Long-Range Repulsion of Colloids Driven by Ion Exchange and Diffusiophoresis

Long-range repulsion of colloids driven by ion exchange and diffusiophoresis

Daniel 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 suspension also been suggested as a possible origin of the behavior (3–5). are usually limited to distances smaller than 1 μm. However, in a However, to date it is still unclear whether any of these hypotheses, range of studies from different disciplines, repulsion of particles has individually or in combination, can fully account for the observed been observed over distances of up to hundreds of micrometers, in behavior, as existing comparisons between experiments and theo- the absence of any additional external fields. Although a range of retical predictions are still unable to clearly discern between the hypotheses have been suggested to account for such behavior, the different hypotheses. An understanding of the physical origins of physical mechanisms responsible for the phenomenon still remain EZ formation is thus still lacking. unclear. To identify and isolate these mechanisms, we perform de- In this paper, we study EZ formation in detail by measuring tailed experiments on a well-defined experimental system, using the time and position dependence of forces acting on the par- a setup that minimizes the effects of gravity and convection. Our ticles in multiple configurations. Our experimental results enable experiments clearly indicate that the observed long-range repulsion us to clearly identify a physical explanation for this intriguing is driven by a combination of ion exchange, ion diffusion, and dif- phenomenon, which we expect to apply also to a wide range of fusiophoresis. We develop a simple model that accounts for our other systems exhibiting similar long-range repulsion. To ratio- data; this description is expected to be directly applicable to a wide nalize the observed behavior, we develop a simple model that range of systems exhibiting similar long-range forces. quantitatively accounts for our experimental data. exclusion zone | Nafion | chemotaxis | unstirred layer | solute-free zone Results To systematically elucidate the EZ phenomenon, we choose the xclusion zone (EZ) formation is a phenomenon where col- perfluorinated polymer membrane material Nafion 117 as a model Eloidal particles in an aqueous suspension are repelled from system, as EZ formation around this material has already been an interface over distances of up to hundreds of micrometers, widely studied (4). Moreover, the material’s physical and chemical leading to the formation of a particle-free zone in the vicinity of properties have been characterized in detail (13, 14) due its wide- the interface. Such peculiar behavior has been observed by spread use as a proton-conducting membrane in polymer fuel cells researchers from different disciplines for a wide range of materi- (13, 14) and other electrochemical applications, where it is used in als, including biological tissues such as rabbit cornea (1), white the production of NaOH, KOH, and Cl2 (14). As a colloidal sus- blood cells (2), polymer gels (3), ion-exchange membranes (4), or pension we use uniformly sized polystyrene particles in aqueous metals (5). Depending on the field of research, different terms solutions of monovalent salt, providing well-controlled and simple have been used to refer to the behavior. In biological systems, experimental conditions. already in the early 1970s, EZs observed close to the surface of Although the origin of the force driving the EZ formation is biological tissues such as stratum corneum were referred to as still unclear, particles generally migrate away from the surface, unstirred layers (1), as these colloid-free layers persisted even suggesting the existence of a force perpendicular to the surface, when the suspensions were stirred. In later studies, the formation acting on the particles. To study this driving force in detail, we of similar EZs, where Indian ink particles were excluded from the vicinity of leukocyte cells (2), was referred to as aureole formation. Significance The observed EZ formation is highly surprising, as the forces acting on the colloidal particles can extend over distances of hundreds of micrometers (1–5). Long-range interactions acting on The ability to displace particles or solutes relative to a back- colloidal particles are generally of electrostatic nature (6, 7), with ground liquid is of central importance to technologies such a range set by the thickness of the electrical double layer sur- as filtration/separation, chromatography, and water purification. rounding a charged colloidal particle, the Debye length λ . Such behavior is observed in so-called exclusion zone formation, D an effect where particles are pushed away from a surface over Whereas in low-polar solvents, these electrostatic interactions can long distances of up to hundreds of micrometers. However, it is act over tens of micrometers (8), in aqueous suspensions these still unclear which physical mechanisms are responsible. Our forces are limited to length scales of typically less than 1 μm (6, 7). work provides a detailed understanding of this exclusion zone A range of hypotheses have been formulated to account for EZ formation, enabling a precise control of the behavior. This could formation, including the emergence of excited coherent vibration be exploited, for instance, for sorting in microfluidic devices, in modes of molecules in the membrane or the surrounding water that advanced antifouling coatings, or for elucidating biological pro- could create large dipole oscillations (9). Deryagin offered a similar cesses where it is likely to play an important, yet unexplored role. explanation, by attributing the aureole formation around cells to

long-range forces originating from electromagnetic vibrations; he Author contributions: D.F. and H.M.W. conceived and designed the experiments; D.F., S.M., also mentioned as a possible explanation forces of a diffusiopho- J.M.R.H., and H.M.W. designed research; D.F. performed research; D.F. and H.M.W. analyzed retic nature arising in the presence of an electrolyte concentration data; and D.F. and H.M.W. wrote the paper. gradient, but dismissed these in favor of the electromagnetic vi- The authors declare no conflict of interest. bration hypothesis (10). Recently, a chemotaxis hypothesis similar This article is a PNAS Direct Submission. to diffusiophoresis has been suggested and theoretically inves- 1To whom correspondence should be addressed. E-mail: [email protected]. tigated to account for the EZ formation (11, 12). Effects assuming This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. a long-range structuring of water near hydrophilic surfaces have 1073/pnas.1322857111/-/DCSupplemental.

6554–6559 | PNAS | May 6, 2014 | vol. 111 | no. 18 www.pnas.org/cgi/doi/10.1073/pnas.1322857111 Downloaded by guest on September 28, 2021 wish to isolate its effects from those of other forces acting on the When plotting dEZðtÞ in a double-logarithmic plot, we observe particles, such as those induced by gravity or by advection of the a simple power-law scaling with slope of 1=2, as shown in Fig. 1F. suspension. Previous studies of the EZ phenomenon have always As expected, the motion only due to gravity, shown as green used a horizontal setup, where the driving force FEZ is perpen- diamonds, exhibits a purely linear behavior, corresponding to dicular to the gravitational force, which can lead to the buildup a slope of unity. The simple square-root-of-time scaling of dEZðtÞ of complex flow patterns (15), an effect that becomes even more is highly reproducible, as illustrated by Fig. 1G, where we display pronounced in a setup with Nafion at the bottom, as shown in results from five independent experiments performed for par- Movie S1. This has made it difficult to isolate the effects of FEZ ticles in an aqueous solution of 1 mM NaCl. The data coincide on EZ formation, thus preventing a detailed analysis of the ki- almost perfectly, as captured by the average of these five in- netics of the process. dependent experiments and the corresponding SDs, presented in H To circumvent these problems, we perform experiments in Fig. 1 . In fact, we observe a square-root-of-time scaling in all a vertical sample cell, where the Nafion material is placed at the our experiments, independent of suspension properties such as top. This ensures that all forces acting on the particles point in the concentrations and types of salt or colloidal particles used. F the same direction. We thus expect that particle motion in the Moreover, our experiments also indicate that the force EZ is not system occurs along purely vertical trajectories that are perpen- significantly affected by parameters such as the illumination, the dicular to the Nafion surface; this enables us to readily subtract thickness of the Nafion layer inside the capillary, or by artifacts such as bubble formation near the surface or irregularities in the the effects of gravity, thereby isolating the effects of the force SI Text that is induced by the Nafion surface. We therefore choose this shape of the interface (Figs. S1 and S2 and ). The observed square-root-of-time dependence thus appears to be a re- geometry as our standard experimental setup to study the ki- markably robust feature of the EZ formation process. netics of diffusion zone formation. The EZ remains horizontal, This scaling indicates that a diffusive process is important in as shown for a typical experiment in the sequence of microscopy determining EZ formation. Motivated by this observation, we images displayed in Fig. 1 A–C. To follow the kinetics of the extract an effective diffusion coefficient from the data in Fig. 1E, process in detail, we take snapshots of the sample every 5 s. As yielding a value of D = 1:26 × 10−5 cm2=s; this is orders of explained in Materials and Methods, we compose a time–space eff D magnitude larger than the diffusion coefficient expected for our diagram from these acquired images (Fig. 1 ), capturing the particles in water, as obtained from the Stokes–Einstein relation, development of the EZ profile as a function of time. −9 2 Dp = kBT=6πηr = 4:36 × 10 cm =s, where η = 0:001 Pa is the vis- This diagram enables us to readily extract the distance dEZðtÞ cosity of water, r = 0:5 μm the particle radius, kB the Boltzmann from the Nafion surface to the edge of the EZ, which we plot in Fig. constant, and T = 298 K the ambient temperature. However, the 1E as red circles. From a separate experiment without Nafion value of Deff is within the range of typical diffusion coefficients surface we extract the speed of sedimentation of the particles solely of monovalent ions in aqueous solution, with typical values + due to gravity. As expected, the sedimented distances from the top ranging from 1:03 × 10−5 cm2=sforLi ions to 9:31 × 10−5 cm2=s + of the cell, shown in Fig. 1E as green diamonds, are considerably for H . This suggests that the physical mechanism that governs smaller than for the system where Nafion is present (SI Text). To EZ formation is directly related to the diffusion of ions.

F SCIENCES isolate the effects of the force EZ, we subtract the gravity-induced In fact, Nafion interacts strongly with ions in aqueous sol-

particle displacement from the observed EZ distance, shown in Fig. utions; it is a cation exchange material. When an aqueous solu- APPLIED PHYSICAL 1E as blue squares. Initially, the displacement of particles proceeds tion containing cations comes in contact with Nafion, as a result rapidly, with a typical velocity of 0.9 μm=s, 300 s after first contact of a higher affinity of the material for these cations, these ions + of the suspension with the Nafion surface. The speed of the par- are replaced with H ions initially bound inside the Nafion (14). ticles at the edge of the EZ then decreases continuously as time This ion-exchange process will thus lead to an inhomogeneous progresses, reaching a typical value of 0.29 μm=s after 1 h. distribution of ions in the liquid. As a result, a time-dependent

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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) at different 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 of the 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 and Methods). (E) EZ distance dEZ (red circles) extracted from the space–time diagram in D. The displacement only due to gravity, from an experiment without Nafion, is shown as green diamonds. The blue squares show the EZ distance with the influence of gravity subtracted, isolating the displacement due to the EZ forming 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 same conditions in suspensions of 1 μm polystyrene particles in a 1-mM NaCl aqueous solution. (H) Average of the five independent EZ position measurements with the corresponding SD, displayed in a double-logarithmic plot. A power-law fit to the data yields an exponent of 0.495.

Florea et al. PNAS | May 6, 2014 | vol. 111 | no. 18 | 6555 Downloaded by guest on September 28, 2021 concentration profile should develop for each ionic species. We calculations, taking into account the balance between the hy- hypothesize that this inhomogeneous ion distribution is directly drostatic pressure and the electrostatic stress (19), result in related to the process of EZ formation and the corresponding a constant particle velocity U, proportional to the gradient of the ∇ C force FEZ acting on the particles. logarithm of the ionic concentration log ,as Indeed, there is a physical mechanism that results in a drift of particles in liquids with inhomogeneous ion concentrations. This U = DDP∇log C; [1] effect, termed diffusiophoresis, was first theoretically described D by Deryagin et al. (16) and later confirmed and studied in more where DP is the so-called diffusiophoresis constant that de- λ ζ detail both experimentally (17, 18) and theoretically (19–21); the pends on the (average) Debye length D, the zeta potential , theoretical description of diffusiophoresis is in excellent agree- the viscosity η, the temperature T, and the diffusion coefficients ment with experiments, as recently illustrated by a study using D+ and D− of the cations and the anions, respectively (17, 19, microfluidic coflow devices (18). 21). If indeed diffusiophoresis is the physical mechanism respon- To obtain an intuitive understanding of this effect, we consider sible for the observed EZ formation, then the particle velocities a charged particle immersed in an electrolyte solution with a salt and thus also the velocity of the edge of the EZ should always ∇ C gradient ∇C, as shown schematically in Fig. 2A. The concen- remain proportional to log . tration of excess counterions decays with distance from the To test our hypothesis of diffusiophoresis as the underlying −x=λ D mechanism leading to EZ formation, we thus wish to predict the surface as C+ = C+0 exp , where the Debye length λD scales −1=2 concentration profile of ionic species in the system as a function with salt concentration approximately as λD ∝ C . As a result, the presence of a salt gradient ∇C also implies a variation of the of time. In doing so, we consciously neglect some of the rich Debye screening length along the particle surface, as shown in properties of Nafion, where effects such as interdiffusion of a schematic close-up of the particle surface region in Fig. 2B. species inside the material (22), swelling behavior (23), or mi- Within the Debye layer, any small subvolume of fluid contains crostructural changes during hydration (13) are known to occur. an excess of counterions and thus possesses a net charge. As We focus instead on the well-known ion-exchange properties of a result, each subvolume experiences electrostatic forces under the material (14) and assume that the material acts as a perfect drain for cations, where all cations that come in contact with the the influence of the local electric field. For a uniform Debye + length, the charge distribution around the particle is spherically material are exchanged for H ions. The concentration of ex- symmetric, and the resulting stresses occurring in the fluid can be changeable protons within the Nafion material is on the order of fully balanced by hydrostatic pressure. However, in the presence 4 M (14), far exceeding the cation concentrations used in our of a salt gradient, this is no longer the case; the counterion cloud samples. We thus expect that the ion exchange is limited purely by diffusion, rather than by the availability of protons in the Nafion around the particle is anisotropic, with λD decreasing toward the higher salt concentrations. The corresponding asymmetry of the material (24). The problem of predicting the ion concentration profiles thus reduces to solving a set of coupled diffusion equa- charge distribution leads to both a gradient in the hydrostatic tions for all of the involved ionic species under these particular pressure and nonzero components of the local electric field boundary conditions. Whereas the diffusion equation is readily tangential to the particle surface, as schematically shown in Fig. solved for the case of a single electrolyte, for the case of multiple 2B. Both these effects lead to stresses in the fluid tangential to electrolytes, the diffusion problem is difficult to solve analytically. the surface that cannot be balanced by hydrostatic pressure, thus It is known that the diffusion of one species is strongly affected by leading to a flow of fluid along the surface of the particle, gen- the presence of others (25); to correctly describe the process we erally toward the direction of lower salt concentration. As the thus need to account for the diffusion of all four ionic species overall system is incompressible, this flow must be balanced by + − + − present in the solution (X ,Y ,H ,andOH ). a corresponding motion of the particle in the opposite direction – C For each of these ionic species we can write a Nernst Planck (19, 21), as shown schematically in Fig. 2 . Detailed theoretical equation, yielding a system of four coupled equations. However, local electroneutrality and the chemical equilibrium of water ! + − ðH2O H + OH Þ reduce these four equations to an analytically tractable system of just two equations for the two elec- AB trolytes (XY and HY) (25), as low salt high salt ∂C XN i = D ∇2C ; [2] ∂t ij j j= C 1 where the Dij is the matrix of diffusivity coefficients for the electrolytes that can be obtained experimentally (25, 26) but also can be theoretically approximated (26). This treatment considers the Excess counter-ions Force on (charged) fluid volume concentration profiles of the two electrolytes rather than those of Iso-lines of electrostatic potential Debye length (from surface) the separate ions, thus implicitly satisfying charge neutrality. As SI Text Fig. 2. Schematic illustration of diffusiophoresis. (A) A particle in a macro- explained in more detail in , it has been shown that a sys- scopic salt gradient, with the salt concentration increasing from left to tem of this kind can be solved analytically, yielding a solution for right. (B) Zoom-in of the boxed region in A: The particle is surrounded by a the concentration Ciðx; tÞ of the simplified form layer containing excess counterions, shown as red circles. The characteristic λ thickness of this layer, the Debye length D, decreases from left to right, as Ciðx; tÞ = f Cinitial; Dij; FðD; x; tÞ ; [3] indicated by the dashed line. As a result, the electrostatic forces between the surface and the charged fluid within the Debye layer are stronger in the where Cinitialðx; tÞ is the initial concentration of electrolytes, and high-salt regions (indicated as solid red arrows for representative volumes of FðD; x; tÞ equal charge in the low- and high-salt regions). A pressure difference results, is the solution for the binary case for the given bound- leading to a flow of fluid along the particle surface (indicated by the light ary conditions (27). This solution thus yields detailed predictions gray arrow), generally toward the regions of lower salt concentration. (C)As for the ion concentration profiles. a result, the particle must move in the opposite direction, as both the fluid We plot in Fig. 3A the resulting ion concentrations for a 1-mM and the particle are incompressible. NaCl solution at a time t = 300 s. As prescribed by the boundary

6556 | www.pnas.org/cgi/doi/10.1073/pnas.1322857111 Florea et al. Downloaded by guest on September 28, 2021 A B C D

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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 the _ analytical 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 ∇ ð ð ð Þ ÞÞ log C dEZ t ,t displays linear behavior; the slope corresponds to the diffusiophoresispffiffiffiffiffiffiffiffiffi coefficient DDP.(C) Master curve for the salt concentration profile, ð Þ= = obtainedpffiffiffiffiffiffiffiffiffi by plotting C x,t C0 asp affiffiffiffiffiffiffiffiffi function of the rescaled position x t0 t.(D) Master curve for the concentration gradient, obtained by plotting = ∇ ð Þ = t t0 log C x,t as a function of x t0 pt.(ffiffiffi E) Predicted trajectories for particles with different starting positions. Particles accumulate on a single trajectory, representing the edge of the EZ, dEZ = a t.(F) Space–time diagram of EZ formation, obtained from our simple model based on the calculated salt profile and the 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).

conditions, the concentration of NaCl (red circles) vanishes close Cðx; tÞ is governed by diffusion, any characteristic length scales in

to the interface and increases with increasing distance from the the profile have to scale with the square root of time. Further, as SCIENCES x = x = ∞ surface, reaching a plateau at a distance of several millimeters. the boundary conditions at 0 and are fixed, its magni- APPLIED PHYSICAL Conversely, the concentration of HCl (blue squares) exhibits tude remains unchanged. To illustrate this scaling (full anal- a plateau at small separation distances, whereas with increasing ysis in SI Text), we plotpffiffiffiffiffiffiffi the calculated Cðx; tÞ for different times separation it decreases toward zero. The total electrolyte con- t as a function of x t0=t, resulting in a single master curve, centration profile (green diamonds), derived by adding these two plottedinFig.3C. This shows that forp allffiffiffiffiffiffiffi times t, the concen- profiles, exhibits a distinct maximum. This can be explained by tration profile is given by Cðx; tÞ = Cðx t0=t; t0Þ,wheret0 is an the fact that the flux of HCl from the surface has to match the arbitrary reference time. Similarly, the gradient of the profile is t flux of NaCl to the surface; however, HCl diffuses faster than obtained directlypffiffiffiffiffiffiffi from the gradientpffiffiffiffiffiffiffi at a reference time 0 as NaCl, thus leading to a region of lower total electrolyte con- ∇log Cðx; tÞ = t0=t∇log C ðx t0=t; t0Þ, as shown by the master D centration near the surface and a corresponding region of in- curve shownpffiffi in Fig. 3 . In ourp experimentsffiffi we have observed creased electrolyte concentration farther away from the surface, d = a t d_ = a= t EZ and thus EZ 2 . The scaling propertiespffiffi of which is required to conserve the total number of electrolytes in Cðx; tÞ also imply that ∇log C ðd ðtÞ; tÞ = const:= t, thus pre- the solution. EZ dicting that the diffusiophoresis condition dEZ = DDP∇log C is If EZ formation is indeed governed by diffusiophoresis, then fulfilled for particles at the edge of the EZ. The diffusiophoresis particles should migrate at a velocity proportional to the calcu- D ∇ C 1 coefficient DP can thus be directly obtained from the magnitude lated log , as predicted by Eq. . To test this hypothesis, we a of the EZ trajectory as consider the particles at the edge of the EZ, for which we can d a readily extract the position EZ directly from our experiments. D = pffiffiffiffi : [4] Uðd ðtÞ; tÞ DP The corresponding particle velocities EZ should then be 2 t0∇log CðdEZðt0Þ; t0Þ directly proportional to the calculated ∇log C at the same position and time. Particles in front of this trajectory will move slower and particles Uðd ðtÞ; tÞ Indeed, when plotting the experimental EZ as a farther back will move faster, as shown in Fig. 3E, where we plot function of the calculated ∇log C, we find an almost perfectly the predicted trajectories of particles with different starting posi- linear behavior, as shown in Fig. 3B for a system of polystyrene tions; this leads to the observed accumulation of particles at the particles in a 1-mM NaCl solution. This behavior is in full edge of the EZ. agreement with our hypothesis of diffusiophoresis; we thus in- The scaling properties of Cðx; tÞ thus enable us to predict the terpret the slope of this curve as the diffusiophoresis coefficient, detailed motion of particles and thereby the formation of the −5 2 determined by a linear fit as DDP = 2:81 × 10 cm =s. EZ, based on a single ion profile, calculated at one single ref- Our experiments show that the EZ formation exhibits a robust erence time t0, with the diffusiophoresis coefficient DDP as the square-root-of-time dependence. This is in fact a direct conse- only free parameter. Doing so, we use a simple one-dimensional quence of the scaling properties of the time-dependent salt model to predict the detailed distribution of particles in the concentration profile Cðx; tÞ. Because the time evolution of sample. In this model, point particles are initially uniformly

Florea et al. PNAS | May 6, 2014 | vol. 111 | no. 18 | 6557 Downloaded by guest on September 28, 2021 distributed and their trajectories are calculated based on the Conclusions calculated salt concentration profile. To visualize the predic- We have performed detailed experiments to study the origin of EZ tions of this simple model, in analogy to Fig. 1D,weconstruct formation. By minimizing the influence of convection and other a space–time diagram, displayed in Fig. 3F, using the DDP value forces acting on the particles, we isolate the force responsible for obtained from Fig. 3B. Comparing this diagram to the experi- the observed long-range interactions. We observe a robust square- mental diagram shown in Fig. 3G, we observe that the scaling of root-of-time scaling of the EZ distance, which we link directly to the edge of the EZ is captured very well, again displaying the the diffusion of ions in the system. We have used a theoretical characteristic square-root-of-time dependence. However, the description of diffusion in a ternary system with boundary con- edge of the EZ appears much sharper than in the experiment. ditions set by the ion-exchange properties of the Nafion material. We attribute this discrepancy to the fact that our simple model Comparing this model to the observed particle kinetics, we show ∇ C does not consider the diffusion of particles or the effects of flow that the particle velocity is proportional to log , the scaling instabilities driven by gradients in particle concentration that are predicted for diffusiophoresis of colloids in a salt gradient. Our built up during the process (Movie S2). experiments thus indicate that EZ formation is caused by the Our experiments thus show that ion exchange and diffusiopho- buildup of a salt gradient near the interface, which in turn leads to resis can fully account for the phenomenon of EZ formation. Using a diffusiophoretic migration of particles. A simple model that combines these ion diffusion and diffusio- a single parameter, the diffusiophoresis coefficient D , the de- DP phoresis effects yields predictions for EZ formation that account for velopment of the EZ with time is predicted with high accuracy, as H our experiments. Further evidence for diffusiophoresis as the origin shown in Fig. 3 . However, so far we have not considered the of EZ formation is the fact that the kinetics of the process can be dependence of the diffusiophoresis coefficient on particle and so- precisely tuned by varying the properties of the surface material or lution properties such as the zeta potential, the Debye screening the solution, such as the ionic species or the ionic strength. length, and ionic mobilities (17, 19, 21). To further scrutinize our Furthermore, we expect that diffusiophoresis may be responsible hypothesis, we thus vary the suspension properties to test whether, also for EZ formation for the broad range of materials discussed in as predicted, they affect the dynamics of EZ formation. To do so, the Introduction, where a gradient in the concentration of ions or we make use of the differences between the ionic mobilities of SI Text − other solutes can be expected to occur near the interface ( ). monovalent Cl -based electrolytes. We prepare a series of samples Interestingly, due to the dependence on ∇log C = ∇C=C rather containing LiCl, NaCl, KCl, and CsCl where the diffusivity of the than just on ∇C, only relative changes in concentration are rele- cation continuously increases from Li to Cs. As shown in Fig. 4A, vant, and even at very low ion concentrations diffusiophoresis still whereas in all cases EZ formation still exhibitspffiffi the characteristic causes EZs similar to those observed at higher salt concentrations. square-root-of-time dependence dEZ = a t, we indeed find a sys- Since its discovery in the 1940s, diffusiophoresis has often tematic change of the kinetics of EZ formation in this series of been overlooked in the study of colloidal systems. Although it samples, as the parameter a decreases consistently with increasing has recently gained renewed attention, for instance as a driving diffusivity of the cations (28). This clearly indicates that the diffu- force for so-called self-propelled particles (29, 30) or in micro- sivity of the ions directly affects the kinetics of EZ formation, thus fluidic applications to control the distribution of colloids in further strengthening our hypothesis. a channel (18), the effect is still rarely studied or used in appli- The diffusiophoresis coefficient should also depend on salt cations. However, the effects studied here could have important concentration C, which affects both the Debye screening length implications also in biological systems, for instance in the study and the zeta potential. To test this, we prepare samples containing of biological chemotaxis, where a migration of cells is observed in response to chemical gradients. Diffusiophoresis could also be different concentrations of NaCl salt, ranging from C = 1mMto the nonspecific repellant force creating phagosomes (31) and C = 100 mM. As shown in Fig. 4B, as a function of salt concen- vacuoles (32) surrounding bacteria (33) or parasites (34–36) tration we observe a systematic change in the kinetics of EZ attacking or being attacked by living cells. The results of our formation, whereas the characteristic square-root-of-time scaling study suggest other potential applications, as they offer a de- d C of EZ with time is again preserved. With increasing we observe tailed understanding of how EZ formation can be precisely d a shift toward lower EZ, corresponding to a decrease of the pa- controlled by tuning the properties of the involved interfaces and a rameterpffiffi that describes the time dependence of the EZ position solutions. This understanding could be exploited, for instance, d = a t EZ . for the creation of novel antifouling materials or for the sorting of cells or particles in microfluidic devices. Materials and Methods ABSample 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-μm diameter, and a monovalent salt (Sigma Aldrich). The capillary tubes were fitted at one end with Nafion by stamping the tube for 20 s on a Nafion sheet placed on a glass plate, preheated to 265 8C. This process ensured that the Nafion material uniformly filled and sealed the capillary. To prevent evaporation, the capillary was further sealed using a UV curable glue (Norland 63; Norland Products), which closely matches the refractive index of glass. To prevent optical distortion we placed a second coverslip glass (size 1) on top of the capillary and glue, thereby creating a uniform layer of index-matching materials. The glue was then cured under a UV lamp for 24 h. At the start of an experiment the Fig. 4. EZ formation as a function of solution properties. (A) For different capillary was filled with a PS suspension, with this moment referred to as the monovalent salts with mobilities increasing from LiCl (red circles), NaCl (blue starting time of the experiment. The open end of the capillary was then sealed squares), and KCl (green diamonds) to CsCl (black crosses); C = 1 mM. (B) with a 90-s epoxy (Bison) to prevent evaporation of the solution. dEZðtÞ as a function of time for different salt concentrations of NaCl solutions, 1 mM (red circles), 10 mM (blue squares), and 100 mM (green dia- Imaging. The final sample cell was imaged in a vertical setup consisting of (i) monds). The solid lines have a slope of 1/2. an XYZ stage used to manipulate the supporting microscope slide, (ii)An

6558 | www.pnas.org/cgi/doi/10.1073/pnas.1322857111 Florea et al. Downloaded by guest on September 28, 2021 inverted microscope (Zeiss Axiovert A200 and Motic BA 310) capable of imag- differences in optical path length through the setup. EZ trajectories were ing in a vertical geometry by use of an attached microscope objective inverter obtained by digital image analysis, where the position of the Nafion interface (InverterScope Objective Inverter; LSM TECH) fitted with an objective of 1× was identified as the point where the first derivative of the intensity reached magnification, and (iii) a 100-W Olympus lamp in combination with a Motic a maximum. condenser (N.A. = 0.55, walking distance = 70 mm). The images were recorded × using a monochromatic CCD camera (Imaging source, DMK21AU04, 640 ACKNOWLEDGMENTS. We thank Y. Gao, P. Voudouris, Z. Fahimi, P.D. Anderson, 480 pixels). J. Mattsson, and V. Trappe for valuable discussions. S.M. gratefully acknowledges financial support from the Stichting voor de Technische Wetenschappen (STW), the Digital Image Analysis. To construct the time–space diagram shown in Fig. 1D, technological branch of the Netherlands Organisation of Scientific Research, and horizontally averaged vertical lines from each frame were combined into the Dutch Ministry of Economic Affairs under Contract 12538, Interfacial effects in a single image. The diagram was then corrected to remove the influence of ionized media. D.F. and H.M.W. are grateful for financial support from the Institute the flickering of the lamp, inhomogeneous illumination, glass thickness, and for Complex Molecular Systems (ICMS) at Eindhoven University of Technology.

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