2− 40 [κ−1 = 13.6 nmwhen c= ðci + coÞ=2≈ 1 mM], indicating that size effects can besignificant.The diffusiophoresis of particles with different sizes in a dead-

end channel is shown in Fig. 2. Owing to the broad range of κa,we observe strong size-dependent particle focusing, where thelarger particles tend to reside deeper in the channel, and they

A

B C

D E

Fig. 2. Size-dependent particle focusing driven by a solute gradient (NaCl: ci =2 mM, co = 0.02 mM). (A) Fluorescent images and (B) intensity distributions ofcolloids with different diameters ranging from 0.06 to 1.01 μm at t = 300 s. Theintensity I is normalized by the maximum intensity Imax. (C) Theoretical pre-diction for colloid density profiles with different diameters at t = 300 s. Plot of(D) peak position (xp) and (E) focus magnitude (Imax/Imin, Imin is the minimumintensity near the inlet) for various particles at t = 300 s obtained from B. Blackcurves represent theoretical predictions. (Scale bar: A, 50 μm.)

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generally tend to concentrate more (Fig. 2A and Movie S3). Forinstance, the smallest particles (2a= 0.057  μm) show a focusingratio of about 2, and the terminal peak position xp is about 50%of the channel length L at t= 300 s, whereas, the largest particles(2a= 1.01  μm) show a focusing ratio as large as 100, and theparticles travel nearly to the end of the channel without notice-able dispersion (Fig. 2 D and E).For a given zeta potential (−70 mV), Γp is estimated to vary by

almost a factor of 5 based on Eq. 2 for our range of particle sizes(Fig. S3). Considering that Dp also changes significantly with sizebased on the Stokes–Einstein relation, a combination of dif-fusiophoresis and Brownian diffusion leads to a strong size ef-fect, which is confirmed from a good agreement between theexperiment and the theory based on Eqs. 1 and 2 (Fig. 2 B andC). The discrepancy near the entrance of the channel for smallparticles shown in Fig. 2C mainly comes from the penetration ofthe fluorescent signal coming from the main channel because ofthe strong optical density required for small particles.Other possible size-dependent phenomena include particle–

particle interactions (5), wall interactions (21), hydrodynamicdispersion (22), gravimetric effects (21), electrokinetic lift (23), etc.,but we argue that these effects are negligible compared with thefinite κa effect (detailed discussion in Supporting Information).Note that the presence of the fluid advection does not significantlyinfluence the particle transport, but only contributes to the lateralinhomogeneity in the early stage, which is confirmed by comparingthe theoretical results with and without the solute gradient-inducedfluid advection (Figs. S4 and S7; Supporting Information).All of the demonstrated results so far were driven by a single solute

gradient with a contrast of 100 (NaCl: ci = 2 mM, co = 0.02 mM),which may be limited for practical applications, especially within vivo drug delivery where a strong solute gradient may leadto an osmotic shock. Thus, to seek broader insights of dif-fusiophoresis so as to gain further control over the movement ofcolloidal particles, we focus on the role of individual mechanismsthat contribute to the overall transport. Recall that diffusiophoresishas two contributions, one from chemical potential differences(chemiphoresis), and the other from differences between the dif-fusivities of anions and cations, which creates a local electric field(electrophoresis). Considering that the diffusivity of K+ is nearlyidentical to that of Cl− (DK + /DCl− = 0.97, cf. DNa+ /DCl− = 0.66; ref.24), we suggest that there can be an alternative strategy to controlcolloid transport besides a single solute gradient: by displacing asolute with another solution containing different species. Analo-gous to a liquid junction potential (25, 26), the interdiffusion ofmulticomponent solutes having equimolar concentration alsogenerates a spontaneous electric field that can allow electropho-resis. For instance, the interdiffusion of NaCl and KCl solutions

can generate an electric field of ∼40 V/m, which is comparable tothe single solute (NaCl) gradient case (Supporting Information).Using this strategy, we can create a local electric field that

gives rise to the electrophoresis of the particles in the channel-wise direction so that the colloid transport into the channel isenhanced by locating NaCl inside the dead-end channel and dis-placing it with KCl, which flows through the main channel. Asillustrated in Fig. 3 A and B (Movie S4), we show that this strategyallows particle focusing and fast colloidal transport that is com-parable to the single solute gradient case (black curves). This“displacement” strategy is desired when a constant osmolarity ofthe solution is required, such as with in vivo transport systems.Likewise, if we place NaCl and KCl in the opposite configu-

ration, the electric field is now generated in a reverse direction,which is expected to slow down the colloid transport. Indeed,such an initial distribution of electrolyte completely prevents thecolloidal particles from going into the channel until the electricfield has vanished (blue curve in Fig. 3B; Movie S4). This effectlasts for over a minute, which is the same order of magnitude asthe solute diffusion time, L2=Ds ∼ 100 s. This approach may havea significant implication in programmable drug delivery appli-cations where such an electric field can serve as a trigger for drugrelease (27).

A B

Fig. 3. Interdiffusion of equimolar multicomponent solutes (ci = co = 2 mM)for inducing diffusiophoresis in a dead-end channel under constant osmo-larity. (A) Fluorescent intensity distribution at 300 s and (B) trace of frontposition, xf = xði= 0.2IiÞ, for different solute configurations (blue: inner =KCl, outer = NaCl; red: inner = NaCl, outer = KCl). Results of a single solutewith a gradient (NaCl: ci = 2 mM, co = 0.02 mM) are presented in black forcomparison.

A B C D

E F G

Fig. 4. Control of colloidal particles in dead-end channels via a solutegradient for various applications. (A–D) Size-dependent particle sorting frommixture of particles in a dead-end channel driven by a solute gradient (NaCl:ci = 2 mM, co = 0.02 mM). The mixture consists of polystyrene particleshaving diameters of 0.21 and 1.01 μm dyed with different fluorophores.(A–C ) Fluorescent images of (A) diameter = 0.21 μmparticles and (B) 1.01 μmparticles at t = 300 s, and (C) intensity distributions of A and B along thechannel. D is an overlaid image of A and B. (E–G) Control of lipid vesicles fordrug delivery applications. Fluorescent images of (E) SUVs (mean diameter ≈56 nm) and (F) LUVs (mean diameter ≈ 861 nm), and (G) intensity distri-butions of the vesicles along the channel at t = 300 s. (Scale bars: 50 μm.)

260 | www.pnas.org/cgi/doi/10.1073/pnas.1511484112 Shin et al.

We have demonstrated an effective way of delivering colloidalparticles into dead-end channels by imposing solute gradients, ei-ther by a single solute gradient, or by multispecies interdiffusion.We further demonstrated that size-dependent diffusiophoresis canbe obtained by controlling κa, a fact that is commonly ignored. Akey observation regarding the size-dependent diffusiophoresis isthat there is a tendency for the larger particles to focus more andtransport farther into the channel, which suggests many techno-logical implications such as particle sorting that are otherwise dif-ficult to achieve in dead-end geometries. As demonstrated in Fig. 4A–D (Movie S5), particle sorting from a mixture of two (or more)particles having different sizes can be simply achieved. This featureis enabled by the fact that particle–particle interactions are veryweak for phoretically driven particles such that the particles areeasily separated without interfering with neighboring particles (5).In addition to particle sorting, size-dependent diffusiophoresis

can be important especially in pharmaceuticals because it makesit possible to manipulate the final concentration and location ofparticles within deep pores with controlled dispersion based ontheir sizes. As an example, delivery of lipid vesicles (lipid com-position is provided in Materials and Methods) in deep pores canbe manipulated via imposing a solute gradient across the pore,where the peak penetration depth and the concentration ratio ofthe vesicles are controllable based on the size of the vesicles (Fig.4 E–G and Movie S6). This demonstration implies the possibleapplication of size-dependent diffusiophoresis in site-specificdelivery systems where localized targeting of particles withminimal dispersion to the periphery is desired (14).

Materials and MethodsMaterials. Fluorescent polystyrene latex particles were purchased from Bang-slab. NaCl, KCl, EDTA, Hepes, NaOH, and chloroform were purchased fromSigma Aldrich. 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), 1,2-dio-leoyl-sn-glycero-3-phospho-L-serine (sodium salt) (DOPS), and 1,2-dipalmi-toyl-sn-glycero-3-phosphoethanolamine-N-(lissamine rhodamine B sulfonyl)(ammonium salt) (Rh-DPPE) were purchased from Avanti Polar Lipids.Polydimethylsiloxane (PDMS) was purchased from Dow Corning (Sylgard 184).

Sample Preparations. The polystyrene latex particles were dispersed in eitherNaCl or KCl solutions prepared from deionized water with concentration of0.05 wt%. Lipid vesicles were prepared from a lipid mixture of DOPC/DOPS/Rh-DPPE in a 89.5/9.5/1 mol% ratio dissolved in chloroform. Small unilamellarvesicles (SUVs) were prepared via the standard sonication method (28); 1 mgof lipid mixture was dried overnight under vacuum. Then, the dried lipidswere rehydrated with a buffer solution (0.5 mM Hepes, 0.5 mM EDTA, ad-justed with 1 M NaOH to pH≈ 8) followed by sonication using a tip sonicatorfor 10 min. Large unilamellar vesicles (LUVs) were prepared via the elec-troformation method (29). Fifty μg of lipid mixture was spread on the sur-faces of indium tin oxide (ITO) glass slides, then dried overnight undervacuum. The ITO glass slides were separated by a PDMS gasket, which wasfilled with Hepes buffer solution. An AC electrical field (1.7 Vpp, 10 Hz) wasapplied to the ITO glass slides for 30 min to form LUVs. The diameter and thezeta potential of the polystyrene particles and lipid vesicles were measuredusing Zetasizer Nano-ZS (Malvern Instruments).

Experimental Setup. All channels were made from PDMS with a two-step li-thography process for making a thin dead-end channel (∼10 μm) that isconnected to the thick main channel (∼100 μm). The ratio between theelastomer base and the cross-linker was 10:1.5 to enhance the stiffness ofthe channels. The channels were sealed with the same PDMS slab to ensurethat all of the surfaces are the same. The fabricated PDMS channels wereimmersed in aqueous solution overnight before the experiments to preventwater permeation. The flow rate was controlled using a feedback-controlledpump (MFCS-EZ; Fluigent) that is equipped with a precise flow meter (Flow-unit; Fluigent). Under the low Reynolds number flow conditions that applyhere, the main channel flow penetrates a distance of order ∼10 μm into theside channel. The Péclet number at the entrance, Uw=Ds, where U is flowvelocity near the entrance of the dead-end channel (≈10 μm/s), w is the widthof the channel (40 μm), and Ds is the solute diffusivity (≈1.5×10−9 m2/s),can be reduced below 1, so the advection from the main flow canbe neglected.

ACKNOWLEDGMENTS. We thank Jie Feng for help with zeta potential andparticle size measurements, and Janine Nunes, Orest Shardt, and Anne-Florence Bitbol for valuable discussions. We also thank the anonymousreferees for valuable comments. We acknowledge Unilever Research forsupport.

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