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CXXXX American Chemical Society

Diffusive Dynamics of Nanoparticlesin Arrays of NanopostsKai He,† Firoozeh Babaye Khorasani,† Scott T. Retterer,‡ Darrell K. Thomas,§ Jacinta C. Conrad,†,* and

Ramanan Krishnamoorti†,*

†Department of Chemical and Biomolecular Engineering, University of Houston, Houston, Texas 77204-4004, United States, and ‡Center for Nanophase MaterialsSciences and BioSciences Divisions and §Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37934, United States

The transport properties of nanoparti-cles in complex confined media playa significant role in biomedical, envi-

ronmental applications, and in oil and gasexploration and production. For example,therapeutic drugs encapsulated as nano-particles must be delivered1 and trans-ported through the extracellular matrix2

to reach targeted cells.3,4 Similarly, the fateof engineered nanoparticles and nano-scale contaminants in aquatic systems de-pends in part on their transport propertiesin environmental porous media such assaturated soils5 or biofilms.6,7 Finally, nano-particles developed for use in enhanced oilrecovery applications8,9 must be designedfor transport through highly confinedpore throats, especially in low permeabilityreservoirs.Confining a micrometer-sized colloidal

particle changes its transport properties: par-ticles near walls10 or in enclosed chambers11

diffuse more slowly than those in bulk. Asthe size of the particles is decreased belowone micrometer, however, the effectsof electrostatic12 and hydrodynamic13,14

interactions on dynamics become morepronounced. Moreover, natural porousmedia are extremely heterogeneous: pore

diameters range from nanometers to milli-meters, pores can be connected or dis-connected, and the surface propertiesof the pores can vary in chemistry andin wettability. As a result, heterogeneitiesin confinement, pore connectivity, andporous media�nanoparticle interactionscan introduce different physical mechan-isms for transport that generate a widevariety of transport properties observedfor nanomaterials in complex media.15�23

One route toward an improved funda-mental understanding of transport me-chanisms is to create simplified modelsof porous media.24 Specifically, recent ad-vances in self-assembly and microfabri-cation have generated model media inwhich the pore structure and wettabilityare precisely controlled, for example includ-ing inverse opals25,26 and microfabricated27

and nanofabricated28 post arrays. Thesemodel media exhibit uniform and mono-dispersed pore sizes and spacings. By elim-inating variations in pore connectivity andsurface chemistry, these simplified modelmedia reduce heterogeneity and therebyenable microscopic studies that isolate theeffects of well-defined confinement on thetransport properties of nanoparticles.

* Address correspondence toramanan@uh.edu;jcconrad@uh.edu.

Received for review February 12, 2013and accepted May 14, 2013.

Published online10.1021/nn4007303

ABSTRACT The diffusive dynamics of dilute dispersions of nanoparticles

of diameter 200�400 nmwere studied inmicrofabricated arrays of nanoposts

using differential dynamic microscopy and single particle tracking. Posts of

diameter 500 nm and height 10 μmwere spaced by 1.2�10 μm on a square

lattice. As the spacing between posts was decreased, the dynamics of the

nanoparticles slowed. Moreover, the dynamics at all length scales were best

represented by a stretched exponential rather than a simple exponential. Both

the relative diffusivity and the stretching exponent decreased linearly with

increased confinement and, equivalently, with decreased void volume. The slowing of the overall diffusive dynamics and the broadening distribution of

nanoparticle displacements with increased confinement are consistent with the onset of dynamic heterogeneity and the approach to vitrification.

KEYWORDS: diffusion . porous media . non-Gaussian dynamics . nanoparticles

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Here, we focused on transport at zero shear rate andinvestigated the dynamics of nanoparticles diffusing innanopost arrays. We fabricated arrays of nanopostswith a diameter of 500 nm arranged on a square latticewith spacing of 1.2�10 μm using an electron-beamlithography process. The dynamics of nanoparticlesof diameter ranging from 200 to 400 nm diffusing inthesemediaweremeasured using differential dynamicmicroscopy and single-particle tracking. As the nano-particles were increasingly confined by the array ofposts, we found that the image structure function couldnotbe fit by a single exponential but insteadwas bestfitby a stretched exponential function. The relative diffu-sivity and the stretching exponent extracted from thestretched exponential fits decreased approximatelylinearly as the void fraction of the nanopost arraywas decreased or as the ratio between the particlediameter and nanopost spacing was increased. Weconfirmed this behavior in direct-space measurementsusing single-particle tracking. The slowing of dynamicsis consistent with confinement of the nanoparticleby the cylindrical posts. The emergence of a spectrumof relaxation times suggests either a heterogeneousenvironment on a length scale comparable to theparticle size or an analogy to vitrification29 of the nano-particles in confined geometries.

RESULTS AND DISCUSSION

We collected time-resolved fluorescence opticalmicrographs for the 200 to 400 nm nanoparticlesdiffusing in the bulk and in microfabricated post arrays(Figure 1) with post spacing ranging from 1.2 to 10 μmcorresponding to a void fraction (θ) range of 0.76 to1 and a confinement parameter (ζ) range of 0 to 0.35.From time series of fluorescence images, we calculatedthe delay-time dependence of the azimuthally aver-aged image structure function D(q,Δt).For each wavevector q, the magnitude of D(q,Δt)

increased with increasing delay time Δt, as shown inFigure 2, indicating that the positions of nanoparticlesbecame increasingly uncorrelated from their originalpositions with increasing Δt. For particles diffusing inthe bulk and in large spacing post arrays (Figure 2a�c),D(q,Δt) reached a plateau at longΔt, whereas for thosediffusing in the small spacing post arrays (Figure 2d,e),no plateau was observed even at the longest delaytimes accessed in these experiments. Nonetheless, thetemporal evolution of the structure function forthese confined cases exhibited at least two distinctivedynamic events. We attributed the fastest of theseevents to the diffusive dynamics of the nanoparticles inthe confined media. We were unable to reasonablycapture the longer dynamics in the structure functiondata in these experiments. Because the slower dynamicalevent was not observed in confined media at smallvalues of the wavevector q (Figure 2f), we attributedthe changes in D(q,Δt) on long time scales, observed at

large q for confined samples, to the dynamics corre-sponding to the jump between two “caged” domains or

Figure 1. (a) Schematic of cylindrical post arrays filledwith polystyrene nanoparticle dispersions. (b, c) Scanningelectron micrographs of posts of diameter dp = 500 nm,(b) height H = 10 μm and spacing S = 6 μm, and (c) heightH = 11.9 μm and spacing S = 1.6 μm.

Figure 2. Structure function D(q,Δt) as a function of delaytimeΔt at q= 8 μm�1 (a, d), q= 5 μm�1 (b, e), and q = 2 μm�1

(c, f) for 400 nm diameter nanoparticles diffusing in twodifferent post arrays using fluorescenceDDM. Panels (a), (b),and (c) present data obtained for a post spacing S=8μm, ζ=0.051; panels (d), (e), and (f) present data obtained for S =1.6 μm, ζ = 0.270. Dashed blue lines (notated as E) representfits to a simple exponential model eq 1), and solid red lines(notated as SE) represent fits to a stretched exponentialmodel (eq 2).

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to the weak, possibly apolar or weakly polar, interactionsbetween the particles and the post arrays.We previously showed30 that the image structure

functions D(q,Δt) obtained using brightfield and fluo-rescence DDM for 200�400 nm diameter nanoparti-cles in unconfined Newtonian media were excellentlyfitted using a single exponential model:

D(q,Δt) ¼ A(q) 1 � exp� Δt

τ(q)

� �" #þ B(q) (1)

where τ(q) is a q-dependent relaxation time and A(q)and B(q) are fitting parameters that depend on thewavevector q and on the optical train. This functionalform also produced good fits of the short time scaleimage structure functionD(q,Δt) for particles that wereonly weakly confined by posts (S = 8 μm), as shown bythe dashed lines in Figure 2a�c. By contrast, the delaytime dependence of D(q,Δt) for particles diffusing inhighly confined post arrays (S = 1.6 μm) could not becaptured by the single exponential model, as shown bythe dashed lines (single-exponential fit) in Figure 2d�f.These data required a new fitting model. We notethat the Kohlrausch�Williams�Watts function, whichincorporates a stretched exponential term (i.e.,exp[�((Δt)/(τ(q))r(q))]) instead of the single exponentialterm applied in eq 1, has been successfully used todescribe the slowing of dynamics in complex glassy31

and polymeric32 systems. To characterize the fastdynamic process, we thus fitted all structure functionsfor particles diffusing in confined nanopost arrays overa limited range of delay times to a single stretchedexponential model,

D(q,Δt) ¼ A(q) 1 � exp� Δt

τ(q)

� �r(q)" #

þ B(q) (2)

and extracted four parameters: the signal prefactorA(q), the background B(q), the q-dependent relaxationtime τ(q), and the stretching exponent r(q). We choseto fit the data over a delay time range that extendedup to five times the relaxation time as a typical cutoffvalue; using higher cutoff values did not substantiallyalter the results that we report here. Using this proto-col, the short time dynamics for all samples could bewell fitted by the stretched exponential model. Fromthe fits to the model in eq 2, we focused on the twofitting parameters τ(q) and r(q) that are most closelyassociated with the dynamics. As noted in a previouspublication,30 the signal background term B(q) wasonly dependent on the details of the optical train.To characterize the dynamics of nanoparticles in

post arrays, we first examined the wave-vector depen-dence of the relaxation time τ(q), obtained by fittingthe image structure function to eq 2. Over a narrowrange of q values (i.e., 6 μm�1 < q < 8 μm�1) corre-sponding to length scales shorter than the spacingbetween posts, τ(q) scaled as q�2 for all post spacings,

as shown for representative data from 400 nm nano-particles in Figure 3. This result indicates that thedynamics of the nanoparticles remained diffusivewhen moderately confined by cylindrical posts. Never-theless, the relaxation time τ(q) at a fixed value of qincreased in magnitude as the spacing between postswas decreased and the nanoparticles were increasinglyconfined, indicating that cylindrical obstacles (or in-crease in confinement) slowed the diffusive dynamicsof the nanoparticles. The qualitative trends shownhere for the 400 nm nanoparticles dynamics were alsoobserved for smaller nanoparticles of diameter 200and 300 nm (Supporting Information, Figures S2�S5).We verified the dynamical measurements obtained

usingDDMby analyzing the dynamics of 400 nmnano-particles diffusing in post arrays in real space usingsingle particle tracking. The ensemble-averaged MSDof the nanoparticles, calculated from the trajectories,was linearly proportional to delay time Δt across theentire accessible range of delay times, as shown inFigure 4, confirming that the particle dynamics werediffusive in post arrays. We therefore extracted theaverage diffusion coefficient D via

ÆΔx2(Δt)æ ¼ 2DΔt (3)

from the MSD. We note that our MSD measurementsprobed length scales from 0.09 to 1.2 μmand thusmea-sured thedynamics of nanoparticleswithin thepores forall post spacings studied here. Furthermore, the MSDdecreased as particles were increasingly confined bythe posts, indicating that the particle dynamics wereslowed down by the presence of obstacles or by theconfinement of the nanoparticles by the posts. Theseobservations are consistent with the results obtainedfrom the DDM analysis in reciprocal space.To obtain the diffusion coefficients D for the nano-

particles, we fitted the image structure function ob-tained via DDM to eq 2 (for nanoparticles of diameter

Figure 3. Relaxation time τ(q) (in seconds) as a function ofthe magnitude of the wave vector q (in μm�1) for 400 nmnanoparticles diffusing in the bulk (black squares) and inpost arrays with S = 4 μm, ζ = 0.105 (orange diamonds);S = 1.8 μm, ζ = 0.237 (olive right triangles); and S = 1.2 μm,ζ = 0.354 (navy stars). The inset shows that τ(q) scales asq�2 over the range of wavevectors from q = 6 to 8 μm�1.The error bars for τ(q) are smaller than the symbols.

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200, 300, and 400 nm) or the MSD obtained via particletracking to eq 3 (for nanoparticles of diameter 400 nm).We compared the coefficients of nanoparticles of dif-ferent diameters as a function of void fraction θ andconfinement parameter ζ by normalizing all diffusioncoefficients by D0, the diffusion coefficient for nano-particles diffusing freely, as shown in Figure 5. Wefound that the diffusion coefficients extracted fromthe MSD were in excellent agreement with thoseobtained from the DDM measurement on 400 nmparticles, confirming that the two analyses of themicroscopy data probed the same diffusive behavior.The relative diffusivity (D/D0) scaled linearly with θ andζ and collapsed onto a single curve for the threedifferent nanoparticles. The linear dependence of re-lative diffusivity on void fraction persisted even whenthe distance between posts was only of the orderof three times the particle size. A similar reductionin diffusion coefficientwith increase in obstacle densityand linear relationship between the relative diffusivityand void fraction was observed for several casesof probe diffusion in crowded media;33,34 by contrast,in periodic 3-D porous media a nonlinear relation-ship between relative diffusivity and the confinement

parameter was observed over a similar confinementrange.26

The diffusivities reported in Figure 5 were derivedfrom averaged measurements of relaxation time orensemble-averaged mean-squared displacements.The image structure function D(q,Δt), however, wasnot adequately represented by a single exponentialfunction, as would be expected for a relaxation processwith a single time scale (e.g., for nanoparticles diffusingin a homogeneous medium), but was instead bestmodeled as a stretched exponential eq 2with a stretch-ing exponent r(q). To gain further insight into theeffect of confinement induced by the post arrays onthe distribution of dynamics of the nanoparticles, wetherefore examined the behavior of r(q). Typically,stretched exponential behavior (i.e., r(q) < 1) is attrib-uted to the underlying presence of several relaxationprocesses with distinct relaxation times that act addi-tively to provide the collective response observed.We found that, for a given nanoparticle diameter anda given post spacing, r(q) was nearly constant over1 order of magnitude in wavevector q (SupportingInformation, Figure S6). We therefore report a singlestretching exponent Æræ, which is a function only of thediameter of the nanoparticles and the spacing be-tween the posts, as a function of the void fractionand confinement scale. We found that the stretchingexponent Æræ describing the dynamics of nanoparticlesin post arrays became smaller as the spacing betweenthe posts was decreased (that is, at smaller void frac-tion θ and larger confinement parameter ζ collapsedon to a single scaling curve with θ and ζ irrespective ofparticle size) and followed a roughly linear relationshipwith q and ζ (Figure 6). These results indicate thatconfinementmodifies the distribution of the relaxationprocesses within the sample. A detailed descriptionof the mathematical consequences of the relaxationdescribed by a stretched exponential functional form35

is given in the Supporting Information.Analysis of trajectories of single particles revealed

a more direct manifestation of the distribution of

Figure 4. Mean squared displacement ÆΔx2(Δt)æ as a func-tion of delay time Δt for 400 nm nanoparticles diffusingfreely (black square) and in post arrays with S = 4 μm, ζ =0.105 (orange diamond); S = 1.8 μm, ζ = 0.237 (olive righttriangle); and S = 1.2 μm, ζ = 0.354 (navy star). ÆΔx2(Δt)æscales linearly asΔt across the range of delay times probed.

Figure 5. Relative diffusivity D/D0 as a function of (a) void fraction θ and (b) confinement parameter ζ (=dNP/S), measured byDDM (filled symbols) and single particle tracking (open symbols), for aqueous dispersions of nanoparticles of diameter of400 (black square), 300 (red circle), and 200 nm (violet triangle). The relative diffusivity scales linearly with θ and ζ.

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relaxation processes through the one-dimensionalprobability distribution of displacements Gs(Δx,Δt),shown for representative distributions at three differ-ent delay times (Δt = 0.183, 0.517, and 0.85 s) and threedifferent confinements in Figure 7 and in Figure S7 inthe Supporting Information. For dilute uncorrelatednanoparticles in an unconfined and Newtonian medi-um performing a random walk, Einstein's analysisof Brownian motion indicates that the displacementsof particles should follow a Gaussian distribution.36 Wefound, however, that the distributions of displacementfor the 400 nmdiameter nanoparticles diffusing in postarrays deviated from a simple Gaussian, as expectedfor ordinary diffusion. We additionally found thatGs(Δx,Δt) did not simply scale as a stretched Gaussiandecay (i.e., Gs(Δx,Δt) ∼ exp[�(|Δx|/γ(Δt))β]). Instead,Gs(Δx,Δt) was best fit to a sum of a stretched Gaussian(for long displacements) and a simple Gaussian model(for short displacements):

Gs(Δx,Δt) ¼ C1 exp � j Δx jγ(Δt)

� �β" #

þ C2 exp � Δx

λ

� �2" #

(4)

where C1 and C2 are prefactors and γ(Δt) and λare the decay lengths for stretched Gaussian andsimple Gaussian, respectively. We found that the sim-ple Gaussian (G) exhibited a nearly constant width(Figure S8 in Supporting Information). Moreover, tominimize the number of parameters used to fit theone-dimensional probability distribution we notedthat the best fits were obtained for all values of Δtwith a stretching exponent β that was ∼2Æræ. Wetherefore fixed the Gaussian stretching exponent β at2Æræ and the decay length λ of the Gaussian thatdescribed the short displacement data at 0.48 μmand thereby systematically fitted the data to a singleparameter model and obtained the delay time depen-dent decay length γ(Δt). We found that this protocolallowed us to obtain good fits to Gs(Δx,Δt) acrossthe range of delay times probed in the experimentand moreover allowed us to resolve relatively smallchanges in the value of Æræ, as shown in Figure S8 in theSupporting Information.

The decay length γ(Δt) extracted from the fits toeq 4 increased as the square root of the delay time,i.e., γ(Δt)∼(Δt)1/2, as shown in Figure 8. We note thatthe stretched Gaussian can be interpreted as a sum ofGaussian distributions, with a distribution of relaxationtimes and processes,35,37 with the reported diffusioncoefficient an averaged quantity over this distribution

Figure 6. Stretching exponent Æræ as a function of (a) void fraction θ and (b) confinement parameter ζ (=dNP/S), measured byDDM for aqueous dispersions of nanoparticles of diameter of 400 (black squares), 300 (red circles), and 200 nm (violet triangles).

Figure 7. Probability distribution of particle displacementsGs(Δx,Δt) for 400 nm particles diffusing freely and in postarrays with S = 8 μm, ζ = 0.051 and S = 1.2 μm, ζ = 0.354 atdelay times Δt of (a) 0.183 s, (b) 0.517 s, and (c) 0.85 s,respectively. Lines indicate fits to eq 4.

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(as shown explicitly in the Supporting Information).The decay length scale γ(Δt) thus represents a meancharacteristic length scale that is an average of thelength scales associated with each of the relaxationprocesses that contribute to the stretched exponentialdistribution of displacements. The observed delay timescaling of γ(Δt) suggests that each relaxation processis diffusive in origin, with the distribution of relaxa-tion times reflecting the heterogeneity in the environ-ment,38,39 perhaps due to proximity of the posts. Thisargument is consistent with the diffusive characterof the average dynamics of the nanoparticles, exem-plified by the linear scaling of the MSD with delaytime and by the q�2 dependence of the relaxation timeobtained from the image structure function. Moreover,the averaged relative decay length Æγ(Δt)/γ0(Δt)æ,where γ0(Δt) corresponds to the decay length forthe free diffusion case and the averaging is over thedelay times studied here, scaled nearly linearly withvoid fraction and confinement parameter (Figure 9), asalso found for the relative diffusivity in Figure 5. Finally,we observe that the magnitude of the decrease in theaveraged relative decay length is quantitatively similarto that in the relative diffusivity.

The slowed dynamics of the nanoparticles, thestretched exponential behavior of the relaxation pro-cess, the linear decrease of the stretching exponent,and the linear decrease of the averaged relative decaylength with increased confinement are manifestationsof the impact of the cylindrical post arrays on thedynamics of the nanoparticles. The decrease in diffu-sivity follows linearly with the increase in confinement,and thus, confinement effects alone are sufficientto explain the slowing of dynamics. For even thesmallest post spacing (∼1.2 μm) and the largest parti-cles (400 nmdiameter) studied, however, the dynamicsremained Fickian in nature. The observation of Fickianyet anomalous diffusion is similar to that of Granick andco-workers for two classes of systems: colloidal parti-cles diffusing along linear phospholipid bilayer tubeswith radii similar to that of the colloidal particles, andcolloidal beads diffusing in entangled fiber networks.38

Granick and co-workers attributed non-Gaussian yetFickian dynamics in these two systems to slowly vary-ing fluctuations in a heterogeneous environment.39

In the systems described in this paper, however,we cannot categorically determine the origins of theanomalously stretched dynamics. We expect that bothheterogeneous environments (i.e., varying proximity ofthe nanoparticles to the nanoposts arranged on asquare lattice), on the scale of a few particle diameters,and caging-induced confinement and approach tovitrification, on the scale of the particle size, give riseto these dynamics. While the current work provides asystematic study of the changes in the relaxationspectra induced by confinement heterogeneity andby the approach to a confinement-induced vitrifica-tion, it does not allow us to distinguish between thesetwo phenomena.

CONCLUSIONS

Our measurements, summarized in Figures 5�8,revealed two characteristic features of nanoparticlediffusion in post arrays: slowing of dynamics, shownby the decrease in the average diffusion coefficient,and the emergence of multiple relaxation processes

Figure 8. Decay length γ(Δt) as a function of the delay timefor 400 nm nanoparticles diffusing freely (black square) andinpostarrayswithS=8μm,ζ=0.051 (redup triangle),S=4μm,ζ = 0.105 (orange diamond), S = 1.8 μm, ζ = 0.237 (olive righttriangle) and S = 1.2 μm, ζ = 0.354 (navy star). γ(Δt) scales asΔt0.5. The error bars are smaller than the symbols.

Figure 9. Averaged relative decay length Æγ(Δt)/γ0(Δt)æ as a function of (a) void fraction θ and (b) confinement parameterζ(=dNP/S), measured by single particle tracking for aqueous dispersions of nanoparticles of diameter of 400 nm. The relativedecay length scales linearly with θ and ζ.

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with a spectrum of relaxation times, shown by thedecrease in the stretching exponent. These featuressuggest an intriguing connection between confineddynamics of nanoparticles and their relationship to thesystematic variation in confinement heterogeneityand to confinement-induced vitrification.29 In ref 29,the anomalous and non-Gaussian dynamics of denseliquids confined in porous media could be describedby mode-coupling theory, which is commonly used todescribe the slowing of dynamics near the glass transi-tion. In this interpretation, the porousmedium acts as afrozen host structure, and the anomalous slowed and

stretched dynamics reflect the interplay betweenthis underlying spatial structure and an approachto vitrification.29 In our experiments, in which theconfinement was relatively modest, the diffusive dy-namics became slower (by ∼25%) and the stretchingexponent decreased (to 0.7). We expect that similarmeasurements in more extreme confinements, aswell as those in which particle�surface interactionsor pore connectivity are varied, will distinguish be-tween heterogeneity and vitrification and therebyprovide unique insight into the changing nature ofthe dynamics of nanoparticles in complex media.

METHODSNanoparticle Dispersions. Fluo-max dyed red aqueous fluores-

cent polystyrene particles with diameter (dNP) of 200, 300,and 400 nm were purchased from Thermo Fisher ScientificCDD. The as-received dispersions (with 1 wt % of nanoparticles)were diluted with deuterium oxide (Sigma-Aldrich) to a volumefraction of j = 1 � 10�4, corresponding to number densitiesof 2.2� 1010 mL�1, 6.7� 109 mL�1, and 2.8� 109 mL�1 for 200,300, and 400 nm nanoparticles, respectively. A detailed descrip-tion of thesematerials and the sample preparation protocol canbe found in ref 30.

Fabrication and Characterization of Cylindrical Nanopost Arrays. Tosystematically explore the effect of confinement by nanoposts,we fabricated individual arrays of posts of area of 250� 250 μm2

in which the post spacing was held constant. Rectangularcylindrical post arrays of constant diameter (dp) 500 nmseparated by spacing (S) of 1.2 to 10 μm, as shown in Figure 1,were prepared on silicon wafers using methods developedpreviously.28 Briefly, ZEP520A photoresist was coated on a4 in. p-type silicon wafer Æ100æ at 6000 rpm for 45 s and bakedat 180 �C for 2 min. Square arrays with post patterns wereproduced by electron beam lithography (JBX-9300 FS). The postpatterns were then developed in xylene for 30 s, rinsed withisopropyl alcohol (IPA), and dried with N2. A 15 nm layer of Crwas deposited using a dual gun electron beam evaporationvacuum chamber (Thermionics, Port Townshend, WA) at themetal evaporation rate of 1 Å/s. Photoresist lift off (by sonicationof the wafer in acetone for 2 min) was performed to ensure theremoval of the resist and the majority of the Cr film. Cylindricalposts of 10 μm height were produced using cryogenic siliconetching by an Oxford Plasmalab system 100 (etched by SF6 andO2. Plasma at�110 �C, with an optimized flow rate of 18 sccmofO2) for 4 min. Finally, a uniform 10 nm thick SiO2 layer wasdeposited by atomic layer deposition. This fabrication processresulted in a hydroxyl coated silica posts arranged in a regulararray that did not offer strong specific interactions with the

polystyrene nanoparticles that had a small extent of carboxylatefunctionality to help disperse in water.

The geometrical characteristics of the post array (i.e., thepost spacing and height) were measured by scanning electronmicroscopy and are summarized in Table 1. We additionallyreport two metrics that parametrize the confinement experi-enced by the nanoparticles. First, we calculated the area-basedvoid fraction θ for each of the dilute nanoparticle dispersions as

θ ¼(Sþdp)

2 � π

4(dpþdNP)

2

(Sþdp)2 (5)

where dp is the diameter of the post, dNP is the diameter ofthe nanoparticles, and S is the spacing between posts. Thesecond term in eq 5 accounted for the area within half a particlediameter of each post that was inaccessible to the centers of thenanoparticles. Second, we calculated a confinement parameterζ = dNP/S, the ratio of the nanoparticle diameter to the spacingbetween posts, to capture the topological confinement ofthe particles in the array. A value of 0 for ζ corresponds to theunconfined case, while a value of 1 corresponds to completeconfinement.

Experimental Procedure for Study of Nanoparticle Diffusion in PostArrays. Dispersions of nanoparticles were introduced into sili-con-based microchannel arrays using a pipet. A rectangularcover glass with dimensions of 48 � 65 mm2 (thickness 0.13�0.17mm,Gold Seal) and an epoxy-based adhesive (Devcon)wasused to seal the microchannel and thereby form a hermeticsystem. Nanoparticles diffusing in the sealed systems wereimaged on a Leica DM4000 inverted microscope with a 100�oil immersion objective (HCX PL APO, numerical apertureof 1.40) and a pixel size of 0.195 ( 0.002 μm/pixel using a highspeed AOS Camera (AOS Technologies AG). We used reflectedlight to precisely locate each rectangular post array that waspatterned on the microchannel, as the opacity of the siliconwafer precluded the use of transmitted light. We then switched

TABLE 1. Post Spacing, Height, and Void Fraction for Nanoparticles Diffusing in Different Post Arrays Described in This

Paper

void fraction θ confinement ζ = dNP/S

designed spacing (μm) measured spacing (μm) height (μm) 200 nm 300 nm 400 nm 200 nm 300 nm 400 nm

10 9.79 ( 0.05 10.0 ( 0.1 0.996 0.995 0.993 0.021 0.031 0.0418 7.79 ( 0.06 10.3 ( 0.1 0.994 0.993 0.990 0.026 0.039 0.0516 5.86 ( 0.03 10.0 ( 0.1 0.990 0.988 0.984 0.034 0.051 0.0684 3.82 ( 0.04 10.2 ( 0.1 0.979 0.973 0.965 0.052 0.079 0.1052 1.93 ( 0.06 11.6 ( 0.1 0.935 0.915 0.891 0.104 0.155 0.2071.8 1.69 ( 0.07 11.8 ( 0.1 0.920 0.895 0.867 0.118 0.178 0.2371.6 1.48 ( 0.15 11.9 ( 0.1 0.901 0.871 0.836 0.135 0.203 0.2701.2 1.13 ( 0.10 12.1 ( 0.1 0.855 0.811 0.760 0.177 0.265 0.354

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to fluorescence mode to image the nanoparticle dispersions.In the microscopy experiments, we collected 4200 images overan area of 124.8 � 93.6 μm2 (corresponding to 640 pixels �480 pixels) at a frame rate of 63 (for 200 nm diameternanoparticles) or 120 (300 and 400 diameter nanoparticles)frames per second (fps). This imaging protocol allowed us toaccess delay-time scales ranging from 0.0083 to 35 s (at 120 fps)for the 200 diameter NPs and 0.0158 to 67 s (at 63 fps) for thelarger NPs. Because the field of view was smaller than the postarray and care was taken to center the field of view with that ofthe array, edge effects were precluded from our data collectionand analysis.

Image Analysis. To measure the diffusive dynamics of thenanoparticles in cylindrical post arrays with varied postspacing, we implemented a differential dynamic microscopy(DDM) algorithm to analyze the fluctuations in light intensityin a time series of fluorescence micrographs.40,41 We previouslyshowed30 that fluorescence- and brightfield-based DDM pro-vided equivalent data to those obtained from dynamic lightscattering and enabled measurements over a broader concen-tration range of scatterers and over a broader range of themagnitude of the scattering vector (q). Briefly, we calculated theazimuthally averaged image structure function (ISF) D(q,Δt)from the Fourier analysis of a delay-time series of differenceimages. The Fourier transforms of the difference images werenearly isotropic for all delay times studied in this work, therebyvalidating the use of an azimuthally averaged D(q,Δt) anda single particle diffusion coefficient (Supporting Information,Figure S1). We fitted the structure factor to a generalizedmodel,D(q,Δt) = A(q)[1 � f(q,Δt)] þ B(q), and extracted a q-dependentrelaxation time τ(q) from the intermediate scattering functionf(q,Δt). Nonlinear least-squares fitting was performed using theLevenberg�Marquardt algorithm, as implemented in Origin(OriginLab, Northampton, MA). We calculated the particle dif-fusivity as Dm = 1/τ(q)q2 from the slope of τ(q) versus q2.

To verify our DDMmeasurements, we used a single particle-tracking (SPT) algorithm to track the 400 nm diameter nano-particles as they diffused in the post arrays.42 Briefly, we locatedthe particle centroids to within 40 nm and linked the particlepositions into trajectories. From the trajectories, we calculatedthe one-dimensional ensemble-averaged mean squared displa-cement (MSD)ÆΔx2(Δt)æ = Æ(x(tþΔt)�x(t))2æ of the particles overtime, where the brackets indicate an average over all particles inthe system, and the probability distribution of particle displace-ments Gs(Δx,Δt).

Conflict of Interest: The authors declare no competingfinancial interest.

Acknowledgment. This publication is based on work sup-ported in part by Award No. KUS-C1-018-02, made by KingAbdullah University of Science and Technology (KAUST). R.K.and K.H. acknowledge the partial support of the Gulf of MexicoResearch Initiative (Consortium for Ocean Leadership Grant SA12-05/GoMRI-002). J.C.C. is supported by the American Chemi-cal Society Petroleum Research Fund (52537-DNI7) and theNational Science Foundation (DMR-1151133). A portion of thisresearch was conducted at the Center for Nanophase MaterialsSciences, which is sponsored at Oak Ridge National Laboratoryby the Scientific User Facilities Division, Office of Basic EnergySciences, U.S. Department of Energy.

Supporting Information Available: Diffusion data for 200 and300 nm particles in the post arrays and a mathematical descrip-tion of the consequences of the stretched exponential modelare presented. This material is available free of charge via theInternet at http://pubs.acs.org.

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