Evolution of a Colloidal Critical State in an Optical Pinning Potential Landscape

Pamela T. Korda,1 Gabriel C. Spalding,2 and David G. Grier1

1Dept. of Physics, James Franck Institute, and Institute for Biophysical DynamicsThe University of Chicago, Chicago, IL 60637

2Dept. of Physics, Illinois Wesleyan University, Bloomington, IL 61702(Dated: March 28, 2002)

As a step toward isolating the influence of a modulated substrate potential on dynamics and phasetransitions in two dimensions, we have studied the behavior of a monolayer of colloidal spheres drivenby hydrodynamic forces onto a large array of holographic optical tweezers. These optical traps con-stitute an effective substrate potential whose symmetry, separation and depth of modulation all canbe varied independently. We describe a particular set of experiments in which a colloidal monolayerinvades the optical pinning potential much as magnetic flux lines invade type-II superconductors,including cooperative avalanches, streaming motion and a symmetry-altering depinning transition.The kinetically hindered intermediate state in this process resembles the critical state long associ-ated with flux entering zero-field-cooled superconductors. By tracking the particles’ motions, weare able to determine the microscopic processes responsible for the evolution of the colloidal criticalstate and compare these with recent simulations of flux line dynamics.

In the absence of an externally imposed potential en-ergy landscape, the behavior of two-dimensional systemsis wholly determined by interactions among the con-stituent particles. Examples of such systems includeelectrons on the surface of liquid helium, vortices inclean type-II superconductors, and colloidal monolayers.Colloidal monolayers in particular have been studied asmodel systems whose phase behavior offers insights intothe general mechanisms of structural phase transitions inreduced dimensionality1.

In contrast, the majority of two-dimensional systemsare strongly influenced by their constituent particles’ in-teractions with substrate potentials. Adatoms adsorbedon crystal surfaces2, magnetic flux lines pinned in de-fected and patterned3–8 type-II superconductors, atomsimbibed into graphite intercalation compounds9, andcharge density waves10 all reflect such an influence withthe appearance of new thermodynamic, dynamic, and ki-netic phases and states.

Understanding how substrates modify two-dimensionalphase behavior is complicated in experimental studiesby the comparative difficulty of quantifying and control-ling substrates’ properties. However, colloids’ interactionwith easily controlled external forces offers some possi-bilities. For example, a physically textured surface withmicrometer-scale features can influence the free energy ofa colloidal overlayer, either through electrostatic interac-tions or through entropic depletion attractions mediatedby small dispersed particles11.

Alternatively, a pattern of light applied to the sam-ple can create a modulated potential energy landscapewith which the colloid can interact12–15. We have uti-lized this approach by applying the holographic opticaltweezer technique16–18 to create tailored optical potentiallandscapes for colloidal particles. This Article describesexperimental observations of convection-driven colloidaltransport into a large, initially empty array of discreteholographic optical traps. The resulting behavior is rem-iniscent of magnetic flux lines’ invasion of a zero-field-

cooled type-II superconductor.Each potential well in our controlled potential en-

ergy landscape is a discrete optical tweezer19, a three-dimensional optical trap for dielectric particles formedwhen an intense beam of light is brought to a diffraction-limited focus by a high numerical aperture lens. Ratherthan using N separate beams to produce N tweezers,we use a computer-designed diffractive optical element(DOE)18 to modulate the wavefront of a single beam sothat it reproduces the interference pattern of N beams allpassing through the DOE’s plane. A telescope projectsthe modulated wavefront onto the back aperture of a100× NA 1.4 S-Plan Apo oil immersion objective lenswhich then focuses the light into an array of optical trapsin its focal plane16,18. Figure 1 schematically depicts theoptical train used in this study.

Diffractive

ElementOptical

TweezersHolographic Optical

Sample VolumeIlluminator

Laser

Video Camera

Telescope

Dichroic Mirror

Objective

10 mµ

FIG. 1: Schematic diagram of holographic optical tweezer sys-tem projecting a 16× 20 array of optical traps into a samplevolume containing a colloidal monolayer. The inset is a videomicrograph of light from the optical tweezers reflected by thelower glass-water interface. The superimposed rectangle indi-cates the array’s domain in subsequent figures.

2

Applying such an array of optical traps to a low-densitycolloidal suspension generally causes particles in the im-mediate neighborhood of traps to become immobilized,and leaves those farther from the array unaffected. Inthis sense, each trap is analogous to a pinning centerfor flux lines in a type-II superconductor, and thus theholographic optical tweezer array can be considered an“optical pinscape” for colloidal particles.

The samples used in this study consist of silica spheresof radius a = 0.75 µm (Bangs Labs #4258) suspended indeionized water with a total ionic concentration aroundn = 10−6 M. The suspension is confined between twoglass plates separated by 40 µm. All parts are cleanedstringently before assembly and flushed copiously withdeionized water afterwards to minimize contaminationby stray ions. Experiments are performed at an ambi-ent temperature of T = 295 ± 1 K. The sample cell ismounted on the stage of an inverted optical microscopeand imaged in bright field onto an attached CCD videocamera. The objective lens of the microscope both formsthe array of optical tweezers and creates images of thecolloid. Images of spheres in a 70× 55 µm2 field of viewnear the center of the sample volume are recorded be-fore being digitized and analyzed with precision particletracking algorithms20. Using these methods, we locateindividual spheres’ centroids to within 20 nm in the planeat 0.5 sec intervals.

Because silica’s density of 2.2 g/cm3 greatly exceedsthat of water, the spheres sediment to the bottom of thesample volume, forming an essentially two-dimensionallayer of areal density 3×10−3 µm−2. Gravity is opposedby the charged spheres’ electrostatic repulsion from thesimilarly charged lower glass-water interface. Dissocia-tion of terminal silanol groups endows each sphere withan effective surface charge number of roughly Z = 5000electron equivalents21,22 and imbues the glass wall witha comparable surface charge density of −2000 e0 µm−2,where e0 is the elementary charge22. The monolayerachieves equilibrium at a center-to-surface height ofabout h = 0.9 µm, as determined by optical microscopy.

To such monolayers, we apply an optical pinscape com-posed of a 20 × 16 square array of traps, with latticespacing 1.8 µm. The inset to Fig. 1 shows light reflectedwhen the tweezer array is focused on the lower glass-water interface at low intensity. For the experiments,the array is focused 0.9 µm above the lower glass walland powered by 1.75 W of laser light at 532 nm froma diode-pumped frequency-doubled Nd:YVO4 laser (Co-herent Verdi). Each trap in this array is capable of local-izing a single sphere in three dimensions against randomthermal forces without appreciably changing the sphere’sheight above the wall.

Light from the trap array also impinges on a smallregion of gold film evaporated onto the upper glasswall. The resulting inhomogeneous local heating drivesa toroidal convection roll which spreads out along theupper wall and returns along the bottom. This flow ad-vects spheres along the bottom of the sample cell to-

ward the array of traps from a region extending for hun-dreds of micrometers in all directions, and at speeds inthe range of v = 7.9 ± 0.1 µm/sec in the field of view.No-flow boundary conditions minimize the flow’s out-of-plane component at the lower wall so that the convection-driven spheres remain sedimented into a monolayer. Ad-vection towards the pinscape only occurs when the op-tical traps are illuminated and ceases immediately oncethey are extinguished.

Before the laser is turned on, the sample is uniformlydilute, with only a few particles in the field of view at anytime. When the pinscape is activated and the toroidalflow is established, spheres flow rapidly toward the illu-minated region, dramatically increasing the local arealdensity. The first arrivals occupy the perimeter of thetweezer array, impeding flow into the interior. Additionalspheres collect outside of the array, forming a domain oftriangular crystal with a nearest neighbor separation of2.5±0.1 µm. This can be seen in Fig. 2, where the regionoccupied by the optical pinscape is outlined in white.

The exterior triangular crystal is stabilized by the samehydrodynamic pressure which drives spheres towards thepinscape. Turning off the laser at this point leaves theexterior crystal in an unstable superheated state whichmelts immediately. This melting process shows no signof the anomalous long range attractions23,24 which havebeen observed in colloidal interaction measurements onconfined polystyrene spheres24–27. Instead, it is consis-tent with recent measurements of screened-Coulomb re-pulsions among similar colloidal silica spheres near a sin-gle glass wall under comparable conditions21. We ex-pect, therefore, that the spheres in the present experi-ment also repel each other according to the conventionalDLVO theory21. Thus the electrostatic force between twospheres at center-to-center separation r has the form28

F (r) =Z2e2

0

ε

[exp(−κa)

1 + κa

]2 (1r

+ κ

)exp(−κr)

r, (1)

where the Debye screening length κ−1 is set bythe concentration n of dissolved ions through κ2 =4πe2

0n/(εkBT ) in an electrolyte of dielectric constant ε.In this case, the Stokes drag29

Fh =6πηa v

1 + 916

ah + 1

8 ( ah )3 − 45256 ( ah )4 − 1

16 ( ah )5(2)

= 0.3 pN

exerted on each stationary sphere at the edge of the crys-tal by the flowing water (η = 1 cP) is balanced by anearest-neighbor electrostatic repulsion. The observednearest-neighbor separation of r = 2.5 µm at the sur-face of the crystal therefore is consistent with a screen-ing length of κ−1 ≈ 200 nm, and thus with the ionicstrength of n ≈ 10−6 M. These values agree with di-rect interaction measurements on comparably preparedsuspensions20,21,30. Similarly, the optical tweezers’ abil-ity to trap spheres at least marginally against both hydro-dynamic and nearest-neighbor forces reveals their maxi-mum trapping force to be roughly 0.6 pN.

3

Figure 2 shows how a typical colloidal monolayerevolves under the combined influence of in-plane hy-drodynamic pressure, particle-particle interactions, andthe optical pinscape’s static trapping potential. Ratherthan flowing continuously into the array of traps, mostspheres remain pinned near the edge for long periodsof time, as in Fig. 2(a). This kinetically hindered con-figuration closely resembles the critical state in type-IIsuperconductors31,32. In that state, magnetic flux linesbecome immobilized on defects in the superconductorand are forced by nearest-neighbor and body forces toestablish density gradients.

Spheres on the periphery of the trap array popu-late its interior via two distinct mechanisms: thermally-activated single-particle hopping and punctuated burstsof collective rearrangements reminiscent of avalanches ingranular materials, vortex matter, and other jammed,pinned or otherwise kinetically hindered systems33. Al-though the individual spheres’ diameters are smaller thanthe separation between optical tweezers, their electro-static repulsion prevents them from filling every trap inthe array. Hence, the spheres occupy every other latticesite, forming a

√2×√

2 superlattice rotated at 45◦ withrespect to the traps’ axes. Figures 2(a) and (b) show thesystem in this state after 273 sec and 363 sec of illumi-nation, respectively. Comparable superlattice structureshave been observed for flux lines occupying square arraysof magnetic pinning centers patterned onto conventionalsuperconductors34.

Once the superlattice is complete, subsequent fillingcauses the monolayer to depin from the trap array, atwhich point it undergoes a martensitic transition to afloating triangular crystal, as shown in Fig. 2(c). Thisfinal observation is somewhat surprising. If combinedhydrodynamic and nearest-neighbor forces can depin in-dividual spheres, why does the monolayer invade as acommensurate superlattice rather than as an incommen-surate triangular array? While the particles’ measuredtrajectories tell the whole story, three characteristics oftheir collective structure and cooperative motions high-light more generally relevant trends. In particular, weconsider the local areal density ρ(t), the particles’ meanspeed 〈v(t)〉, and the m-fold bond-orientational orderparameters1 whose evolution in time appears in Fig. 3.

We measure the local areal density by calculating thearea of each particle’s nearest neighborhood or Wigner-Seitz cell35. These polygonal areas can be combined todefine regional areal densities inside and outside the do-main of the pinscape, as shown in Fig. 3(a). Spheres onthe outer edge of the cluster, whose local density is notwell defined, are not considered in computing the outerareal density. The outer areal density remains essentiallyconstant even as spheres move into the pinscape and ad-ditional spheres arrive at the crystal’s outer edges. Thisobservation supports the conjecture that the toroidal flowestablishes an effectively constant pressure at the mono-layer’s edges.

Like ρ(t), the particles’ mean speed 〈v(t)〉 gauges the

a)

b)

c)

10µm

FIG. 2: Colloid invades the pinscape. (a) t = 273 sec afterthe tweezer array was activated, many spheres have assembledoutside the pinscape. Spheres pinned at the edges block accessto the interior, although some have penetrated by means ofthermal hopping. (b) t = 363 sec: The array is mostly filled,through single-particle hopping and multi-particle avalanches.Spheres inside the pinscape exhibit four-fold order. (c) t =635 sec: The particles have depinned and the monolayer hasundergone a transition from four-fold to six-fold order. In allimages, the white box indicates the region occupied by thetweezer array.

influx of spheres into the pinscape. It also captures localrearrangements which do not affect the density. Peaksin 〈v(t)〉 are analogous to bursts of electrical activity ob-served in studies of superconducting vortex avalanches36,and indicate collective motion of a significant fraction ofthe particles inside the pinscape. Figure 3(b) shows themean speed for particles within the pinscape averaged

4

over 1 sec intervals.To quantify how order evolves as spheres invade,

we compute the local m-fold bond orientational orderparameters1

ψm(~rj) =1Nj

Nj∑k=1

exp(im θjk), (3)

for m = 6 and 8, where ~rj is the location of particle j,whose Nj nearest neighbors, labelled by k, are arrayed atangles θjk with respect to a reference axis. Each ψm(~rj)achieves its maximum magnitude of unity if the neigh-borhood around particle j is perfectly m-fold ordered.The mean-squared magnitude

Ψm(t) =1

N(t)

N(t)∑j=1

|ψm(~rj(t))|2 (4)

for the N(t) particles within the pinscape’s domain attime t measure the degree of m-fold order displayed bythe system. While Ψ6(t) measures six-fold order accu-rately, Ψ8(t) is preferable to Ψ4(t) for measuring four-foldorder because of the former’s ability to account for diag-onal nearest-neighbor bonds in triangulations of squarelattices. These order parameters appear in Fig. 3(c).

Taken together, the data in Fig. 3 reveal that thesystem’s evolution proceeds in three phases. At first(0 < t < 276 sec), the array fills slowly, as indicatedby the gradually-increasing inner density in Fig. 3(a).Nonetheless, the mean particle speed is high because ofparticles arriving at the pinscape’s periphery from allsides. As these accessible sites become full, the spheres’inward motion becomes blocked and 〈v(t)〉 decreases.During this period, the pinscape is mostly empty, so thatΨ6(t) and Ψ8(t) are consistent with a random distribu-tion of points.

A large avalanche beginning at t = 250 sec initiatesthe invasion’s second stage. Between t = 276 sec andt = 300 sec, the number of spheres inside the pinscapeincreases sharply. This is indicated by a sharp rise inρ(t) and a corresponding peak in 〈v(t)〉. The sudden in-crease in density leads to a rapid growth in both Ψ6(t)and Ψ8(t). Almost as soon as the avalanche ends, how-ever, Ψ6(t) declines while Ψ8(t) continues to rise. Theinvading triangular crystal actually anneals into four-folddomains of the

√2×√

2 superlattice. This state is shownin Fig. 2(b). Even more surprising is the observation thatthe mean density within the pinscape actually decreasesduring this period.

Another avalanche at t = 380 sec ends the growth infour-fold order and ushers in the final phase of the in-vasion. As the density once again begins to increase,the monolayer disengages from the array of optical trapsand reorganizes itself into a six-fold-ordered triangularcrystal. This process requires collective rearrangementsof particles as well as plastic and elastic distortion of thesurrounding unpinned triangular crystal. From Fig. 3(b),

4−foldOrder

Transition

0.12

0.14

0.16

0.18

0.20

0.22a)

Inner Density

Outer Density

Array mostly empty

0.80

0.60

0.50

0.40

b)

0.70

0.20

0.40

0.60

0.80

1.00

0.000 100 200 300 400 500 600

c)

5(a)

5(b)

5(c)

ρ(µm−

2)

〈v(t)〉(µm/sec)

t (sec)

Ψn(t)

Ψ6

Ψ8

FIG. 3: (a) Areal density of spheres inside and outside the pin-scape. Squares indicate times at which the images in Fig. 2were obtained. (b) Average speed of particles within thepinscape over 1 second intervals. Arrows near the primaryavalanche indicate the periods over which the trajectory datain Fig. 5 were obtained. (c) Mean-square magnitudes of thesix- and eight-fold bond orientational order parameters forparticles within the pinscape.

we can see that these rearrangements occur through a se-ries of avalanches, each signalled by a peak in 〈v(t)〉. Themonolayer’s evolution ends with the pinscape completelyfull of triangular crystal as shown in Fig. 2(c) so that noroom is left for further incursions. The interior densityeventually exceeds the outer density, presumably becauseof flow-induced body forces directing spheres inward.

Having used ensemble-averaged measures to establishthe sequence of events by which spheres first occupy andthen disengage from the optical trap array, we can seekexplanations for these events in the microscopic trajec-tories of individual spheres. In particular, we would liketo understand why spheres initially invade the pinscapeas a

√2×√

2 superlattice if they reach equilibrium as afloating triangular crystal. Examining the particles’ mo-

5

tion during the three phases of the invasion sheds lighton the matter.

Particle trajectories20 from the invasion’s early phasereveal that spheres enter the array’s interior by meansof single-particle hops. Figure 4 shows the well-localizedtrajectories of 14 particles near the edge of the opticaltweezer array obtained over 1 sec. Grid crossings inFig. 4 indicate individual traps’ positions. Three parti-cles to the left of the trap array are part of the unpinnedcrystalline reservoir and fluctuate about their equilibriumlattice positions more vigorously than do their neighborspinned on optical traps. Of the pinned particles, only theones labelled A and B are not centered on their tweez-ers. These two spheres are most closely associated withtraps which are not part of the

√2×√

2 domain. If theywere actually centered on these traps, the misfit sphereswould be far closer to their neighbors than other spheresin the domain. Strong nearest-neighbor repulsions dis-place them from these minima. The barrier to hoppingis thus reduced, and random thermal fluctuations even-tually conspire with steady hydrodynamic drag to forcesphere A into the nearest trap in the superlattice domain.Similar processes are observed in simulations of vortexdynamics in periodically pinned superconductors37.

B

A

mµ2

FIG. 4: Trajectories of individual colloidal particles localizedin optical traps and in the crystal outside the trap array mea-sured at 1

30sec intervals over 1 sec. Particle A can be seen

hopping from a site misaligned with the√

2×√

2 domain toa commensurate site. The optical pinscape represented bycrossings in the grid lines lies to the right of the dashed line.

Having observed individual spheres hopping preferen-tially onto optical tweezers at commensurate

√2 ×√

2superlattice sites, we are in a position to explain the tran-sient growth in Ψ8(t) and the corresponding decrease inρ(t) after the principal avalanche at t = 280 sec. Again,insight is gained by looking at the particle tracks. Figure5 shows short trajectories over the entire field of view justbefore, during, and after this avalanche. Despite the ini-tial increase in 〈v(t)〉 at t = 250 sec, the pinned domain’sdensity does not begin to increase until roughly 20 sec-onds later. Figure 5(a) reveals that most particle motionsin this early stage involve collective rearrangements in thesurrounding triangular crystal, with only small streams

of particles filing into the array. Streaming through pin-ning arrays also has been identified in simulations38,39

and time-resolved imaging40 of relaxation in the super-conducting critical state.

The main part of the avalanche, shown in Fig. 5(b),involves highly cooperative translation of an entire trian-gular domain into the pinscape. Within 10 seconds, how-ever, most of the pinned crystal has recovered its four-fold superlattice structure, as shown in Fig. 5(c), largelythrough single-particle rearrangements of the kind shownin Fig. 4. Even if avalanches tend to force dense triangu-lar crystal into the array, crowded spheres can advanceinto less dense and energetically more favorable superlat-tice configurations.

The resulting superlattice crystal consists of two in-compatible domains. These compete for the pinscape bydisplacing spheres from their traps at the domain bound-aries. Progress in domain coarsening involves pushingdisplaced spheres out of the pinned crystal and back intothe surrounding reservoir of triangular crystal. This ex-plains the transient decline in the pinned crystal’s densityafter the avalanche.

Eventually, avalanches of spheres drive enough addi-tional incommensurate crystal into the pinned domain’sedges that the interior, now almost completely occupied,can no longer accommodate further single-particle relax-ation events. The advance of six-fold order then proceedsthrough local shearing rearrangements similar to thosewhich characterize the one-layer to two-layer martensitictransition in colloid confined between parallel repulsivewalls41. Locally shearing the square domains into trian-gular domains leads to the formation of misaligned trian-gular grains, as can be seen in Fig. 2(c). Once depinned,the incommensurate six-fold monolayer increases its den-sity by annealing away the resulting grain boundaries, avery slow process involving rare cooperative motions oflarge numbers of spheres.

The same sequence of events was obtained repro-ducibly for separate runs of this experiment. The pat-tern of avalanches which first produces commensuratesquare superlattice coverages and eventually drives asymmetry-altering depinning transition proceeded in allcases through the same microscopic mechanisms observedfor the particular run described above.

Qualitatively similar flux transport by hopping,streaming and avalanches has been observed directly inthin superconducting films through Lorentz microscopy40

and in microfabricated superconducting arrays by scan-ning probe microscopy37. Such mechanisms alsohave been inferred from transport measurements onconventional36,42 and high-temperature superconductors.Order-order transitions similar to the one observed inthis experiment have been inferred indirectly from trans-port measurements on patterned superconductors43–46

and Josephson junction arrays47.This study constitutes the first demonstration of holo-

graphic optical tweezers’ utility for studying cooperativephenomena in strongly coupled systems. As a model

6

a)

b)

c)

10 µm

FIG. 5: Trajectories of colloidal particles over 3 second inter-vals. Filled dots indicate the particles’ final positions. Rect-angles indicate the domain of the optical trap array. (a) Tra-jectories from the early stage of the principal avalanche. Mostparticles inside the pinscape are organized into four-fold do-mains. (b) The avalanche drives particles into the pinscapeas a domain of six-fold order. (c) Four-fold order returns viasingle-particle hopping.

system, HOT-modulated colloidal monolayers offer theunique opportunity to both accurately measure and con-tinuously control the particles’ interactions with eachother and with the substrate potential. Such control willmake possible systematic studies of effects induced byvarying the strength, scale, and symmetry of the pinningpotential landscape, and could even allow for the study ofthe influence of controlled disorder deliberately encodedinto the tweezer array. The laser-induced toroidal flowtechnique also introduced for this study is useful for es-tablishing an essentially isotropic hydrostatic pressure onthe pinned colloid. Still other insights can be gained bystudying colloidal transport in linear flows. This work isongoing and will be published elsewhere.

We are grateful to Heinrich Jaeger, Tom Witten,Franco Nori and Charles Reichhardt for enlightening con-versations. This work was supported in part by theMRSEC Program of the National Science Foundationthrough Grant Number DMR-980595, in part throughNSF Grant Number DMR-978031, and in part by a Fel-lowship in Science and Engineering from the David andLucile Packard Foundation. Additional funding was pro-vided by a grant from Arryx, Inc. The diffractive opticalelement used in this study was fabricated by MatthewDearing and Steven Sheets to a design by Eric Dufresneusing the methods of Ref. 18.

1 C. A. Murray, in Bond-Orientational Order in CondensedMatter Systems, edited by K. J. Strandburg (Springer-Verlag, Berlin, 1991).

2 I. Lyuksyutov, A. G. Naumovets, and V. Pokrovsky, Two-Dimensional Crystals (Academic Press, San Diego, 1992).

3 V. V. Metlushko et al., Solid State Comm. 91, 331 (1994).4 C. Reichhardt, C. J. Olson, and F. Nori, Phys. Rev. B 57,

7937 (1998).5 C. J. Olson, C. Reichhardt, and F. Nori, Phys. Rev. Lett.

81, 3757 (1998).

6 C. Reichhardt and F. Nori, Phys. Rev. Lett. 82, 414(1999).

7 C. Reichhardt, R. T. Scalettar, G. T. Zimanyi, and N.Grønbech-Jensen, Phys. Rev. B 61, R11914 (2000).

8 D. J. Morgan and J. B. Ketterson, J. Low Temp. Phys.122, 37 (2001).

9 Intercalation in Layered Materials, edited by M. S. S. Dres-selhaus (Plenum Press, New York, 1986).

10 Charge density waves in solids, edited by L. P. Gorkov andG. Gruner (North-Holland, New York, 1989).

7

11 A. D. Dinsmore, A. G. Yodh, and D. J. Pine, Nature 383,239 (1996).

12 B. J. Ackerson and A. H. Chowdhury, Faraday Discuss.Chem. Soc. 83, 1 (1987).

13 M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, Sci-ence 249, 749 (1990).

14 K. Loudiyi and B. J. Ackerson, Phys. A 184, 1 (1992).15 Q.-H. Wei, C. Bechinger, D. Rudhardt, and P. Leiderer,

Phys. Rev. Lett. 81, 2606 (1998).16 E. R. Dufresne and D. G. Grier, Rev. Sci. Instr. 69, 1974

(1998).17 D. G. Grier and E. R. Dufresne, U. S. Patent 6,055,106,

The University of Chicago (2000).18 E. R. Dufresne et al., Rev. Sci. Instr. 72, 1810 (2001).19 A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu,

Opt. Lett. 11, 288 (1986).20 J. C. Crocker and D. G. Grier, J. Colloid Interface Sci.

179, 298 (1996).21 S. H. Behrens and D. G. Grier, Phys. Rev. E 64, R401

(2001).22 S. H. Behrens and D. G. Grier, J. Chem. Phys. 115, 6716

(2001).23 A. E. Larsen and D. G. Grier, Phys. Rev. Lett. 76, 3862

(1996).24 A. E. Larsen and D. G. Grier, Nature 385, 230 (1997).25 G. M. Kepler and S. Fraden, Phys. Rev. Lett. 73, 356

(1994).26 J. C. Crocker and D. G. Grier, Phys. Rev. Lett. 77, 1897

(1996).27 M. D. Carbajal-Tinoco, F. Castro-Roman, and J. L.

Arauz-Lara, Phys. Rev. E 53, 3745 (1996).28 W. B. Russel, D. A. Saville, and W. R. Schowalter, Col-

loidal Dispersions, (Cambridge University Press, Cam-bridge, 1989).

29 J. Happel and H. Brenner, Low Reynolds Number Hydro-dynamics (Kluwer,Dordrecht, 1991).

30 J. C. Crocker and D. G. Grier, Phys. Rev. Lett. 73, 352(1994).

31 C. P. Bean, Rev. Mod. Phys. 36, 31 (1964).32 Y. B. Kim, C. F. Hempstead, and A. R. Strnad, Rev. Mod.

Phys. 36, 43 (1964).33 A. J. Liu and S. R. Nagel, Nature 396, 21 (1998).34 M. J. Van Bael, K. Temst, V. V. Moshchalkov, and Y.

Bruynseraede, Phys. Rev. B 59, 14674 (1999).35 F. P. Preparata and M. I. Shamos, Computational Geom-

etry (Springer-Verlag, New York, 1985).36 S. Field, J. Witt, F. Nori, and X. Ling, Phys. Rev. Lett.

74, 1206 (1995).37 C. Reichhardt et al., Phys. Rev. B 54, 16108 (1996).38 F. Nori, Science 271, 1373 (1996).39 C. J. Olson, C. Reichhardt, and F. Nori, Phys. Rev. Lett.

80, 2197 (1998).40 K. Harada et al., Science 274, 1167 (1996).41 J. A. Weiss, D. W. Oxtoby, D. G. Grier, and C. A. Murray,

J. Chem. Phys. 103, 1180 (1995).42 K. Behnia, C. Capan, D. Mailly, and B. Etienne, Phys.

Rev. B 61, R3815 (2000).43 V. Metlushko et al., Phys. Rev. B 60, R12585 (1999).44 D. J. Morgan and J. B. Ketterson, Phys. Rev. Lett. 80,

3614 (1998).45 A. Terentiev et al., Phys. Rev. B 61, R9249 (1999).46 J. I. Martın et al., Phys. Rev. B 62, 9110 (2000).47 V. I. Marconi and D. Domınguez, Phys. Rev. Lett. 82,

4922 (1999).