Long-range repulsion of colloids driven by ionexchange and diffusiophoresisDaniel Floreaa,b, Sami Musaa, Jacques M. R. Huyghea, and Hans M. Wyssa,b,1

aDepartment of Mechanical Engineering and bInstitute for Complex Molecular Systems, Eindhoven University of Technology, 5612 AZ, Eindhoven,The Netherlands

Edited by Monica Olvera de la Cruz, Northwestern University, Evanston, IL, and approved March 12, 2014 (received for review December 10, 2013)

Interactions between surfaces and particles in aqueous suspensionare usually limited to distances smaller than 1 μm. However, in arange of studies from different disciplines, repulsion of particles hasbeen observed over distances of up to hundreds of micrometers, inthe absence of any additional external fields. Although a range ofhypotheses have been suggested to account for such behavior, thephysical mechanisms responsible for the phenomenon still remainunclear. To identify and isolate these mechanisms, we perform de-tailed experiments on a well-defined experimental system, usinga setup that minimizes the effects of gravity and convection. Ourexperiments clearly indicate that the observed long-range repulsionis driven by a combination of ion exchange, ion diffusion, and dif-fusiophoresis. We develop a simple model that accounts for ourdata; this description is expected to be directly applicable to a widerange of systems exhibiting similar long-range forces.

exclusion zone | Nafion | chemotaxis | unstirred layer | solute-free zone

Exclusion zone (EZ) formation is a phenomenon where col-loidal particles in an aqueous suspension are repelled from

an interface over distances of up to hundreds of micrometers,leading to the formation of a particle-free zone in the vicinity ofthe interface. Such peculiar behavior has been observed byresearchers from different disciplines for a wide range of materi-als, including biological tissues such as rabbit cornea (1), whiteblood cells (2), polymer gels (3), ion-exchange membranes (4), ormetals (5). Depending on the field of research, different termshave been used to refer to the behavior. In biological systems,already in the early 1970s, EZs observed close to the surface ofbiological tissues such as stratum corneum were referred to asunstirred layers (1), as these colloid-free layers persisted evenwhen the suspensions were stirred. In later studies, the formationof similar EZs, where Indian ink particles were excluded from thevicinity of leukocyte cells (2), was referred to as aureole formation.The observed EZ formation is highly surprising, as the forces

acting on the colloidal particles can extend over distances ofhundreds of micrometers (1–5). Long-range interactions acting oncolloidal particles are generally of electrostatic nature (6, 7), witha range set by the thickness of the electrical double layer sur-rounding a charged colloidal particle, the Debye length λD.Whereas in low-polar solvents, these electrostatic interactions canact over tens of micrometers (8), in aqueous suspensions theseforces are limited to length scales of typically less than 1 μm (6, 7).A range of hypotheses have been formulated to account for EZ

formation, including the emergence of excited coherent vibrationmodes ofmolecules in themembrane or the surrounding water thatcould create large dipole oscillations (9). Deryagin offered a similarexplanation, by attributing the aureole formation around cells tolong-range forces originating from electromagnetic vibrations; healso mentioned as a possible explanation forces of a diffusiopho-retic nature arising in the presence of an electrolyte concentrationgradient, but dismissed these in favor of the electromagnetic vi-bration hypothesis (10). Recently, a chemotaxis hypothesis similarto diffusiophoresis has been suggested and theoretically inves-tigated to account for the EZ formation (11, 12). Effects assuminga long-range structuring of water near hydrophilic surfaces have

also been suggested as a possible origin of the behavior (3–5).However, to date it is still unclear whether any of these hypotheses,individually or in combination, can fully account for the observedbehavior, as existing comparisons between experiments and theo-retical predictions are still unable to clearly discern between thedifferent hypotheses. An understanding of the physical origins ofEZ formation is thus still lacking.In this paper, we study EZ formation in detail by measuring

the time and position dependence of forces acting on the par-ticles in multiple configurations. Our experimental results enableus to clearly identify a physical explanation for this intriguingphenomenon, which we expect to apply also to a wide range ofother systems exhibiting similar long-range repulsion. To ratio-nalize the observed behavior, we develop a simple model thatquantitatively accounts for our experimental data.

ResultsTo systematically elucidate the EZ phenomenon, we choose theperfluorinated polymer membrane material Nafion 117 as a modelsystem, as EZ formation around this material has already beenwidely studied (4). Moreover, the material’s physical and chemicalproperties have been characterized in detail (13, 14) due its wide-spread use as a proton-conducting membrane in polymer fuel cells(13, 14) and other electrochemical applications, where it is used inthe production of NaOH, KOH, and Cl2 (14). As a colloidal sus-pension we use uniformly sized polystyrene particles in aqueoussolutions of monovalent salt, providing well-controlled and simpleexperimental conditions.Although the origin of the force driving the EZ formation is

still unclear, particles generally migrate away from the surface,suggesting the existence of a force perpendicular to the surface,acting on the particles. To study this driving force in detail, we

Significance

The ability to displace particles or solutes relative to a back-ground liquid is of central importance to technologies suchas filtration/separation, chromatography, and water purification.Such behavior is observed in so-called exclusion zone formation,an effect where particles are pushed away from a surface overlong distances of up to hundreds of micrometers. However, it isstill unclear which physical mechanisms are responsible. Ourwork provides a detailed understanding of this exclusion zoneformation, enabling a precise control of the behavior. This couldbe exploited, for instance, for sorting in microfluidic devices, inadvanced antifouling coatings, or for elucidating biological pro-cesses where it is likely to play an important, yet unexplored role.

Author contributions: D.F. and H.M.W. conceived and designed the experiments; D.F., S.M.,J.M.R.H., and H.M.W. designed research; D.F. performed research; D.F. and H.M.W. analyzeddata; and D.F. and H.M.W. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. E-mail: H.M.Wyss@tue.nl.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1322857111/-/DCSupplemental.

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wish to isolate its effects from those of other forces acting on theparticles, such as those induced by gravity or by advection of thesuspension. Previous studies of the EZ phenomenon have alwaysused a horizontal setup, where the driving force FEZ is perpen-dicular to the gravitational force, which can lead to the buildupof complex flow patterns (15), an effect that becomes even morepronounced in a setup with Nafion at the bottom, as shown inMovie S1. This has made it difficult to isolate the effects of FEZon EZ formation, thus preventing a detailed analysis of the ki-netics of the process.To circumvent these problems, we perform experiments in

a vertical sample cell, where the Nafion material is placed at thetop. This ensures that all forces acting on the particles point inthe same direction. We thus expect that particle motion in thesystem occurs along purely vertical trajectories that are perpen-dicular to the Nafion surface; this enables us to readily subtractthe effects of gravity, thereby isolating the effects of the forcethat is induced by the Nafion surface. We therefore choose thisgeometry as our standard experimental setup to study the ki-netics of diffusion zone formation. The EZ remains horizontal,as shown for a typical experiment in the sequence of microscopyimages displayed in Fig. 1 A–C. To follow the kinetics of theprocess in detail, we take snapshots of the sample every 5 s. Asexplained in Materials and Methods, we compose a time–spacediagram from these acquired images (Fig. 1D), capturing thedevelopment of the EZ profile as a function of time.This diagram enables us to readily extract the distance dEZðtÞ

from theNafion surface to the edge of the EZ, which we plot in Fig.1E as red circles. From a separate experiment without Nafionsurface we extract the speed of sedimentation of the particles solelydue to gravity. As expected, the sedimented distances from the topof the cell, shown in Fig. 1E as green diamonds, are considerablysmaller than for the system where Nafion is present (SI Text). Toisolate the effects of the force FEZ, we subtract the gravity-inducedparticle displacement from the observedEZdistance, shown in Fig.1E as blue squares. Initially, the displacement of particles proceedsrapidly, with a typical velocity of 0.9 μm=s, 300 s after first contactof the suspension with the Nafion surface. The speed of the par-ticles at the edge of the EZ then decreases continuously as timeprogresses, reaching a typical value of 0.29 μm=s after 1 h.

When plotting dEZðtÞ in a double-logarithmic plot, we observea simple power-law scaling with slope of 1=2, as shown in Fig. 1F.As expected, the motion only due to gravity, shown as greendiamonds, exhibits a purely linear behavior, corresponding toa slope of unity. The simple square-root-of-time scaling of dEZðtÞis highly reproducible, as illustrated by Fig. 1G, where we displayresults from five independent experiments performed for par-ticles in an aqueous solution of 1 mM NaCl. The data coincidealmost perfectly, as captured by the average of these five in-dependent experiments and the corresponding SDs, presented inFig. 1H. In fact, we observe a square-root-of-time scaling in allour experiments, independent of suspension properties such asthe concentrations and types of salt or colloidal particles used.Moreover, our experiments also indicate that the force FEZ is notsignificantly affected by parameters such as the illumination, thethickness of the Nafion layer inside the capillary, or by artifactssuch as bubble formation near the surface or irregularities in theshape of the interface (Figs. S1 and S2 and SI Text). The ob-served square-root-of-time dependence thus appears to be a re-markably robust feature of the EZ formation process.This scaling indicates that a diffusive process is important in

determining EZ formation. Motivated by this observation, weextract an effective diffusion coefficient from the data in Fig. 1E,yielding a value of Deff = 1:26× 10−5 cm2=s; this is orders ofmagnitude larger than the diffusion coefficient expected for ourparticles in water, as obtained from the Stokes–Einstein relation,Dp = kBT=6πηr= 4:36× 10−9 cm2=s, where η= 0:001 Pa is the vis-cosity of water, r= 0:5 μm the particle radius, kB the Boltzmannconstant, and T = 298 K the ambient temperature. However, thevalue of Deff is within the range of typical diffusion coefficientsof monovalent ions in aqueous solution, with typical valuesranging from 1:03× 10−5 cm2=s for Li+ ions to 9:31× 10−5 cm2=sfor H+. This suggests that the physical mechanism that governsEZ formation is directly related to the diffusion of ions.In fact, Nafion interacts strongly with ions in aqueous sol-

utions; it is a cation exchange material. When an aqueous solu-tion containing cations comes in contact with Nafion, as a resultof a higher affinity of the material for these cations, these ionsare replaced with H+ ions initially bound inside the Nafion (14).This ion-exchange process will thus lead to an inhomogeneousdistribution of ions in the liquid. As a result, a time-dependent

E F G H

A B C D

Fig. 1. Kinetics of EZ formation in a vertical setup. (A–C) Snapshots from a movie of the sample (polystyrene particle suspension in 1 mM NaCl solution) atdifferent times after contact of the suspension with the Nafion material: t = 5 min (A), t = 30 min (B), and t = 60 min (C). The Nafion material is at the top ofthe image; gray areas at the bottom correspond to points where particles are present, whereas the brightest areas in the middle represent the EZ. (D) Space–time diagram of the evolution, constructed by combining the intensity distributions along the vertical direction into a single image (details in Materials andMethods). (E) EZ distance dEZ (red circles) extracted from the space–time diagram in D. The displacement only due to gravity, from an experiment withoutNafion, is shown as green diamonds. The blue squares show the EZ distance with the influence of gravity subtracted, isolating the displacement due to the EZforming force. (F) The same data represented in a double-logarithmic plot; the black solid and black dashed lines are power laws with slopes of 1/2 and 1,respectively. (G) EZ distance dEZ as a function of time, plotted for five independent experiments, shown as different symbols, performed under the sameconditions in suspensions of 1 μm polystyrene particles in a 1-mM NaCl aqueous solution. (H) Average of the five independent EZ position measurements withthe corresponding SD, displayed in a double-logarithmic plot. A power-law fit to the data yields an exponent of 0.495.

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concentration profile should develop for each ionic species. Wehypothesize that this inhomogeneous ion distribution is directlyrelated to the process of EZ formation and the correspondingforce FEZ acting on the particles.Indeed, there is a physical mechanism that results in a drift of

particles in liquids with inhomogeneous ion concentrations. Thiseffect, termed diffusiophoresis, was first theoretically describedby Deryagin et al. (16) and later confirmed and studied in moredetail both experimentally (17, 18) and theoretically (19–21); thetheoretical description of diffusiophoresis is in excellent agree-ment with experiments, as recently illustrated by a study usingmicrofluidic coflow devices (18).To obtain an intuitive understanding of this effect, we consider

a charged particle immersed in an electrolyte solution with a saltgradient ∇C, as shown schematically in Fig. 2A. The concen-tration of excess counterions decays with distance from thesurface as C+ =C+0 exp−x=λD , where the Debye length λD scaleswith salt concentration approximately as λD ∝C−1=2. As a result,the presence of a salt gradient ∇C also implies a variation of theDebye screening length along the particle surface, as shown ina schematic close-up of the particle surface region in Fig. 2B.Within the Debye layer, any small subvolume of fluid contains

an excess of counterions and thus possesses a net charge. Asa result, each subvolume experiences electrostatic forces underthe influence of the local electric field. For a uniform Debyelength, the charge distribution around the particle is sphericallysymmetric, and the resulting stresses occurring in the fluid can befully balanced by hydrostatic pressure. However, in the presenceof a salt gradient, this is no longer the case; the counterion cloudaround the particle is anisotropic, with λD decreasing toward thehigher salt concentrations. The corresponding asymmetry of thecharge distribution leads to both a gradient in the hydrostaticpressure and nonzero components of the local electric fieldtangential to the particle surface, as schematically shown in Fig.2B. Both these effects lead to stresses in the fluid tangential tothe surface that cannot be balanced by hydrostatic pressure, thusleading to a flow of fluid along the surface of the particle, gen-erally toward the direction of lower salt concentration. As theoverall system is incompressible, this flow must be balanced bya corresponding motion of the particle in the opposite direction(19, 21), as shown schematically in Fig. 2C. Detailed theoretical

calculations, taking into account the balance between the hy-drostatic pressure and the electrostatic stress (19), result ina constant particle velocity U, proportional to the gradient of thelogarithm of the ionic concentration ∇logC, as

U =DDP∇logC; [1]

where DDP is the so-called diffusiophoresis constant that de-pends on the (average) Debye length λD, the zeta potential ζ,the viscosity η, the temperature T, and the diffusion coefficientsD+ and D− of the cations and the anions, respectively (17, 19,21). If indeed diffusiophoresis is the physical mechanism respon-sible for the observed EZ formation, then the particle velocitiesand thus also the velocity of the edge of the EZ should alwaysremain proportional to ∇logC.To test our hypothesis of diffusiophoresis as the underlying

mechanism leading to EZ formation, we thus wish to predict theconcentration profile of ionic species in the system as a functionof time. In doing so, we consciously neglect some of the richproperties of Nafion, where effects such as interdiffusion ofspecies inside the material (22), swelling behavior (23), or mi-crostructural changes during hydration (13) are known to occur.We focus instead on the well-known ion-exchange properties ofthe material (14) and assume that the material acts as a perfectdrain for cations, where all cations that come in contact with thematerial are exchanged for H+ ions. The concentration of ex-changeable protons within the Nafion material is on the order of4 M (14), far exceeding the cation concentrations used in oursamples. We thus expect that the ion exchange is limited purely bydiffusion, rather than by the availability of protons in the Nafionmaterial (24). The problem of predicting the ion concentrationprofiles thus reduces to solving a set of coupled diffusion equa-tions for all of the involved ionic species under these particularboundary conditions. Whereas the diffusion equation is readilysolved for the case of a single electrolyte, for the case of multipleelectrolytes, the diffusion problem is difficult to solve analytically.It is known that the diffusion of one species is strongly affected bythe presence of others (25); to correctly describe the process wethus need to account for the diffusion of all four ionic speciespresent in the solution (X+, Y−, H+, and OH−).For each of these ionic species we can write a Nernst–Planck

equation, yielding a system of four coupled equations. However,local electroneutrality and the chemical equilibrium of waterðH2O �! �H+ +OH−Þ reduce these four equations to an analyt-ically tractable system of just two equations for the two elec-trolytes (XY and HY) (25), as

∂Ci

∂t=

XN

j=1

Dij∇2Cj; [2]

where the Dij is the matrix of diffusivity coefficients for the elec-trolytes that can be obtained experimentally (25, 26) but also canbe theoretically approximated (26). This treatment considers theconcentration profiles of the two electrolytes rather than those ofthe separate ions, thus implicitly satisfying charge neutrality. Asexplained in more detail in SI Text, it has been shown that a sys-tem of this kind can be solved analytically, yielding a solution forthe concentration Ciðx; tÞ of the simplified form

Ciðx; tÞ= f�Cinitial;Dij;FðD; x; tÞ�; [3]

where Cinitialðx; tÞ is the initial concentration of electrolytes, andFðD; x; tÞ is the solution for the binary case for the given bound-ary conditions (27). This solution thus yields detailed predictionsfor the ion concentration profiles.We plot in Fig. 3A the resulting ion concentrations for a 1-mM

NaCl solution at a time t= 300 s. As prescribed by the boundary

low

salt

high

salt

Excess counter-ionsIso-lines of electrostatic potential Debye length (from surface)

Force on (charged) fluid volume

A B

C

Fig. 2. Schematic illustration of diffusiophoresis. (A) A particle in a macro-scopic salt gradient, with the salt concentration increasing from left toright. (B) Zoom-in of the boxed region in A: The particle is surrounded by alayer containing excess counterions, shown as red circles. The characteristicthickness of this layer, the Debye length λD, decreases from left to right, asindicated by the dashed line. As a result, the electrostatic forces between thesurface and the charged fluid within the Debye layer are stronger in thehigh-salt regions (indicated as solid red arrows for representative volumes ofequal charge in the low- and high-salt regions). A pressure difference results,leading to a flow of fluid along the particle surface (indicated by the lightgray arrow), generally toward the regions of lower salt concentration. (C) Asa result, the particle must move in the opposite direction, as both the fluidand the particle are incompressible.

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conditions, the concentration of NaCl (red circles) vanishes closeto the interface and increases with increasing distance from thesurface, reaching a plateau at a distance of several millimeters.Conversely, the concentration of HCl (blue squares) exhibitsa plateau at small separation distances, whereas with increasingseparation it decreases toward zero. The total electrolyte con-centration profile (green diamonds), derived by adding these twoprofiles, exhibits a distinct maximum. This can be explained bythe fact that the flux of HCl from the surface has to match theflux of NaCl to the surface; however, HCl diffuses faster thanNaCl, thus leading to a region of lower total electrolyte con-centration near the surface and a corresponding region of in-creased electrolyte concentration farther away from the surface,which is required to conserve the total number of electrolytes inthe solution.If EZ formation is indeed governed by diffusiophoresis, then

particles should migrate at a velocity proportional to the calcu-lated ∇logC, as predicted by Eq. 1. To test this hypothesis, weconsider the particles at the edge of the EZ, for which we canreadily extract the position dEZ directly from our experiments.The corresponding particle velocities UðdEZðtÞ; tÞ should then bedirectly proportional to the calculated ∇logC at the same posi-tion and time.Indeed, when plotting the experimental UðdEZðtÞ; tÞ as a

function of the calculated ∇logC, we find an almost perfectlylinear behavior, as shown in Fig. 3B for a system of polystyreneparticles in a 1-mM NaCl solution. This behavior is in fullagreement with our hypothesis of diffusiophoresis; we thus in-terpret the slope of this curve as the diffusiophoresis coefficient,determined by a linear fit as DDP = 2:81× 10−5 cm2=s.Our experiments show that the EZ formation exhibits a robust

square-root-of-time dependence. This is in fact a direct conse-quence of the scaling properties of the time-dependent saltconcentration profile Cðx; tÞ. Because the time evolution of

Cðx; tÞ is governed by diffusion, any characteristic length scales inthe profile have to scale with the square root of time. Further, asthe boundary conditions at x= 0 and x=∞ are fixed, its magni-tude remains unchanged. To illustrate this scaling (full anal-ysis in SI Text), we plot the calculated Cðx; tÞ for different timest as a function of x

ffiffiffiffiffiffiffit0=t

p, resulting in a single master curve,

plotted in Fig. 3C. This shows that for all times t, the concen-tration profile is given by Cðx; tÞ=Cðx ffiffiffiffiffiffiffi

t0=tp

; t0Þ, where t0 is anarbitrary reference time. Similarly, the gradient of the profile isobtained directly from the gradient at a reference time t0 as∇logCðx; tÞ= ffiffiffiffiffiffiffi

t0=tp

∇logC ðx ffiffiffiffiffiffiffit0=t

p; t0Þ, as shown by the master

curve shown in Fig. 3D. In our experiments we have observeddEZ = a

ffiffitp

and thus _dEZ = a=2ffiffitp

. The scaling properties ofCðx; tÞ also imply that ∇logC ðdEZðtÞ; tÞ= const:=

ffiffitp

, thus pre-dicting that the diffusiophoresis condition dEZ =DDP∇logC isfulfilled for particles at the edge of the EZ. The diffusiophoresiscoefficient DDP can thus be directly obtained from the magnitudea of the EZ trajectory as

DDP =a

2ffiffiffiffit0p

∇logCðdEZðt0Þ; t0Þ: [4]

Particles in front of this trajectory will move slower and particlesfarther back will move faster, as shown in Fig. 3E, where we plotthe predicted trajectories of particles with different starting posi-tions; this leads to the observed accumulation of particles at theedge of the EZ.The scaling properties of Cðx; tÞ thus enable us to predict the

detailed motion of particles and thereby the formation of theEZ, based on a single ion profile, calculated at one single ref-erence time t0, with the diffusiophoresis coefficient DDP as theonly free parameter. Doing so, we use a simple one-dimensionalmodel to predict the detailed distribution of particles in thesample. In this model, point particles are initially uniformly

H

DCBA

E F G

Fig. 3. (A) Concentration for all of the electrolyte species (NaCl, red circles; HCl, blue squares) and their total (green diamonds) at time t = 5 min, using theanalytical solution of the general form given by Eq. 3 for the system NaCl suspension–Nafion. (B) The velocity of the EZ position _dEZðtÞ as a function of∇logðCðdEZðtÞ,tÞÞ displays linear behavior; the slope corresponds to the diffusiophoresis coefficient DDP. (C) Master curve for the salt concentration profile,obtained by plotting Cðx,tÞ=C0 as a function of the rescaled position x

ffiffiffiffiffiffiffiffiffit0=t

p. (D) Master curve for the concentration gradient, obtained by plottingffiffiffiffiffiffiffiffiffi

t=t0p

∇logCðx,tÞ as a function of xffiffiffiffiffiffiffiffiffit0=t

p. (E) Predicted trajectories for particles with different starting positions. Particles accumulate on a single trajectory,

representing the edge of the EZ, dEZ = affiffiffitp

. (F) Space–time diagram of EZ formation, obtained from our simple model based on the calculated salt profile andthe DDP obtained in B. (G) Corresponding experimental space–time diagram extracted from microscopic images of EZ formation in a 1-mM NaCl suspension.(H) Comparison between the EZ distances dEZ obtained from the simple model (red circles) and from the experiments (blue squares).

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distributed and their trajectories are calculated based on thecalculated salt concentration profile. To visualize the predic-tions of this simple model, in analogy to Fig. 1D, we constructa space–time diagram, displayed in Fig. 3F, using the DDP valueobtained from Fig. 3B. Comparing this diagram to the experi-mental diagram shown in Fig. 3G, we observe that the scaling ofthe edge of the EZ is captured very well, again displaying thecharacteristic square-root-of-time dependence. However, theedge of the EZ appears much sharper than in the experiment.We attribute this discrepancy to the fact that our simple modeldoes not consider the diffusion of particles or the effects of flowinstabilities driven by gradients in particle concentration that arebuilt up during the process (Movie S2).Our experiments thus show that ion exchange and diffusiopho-

resis can fully account for the phenomenon of EZ formation. Usinga single parameter, the diffusiophoresis coefficient DDP, the de-velopment of the EZ with time is predicted with high accuracy, asshown in Fig. 3H. However, so far we have not considered thedependence of the diffusiophoresis coefficient on particle and so-lution properties such as the zeta potential, the Debye screeninglength, and ionic mobilities (17, 19, 21). To further scrutinize ourhypothesis, we thus vary the suspension properties to test whether,as predicted, they affect the dynamics of EZ formation. To do so,we make use of the differences between the ionic mobilities ofmonovalent Cl−-based electrolytes. We prepare a series of samplescontaining LiCl, NaCl, KCl, and CsCl where the diffusivity of thecation continuously increases from Li to Cs. As shown in Fig. 4A,whereas in all cases EZ formation still exhibits the characteristicsquare-root-of-time dependence dEZ = a

ffiffitp

, we indeed find a sys-tematic change of the kinetics of EZ formation in this series ofsamples, as the parameter a decreases consistently with increasingdiffusivity of the cations (28). This clearly indicates that the diffu-sivity of the ions directly affects the kinetics of EZ formation, thusfurther strengthening our hypothesis.The diffusiophoresis coefficient should also depend on salt

concentration C, which affects both the Debye screening lengthand the zeta potential. To test this, we prepare samples containingdifferent concentrations of NaCl salt, ranging from C = 1 mM toC = 100 mM. As shown in Fig. 4B, as a function of salt concen-tration we observe a systematic change in the kinetics of EZformation, whereas the characteristic square-root-of-time scalingof dEZ with time is again preserved. With increasing C we observea shift toward lower dEZ, corresponding to a decrease of the pa-rameter a that describes the time dependence of the EZ positiondEZ = a

ffiffitp

.

ConclusionsWe have performed detailed experiments to study the origin of EZformation. By minimizing the influence of convection and otherforces acting on the particles, we isolate the force responsible forthe observed long-range interactions. We observe a robust square-root-of-time scaling of the EZ distance, which we link directly tothe diffusion of ions in the system. We have used a theoreticaldescription of diffusion in a ternary system with boundary con-ditions set by the ion-exchange properties of the Nafion material.Comparing this model to the observed particle kinetics, we showthat the particle velocity is proportional to ∇logC, the scalingpredicted for diffusiophoresis of colloids in a salt gradient. Ourexperiments thus indicate that EZ formation is caused by thebuildup of a salt gradient near the interface, which in turn leads toa diffusiophoretic migration of particles.A simple model that combines these ion diffusion and diffusio-

phoresis effects yields predictions for EZ formation that account forour experiments. Further evidence for diffusiophoresis as the originof EZ formation is the fact that the kinetics of the process can beprecisely tuned by varying the properties of the surface material orthe solution, such as the ionic species or the ionic strength.Furthermore, we expect that diffusiophoresis may be responsible

also for EZ formation for the broad range of materials discussed inthe Introduction, where a gradient in the concentration of ions orother solutes can be expected to occur near the interface (SI Text).Interestingly, due to the dependence on ∇logC=∇C=C ratherthan just on ∇C, only relative changes in concentration are rele-vant, and even at very low ion concentrations diffusiophoresis stillcauses EZs similar to those observed at higher salt concentrations.Since its discovery in the 1940s, diffusiophoresis has often

been overlooked in the study of colloidal systems. Although ithas recently gained renewed attention, for instance as a drivingforce for so-called self-propelled particles (29, 30) or in micro-fluidic applications to control the distribution of colloids ina channel (18), the effect is still rarely studied or used in appli-cations. However, the effects studied here could have importantimplications also in biological systems, for instance in the studyof biological chemotaxis, where a migration of cells is observed inresponse to chemical gradients. Diffusiophoresis could also bethe nonspecific repellant force creating phagosomes (31) andvacuoles (32) surrounding bacteria (33) or parasites (34–36)attacking or being attacked by living cells. The results of ourstudy suggest other potential applications, as they offer a de-tailed understanding of how EZ formation can be preciselycontrolled by tuning the properties of the involved interfaces andsolutions. This understanding could be exploited, for instance,for the creation of novel antifouling materials or for the sortingof cells or particles in microfluidic devices.

Materials and MethodsSample Preparation. Our basic system consisted of a Nafion 117 membrane(Nafion; Sigma Aldrich) fitted inside a capillary of rectangular cross section(length, 50 mm; width, 4 mm; height, 400 μm; Vitrocom), deionized water(MilliQ; resistivity 18.2 MΩ), uniformly sized polystyrene (PS) beads of 1-μmdiameter, and amonovalent salt (SigmaAldrich). The capillary tubeswere fittedat one end with Nafion by stamping the tube for 20 s on a Nafion sheet placedon a glass plate, preheated to 265 8C. This process ensured that the Nafionmaterial uniformly filled and sealed the capillary. To prevent evaporation, thecapillary was further sealed using a UV curable glue (Norland 63; NorlandProducts), which closelymatches the refractive indexof glass. Toprevent opticaldistortion we placed a second coverslip glass (size 1) on top of the capillary andglue, thereby creating a uniform layer of index-matching materials. The gluewas then cured under a UV lamp for 24 h. At the start of an experiment thecapillary was filled with a PS suspension, with this moment referred to as thestarting time of the experiment. The open end of the capillary was then sealedwith a 90-s epoxy (Bison) to prevent evaporation of the solution.

Imaging. The final sample cell was imaged in a vertical setup consisting of (i)an XYZ stage used to manipulate the supporting microscope slide, (ii) An

A B

Fig. 4. EZ formation as a function of solution properties. (A) For differentmonovalent salts with mobilities increasing from LiCl (red circles), NaCl (bluesquares), and KCl (green diamonds) to CsCl (black crosses); C = 1 mM. (B)dEZðtÞ as a function of time for different salt concentrations of NaCl sol-utions, 1 mM (red circles), 10 mM (blue squares), and 100 mM (green dia-monds). The solid lines have a slope of 1/2.

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inverted microscope (Zeiss Axiovert A200 and Motic BA 310) capable of imag-ing in a vertical geometry by use of an attached microscope objective inverter(InverterScope Objective Inverter; LSM TECH) fitted with an objective of 1×magnification, and (iii ) a 100-W Olympus lamp in combination with a Moticcondenser (N.A. = 0.55, walking distance = 70 mm). The images were recordedusing a monochromatic CCD camera (Imaging source, DMK21AU04, 640 ×480 pixels).

Digital Image Analysis. To construct the time–space diagram shown in Fig. 1D,horizontally averaged vertical lines from each frame were combined intoa single image. The diagram was then corrected to remove the influence ofthe flickering of the lamp, inhomogeneous illumination, glass thickness, and

differences in optical path length through the setup. EZ trajectories wereobtained by digital image analysis, where the position of the Nafion interfacewas identified as the point where the first derivative of the intensity reacheda maximum.

ACKNOWLEDGMENTS. We thank Y. Gao, P. Voudouris, Z. Fahimi, P.D. Anderson,J. Mattsson, and V. Trappe for valuable discussions. S.M. gratefully acknowledgesfinancial support from the Stichting voor de TechnischeWetenschappen (STW), thetechnological branch of the Netherlands Organisation of Scientific Research, andthe Dutch Ministry of Economic Affairs under Contract 12538, Interfacial effects inionizedmedia. D.F. andH.M.W. are grateful for financial support from the Institutefor Complex Molecular Systems (ICMS) at Eindhoven University of Technology.

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