16722 DOI: 10.1021/la102273n Langmuir 2010, 26(22), 16722–16729Published on Web 10/11/2010

pubs.acs.org/Langmuir

© 2010 American Chemical Society

Anisotropic and Hindered Diffusion of Colloidal Particlesin a Closed Cylinder

H. B. Eral,* J. M. Oh, D. van den Ende, F. Mugele, and M. H. G. Duits

Physics of Complex Fluids group, IMPACT institute University of TwentePO Box 217, 7500 AE Enschede, The Netherlands

Received June 4, 2010. Revised Manuscript Received August 19, 2010

Video microscopy and particle tracking were used to measure the spatial dependence of the diffusion coefficient (DR)of colloidal particles in a closed cylindrical cavity. Both the height and radius of the cylinder were equal to 9.0 particlediameters. The number of trapped particles was varied between 1 and 16, which produced similar results. In the center ofthe cavity,DR turned out to be 0.75D0 measured in bulk liquid. On approaching the cylindrical wall, a transition regionof about 3 particle diameters wide was found in which the radial and azimuthal components ofDR decrease to respectivevalues of 0.1D0 and 0.4D0, indicating asymmetrical diffusion. Hydrodynamic simulations of local drag coefficients forhard spheres produced very good agreement with experimental results. These findings indicate that the hydrodynamicparticle-wall interactions are dominant and that the complete 3D geometry of the confinement needs to be taken intoaccount to predict the spatial dependence of diffusion accurately.

1. Introduction

Understanding how confinement affects the diffusive behaviorof colloidal particles is crucial to understanding various dynamicprocesses in biological andmicrofluidic systems.1-3 The first classof examples is given by particles that have to diffuse toward a flatwall before they become immobilized as needed for coatingapplications or for biomedical diagnostics inmicrofluidic chips.4,5

Here, diffusive processes under confinement have been found tohave a direct influence on measurable quantities such as reactionrates and retardation times.6,7 Also, the confinement of particlesin more than one dimension and/or involving wall curvatureoccurs; examples are the synthesis of colloids inside droplets,8 thetrapping of particles inside pores,9,10 and the diffusion or directed

transport of biomolecules or particles inside biological cells(or model systems for these11,12). For each of these cases, under-standing the time-dependent particle dynamics is connected to thequestion of how confinement influences diffusion.13

Surprisingly, amajority of fundamental studies referring to thisproblem have been limited to simple geometries such as a particleapproaching a flat wall or quasi-2D systems.14-17 Theoretically,the classical problem of a particle translating in the vicinity ofa rigid flat wall was first treated in 1907 by Lorentz18 and thenFaxen,19,20 which was later improved by Brenner21 and extendedto the double-wall case by Goldman.22 The drag force along theaxis of the cylinderswas studied theoretically bySano.23 For somegeometries, the quantitative correspondence between differentlyobtained analytical expressions is still an issue, as evidenced byrecent papers onmethods to improve the analytical solution.24-26*To whom correspondence should be addressed. E-mail: h.b.eral@

utwente.nl.(1) Happel, J. ; Brenner, H. Low Reynolds Number Hydrodynamics: with Special

Applications to Particulate Media ; Kluwer Academic Publishers: Dordrecht, TheNetherlands, 1983.(2) Kim, S.; Seppo, S. J. Microhydrodynamics; Dover Publications: New York, 2005.(3) Squires, T. M.; Quake, S. R. Microfluidics: fluid physics at the nanoliter

scale. Rev. Mod. Phys. 2005, 77, 977–1026.(4) van Ommering, K.; Lamers, C. C. H.; Nieuwenhuis, J. H.; van Ijzendoorn,

L. J.; Prins, M. W. J. Analysis of individual magnetic particle motion near a chipsurface. J. Appl. Phys. 2009, 105 (10), 104905-104915.(5) van Ommering, K.; Nieuwenhuis, J. H.; van Ijzendoorn, L. J.; Koopmans,

B.; Prins, M. W. J. Confined Brownian motion of individual magnetic nanopar-ticles on a chip: characterization of magnetic susceptibility. Appl. Phys. Lett.2006, 89 (14), 142511-142525.(6) Squires, T. M.; Messinger, R. J.; Manalis, S. R. Making it stick: convection,

reaction and diffusion in surface-based biosensors. Nat. Biotechnol. 2008, 26, 417–426.(7) Beerdsen, E.; Dubbeldam, D.; Smit, B. Understanding diffusion in nano-

porous materials. Phys. Rev. Lett. 2006, 96, 4.(8) Shestopalov, I.; Tice, J. D.; Ismagilov, R. F. Multi-step synthesis of

nanoparticles performed on millisecond time scale in a microfluidic droplet-basedsystem. Lab Chip 2004, 4, 316–321.(9) Kluijtmans, S.; de Hoog, E. H. A.; Philipse, A. P. Self-diffusion of charged

colloidal tracer spheres in transparent porous glass media: effect of ionic strengthand pore size. J. Chem. Phys. 1998, 108, 7469–7477.(10) Kluijtmans, S.; Dhont, J. K. G.; Philipse, A. P. Dynamics of uncharged

colloidal silica spheres confined in bicontinuous porous glass media. Langmuir1997, 13, 4982–4987.(11) Golding, I.; Cox, E. C. Physical nature of bacterial cytoplasm. Phys. Rev.

Lett. 2006, 96 (9), 098102.(12) Claessens, M.; Tharmann, R.; Kroy, K.; Bausch, A. R. Microstructure and

viscoelasticity of confined serniflexible polymer networks. Nat. Phys. 2006, 2,186–189.

(13) Burada, P. S.; Hanggi, P.; Marchesoni, F.; Schmid, G.; Talkner, P.Diffusion in confined geometries. ChemPhysChem 2009, 10, 45–54.

(14) Diamant, H.; Cui, B.; Lin, B.; Rice, S. A. Correlated particle dynamics inconcentrated quasi-two-dimensional suspensions. J. Phys.: Condens.Matter 2005,17, S4047–S4058.

(15) Carbajal-Tinoco, M. D.; Lopez-Fernandez, R.; Arauz-Lara, J. L. Asymmetryin colloidal diffusion near a rigid wall. Phys. Rev. Lett. 2007, 99, 4.

(16) Acuna-Campa, H.; Carbajal-Tinoco, M. D.; Arauz-Lara, J. L.; Medina-Noyola, M. Collective dynamics in quasibidimensional colloidal suspensions.Phys. Rev. Lett. 1998, 80, 5802–5805.

(17) Pesche, R.; Nagele, G. Stokesian dynamics study of quasi-two-dimensionalsuspensions confined between two parallel walls. Phys. Rev. E 2000, 62, 5432–5443.

(18) Lorentz, , H. Adv. Theor. Phys. 1907, 1 (10), 23-33.(19) Faxen, H. The resistance against themovement of a rigour sphere in viscous

fluids, which is embedded between two parallel layered barriers. Ann. Phys. 1922,4 (10), 79-89.

(20) Faxen, H. Fredholm integral equations of hydrodynamics of liquids I. Ark.Mat., Astron. Fys. 1924, 18 (3), 29-32.

(21) Brenner, H. The slow motion of a sphere through a viscous fluid towards aplane surface. Chem. Eng. Sci. 1961, 16 (19), 242-251.

(22) Goldman, A. J.; Cox, R. G.; Brenner, H. Slow viscous motion of a sphereparallel to a plane wall;I. Motion through a quiescent fluid. Chem. Eng. Sci. 1967,22 (14), 637-651.

(23) Sano, O. Mobility of a small sphere in a viscous-fluid confined in arigidcircular-cylinder of finite length. J. Phys. Soc. Jpn. 1987, 56, 2713–2720.

(24) Frej, N. A.; Prieve, D. C. Hindered diffusion of a single sphere very near awall in a nonuniform force-field. J. Chem. Phys. 1993, 98, 7552–7564.

(25) Prieve, D. C.; Bike, S. G.; Frej, N. A. Brownian-motion of a singlemicroscopic sphere in a colloidal force-field. Faraday Discuss. 1990, 90, 209–222.

(26) Benesch, T.; Yiacoumi, S.; Tsouris, C. Brownian motion in confinement.Phys. Rev. E 2003, 68 (2), 021401.

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DOI: 10.1021/la102273n 16723Langmuir 2010, 26(22), 16722–16729

Eral et al. Article

Experimentally, single-walled and double-walled cases havebeen studied with optical tweezers,27-29 microscopy,15 total inter-nal reflection velocimetry,30 and light scattering.31,32 Althougheach of these techniques has its own challenges, the results of thesestudies coincided reasonably well with each other and with thetheoretical expressions. Cylindrical geometries were also studiedin a few (quasi-1D) cases: macroscopic measurements of the dragcoefficient for amillimeter-sized sphere settling along the axis of aclosed 3D cylinder33,34 and microscopic measurements of diffu-sion along the axis of long cylindrical microchannels.35,36

From this short overview, it is clear that both theoretical andexperimental studies that extend beyond simple confining geo-metries are scarce. On the experimental side, this could perhaps beexplained by the challenges in measuring local diffusion coeffi-cients (or alternatively drag forces) in 3D confinements and inmanufacturing suitable confining structures. However, recentdevelopments in soft lithography37,38 have provided new oppor-tunities for the latter. Three dimensional micrometer-scale con-fining geometries can now be manufactured in a variety ofmaterials.39,40 Moreover, these geometries can be also integratedinto microfluidic systems in which liquid flow is controlled viapumps and valves.37 These new capabilities offer unique oppor-tunities for studying diffusion in 3D confinement because of theflexibility in the design of the confining geometry, active controlover liquid flow, and good optical access to the confined volume.

In this article, we study the spatial dependence of the diffusioncoefficient in 3D closed-cylinder geometry experimentally andcompare our results with numerical simulations. Combination ofsoft lithography and real-time confocal scanning lasermicroscopy(CSLM) allowed us to observe the diffusion of micrometer-sizedcolloidal spheres in cylindrical cavities in which both the heightand radius are 9 particle diameters. Microfluidic chips containingpressure-controlled structured membranes41 were used to definethe liquid environment and trap the particles.Diffusion coefficients

were measured via particle tracking in two horizontal planes, onecrossing the cavity center and one near the bottom. Good opticalcontrast and a large number of observations allowed us to resolvethe radial (Dr) and azimuthal (Dθ) diffusion coefficients as afunction of distance from the wall, with high resolution.

This article is further organized as follows. In section 2, we willexplain how drag forces are calculated from the 3D Stokesequation and translated into diffusion coefficients. In section 3,we describe the colloidal fluid, the design and operation of thecylindrical cavities, the video particle-tracking experiments, andthe data analysis. In section 4, we show how spatially dependentdiffusion coefficients are obtained from image analysis andcompare the diffusive behaviors for 1 and 16 enclosed particlesto each other and to numerical calculations. Conclusions aredrawn in section 5.

2. Numerical Calculations

In the low Reynolds number regime, we have a fundamentallinear relation between the hydrodynamic force (FH) and theparticle velocity (vR) via the diffusion tensor (D).

v ¼ - βD 3FH ð1ÞHere β= 1/kBT and D is a diagonal tensor with componentsDr,Dθ, andDz corresponding to the principal directions in cylindricalcoordinates. The diffusion coefficientDR in each direction can beexpressed in terms of resistance and thermal energy.

DR ¼ kBTfR

, where R∈ fr, θ, zg ð2Þ

where kB and T are the Boltzmann constant and the absolutetemperature, respectively. For a spherical particle in an infinitemedium, eq 2 becomes the well-known Stokes-Einstein equation

D0 ¼ kBT3πμdp

ð3Þ

where μ and dp are themediumviscosity and the particle diameter,respectively. From eqs 2 and 3, it can be shown that the normal-ized diffusion coefficient (DR/D0) is the inverse of the dragcoefficient (cd = FR/F0).

DR

D0¼ 3πdpvR

FR¼ F0

FRfor constant vR ð4Þ

In our numerical simulations, the local diffusion coefficient isobtained by solving the Stokes equation to calculate the dragforce F for a given v. This can be done for a particle at any chosenposition inside the cylinder. By imposing no-slip boundary con-dition on the container wall and giving the particle a velocity vR,the drag forces and diffusion coefficients in the r and θ directionsare obtained separately by applying the velocity in these separatedirections. All simulations were implemented using the commer-cial COMSOLmultiphysics package.42 In these simulations, onlythe single-particle case is considered. Hydrodynamic interactionsbetween multiple particles are not taken into account. Thenumerical calculations were validated by comparing the calcu-lated drag coefficients with analytical solutions for well-knowngeometries: a spherical particle near a single wall and in betweentwo walls.

(27) Lin, B. H.; Yu, J.; Rice, S. A. Direct measurements of constrainedBrownian motion of an isolated sphere between two walls. Phys. Rev. E 2000,62, 3909–3919.(28) Leach, J.; Mushfique, H.; Keen, S.; Di Leonardo, R.; Ruocco, G.; Cooper,

J. M.; Padgett, M. J. Comparison of Faxen’s correction for a microspheretranslating or rotating near a surface. Phys. Rev. E 2009, 79 (2), 026301.(29) Jeney, S.; Lukic, B.; Kraus, J. A.; Franosch, T.; Forro, L. Anisotropic

memory effects in confined colloidal diffusion. Phys. Rev. Lett. 2008, 100, 4.(30) Huang, P.; Breuer, K. S. Direct measurement of anisotropic near-wall

hindered diffusion using total internal reflection velocimetry. Phys. Rev. E2007, 76 (4), 046307.(31) Garnier, N.; Ostrowsky, N. Brownian dynamics in a confined geometry -

experiments and numerical simulations. J. Phys. II 1991, 1, 1221–1232.(32) Holmqvist, P.; Dhont, J. K. G.; Lang, P. R. Colloidal dynamics near a wall

studied by evanescent wave light scattering: experimental and theoretical improve-ments and methodological limitations. J. Chem. Phys. 2007, 126 (4), 044707.(33) Lecoq, N.; Feuillebois, F.; Anthore, N.; Anthore, R.; Bostel, F.; Petipas, C.

Precise measurements of particle wall hydrodynamic interactions at low reynolds-number using laser interferometry. Phys. Fluids A 1993, 5, 3–12.(34) Lecoq, N.; Masmoudi, K.; Anthore, R.; Feuillebois, F. Creeping motion of

a sphere along the axis of a closed axisymmetric container. J. Fluid Mech. 2007,585, 127–152.(35) Cui, B. X.; Diamant, H.; Lin, B. H. Screened hydrodynamic interaction in a

narrow channel. Phys. Rev. Lett. 2002, 89, 4.(36) Xu, X. L.; Rice, S. A. Influence of hydrodynamic coupling on the density

dependence of quasi-one-dimensional diffusion. J. Chem. Phys. 2005, 122 (2), 024907.(37) Unger, M. A.; Chou, H. P.; Thorsen, T.; Scherer, A.; Quake, S. R.

Monolithic microfabricated valves and pumps by multilayer soft lithography.Science 2000, 288, 113–116.(38) Quake, S. R.; Scherer, A. From micro- to nanofabrication with soft

materials. Science 2000, 290, 1536–1540.(39) Becker, H.; Gartner, C. Polymer microfabrication technologies for micro-

fluidic systems. Anal. Bioanal. Chem. 2008, 390, 89–111.(40) Becker, H.; Locascio, L. E. Polymer microfluidic devices. Talanta 2002, 56,

267–287.(41) Vanapalli, S. A.; Wijnperle, D.; van den Berg, A.; Mugele, F.; Duits,

M. H. G. Microfluidic valves with integrated structured elastomeric membranesfor reversible fluidic entrapment and in situ channel functionalization. Lab Chip2009, 9, 1461–1467.

(42) van Schijndel, A. W. M. Modeling and solving building physics problemswith FemLab. Build. Environ. 2003, 38, 319–327.

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3. Experimental Section

3.1. Colloidal Suspension. Polystyrene latex particles with adiameter (dp) of 1.13( 0.05 μm, containing a red fluorescent dyeand carboxylate surface groups, were obtained as an aqueoussuspension fromPolysciences. This suspensionwas sonicated andsubsequently diluted with a 100 μM NaCl solution to a particlevolume fraction of approximately 0.5%. The final salt concentra-tion corresponds to a calculated electric double layer thickness of30 nm, which is small compared to the particle size but largeenough to maintain colloidal stability. Given the size and massdensity (F=1.05 g/mL) of the particles, the sedimentation lengthis l = kT/6πΔFgdp3 ≈ 30 μm, which is large enough to preventsignificant particle settling. The refractive index of the PS-latexparticles is 1.6 whereas that of the aqueous solvent is 1.33. Alldiluted suspensions were studied at 22( 2 �C.3.2. Microfluidic Cylindrical Cavities. Capturing particles

in 3D confinements was achieved by filling amicrofluidic channelwith a suspension and subsequently pressing down a membranewith embedded cylinder-shaped cavities to trap fluid underneath(Figure 1). The number of particles that could be trapped variedbetween 1 and 16. The device used for this is a poly-(dimethylsiloxane) (PDMS)-based multilayer microfluidic chipas described in ref 41. Briefly, the chip consists of two layers ofchannels running in perpendicular directions: one containing thefluid of interest and the other for control. The ceiling of the fluidicchannel is an elastomeric membrane in which cavities are em-bedded. On pressurization of the control channel, the membranedeflects and touches the floor; releasing the pressure restores theoriginal state. Cylinders are visualized through a 170-μm-thick

glass slide covered with a 50 μm PDMS layer. This applicationusing so-called integrated structured elastomeric membranes(iSEMs) allows the trapping of aqueous liquids in cavities ofarbitrary shape anddimensions in themicrometer range.Also, thesimultaneous study of multiple cavities is possible. Our micro-fluidic channelhadaheightandwidthof16and200μm,respectively.The cavities had a height H and diameter 2Rcyl of 10 and 20 μm,respectively.

3.3. Confocal Scanning Laser Microscope. Images wererecorded with an UltraView LCI 10 CSLM system (Perkin-Elmer) consisting of a Nikon Eclipse inverted microscope,a Yokogawa (Nipkow disk) confocal unit, and a Hamamatsu12-bit CCD camera. All recordings were made in fluorescencemode using a 25 mW laser with λ = 641 nm and a 100� oilobjective with NA = 1.3. The pixel size corresponding to theimages was 0.135 μm, and the frame rate was 10 s-1. Differentfocal planes and different numbers of particles per cavity werestudied; for each condition, 6 subsequentmovies of∼8000 frameseach were recorded.

3.4. Particle Tracking. The majority of our image analysis(section 3.5) is based on the accurate localization and tracking ofparticles; this was done using the IDL implementation of thealgorithm from Crocker and Grier.43 Additional procedures thatare commonly used for data inspection and error analysis havebeen described in detail44 and are not repeated here. We will nowdiscuss a few specific aspects of the present study.

Figure 1. (a)Three-dimensional representationofour devicewith integrated structured elastomericmembranes (iSEMs).Thedevice consistsof three major structures: a control channel (a1) that can be pressurized, a membrane (a2) in which cylindrical structures are defined, and afluidic channel (a3) via which particles and solvent are delivered. The control and fluidic channels intersect each other perpendicularly.(b)Applyingpressure to the control channel leads to the deflection of themembrane and the trappingof the particle suspension. (c) Schematicillustration of the experiment in which the particles are imaged in a chosen focal plane of a confocal microscope.

(43) Crocker, J. C.; Grier, D. G. Methods of digital video microscopy forcolloidal studies. J. Colloid Interface Sci. 1996, 179, 298–310.

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Because of the refractive indexmismatch between particles andsolvent, image distortion occurred and the optical contrast alsodeteriorated with increasing depth (Z). However, because of thelow particle concentration, sufficient visibility could be obtainedat allZ values of interest.AtZ=5μmabove the PDMS floor, thelocalization accuracy in the (X,Y) focal planewas∼30nmas in anearlier study.45 The effective focal depth relevant to particletracking was estimated to be 2.0 μm from an intensity versusZ scan of a particle that was stuck to the bottom. Amore detaileddiscussion of the effective focal depth is given in the SupportingInformation. The same immobilized particle also allowed anestimation of the mechanical drift of the cavities; we found it tobe less than 1 pixel per 100 s, which is negligible for our purposes.

3.5. Image Analysis. 3.5.1. Cavity Geometry. To in-spect the 3D geometry of the cavity, we employed two differentmethods. In the first method, a fluorescently dyed aqueous liquidwas pumped into the chip and directly visualized by recordinghorizontal (X, Y) images at various vertical (Z) locations. Thisallowed a fairly accurate measurement of the contour in the hori-zontal plane and an estimate of the cavity height (SupportingInformation). A second, independent check of the (X,Y) contourwas made after replacing the dye solution with a suspension andlocalizing the captured particles for typically 5 � 104 frames.

Making a time projection of all locations where a particle wasfound resulted in a cloud representing the accessible area. Fromthis cloud, a contour line was constructed by connecting theoutermost points. A careful smoothing was applied, which re-sulted in a contour that contained all but a few particles. Theexcluded particles were still very close (∼1 pixel) to the calculatedcontour and reflect the uncertainty in the localization of thecontour. Finally, the contour was inflated by 1 particle radius toaccount for the fact that the localizations pertained to the centersof the particles. For the cavity with 16 particles, the contoursfound with the two methods corresponded well to each other andto the dimensions of the PDMSmaster fromwhich the iSEMshadbeen made (not shown). With only one particle in the cavity, thevisualization with fluorescent liquid worked better.

It turnedout that the cylinders did not display perfectly circularcross sections (see Figure 2 for an example), which necessitated a

descriptionof the radial positionof thewallRmaxas a functionof theangle θ. The transformation from Cartesian (X, Y) to polar (R, θ)coordinates was made by localizing the cavity center and subse-quently decomposing the relative vectors (ΔX, ΔY) into R andθ components. Performing this operation also on the calculatedcontour line resulted in an Rmax(θ) function that turned out tohave a specific dependency for individual cavities. To facilitate thedescription of the contour, we described it with a Fourier series

RmaxðθÞ ¼ Rfit þXml

Am sinðmθÞþBm cosðmθÞ ð5Þ

with a truncation atm=4. The quality of this fit turned out to bevery good.

3.5.2. Diffusive Behavior. As explained in section 2, localdiffusion coefficients DR were calculated from the resistance andthermal energy. Experimentally, they can be measured from themean squared displacement ÆΔrR2æ using the Einstein-Stokes-Sutherland equation

ÆΔrR2æ ¼ 2DRτ where R∈ fr, θ, zg ð6Þwith Δr =ΔRer þ RΔθeθþ Δzez and D = Drererþ Dθeθeθ þDzezez. Here, τ is the lag time and ΔrR is the R component of thevector that describes the 2Ddisplacement of a particle during thattime. In our case, the brackets indicate an averaging over differenttimes andparticles at a given location. In eq2,DR canbe expressedas a diagonal tensor that has components of Dr, Dθ, and Dzcorresponding to the principal directions in cylindrical coordi-nates. Inour experiments,we focused on the in-planeMSD, henceonly Dr and Dθ are measured. The diffusion coefficient wasmeasured from the initial slope corresponding to short-timebehavior.

A fundamental question is how to define the locationwhereDRismeasured, given thatDR itself ismeasured via a displacement. Ifthe length scale over which theMSD can show spatial variation ismuch larger than the typical magnitude ofΔrR (as in ref 46), thenthis is only a minor issue. In the present study, where strongvariations in DR can be expected close to the wall, we optimizedthe spatial resolution by consideringΔrR only for the smallest (i.e.,

Figure 2. Overview of the experiment with 16 particles, with all trajectories superimposed. (a) Diffusion coefficient (D) in the midplane(ZFP=H/2) of the cylindrical cavity as a function of the (X,Y) position.DR has been normalizedwith respect to the diffusion coefficient of afree particle in pure solvent (D0). (b) Corresponding map for the number of times that a particle was found in each (X, Y) pixel.

(44) E. Weeks IDL codes for particle tracking. http://www.physics.emory.edu/∼weeks/idl/.(45) Eral, H. B.; van den Ende, D.; Mugele, F.; Duits, M. H. G. Influence of

confinement by smooth and rough walls on particle dynamics in dense hard-spheresuspensions. Phys. Rev. E 2009, 80.

(46) Duits, M. H. G.; Li, Y. X.; Vanapalli, S. A.; Mugele, F. Mapping ofspatiotemporal heterogeneous particle dynamics in living cells. Phys. Rev. E2009, 79.

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unit) time step τ1 (= 100 ms). This generated a minor secondaryissue: because the exposure time of the camera per image (σ) wasonly slightly smaller than τ1, the magnitude of ÆΔrR2æ that wasfound had to be corrected by subtracting σ/3 from τ as prescribedby eq 14 in ref 47.Our σwasmeasured fromaplot of ÆΔrR2æ versusτ for a bulk viscous liquid (Supporting Information).

Unlessmentioned otherwise, the locationofDRwas assigned tothe midpoint of ΔrR. Rounding the measured X, Y locations todiscrete (pixel) numbers was done to make a pixel map that couldbe presented as an image; for a cavity with 16 particles observedfor ∼5 � 104 frames, a pixel was typically “hit” 20 times. Todistinguish between the fundamentally different diffusion direc-tions in a cylinder and to allow averaging over equivalent pixels,ΔrRwas decomposed into radial (R) and azimuthal (θ) directions.This produced values ofDr andDθ as functions ofR and θ. In ouranalysis, the distance ΔR from the cylindrical wall is a moreappropriate variable thanR. In view of the slight deviations fromperfect cylindrical shape (section 3.5.1), we used the Fourierdescription (eq 5) of Rmax (θ) to calculate ΔW(θ) (or its normal-ized form ξ):

ΔWðθÞ ¼ RmaxðθÞ-R and ξ ¼ ΔW=dp ð7ÞHereafter, eq 7 was used to group data forDr andDθmeasured atthe same ΔR.

4. Results and Discussion

4.1. Spatial Dependence of Diffusion Coefficients. Tostart with an overview, we first show representative maps forthe normalized diffusion coefficient (Figure 2a) and the occur-rence frequency (Figure 2b). Both maps were measured in themidplane (ZFP = H/2) of the cylinder. It turns out that evenfarthest away from the walls,DR is significantly lower than that ofa free particle in bulk liquid.Moreover, it is clearly visible thatDRis progressively reduced as the cylindrical wall is approached. Theoccurrence map looks fairly homogeneous except for a depletionzone in the vicinity of the cylindrical wall (which will be discussedin section 4.2).

A closer inspection of Figure 2b reveals two issues that had tobe addressed. First, at two (X, Y) locations the occurrenceprobability was significantly larger: near the cylindrical wall at(X = 22, Y = 23) and near the center at (18, 15). Both particlesappeared to stick to awall for at least part of the time. The particlenear the center appeared to be in the bottom plane and wasdetected because of the enhanced brightness of such particles. Theother particle was clearly stuck to a cylindrical wall. Overall, suchsticking eventswere rare; in 60h of recording only three such caseswere found. Second, slight temporal fluctuations of the cavitycontour were also observed. This was manifested as small dif-ferences between the contours found in subsequent movies andshows up in the time projection of Figure 2b as the increasedwidth of the particle depletion zones in the lower and right-handparts. Both issues (whenever they occurred) were resolved byexcluding the problemareas from the analysis.Omitting this filteringoperation resulted in significantly different results (in Figures 3-5)in the vicinity of the walls.

4.1.1. Midplane.More detailed information was obtained bybinning data as plotted in Figure 2a according to the distancefrom the cylindrical wall and decomposing the diffusion coeffi-cient into radial and azimuthal components as explained insection 3.5.2. We first consider one particle in the midplane(ZFP = H/2) of the cylinder. The experimental data in Figure 3reveal several aspects. In the center of the cavity, Dr and Dθ are

equal and the totalDR is∼25% lower than the bulk valueD0. Asthe particle approaches the side wall along the radial direction, itsdiffusion coefficient follows a plateau, down to ∼3 particlediameters from the wall where a gradual decrease sets in. Thisdecrease is stronger for Dr, which is qualitatively in line withexpectations from the known diffusive behavior of a particle neara single flat wall: here the perpendicular drag force is alwaysgreater than or equal to the parallel drag force.21

Also shown in Figure 3 are numerical calculations (section 2)that take into account the specific 3Dgeometry of our experiments.The theoretically predicted trends in Dr and Dθ as a function of

Figure 3. Normalized diffusion coefficients of a single particle inthe Z midplane of the cavity, plotted versus the normalizeddistance (ξ) from the cylindrical wall. Squares (open for Dθ andclosed for Dr) show the experiments, and vertical bars representestimated standard deviations. (See the text.) Triangles connectedby dashed lines correspond to the numeric solution of the 3DStokes problem. The inset shows the validation of the numericalmethod against the reference case of a particle near a flat wall. Thediffusion coefficients parallel (D ),O) and perpendicular (D^,b) toa single wall as a function of the distance from the wall (ZFP) areshown. Circles show our calculations, and red dashed lines repre-sent the analytical solution from Brenner.21 Similar quantitativeagreementwas found for the case of two parallelwalls (not shown).

Figure 4. Comparison between diffusive behaviors of 1 particle(gray squares, samedataasFigure 3) and16particles (red circles) inthe midplane of the cavity (Z = H/2). Note that the diffusioncoefficients are now given in absolute numbers. The inset shows(for the case of 16 particles) how the apparent DR/D0 depends onthe assignment of the location where Dr was measured: thebeginning (red triangles), end (green circles), or midpoint (blacksquares) of the displacement step.Open symbols correspond toDθ,and closed symbols correspond to Dr.

(47) Savin, T.; Doyle, P. S. Static and dynamic errors in particle trackingmicrorheology. Biophys. J. 2005, 88, 623–638.

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ξ correspond remarkably well with those of the experiments.Concerning the magnitudes, a 10% discrepancy is found betweenthe model and the experiments. We think that this should mainlybe ascribed to the error introduced by the normalization stepwithD0: the latter quantity may have beenmeasured at a slightly lowertemperature. Another experimental error could be due to theslight uncertainty in the size of the individual particle in the cavity.Finally, we remark that the signal/noise (S/N) ratio of the experi-mental data is fair, even for ∼5 � 104 observations.

The case of 16 particles in the cavity suffers less from poorstatistics, but the question is to what extent the diffusive behavioris the same. The still rather low particle volume fraction (∼0.004)suggests that close-range hydrodynamic interactions (collisions)between particles will rarely occur. A comparison between thecases of 1 and 16 particles per cavity is shown in Figure 4. Thecorresponding curves for Dr and Dθ look fairly similar. The datafor the 16 particles show a slightly less steep decay near the wall,and in the center of the cavity,DR appears to be 5% smaller. Thelatter discrepancy falls within the experimental error.Whether theformer small difference could perhaps reflect interparticle hydro-dynamics remains to be elucidated and would require a differentsimulation method. In any case, the better statistical accuracy ofthe experiment with 16 particles allows us to address severaladditional questions. We first address the accuracy that can bereached inmeasuring local diffusion coefficients with our particle-trackingmethod. Uncertainties exist in both the magnitude ofDRand in the location where it is measured, and the errors can beeither systematic or stochastic.

The first issue is related to the assignment of a location to themeasuredDR. Different choices can be made: the beginning, end,or midpoint of the diffusive displacement ΔrR. The consequencesof this choice are shown in the inset of Figure 4. The differencesare small, which is rationalized by the fact that the typicaldisplacement of τ1 = 0.1 s is only (0.1-0.2)ξ. At the smallest ξ,the value of Dr based on the midpoint is smaller than at the

beginning and end. This can be understood by considering that ofall radial displacements that begin or end in the ring closest to thewall only those that are small will also have the midpoint in thatring. Larger steps will result in midpoints at larger ξ. This biasingeffect applies only to radial displacements in the outer ring.

A second issue is the drift (i.e., nonzero average displacement48)due to the gradient of the diffusion coefficient (∂Dr/∂ξ) in theradial direction, which canmake a contribution to ÆΔrR2æ propor-tional to ξ2.We simulated this effect for the case of a particle neara flat wall and found (Supporting Information) that on using theEinstein-Stokes equation the localDr (perpendicular to the wall)had to be replaced by

Deffr ðξÞ ¼2ξ- 12ξ

� �1þ 4ξ

2 - 3ð2ξ- 1Þ8ξ3ð2ξ- 1Þ τ

0" #

ð8Þ

with τ0 being the normalized lag time (τ0 = τ/t0, where t0 = dp2/

2D0), which amounts to 0.028 in our case. The term inparenthesesincorporates the effect of the wall on the drag coefficient, and theterm in square brackets represents the drift effect. The latter termis always>1but becomes 0.58. Thismeans that thecorrection termplays a role only very close to the wall (∼1 camerapixel in our case). Although the correction is thus only minorin our case, it is also clear from eq 8 that this is due to the smallvalue of τ.

A third issue is how the magnitude of DR is affected bylocalization errors. The measured coordinates of the contourand center of the cavity suffer from inaccuracies that are largerthan that of the particle localizations, and it is obvious that thestrongest influence of these errors is expected near the wall. Weexamined this issue by repeating the image analysis but nowincreasing or decreasing the local contour radiiRmax(θ) by 0.15dp.Also, the X and Y coordinates of the cavity center were varied bythe same amount. Comparing DR values at the same ΔR for thedifferent data sets thus created allowed us to estimate a ΔR-dependent standard deviation. To obtain the total standarddeviation, these errors were combined with the statistical inaccu-racy that arises from the finite number of observations N withineach ΔR ring. The relative error due to this effect amounts to(2/N)1/2,46 and it is also ΔR-dependent. The resulting error barsfor individual cases are shown in Figures 3 and 4. It turns out thatthe typical relative error inDR is 8% for 1 particle per cavity and4% for 16 particles per cavity.

4.1.2. BottomPlane.For the case of 16 particles, the diffusivebehavior at the bottom of the cavity could also be examined,albeit an additional data-filtering step was needed to ensure thatonly particles very close to the bottomare taken into account. Theneed for this step arises from the fact that the Z range in whichparticles are detected is ∼2 μm (Supporting Information). Withthe focus onZFP= 4.5( 1 μm, this effect is unimportant becausethe distance ΔZ of a particle to the nearest wall varies onlybetween 4 and 5 μm.However, atZFP=0.5dp,ΔZ varies between0.5 (contact) and 2 μm. In this range,Dr andDθ can be expected toshow a strong dependence on the Z position.

To exclude particles that are not very close to the bottom, wecollected the intensities of all ∼105 particles tracked at ZFP =0.5dp into a histogram (inset of Figure 5). Two distinctive peaksthat differ in intensity by a factor of ∼3 were found. We thenassumed that only this secondary peak corresponds to particles atthe bottom and restricted our analysis to this subset. The fact that

Figure 5. Normalized radial (Dr) and azimuthal (Dθ) diffusioncoefficientsmeasured at themidplane (ZFP=H/2, red circles) andbottom (ZFP = 0.5dp, blue triangles) for the case of 16 particles inthe cavity. Open symbols correspond to Dθ, and closed symbolscorrespond to Dr. Diamonds connected by solid lines show thenumerical solution of the 3D Stokes equation. Open symbolscorrespond to Dθ, and closed symbols correspond to Dr.The insetshows histograms for the intensity measured atZFP= 0.6dp (solidline) andZFP=H/2 (red dashes). This intensity corresponds to thebrightness of the fluorescent image of a particle, which depends onits distance to the Z-ZFP focal plane as well as on ZFP itself.

(48) Lancon, P.; Batrouni, G.; Lobry, L.; Ostrowsky, N. Drift without flux:Brownian walker with a space-dependent diffusion coeffcient. Europhys. Lett.2001, 54, 28–34.

16728 DOI: 10.1021/la102273n Langmuir 2010, 26(22), 16722–16729

Article Eral et al.

this data selection gave substantially lower values for Dr and Dθ,whereas a similar filtering at ZFP = H/2 had negligible effects,seems to justify our assumption.

The thus-measured dependencies of Dr and Dθ on ξ at thebottom plane are presented in Figure 5, together with the dataobserved in the midplane (and with numerical simulations). Forξ >3, the proximity of the bottom plane is manifested as an∼25% reduction in the DR values as compared to the midplane.As the cylindrical wall is approached (ξ < 3), a transition takesplace to a regime in which the drag force becomes stronglydominated by the cylindrical wall. This is evident from thequantitative correspondence at small ξ of the Dr and Dθ curvesmeasured at the bottom and midplanes.

The comparison to numerical calculations revealed someinteresting aspects. Because the inaccuracy in ZFP was estimatedto be 0.2 μm, calculationswere performed forZFP= (0.6-1.5)dp .The results (shown in the Supporting Information) revealed thatthe plateau levels for Dr and Dθ depended strongly on ZFP andthat thebest agreementwith our experimentswas found forZFP≈0.6 μm. It also turned out that all calculated Dr curves showed avery similar limiting behavior for ξ

DOI: 10.1021/la102273n 16729Langmuir 2010, 26(22), 16722–16729

Eral et al. Article

of the wall by comparing for each movie two sets of Rmax(θ): the“smooth” contour obtained from fittingwith eq 5 and the “noisy”primary data. The difference ΔRmax did not show any systematicdependence on θ, and its distribution P0(ΔRmax) had a standarddeviation σ of 140 nm. This σ, which includes the contribution ofdynamic wall fluctuations, matches the found depletion rangeverywell. To illustrate this further, we took a step functionP(ξ) asexpected for hard spheres and walls and convoluted it with thedistribution P0(ΔRmax). As can be seen in Figure 7, the resultingcurve agrees verywell with the experimentalP(ξ) data. Finally,wealso did a computer simulation in which a screened electrostaticrepulsion between particles and a wall is included. Here it turnedout that the additional change in our P(ξ) due to a potential witha contact value of 8kBT and a Debye length of 30 nm has anegligible effect.

The implications of these findings for the present study are thefollowing: (i) it is corroborated that the thermodynamic interac-tions between particles and wall are essentially like that of hardsurfaces and (ii) fluctuations of the PDMS wall cannot beexcluded, but if so, their amplitude does not exceed 150 nm. Thislatter aspect was already taken into account in the error analysisof D(ΔR) and appears to have a rather small impact on ourmeasurements of the local diffusion coefficient. Interestingly, thecase of a fluctuatingwall could also be found in systems occurringin nature (e.g., emulsion droplets or living cells. Our method of

measuring the diffusive behavior in confinementswith fluctuatingwalls might be suitable for studying such cases as well.

5. Summary and Outlook

In the present study, three novel aspects were combined: themicrofluidic trapping of particles into a closed 3D confinementwith a designed geometry, the application of particle tracking tomeasure both interactions and diffusion coefficients near walls,and the combination of experiments and numerical modeling toexamine the effects of 3Dconfinement on local diffusive behavior.Several conclusions can be drawn: (1) For the first time, thespatial dependence of radial and azimuthal diffusion coefficientshave been measured for particles in a closed cylinder. (2) Numer-ical calculations based on the solution of the 3D Stokes problemfor a single particle correspond to the experimental results re-markably well. For the experiments with 16 particles, slightdiscrepancies fromnumerics, which can be attributed to particle-particle hydrodynamics, are observed. (3) Very close to thecylindrical wall, the curvature of this wall does not have asignificant influence on the local diffusion coefficients. (4) Alongthe central axis of the cylinder, the presence of the top and bottomwalls has a significant influence on the magnitude of the diffusioncoefficient for all axial positions. Hence, to predict the completespatial dependence of the local diffusion coefficient quantita-tively, the complete 3D geometry of the confinement needs to betaken into account.

A current limitation of the iSEM-based cavities is that particlescould not be trapped in large numbers. The ability to reach highervolume fractions inside 3D cavities would be very interestingbecause it would allow us to study at which number of particles orvolume fraction the effect of particle-particle hydrodynamicsbecomesmanifest and how these interactionswill change the localdiffusive behavior. This will be a topic of further study.

Acknowledgment.We thankD.Wijnperle for contributions toiSEM design and manufacturing, D. Mampallil, O. Agbaba, andT. Yigit for early experiments, and R. Nauta for Labview pro-gramming of microfluidic pumps. We are also grateful fordiscussions with S. Vanapalli onmicrofluidics and withM. Ekiel-Jezewska on hydrodynamics. This work was financially sup-ported by the Chemical Sciences division of The NetherlandsOrganization for Scientific Research NWO (an ECHO grant).

Supporting Information Available: The following issuesare discussed: Size and shape of the cavity. Effects of finiteshutter time. Effective depth of focus. Computer simulationson the effect of confining boundaries. Dynamics of particleswith space-dependent mobility. This material is availablefree of charge via the Internet at http://pubs.acs.org.

Figure 7. (Red circles) Probability of finding a particle center as afunction of dimensionless distance from the cylindrical wall. Thismeasurement was made at the Z=H/2 plane, with 16 particles inthe cavity. P(ξ) is normalized to the value measured in the centralarea. The solid black line shows the expectedP(ξ) profile generatedby taking into account the uncertainty in detecting the cylindricalwall. The error distribution corresponding to the latter uncertaintyis given by the solid blue line. The circle indicates the size of theparticle.