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Colloidal Dynamics Near a Particle-Covered Surface H. B. Eral,* F. Mugele, and M. H. G. Duits Physics of Complex Fluids group, Faculty of Science and Technology, IMPACT and MESA+ Institutes, University of Twente, 7500 AE Enschede, The Netherlands ABSTRACT:
How the diffusive dynamics of colloidal spheres changes in the vicinity of a particle-coated surface is of importance for industrial challenges such as fouling and sedimentation as well as for fundamental studies into confinement effects. We addressed this question by studying colloidal dynamics in a partially coated surface layer, using video microscopy. Particle mean squared displacement (MSD) functions were measured as a function of a (local) effective volume fraction (EVF), which was varied by making use of gravity settling. Comparison of MSDs at the bare and coated surfaces for EVF of 0.2 0.4 revealed that at the latter surface the motion amplitudes are strongly reduced, accompanied by a sharp transition from diffusive to nearly caged motion. This clearly indicates that the surface-attached particles cannot be taken into account via volume fraction and that their immobility has a distinct effect. For EVF > 0.45, the caging becomes dominated by the suspended particles, making the dynamics at the bare and coated surfaces similar.
1. INTRODUCTION The Brownian dynamics of colloidal particles in the vicinity of a surface that is covered with immobilized particles is a scarcely studied case, despite its relevance for several physical phenomena, occurring in a wide range of suspension volume fractions. As a first example, a particle-covered surface can be regarded as an intermediate state of fouling. Fouling usually occurs at low suspension volume fractions, initiated by thermodynamically (TD) driven attachment of colloids to surfaces of containers or channels. It is an important problem in various applications requiring clean surfaces1,2 and forms a major obstacle in the development of microfluidics with suspensions.3,4 Besides TD interactions,5 7 also hydrodynamic (HD) interactions between suspended particles and the (clean or fouled) surface play a role in colloidal fouling. This aspect has been much less illuminated, and besides some limiting cases a lot is unknown. Even simplifying to hard spheres in the suspension and on the surfaces, it is not clear how the thermal motion of particles near the surface is modulated by particles attached to the surface. For example, besides a hindered diffusion with lower motion amplitudes, also temporal caging of the suspended particles might occur. Second, a layer of immobilized particles on a substrate can also bear resemblance to the top layer of colloidal sediments or filter r 2011 American Chemical Society
cakes. Here, the particle volume fractions are generally higher than in most fouling scenarios. The way in which particles arriving from the adjacent suspension diffuse and become incorporated in the particle layer again depends on both TD and HD interactions. Because the particle layers now have a 3D character, not only the roughness of the top surface but also the permeability of the underlying material will play a role. To understand this problem, the two contributions need to be separated. Third, a flat surface covered with immobilized particles presents geometry at or via which colloids from the suspension can get confined. The influence of spatial confinement on colloidal dynamics has received considerable interest recently, ranging from the restrictions imposed by one surface8 15 to the 2D or 3D confinement between multiple surfaces.16 22 Especially the high volume fraction regime has been addressed, after it was discovered that confinement can induce glass formation in (hard-sphere) fluids at lower densities than in bulk.16,17,23 This phenomenon is still poorly understood. Besides the global confinement geometry, also the local properties of the surfaces could Received: April 6, 2011 Published: August 09, 2011 12297
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surfaces are compared as a function of an effective local volume fraction (EVF). This volume fraction near the surface is gradually increased during the experiment by starting with a dilute suspension and using the gravity settling.
2. EXPERIMENTAL METHODS
Figure 1. (a) Schematic of the experimental setup with mobile (green) and sintered (red) particles. (b and c) Confocal images of mobile and sintered particles that share the layer.
play a role. For example, surface roughness can modulate the dynamics. Walls with permanently attached particles might even cause lateral caging of the suspended particles. Alternatively, the simultaneous presence of fixed and mobile particles might present an extreme case of dynamic heterogeneities, which have been found in glass-forming systems.24 28 To what extent this presence of immobile particles could bias suspensions toward glass formation is currently unknown. The present limitations in insight into these problems can be attributed to the complexity of the multibody hydrodynamics, which generally requires numeric simulations29 31 to predict behavior. Recently, mode coupling theory has been adapted to confinement; yet it is limited to confinement by two smooth walls.32 Besides that, there is also a lack of experimental studies into colloidal dynamics near surfaces. Existing studies in literature have mostly been focused on the HD interaction of an isolated particle with one or more smooth surfaces.10,11,33 37 Studies at finite suspension concentrations have mainly been carried out in cylindrical channels or cavities, or between two nearby flat surfaces.10,16 19,23 Although in some of these cases the surfaces were rough,18,19,23 it was not always possible to separate the effects of confinement and surface roughness. The dynamics of a dense suspension near a single surface has only been studied for smooth surfaces up to now.14,38 In one of these studies, it was found that at high volume fraction, the HD effect of the wall became similar to that of the particles.38 We present here a study that is specifically focused on colloidal dynamics at a particle-covered surface. The goal of the present work is to assess the effect of a partially coated particle monolayer, on the dynamics of suspended particles occurring in the same layer. This dynamics is quantified via the mean squared displacement (MSD) function, which is measured using confocal video microscopy, and analyzed for both its amplitude (A) and its exponent (R). A and R reveal complementary information, about the magnitude of the hydrodynamic drag and about the extent of caging, respectively. The specific effect of the surface coating is assessed by comparison to the dynamics at a smooth flat surface. The local particle density at the surface is an important parameter in our study. To account for the increased crowding due to the fixed particles, the MSDs at the bare and coated
The experimental setup is shown in Figure 1. A colloidal suspension is allowed to settle onto a glass coverslip containing fixed particles. Using a confocal scanning laser microscope, image-time series are recorded from below while fixing the focal plane at the layer of (both fixed and mobile) particles directly above the glass coverslip. 2.1. Substrate Preparation. The sample chambers are prepared by glueing a glass tube onto 170 μm thick glass slides. For the smooth substrate, the glass coverslip was used as received. The rough substrate was prepared by spin-coating a colloidal suspension on a glass coverslip and letting the solvent evaporate. The particles were 1.2 μm diameter Rhodamin labeled silica spheres with a polydispersity of 8%. After suspending them in a 40:30:30 mixture (by weight) of water, ethanol, and ethylene glycol up to a volume fraction of 15%, 1 mL of the suspension was spin-coated at a rate of 2000 rpm for 30 s. Because of the solvent evaporation, the particles stick to the glass. Drying for 30 min in an oven at 200 C served to remove the last traces of solvent and sinter the particles to the glass. The surface coverage was calculated as the area fraction of the projection of the spheres and amounted to 0.38. 2.2. Colloidal Suspensions. Fluorescein labeled silica particles of radius ap = 0.75 μm (MicroMod GmbH) with 5% polydispersity and a mass density of 1.8 g/mL were suspended in a mixture containing water and glycerol (45:55 by weight, viscosity η = 4.5 ( 0.4 mPa s, mass density 1.11 ( 0.01 g/mL). This allowed particle visualization up to at least Z = 25 μm deep into the suspension. A small amount of concentrated stock suspension was sonicated and subsequently diluted with solvent mixture to a final volume fraction of 0.01. The solvent also contained 100 μM LiCl, corresponding to a calculated Debye length39 of 27 nm. This gives electrostatic interparticle repulsions that are sufficiently short ranged to render a (nearly) hard sphere (and hard substrate) system, while they are strong enough to maintain colloidal stability. The calculated sedimentation length39 of the particle/solvent combination is 0.33 μm. 2.3. Confocal Microscopy. Images were recorded with an UltraView LCI10 CSLM system (Perkin-Elmer) consisting of a Nikon Eclipse inverted microscope, a Yokogawa (Nipkov disk) confocal unit, and a Hamamatsu 12-bit CCD camera. All recordings were done in fluorescence mode using 25 mW lasers with wavelengths of 488 or 561 nm and a 100/oil objective with NA = 1.3. The effective pixel size was 0.135 μm. All microscopy experiments were done at 22 ( 2 C with suspensions in capped bottles as shown in Figure 1a. With the rough surface, first an image was made of the sintered Rhodamin-labeled spheres, using the 561 nm laser line. Next, the 488 nm line was selected to image fluorescein-labeled spheres. Immediately after filling the holder with the suspension, Z-stacks and time-series at constant Z (typically 6 of each) were recorded in alternating fashion, for a total duration of about 1 h. Z-stacks were recorded up to Z = 20 μm in steps of 0.1 μm at 0.2 s per image. Time series were recorded at 5 frames per second for typically 2000 images. Here, the Z-location was fixed at 0.75 μm above the glass bottom, corresponding to the radius of the suspended particles. 2.4. Particle Tracking. The image analysis is based on the accurate localization and tracking of particles by the algorithm from Crocker and Grier40 implemented in IDL. The effective thickness of the focal plane for localizing particles at the surface was estimated to be 1.0 ( 0.3 μm (Z = 0.75 1.75 μm above the glass). Individual (2D) images and (3D) image stacks were processed using publicly available codes.41 Particle trajectories in the xy-plane were converted into the mean squared displacement ÆΔr2æ as a function of lagtime τ. Unless mentioned 12298
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Langmuir otherwise, the MSDs for the individual particles observed at the same time were lumped together into an average MSD. To study the dependence of ÆΔr2æ (τ), on the real time t (the time lapse after starting the experiment), image-time series were divided into segments of 100 frames (corresponding to 20 s each), to represent “time points”. For O(100 1000) particles per image, this gives a reasonable trade-off between the resolution in time (t) and the statistical accuracy of
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the MSD. For each “time point” t, the MSD was fitted to a power law: ÆΔr2æ ≈ A(τ/τ0)R with A the amplitude, τ0 the unit exposure time (0.2 s), and R the exponent. Here, the τ range for fitting was chosen from 3τ0 to 10τ0. A and R provide complementary information on, respectively, the displacement magnitude and the type of motion.42 For thermally driven motion, R can vary between 0 and 1.
3. RESULTS AND DISCUSSION
)
)
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3.1. General Trends. As the mobile particles in the container settle due to gravity, we follow the dynamics and the structure of the suspension at the bottom surface as a function of time. As shown in Figure 2a, for both surfaces the MSD decreases gradually over time as the sedimentation proceeds. Both data sets tend toward a plateau, suggesting an approach of sedimentation equilibrium. The slight increase in MSD (τ0 = 0.2 s) at the rough surface for t > 1500 s might be due to photobleaching, which tends to “filter out” the mobile particles with the longest residence time (these are generally the slowest particles). Also, a slight heating of the sample (via the laser illumination) could have contributed via a lowered solvent viscosity. Looking at the relative magnitudes of the two data sets, it is clear that the MSD at the rough surface is lower at all times. The initial (time-) gradient is also steeper, suggesting a more rapid increase in the crowding at the rough surface. These are the expected trends if one assumes that the motion amplitudes are mainly determined by crowding. To enable quantitative comparisons, the measured MSDs have been divided by calculated MSDs for particles in dilute bulk suspension (MSD = 4D0τ with D0 = kT/6πηap). At the smooth substrate, the normalized MSD amounts to ∼0.45 at the start of the experiment. In the dilute limit, this ratio should become equal to the normalized diffusion coefficient D /D0, where the subscript “ ” indicates diffusion parallel to the surface. For hard spheres that are almost in contact with the surface, D /D0 ≈ 0.4 has been found.43,44 Also, the statistical signature of the erratic particle motions was studied as a function of time, by monitoring the exponent (R) obtained from the MSD at short lagtimes. As Figure 2b shows, at the smooth substrate, the initial R is very close to 1.0. This corroborates the expected diffusive (i.e., Brownian) behavior. Remarkably, for the rough surface, the initial exponent is close to 1.0 as well. This is ascribed to the arrival of individual mobile particles at the relatively large open areas of the virginal rough
Figure 2. Evolution of the particle MSD at the smooth and rough substrates. Panel (a): MSD amplitude, normalized by the result for a dilute bulk suspension. The noise floor amounts to 0.004 Y-units. Inset: MSD in μm2 for τ0 = 0.2 s (circles) and τ0 = 2 s (squares). Panel (b): MSD exponent, measured for τ = 0.6 2 s.
Figure 3. “Solid volume fraction” (jsolid) profiles as a function of the time lapsed after starting the experiment. Panel (a): Particle-coated surface. Panel (b): Smooth surface. The inset shows (for both substrates) the time dependence of the effective volume fraction (jeffective) in the particle layer closest to the surface. See text for further details. 12299
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Figure 4. Normalized MSDs at a given normalized lagtime (a) and the exponent R (b), versus calculated effective volume fractions (jeffective) for smooth and rough surfaces. For comparison, also normalized data for bulk suspensions are included. Here, jeffective is equal to the bulk volume fraction.
substrate (see Figure 1c); apparently, in this case the mobile particles do not yet feel the presence of the sintered particles. As the crowding increases, the exponent becomes subdiffusive (R < 1) for both the rough and the smooth surfaces, indicating a trend toward caging. Interestingly, the exponent decreases more quickly at the rough surface and remains lower at all times. Also, a different rough substrate having a similar surface coverage (∼0.4) by sintered particles but with a smaller typical size of the open areas (i.e., a more homogeneous distribution of the particles) was explored. Here, the same qualitative trend for R was found, but with an initial R ≈ 0.5 (result not shown). 3.2. Volume Fraction Dependence. Clearly, the changing characteristics of the particle dynamics must be related to the local structure and density of the fluid, which evolve as the sedimentation proceeds. To study this relation in more depth, we recorded (fast) Z-scans prior to each new time series and used these scans to localize all suspended particles in three dimensions. Density-versus-Z profiles obtained by integrating over X and Y were convoluted with a sphere that is representative for the particle population, to make the structure in the profiles better visible to the eye. Subsequently, we calculated per Z-slice which fraction of the slice volume was occupied by particles (as in Eral et al.23). This produces a local “solid volume fraction” profile. In case of the rough substrate, also the sintered particles were taken into account, because they contribute to the crowding. The resulting profiles are shown in Figure 3. Comparing Figure 3a and b, it becomes evident that the degree of vertical ordering into layers is influenced by the surface roughness. Clearly, the well-known layering effect that normally occurs45 is disturbed by the sintered particles. Also, the packing of spheres in the surface layer (Z < 2 μm, see below) occurs less efficiently as compared to the smooth surface: the solid volume fractions are slightly lower. This is ascribed to frustrated order: in general, a bidispersed particle mixture can be packed more efficiently.46,47 However, in our case the smaller particles are fixed to the surface, which limits the possibilities for optimizing the packing. This loss of order is transferred to the neighboring layers at larger Z. These differences in structure and density between the layers at the smooth and rough substrates indicate that the MSDs at the two surfaces might be more suitably compared as a function of the local volume fraction rather than the lapsed time (as in Figure 2). Yet the question then arises of how to define the effective volume fraction that is felt by the particles. In other
words, what would be the best definition of the “surface layer”? Because in the present study the roughness features have length scales that are similar to the size of the mobile particles, the thickness of the surface layer should be comparable to that. While the bottom of the layer (Z = 0) should coincide with the glass surface, for the upper Z-value a choice should be made, taking into account the optical Z-resolution (as in section 2.4), the sedimentation length, and the size polydispersity (section 2.2). Besides that, also the “collision geometry” should be taken into account: a particle with its Z-center located at Z = ap can still collide with other mobile particles having their center at Z < 3ap. In our case, the latter contribution is the most important. Therefore, we averaged the volume fraction profiles in Figure 3 between Z = 0 and 2.0 μm ( Δ, taking Δ = 0.3 μm as an estimated error bar. The resulting values for the effective volume fraction (EVF) are shown in the inset of Figure 3. Plotting the MSD amplitudes and exponents of Figure 2 now as a function of the estimated volume fraction results in Figure 4. Besides these measurements, also additional data from other experiments are included: (i) sediment layers prepared at a high initial volume fraction, leading to (quasi) equilibrium states at higher EVFs, and (ii) measurements of MSDs in bulk fluid, reported in an earlier study.23 To allow comparison between data for different particle radii and solvent viscosities, we normalized the lagtime τ and the MSD as τ*= τ/τB with τB = ap2/D0 the Brownian diffusion time, and MSD* = MSD/ap2. For the present study τ* = 0.021, indicating that displacements over length scales smaller than the particle radius are considered. From Figure 4a and b, it becomes clear that the particle dynamics at the rough surface is very different from that at the smooth surface: both A and R undergo a sharp downward transition around EVF = 0.2. As a consequence, the amplitude A becomes up to 2 decades lower than at the smooth surface, while the exponent R approaches 0, corresponding to particle caging. For EVF > 0.4, both A and R appear to decrease more gradually, based on the additional data point. At low EVF (0.45), the differences are smaller. The comparison with data measured in bulk fluid reveals that the MSD amplitudes at the smooth surface and in bulk fluid follow roughly similar dependencies on the volume fraction. The amplitude ratio (which should be ∼0.4 in the dilute limit) varies mostly between 0.2 and 0.5, which is a small difference as compared to the effect of the sintered particles. The MSD exponent of the bulk fluid shows a volume fraction dependence 12300
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Figure 5. Histograms of amplitudes and exponents of MSDs, measured at a normalized lag time τ/τB = 0.021. Each histogram is measured as a function of the time t after starting the experiment. Panels (a,b): exponents at the rough (a) and smooth (b) substrates. Panels (c,d): histogram of the amplitude (A) normalized with respect to bulk, i.e., log10 (A/4D0τ) at the rough (c) and smooth (d) surface.
that is mostly between that at the smooth and rough substrates. For EVF > 0.4, the exponents in bulk and at the smooth surface are similar. These findings clearly point out that specifying the EVF is insufficient to capture the differences in dynamics at the two substrates. This outcome does not depend on the precise choice of the upper Z-coordinate for calculating EVF. Evidently, other aspects besides crowding must play an important role too. 3.3. Heterogeneity of the Dynamics. It should be noted that the above trends reflect the average dynamic behavior of the ensemble of particles at the surface. However, there are ample reasons to expect dynamic heterogeneities. These are known to occur in dense suspensions, for example, in the glassy state.26,27,48,49 For the particle-coated surface, there are additional reasons to expect dynamic heterogeneities, because the surface itself is rather heterogeneous (see Figure 1c). To gain deeper mechanistic insights, one therefore has to consider the dynamics of individual particles (for example, an average R close to zero could mean that many particles are caged while a few others are still diffusing). To enable an analysis at the single particle level, we followed the procedure similar to the one described in Duits et al.42 Individual trajectories were segmented into blocks that were long enough to extract individual MSDs with sufficient accuracy, but still short enough to sample also the fastest particles. Values for A and R obtained from the blocks were then collected into histograms. To measure these histograms as a function of the time (t), this procedure was applied to the first quarter of each movie.
The resulting histograms are shown in Figure 5. A first global observation is that the particle dynamics is much more heterogeneous at the rough surface. This is shown by both the amplitude and the exponent histograms. Clearly, at the rough surface, there is a much stronger distinction between “fast” and “slow” particles. This difference applies at all times, but is strongest at small t, that is, at lower local volume fractions. Plotting the amplitudes and exponents of individual MSDs against each other, we found positive correlations between these two, which were again stronger for small t (result not shown): the particles with the lowest amplitude also had the lowest exponent. A closer inspection of the sub graphs of Figure 5 allows making additional observations (for the rough substrate). In Figure 5a, most of the exponent histograms show two peaks, roughly located near R ≈ 1 and R ≈ 0. This reveals that even in cases where the average R (as displayed in Figure 2b) is significantly above zero, caging can already take place at the level of individual particles. Also, a clear trend with time can be observed: for increasing time (and local volume fraction), the relative importance of the peak at R ≈ 0 grows at the expense of the peak at R ≈ 1. This indicates that more particles become caged, which is in line with expectations. A technical remark on Figures 5a and b is that seemingly unphysical values of R (below zero or above 1) occur. This is due to the larger statistical errors, associated with the analysis of individual trajectory data (see Duits et al.42 for more details on this). In Figure 5c, it is observed that for the smallest t, the amplitude histogram shows two peaks as well. These lowest and highest amplitudes correspond to the lowest and highest exponents in 12301
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Langmuir Figure 5a. It is remarkable that the lowest amplitudes are found in the first measurement, where the EVF was the lowest. We speculate that these particles remain at the surface in the subsequent measurements, but get “lost” due to photobleaching. The trend in Figure 5 that the histograms at the rough and smooth surfaces become more similar (although never very closely) as the local volume fraction is increased indicates that the particle dynamics in the surface layer is increasingly influenced by the suspended particles in the adjacent layer. This corroborates that our system does not resemble a quasi-2D system, even though the mobile particles are of about the same height as the fixed particles. 3.4. Mechanistic Aspects. Taking the foregoing observations together, the following mechanistic picture emerges for the particle dynamics at the rough surface. Particles that have arrived at uncovered areas of the substrate are kept for a finite time in the surface layer, because of their slowed diffusion in the vertical direction and their short sedimentation length. For short lag times, they will therefore perform a quasi-2D surface diffusion, characterized by a linear lag time-dependence of the MSD. This type of motion changes into subdiffusive when obstacles are met, a condition that occurs at longer lag times. These obstacles can be either static (due to fixed particles) or dynamic (due to other mobile particles, occurring either in the surface layer or just above it). This picture depends strongly on the local suspension volume fraction. At low volume fractions, the static obstacles dominate, but it is also possible that they are not encountered by the mobile particles, due to a prior escape from the surface layer via vertical diffusion. At high volume fractions, the mobile particles have more difficulty in escaping from the layer, while the chance of horizontal encounters with other mobile particles is increased. Under these conditions, the caging will be dominated by the mobile particles. This scenario explains why for the present particle/substrate combination, no local volume fraction could be found at which the fixed particles can cage mobile particles strong enough to trigger (either local or global) glass formation. However, this scenario could change for substrates with smaller open areas or with particles having a smaller sedimentation length. Here, a 2D caging can be expected.
4. CONCLUSIONS AND OUTLOOK We compared the dynamics of colloidal hard-sphere suspensions at a smooth and a rough surface as a function of local volume fraction. Roughness was provided by sintered particles having a size similar to that of the suspended particles. The inplane mean squared displacement function is always smaller at the rough substrate. Especially at intermediate EVFs (0.2 0.4), the fixed particles cause a strong reduction in both the magnitude and the exponent of the MSD, indicating that the lack of motion by a fraction of the particles is primarily responsible for the effect, and not the crowding. The magnitude of the additional MSD reduction by the fixed particles is also appreciable when compared to the slowing effect of a flat smooth surface. The exponent of the MSD points at caging by fixed particles, but this caging remains partial and is insufficient to induce glass formation. At high EVF (>0.45), the differences between the MSDs at the smooth and rough surfaces become smaller, although they do remain significant. We believe the insights deduced from this study can help development of new technological applications
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such as novel filtering methods utilizing wetting phenomena and particle surface interactions.50 52
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We would like to thank Dirk van den Ende for fruitful discussions and Jolet de Ruiter for preparation of particle-coated glass slides. Eric Weeks is acknowledged for providing particle tracking software. We thank the Chemical Sciences division of The Netherlands Organization for Scientific Research (NWOCW) for financial support (ECHO grant). ’ REFERENCES (1) Bacchin, P.; Aimar, P.; Sanchez, V. Model for colloidal fouling of membranes. AICHE. J. 1995, 41, 368–376. (2) Klein, G. M.; Meier, J.; Kottke, V. Fouling in membrane apparatus: The mechanisms of particle deposition. Food Bioprod. Process. 1999, 77, 119–126. (3) Marshall, J. S. Particulate Aggragate Formation and Wall Adhesion in Microchannels; American Society of Mechanical Engineers: New York, 2006; pp 519 526. (4) Mukhopadhyay, R. When microfluidic devices go bad - How does fouling occur in microfluidic devices, and what can be done about it? Anal. Chem. 2005, 77, 429A–432A. (5) Walz, J. Y. The effect of surface heterogeneities on colloidal forces. Adv. Colloid Interface Sci. 1998, 74, 119–168. (6) Yiantsios, S. G.; Karabelas, A. J. The effect of colloid stability on membrane fouling. Desalination 1998, 118, 143–152. (7) Hoek, E. M. V.; Agarwal, G. K. Extended DLVO interactions between spherical particles and rough surfaces. J. Colloid Interface Sci. 2006, 298, 50–58. (8) Lorentz, H. Abh. Theoret. Phys. 1907, 1. (9) Faxen, H. Fredholm integral equations of hydrodynamics of liquids I. Ark. Mat. Astron. Fys. 1924, 18. (10) Mittal, J.; Truskett, T. M.; Errington, J. R.; Hummer, G. Layering and position-dependent diffusive dynamics of confined fluids. Phys. Rev. Lett. 2008, 100, ???. (11) Brenner, H. The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Eng. Sci. 1961, 16, 242–251. (12) Goldman, A. J.; Cox, R. G.; Brenner, H. Slow viscous motion of a sphere parallel to a plane wall II. Chem. Eng. Sci. 1967, 22, 653. (13) Goldman, A. J.; Cox, R. G.; Brenner, H. Slow viscous motion of a sphere parallel to a plane wall. I. Motion through a quiescent fluid. Chem. Eng. Sci. 1967, 22, 637. (14) Dullens, R. P. A.; Kegel, W. K. Topological lifetimes of polydisperse colloidal hard spheres at a wall. Phys. Rev. E 2005, 71, 9. (15) Reinmuller, A.; Schope, H. J.; Palberg, T. Transient Moire rotation patterns in thin colloidal crystals. Soft Matter 2010, 6, 5312–5315. (16) Sarangapani, P. S.; Zhu, Y. X. Impeded structural relaxation of a hard-sphere colloidal suspension under confinement. Phys. Rev. E 2008, 77, 010501. (17) Nugent, C. R.; Edmond, K. V.; Patel, H. N.; Weeks, E. R. Colloidal glass transition observed in confinement. Phys. Rev. Lett. 2007, 99, 025702. (18) Nemeth, Z. T.; Lowen, H. Freezing and glass transition of hard spheres in cavities. Phys. Rev. E 1999, 59, 6824–6829. (19) Scheidler, P.; Kob, W.; Binder, K. The relaxation dynamics of a confined glassy simple liquid. Eur. Phys. J. E 2003, 12, 5–9. (20) Oguz, E. C.; Messina, R.; Lowen, H. Helicity in cylindrically confined Yukawa systems. EPL 2011, 94, 28005. 12302
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