APS/123-QED

Quantitative imaging of colloidal flows

Rut Besseling1, Lucio Isa1, Eric R. Weeks2, and Wilson C. K. Poon1

1SUPA (Scottish Universities Physics Alliance) and School of Physics & Astronomy,The University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom.

2Physics Department, Emory University, Atlanta, Georgia 30322, USA.

We present recent advances in the instrumentation and analysis methods for quantitative imagingof concentrated colloidal suspensions under flow. After a brief review of colloidal imaging, we de-scribe various flow geometries for two and and three-dimensional (3D) imaging, including a ‘confocalrheoscope’. This latter combination of a confocal microscope and a rheometer permits simultaneouscharacterization of rheological response and 3D microstructural imaging. The main part of the paperdiscusses in detail how to identify and track particles from confocal images taken during flow. Afteranalyzing the performance of the most commonly used colloid tracking algorithm by Crocker andGrier extended to flowing systems, we propose two new algorithms for reliable particle tracking innon-uniform flows to the level of accuracy already available for quiescent systems. We illustrate themethods by applying it to data collected from colloidal flows in three different geometries (channelflow, parallel plate shear and cone plate rheometry).

PACS numbers: 83.80.Hj, 83.50.Ha, 83.60.La, 64.70.Pf, 61.20.Ne, 61.43.Fs, 82.70.Dd

I. INTRODUCTION

The last two decades have seen a surge of interest inthe behavior of concentrated colloidal suspensions. Thesesystems have long attracted attention because of theirevident practical importance. However, developmentssince the 1980s have shown that well-characterized col-loidal suspensions, in which the size, shape and interac-tion of the particles are known, can serve as experimentalmodel systems for understanding generic phenomena incondensed systems. Initially, this ‘colloids as big atoms’approach has focussed on the use of model colloids tostudy equilibrium phenomena in the bulk such as liquidstructure and phase behavior [1, 2, 3]. Since then, in-terfacial phenomena have been investigated [4], as wellas bulk non-equilibrium phenomena such as phase tran-sition kinetics [5], glassy arrest [6, 7] and gelation [8]. Inall cases, the well-characterized nature of the experimen-tal systems has meant that very direct comparison withtheory and simulations are possible; such synergism givesrise to rapid advances in understanding.

Most recently, the spotlight has been on the use ofmodel colloids to study driven non-equilibrium phenom-ena. In particular, coincident with intense developmentsin a variety of theoretical approaches [9, 10, 11], col-loids are increasingly seen as model systems for study-ing the rheology of arrested matter. Here, perhaps moreso than previously, fundamental interest and immediateindustrial relevance directly coincide. Concentrated par-ticulate suspensions, sometimes known as pastes, havewidespread applications [12], most (if not all) of whichwill involve the suspensions being mechanically drivenfar away from equilibrium either as part of processing(e.g. in ceramics manufacture [13]) and/or during use.Here, as before, the study of well-characterized colloidscan yield fundamental insights, many of which are likelyapplicable to ‘real’ systems with little need of ‘transla-tion’. Moreover, we may expect that driven colloidal

suspensions can, in some respects, be similar to drivengranular materials, themselves the focus of intense studyfor both fundamental and applied reasons [14]. Quan-titative similarities of this kind have indeed been foundrecently for the case of channel flow [15]. If more suchanalogies are found in the future, a unified description ofcolloids and grains may indeed be possible [16].

The elucidation of structure and dynamics have alwaysbeen important goals in the study of colloids in general,and of model colloids in particular. Traditionally, struc-tural and dynamical information in this and other ar-eas of soft matter science is derived from scattering [17].The outputs from such experiments are the static anddynamic structure factors. These average quantities areoften directly calculable from theory, which partly ex-plains the appeal of scattering methods in the first place.

But the upper range of the colloidal length scale is inthe optical domain, and so is amenable to direct imagingin an optical microscope. Given the centrality of imagingin Perrin’s pioneering (and Nobel Prize winning) work us-ing colloids to prove the existence of atoms [18], it is atfirst sight surprising that optical imaging played almostno further role in the study of colloids until the last twodecades of the 20th century. But the imaging of all butthe most dilute suspensions had to await two develop-ments.

First, model systems are needed in which the refrac-tive index of the particles can be closely, if not perfectly,matched to that of the surrounding solvent; otherwiseconcentrated suspensions of large particles that are inprinciple optically resolvable are turbid, and thus notamenable to optical imaging. A number of such sys-tems have been developed since the 1980s. (The devel-opment of such systems also benefits the use of lightscattering, which also requires index matching.) Sec-ondly, an imaging method needs to be found that candeal with at least a certain degree of translucency in sam-ples. Such a method, confocal microscopy, was invented

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(and patented) by Marvin Minsky in 1955. The develop-ment of the methodology in the first few decades sinceits invention was driven largely by the requirements ofbiologists. Since the mid-1990s, however, there has beena surge in interest in applying confocal microscopy to thestudy of model concentrated colloidal suspensions. Ini-tially, this interest was focussed on quiescent systems [19].In the last few years, however, it has been demonstratedthat confocal microscopy can also be used with profit tostudy flowing colloids, and thus yield unique insights intothe rheology of pastes. The purpose of this work is to setout in detail, for the first time, how this can be done.

The rest of the paper is organized as follows. We firstbriefly review the use of imaging methods to study col-loids. We then describe in Section III new hardware thatwe have developed to image colloidal flows in variousgeometries. In Section IV we review the basics of fea-ture identification in (confocal) images and discuss var-ious limits to particle identification in 2D and 3D flowfields. The core of the work is Section V, where we turnto particle tracking. In Section VA we evaluate for thefirst time, using data from simulations, the applicabilityof a classic (and widely applied) tracking algorithm [20]in quiescent and sheared systems where the average mo-tion is zero. We then describe in detail, Sections. V Band VC, our new methods to track particles in the pres-ence of flow. Finally, in Section VI we demonstrate theapplicability of these methods to imaging colloidal flowsin various geometries, including home built environmentsand a commercial rheometer.

II. IMAGING COLLOIDAL SUSPENSIONS

The imaging of a single layer of colloids has been usedto great effect to study fundamental processes in 2D asillustrated for example by the work of Maret and co-workers e.g. [21, 22, 23]). Despite being less problem-atic than three-dimensional imaging, 2D (bright-field)imaging may nonetheless have its specific challenges, e.g.when analyzing imaged objects that have come into veryclose proximity [24].

The use of conventional (non-confocal) optical mi-croscopy to study concentrated colloidal suspensions in3D has been reviewed before [25]. In nearly index-matched suspensions, contrast is generated using eitherphase contrast or differential interference contrast (DIC)techniques. One advantage of conventional microscopy isspeed: image frames can easily be acquired at video rate.But it has poor ‘optical sectioning’ due to the presence ofsignificant out-of-focus information, so that particle coor-dinates in concentrated systems cannot be reconstructedin general, although structural information is still obtain-able under special circumstances [26].

Compared to conventional microscopy, confocal mi-croscopy delivers superior optical sectioning by using apinhole in a plane conjugate with the focal (xy) plane. Itallows a crisp 3D image to be built up from a stack of 2D

images. But each 2D image needs to be acquired by scan-ning, which imposes limits on its speed. The techniquehas been described in detail before [27].

The use of confocal microscopy in the study of con-centrated colloidal suspensions was pioneered by vanBlaaderen and Wiltzius [28], who showed that the struc-ture of a random-close-packed sediment could be recon-structed at the single particle level. Confocal microscopyof colloidal suspensions in the absence of flow has been re-viewed recently [19, 29, 30, 31], and we refer the reader tothese reviews for details and references. Here, we simplynote that this methodology gives direct access to localprocesses, such as crystal nucleation [32] and dynamicheterogeneities in hard-sphere suspensions near the glasstransition [33, 34].

In this work, we focus on the use of confocal microscopyfor imaging colloids under flow, or confocal rheo-imaging(reviewed in [35]). Conventional rheology studies the me-chanical response of bulk samples. As far as the studyof concentrated, model suspensions is concerned, muchattention has been given in the last few years to non-linear rheological phenomena, e.g., the different waysin which repulsion- and attraction-dominated colloidalglasses yield [36, 37]. The bulk rheological data are con-sistent with the former yielding by a single-step processof cage breaking, and the latter yielding in two steps, firstbreaking interparticle bonds, and then breaking nearest-neighbor cages. Confocal imaging can play a decisiverole in the verification of such microscopic interpretation,which inevitably makes reference to local processes on thesingle-particle level. Moreover, direct imaging can clearlyshed light on complicating factors in conventional rheo-logical measurements such as wall slip [38, 39] and flownon-uniformities such as shear banding [40]. Here, sig-nificant progress can be made without imaging at single-particle resolution, by using various coarse-grained ve-locimetry methods. Traditional Particle Image Velocime-try (PIV) [41] requires transparent samples. This tech-nique has recently been used in a rheometer to give im-portant information on slip in emulsions [42, 43].

Other methods for velocimetry with no requirementfor transparency have been developed, such as hetero-dyne light-scattering [44] and ultrasonic velocimetry [45].The latter has been applied to characterize slip and flownonlinearities in micelles and emulsions [46, 47].

A robust method for velocimetry which can also pro-vide additional information on the density profiles is Nu-clear Magnetic Resonance Imaging (NMRI) [48, 49, 50,51]. The technique has spatial resolution down to ∼20 µm and has been combined with rheometric setups torelate velocity profiles to macroscopic rheology [52, 53].This approach has been used to investigate the occur-rence of shear bands [54] and shear thickening [55].

Thus, both PIV and NMRI give additional insight un-available from bulk rheology alone. But to build up acomplete picture of colloidal flow, it is desirable also tohave information on the single particle level. For this pur-pose, a method related to PIV and particle tracking has

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been applied to non-Brownian suspensions and allowedthe measurement of non-affine particle motion and diffu-sivity [56, 57]. But direct imaging of the microstructureduring flow is needed to give complete microstructuralinformation. Optical microscopy has this capability.

It is possible to use conventional (non-confocal) videomicroscopy to study shear effects in 3D [58, 59, 60, 61].But the poor optical sectioning hinders complete, quanti-tative image analysis. Confocal microscopy significantlyimproves sectioning, and permits in principle the extrac-tion of particle coordinates. But the need for scanninginitially meant rather slow data acquisition rates, so thatobservations in real time (i.e. during shear) producedblurred images that again limited the potential for quan-titative analysis [62]. A common solution was to applyshear, and then image immediately after the cessation ofshear, both in 2D [63, 64] and in 3D [62, 65, 66, 67]. (Ear-lier work using conventional video microscopy [58, 59]resorted to the same strategy.)

More recently, the availability of fast confocal systems(see Section III B) means that nearly-real-time recon-struction of structure during flow in 3D at single-particleresolution has become possible. Such experiments facetwo key challenges: sample environment and data anal-ysis. First, the flow geometry used clearly must be com-patible with the optical requirements of simultaneousconfocal imaging. A number of different arrangementshave been demonstrated to date. Derks and co-workerscarried out a first experimental study by using a counter-rotating cone and plate shear cell combined with a fastconfocal microscope [68] and obtained particle coordi-nates and tracks in the zero-velocity plane as well as ve-locity profiles across the geometry gap. The same grouphas recently produced a more sophisticated set up whichuses a parallel plate shear cell [69] capable of spanning avast range of shear rates and frequencies which they usedto study crystallization of colloids under shear. A paral-lel plate shear cell has also been used by Besseling andco-workers [70] to study the shear-induced relaxation inhard-sphere colloidal glasses, while recent experimentsby Isa and co-workers [15] have elucidated the behav-ior of colloidal sediments flowing into micro-channels. Inthis work, we give the details for two of these geome-tries [15, 70], and describe and demonstrate a new one:the coupling of a fast confocal scanner to a commercialrheometer, which allows simultaneous confocal imagingand full rheological characterization of the same sample.

The second challenge is data analysis: how to extractaccurate particle coordinates from raw image stacks. Inparticular, special methods are needed for reliable track-ing, since the large displacements from frame to frameimposed by flow may inhibit correct identification of par-ticles between frames. The same problem confronts theuse of imaging to study granular flows [71]. In this work,we describe in detail a new method for tracking particlesfrom confocal images acquired during flow. We demon-strate its correctness and measure its limitations by usingdata from computer simulations, as well as illustrate its

use with real experimental data.

III. MATERIALS AND INSTRUMENTATION

A. The colloidal particles

Our goal is to perform confocal imaging in real time atsingle-particle resolution of colloidal suspensions underflow. Confocal microscopy is, in principle, able to imag-ing inside slightly turbid systems, but the image qualitydeteriorates with sample turbidity. In order to obtain assharp images as possible to test the limits of our method-ology, we performed experiments using a index-matchedsuspension that is optically clear.

The particles were poly–methyl–methacrylate(PMMA) spheres, sterically-stabilized by chemically-grafted poly–12–hydroxy stearic acid (PHSA) [72].The particles can be dyed with a fluorophore (NBD, 4chloro–7 nitrobenz–2 oxa 1,3 diazole), which is excited at488 nm and emits at 525 nm. Particles can be suspendedin a mixture of decalin (mixed-decahydronaphtalene,Sigma–Aldrich, ndecalin = 1.4725 ± 0.0005) andtetralin (tetrahydronaphtalene, Sigma–Aldrich,ntetralin ' 1.5410±0.0005) to achieve full refractive indexmatching of the solvent and the particles (nsusp ' 1.5).This matching ensures hard-spheres interactions [73]and also limits scattering of both the laser and theexcited light during confocal microscopy. However, thedecalin-tetralin mixture has a lower density than PMMA(1.188g/cm3). To achieve buoyancy-matching, particlescan be suspended in a mixture of cyclo-heptyl-bromide(CHB) and mixed-decalin [74]. The buoyancy-matchingcomposition also closely matches the refractive indexof the suspension (nsusp = 1.494) [29]. The addition ofCHB to a hard-spheres suspension induces charge on theparticles, which can be screened by adding a suitableamount (4 mM) of salt [74] (tetrabutylammoniumchloride, (C4H9)4NCl, MW = 277.92g/mol, Fluka).

The buoyancy-matching is very sensitive to temper-ature changes; the thermal expansion coefficient of thesolvent exceeds that of PMMA by about a factor tenand a decalin-CHB mixture of a given composition willtherefore match the particle density only in a very narrowtemperature range [75]. We exploit this fact to preparesuspensions of different volume fractions by centrifugingthe suspension at a temperature T & 35◦ C, above thebuoyancy matching temperature, to create a sedimentwhich can subsequently be diluted. Finally, imaging caneither be performed on a fully fluorescent sample or onrefractive index-matched systems seeded with fluorescentparticles. In the course of our description we shall specifythe details of the system used in each example.

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zx

FIG. 1: Projection of a raw 3D image stack of x × y × z ∼29× 29× 15 µm3 (256× 256× 76 voxels) taken in ∼ 1 s. Theparticle radius is 850 nm.

B. The confocal microscope

To perform confocal imaging during flow, high acqui-sition rates and thus fast laser scanning methods arerequired. Among these are spinning disk systems [27](with possible micro-lens array extension) or laser scan-ning by resonant galvanometric mirrors. The confocalscanner we use (VT-Eye, Visitech International, witha solid state 488 nm laser) employs a combination ofa standard galvanometer and an acousto-optic deflector(AOD) [96]. The former positions the laser beam at a cer-tain y-position, while the AOD much more rapidly scansa line along x. The acquisition rate is thus mainly deter-mined by the ‘slow’ galvanometer. Typical frame ratesfor 2D image series range from fscan = 5 Hz for imagesof 1024× 1024 pixels to fscan = 100− 200 Hz for imagesof 256× 256 pixels. The upper limits on colloid diffusiv-ity or flow speed imposed by these acquisition rates aredescribed in Sec. IVB.

We have imaged flow in a parallel plate shear cell andin square capillaries using the ‘standard configuration’,where the confocal scanner is coupled to a Nikon TEEclipse 300 inverted microscope, with a 100× or 60×magnification, oil-immersion objective with a numericalaperture (NA) of 1.4. The depth of the focal plane, z,is controlled by a piezo-element mounted on the micro-scope nosepiece. For 3D imaging, a z-stack of 2D im-ages is collected (Fig. 1) by rapid variation of the heightof the piezo and synchronized 2D acquisition at each z.The corresponding 3D acquisition time is Nz/fscan withNz the number of 2D slices. We have also coupled theconfocal scanner to a commercial rheometer, to enablesimultaneous imaging and rheological measurements onthe same sample.

Z scany

z

x

FIG. 2: Sketch of the shear cell. The sample is positionedbetween two parallel slides; the top one is driven by a me-chanical actuator while the bottom one can either be fixed orbe translated in the opposite direction (dashed arrow). Thesuspension is imaged from below (image volume highlighted)with a confocal microscope the focal depth of which is con-trolled by a piezo-electric element.

C. Shear cell

Initial experiments on shear flow were performed witha linear parallel plate shear cell (plate separation Zgap ∼400 − 800 µm, parallel to ±5 µm over a lateral distanceof 2 cm) where the top plate is driven at 0.05− 10 µm/sby a mechanical actuator with magnetic encoder. Wedenote the velocity, vorticity and velocity gradient direc-tion by x, y, and z, respectively, as shown in Fig. 2. Themaximum relative plate translation along x is Ls ∼ 1 cm,so that steady shear can be applied up to a total accu-mulated strain ∆γ = Ls/Zgap & 1000%. The cell can beoperated either with the bottom plate fixed or with theplates counter propagating via an adjustable lever sys-tem, which allows the height of the zero velocity plane tobe set at any distance from the bottom plate. A drop ofsuspension (covering an area of ∼ 200 mm2) is confinedbetween the plates by surface tension. A solvent bathsurrounding the plates minimizes evaporation. Wall slip,prominent in glassy systems [76], and wall-induced or-dering were prevented by sintering a concentrated, disor-dered layer of particles – obtained by spincoating a sus-pension with volume fraction φ ∼ 30%– onto the glasssurfaces, Figure 3(a). We typically image a 30 µm × 30µm × 15 µm volume in the drop (with ∼ 3000 particles),up to ∼ 40 µm above the coverslide.

D. Capillary flow

We have also studied the flow of concentrated colloidalsuspensions in glass micro-channels [15, 77]. Figure 4(a) shows a sketch of a sample cell for such experiments.It is assembled by gluing a glass capillary onto a micro-scope coverslide with UV curing glue (Norland OpticalAdhesive) by exposing it to UV light for a few minutes.Once the glass channel is attached, a glass vial (1.5 cmdiameter), the bottom of which is removed, is glued on

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FIG. 3: Confocal images of the coating (a) on a cover slide forshear flow imaging (image size: 56 µm × 56 µm), (b) on theinner surface of a 50 µm × 50 µm glass capillary (image size:43 µm × 43µm. The larger particles in (b) form the coating,the smaller ones are suspended and flowing.

b)

Sample cell

Glass coverslide

Capillary

Flow directionImaging direction

Bottom view

a)

Capillary

Microscopeobjective

Image plane

FIG. 4: (a) Sketch of the sample cell for capillary flow. Thecapillary is not drawn to scale. The construction is placedon the microscope stage plate and the flow is imaged frombelow. (b) Close up. Only the capillary and objective aredrawn. The flow (green arrow) is imaged from below and 2Dxy slices are collected at a depth z.

top of one end of the channel also with UV glue (2 hoursexposure). Finally a PVC tube (1 mm internal diame-ter) is connected to the free end of the capillary and theconnection is sealed with epoxy glue. The sample cell ispositioned onto the microscope stage with the cover slide,forming the bottom of the cell, in direct contact with themicroscope objective via the immersion oil.

All channels are borosilicate glass capillaries (Vit-rocom) with rectangular (20 × 200µm2, 30 × 300µm2,40 × 400µm2) or square cross sections (50 × 50µm2,80× 80µm2, 100× 100µm2) and a length of 10 cm. Thecapillaries can either be used untreated, i.e. smooth onthe particle scale, or have their inner walls coated witha sintered disordered layer of PMMA particles to ensurerough boundaries (Figure 3(b)). This is achieved by fill-ing the capillary with a 15–20% suspension of similar par-ticles and subsequent drying in a vacuum oven at 110-120◦C.

The imaging geometry is sketched in Fig. 4(b). Theflow is along the channel in the x direction; by adjust-ing the image size we can capture the entire cross sec-tion of the capillary. Using a modified microscope stageplate with a long rectangular slot instead of the stan-dard circular aperture, the imaging can be performedat different positions along the channel, over a range of. 5 cm. The suspension is first loaded into the samplecell and then driven along the channel by a constant pres-sure difference achieved by displacing a syringe plungerconnected to the PVC tubing. During imaging, the pres-sure is monitored with a pressure gauge (MKS Series 902Piezo Transducer).

E. Confocal rheoscope

In order to perform confocal imaging of the flowand simultaneously obtain the global suspension rheol-ogy, we have combined the fast confocal scanner with astress-controlled rheometer (AR2000, TA Instruments),Figs. 5(a),(b). The rheometer has a custom-built, openbase construction, mounted on its normal force sensor,with pillars providing space for a mirror and objectivemounted on a piezo-element, which makes it possible tovary the depth of the focal plane. An aluminium platewith imaging slit is mounted on the pillars, and can beaccurately levelled via three adjustment screws. A glassslide (radius 2.5 cm, thickness ∼ 180 µm), mounted onthe plate, forms the bottom surface of the measurementgeometry through which the imaging is performed. Therheometer can thus be operated in plate-plate or coneplate geometry, but generally we used a stainless steelcone of radius rc = 20 mm and cone angle θ = 1◦. Boththe glass slide and the cone can be made rough on theparticle scale using the spincoating and sintering method.Evaporation can be minimized by a solvent trap, butwe found superior reproducibility of the rheology of themost concentrated suspensions by slightly under-loadingthe geometry and applying a small rim of immiscible liq-uid (glycerol) around the geometry edge. Through theimaging window we can directly measure the width ofthe rim (typically ∼ 1 mm) as well as the geometry areacovered by the suspension. The contribution of the glyc-erol to the overall stress can thus be calculated; e.g. at ashear rate of 10 s−1 it is ∼ 2 mPa, negligible comparedto the stress levels of our suspensions, see Fig. 20. Fur-thermore, careful loading ensures that there is no directcontact between the sample and the glycerol rim, avoid-ing contamination and spurious effects due to a possibleinterface. Finally, we checked that bending of the coverslide was negligible, see Sec. VIC.

The confocal scanner is coupled to the optics underthe plate. In our setup, the scanner remains fixed in thestandard configuration, connected via a C-mount to theNikon microscope, see Fig. 5(c). To provide the coupling,we altered the optical path of the laser and the excitedlight. By positioning a movable mirror, the beam exits

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through the rear of the microscope (Fig. 5(c)) and thenpasses through additional mirrors and two lenses (acro-mat lenses, broadband coated, focal length f = 18 cm,diameter = 40 mm); one of the lenses is mounted on amechanical arm, at the end of which, situated below thetransparent plate, a final mirror and the piezo objectivemount are located. The two lenses provide one-to-oneimaging of the back aperture of the objective in its stan-dard position on the nosepiece of the Nikon, onto thatof the objective in its new position under the transpar-ent rheometer plate. During imaging, the piezo-elementis controlled by the software of the confocal microscope,providing the same 3D imaging capability as in normaloperation.

FIG. 5: (a) Schematics of the confocal rheoscope. The toparrow marks translation of the rheometer head to adjust thegeometry gap, the horizontal arrow indicates translation ofthe arm supporting the objective to image at different radialpositions. (b) Close up of the central part of the confocalrheoscope. The position of the mirror, directly underneaththe piezo, is indicated as B in the right figure. (c) Globalsketch of our implementation. The optical path (colored) fromthe confocal (CF) is guided from the back of the microscope(Nikon) via mirrors and two lenses L1 and L2, to positionB corresponding to the mirror in (a). The distance betweenlenses L1 and L2 is 2f + d, with f the focal length and | d |<0.5 cm a small displacement to allow for lateral positioningof the objective underneath the plate. The distance betweenthe standard position of the objective back aperture (aboveA) and the new one (above B) is therefore 4f + d.

IV. PARTICLE LOCATION AND ITSLIMITATIONS

A. Locating and identifying the particles

The first step to obtain quantitative information onparticle dynamics from the images is to locate the par-ticles. The most widely used algorithm for this purposeto date in colloid science is that of Crocker and Grier(CG) [20], with relevant software in the public domain[78].

Three main assumptions are needed in order to locateand identify the particles. The features must appear asbright objects onto a dark background, we assume thatthey are spherical in shape [97] , and that the maximumin the brightness of a feature corresponds to its center.The concepts at the basis of feature location are stillapplicable to objects which do not follow these require-ments but the practical algorithm for locating them willbe different and generally more complicated, see e.g. [79].

Since the particle centers are identified in terms of theirintensity, undesired intensity modulations which can giverise to mistakes in particle location need to be eliminated.This is achieved via image filtering using a spatial band-pass filter. This eliminates any long wavelength contrastgradients and also short wavelength pixel to pixel noise.

The coordinates of the centers of the features are ini-tially obtained by locating the local intensity maximain the filtered images. A pixel corresponds to a parti-cle center if no other pixel has a higher intensity withina given distance to it; typically this distance is slightlylarger than the average particle radius. These coordi-nates are then refined to get the positions of the particlecenters with a higher accuracy by applying a centroidingalgorithm which locates the brightness-weighted centerof mass (centroid) of the particles. With this refinementprocedure the coordinates of the particle centers can beobtained with sub-pixel resolution down to less than 1/10of the pixel size [20]. The centroiding procedure provesitself effective but has some limitations. Correctly locat-ing the particles becomes more difficult as the system be-comes more concentrated and individual particle imagesmay start to overlap. This difficulty can be dealt with byfluorescent labelling only the particle cores [33] so thatimages are well separated even at the highest densities.When such particles are not available, improvements incoordinate refinement may be required. Such an improve-ment, based on fitting the intensity profile of the particleto the ‘sphere spread function’ (SSF), has been devisedby Jenkins [80]. We have used this method success-fully for our 3D images described in Sec. VIB. Finally,to avoid edge effects, particle centers identified within aradius from the image edge are ignored.

As noted above, the coordinates of particle centers areoften found to an accuracy tied to the pixel size. Thiscorrectly implies that modifying the microscope opticsso that the size of a pixel is smaller will improve the lo-cation of particle centers. In general, if the image of a

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particle is N pixels across, the center of that particle canbe found to an accuracy of (pixel size)/N . It is importantto recognize that this accuracy is different from, and of-ten better than, the optical resolution of the microscope.The optical resolution relates to telling the difference be-tween two closely positioned bright objects: if they arecloser together than the resolution, then their diffraction-limited images blur together in the image. The resolutionlimit for an optical microscope is given by the wavelengthof light used and the numerical aperture (NA) of the ob-jective lens, as λ/(2NA), and the best resolution for anoptical microscope is about 0.2 µm. This figure reflectsthe wave nature of light. The accuracy with which par-ticle centers can be located is set by different physicalconstraints, e.g. the fact that particles cannot physicallyoverlap (although their images may) [20], and the knowl-edge that they are spherical.

B. Limitations

The above mentioned accuracy of locating particles isintrinsic, and applies even for particles frozen in the im-age. It can be estimated from the plateau value of themean squared frame to frame displacements (which wedenote as MSFD) measured in a close packed sediment,where particles are essentially immobile. In general how-ever, additional errors on the exact center position arisefrom short time diffusive motion and flow advection dur-ing acquisition of the particle image, if the images arefrom a scanning system (such as a confocal microscope)which does not acquire each pixel simultaneously.

Let us estimate these effects for 2D and 3D imaging.For the 2D case, given the frame acquisition rate fscan

and the image size in pixels (n×m), the time required toscan one line is ' 1/(nfscan). If a is the particle radiusin pixels, then the time required to image a particle is:

t2Dim = 2a/(nfscan). (1)

With our imaging system, a 256 × 256 pixel image canbe taken at fscan = 90 Hz. Using a 100× magnificationobjective, for which the xy pixel size is∼ 0.2 µm, the timeto acquire a 2D image of a particle with radius a = 1 µmis therefore t2D

im ' 0.4 ms.For 3D images, acquired as a z-stack of 2D slices, the

limiting factor is the speed at which the particle is im-aged in the z-direction. The voxel size in the z direction(i.e. the z-spacing between the 2D slices) may differ fromthat in the x and y direction. Denoting the particle ra-dius in z-pixels by az, the time required for a 3D particleimage is:

t3Dim = 2az/fscan, (2)

with fscan the acquisition rate for a complete 2D image asbefore. Thus, for the 3D case, using again fscan = 90 Hzand a typical value of 0.2 µm for the z-pixel size, theacquisition time for our a = 1 µm particle is t3D

im ' 0.1 s.

We first consider the (short time) diffusive motion ofcolloids in a suspension on these time scales. In the di-lute limit, the diffusion constant is Ds,0 = (kBT )/(6πηa),with kB the Boltzmann constant and η the solvent vis-cosity. For concentrated suspensions the short time diffu-sion constant is reduced due to hydrodynamic hindering,Ds(φ) = Ds,0H(φ) with H(φ) < 1 [81, 82, 83, 84, 85].The average motion in one direction during the acqui-sition time is

√2Ds(φ)tim, i.e. the additional error is

δ =√

Ds(φ)tim/2. Using η = 2.7×10−3 Pa·s for our sol-vent (decalin) and T = 300 K, Ds,0 = 8.13× 10−2µm2/sfor the a = 1 µm colloid. The resulting error due tothermal displacement during 2D imaging of a dilute sys-tem is then δ2D ' 2 nm, while for the 3D case, usingthe same frame rate and z-pixel size as above, we haveδ3D ' 35 nm. While the former is considerably smallerthan the intrinsic 1/10 pixel accuracy (∼ 20 nm), for3D the thermal motion is the limiting factor. Note thatthese considerations apply to hard sphere systems only;when additional interactions limit the short time dis-placements, the intrinsic 1/10 pixel limit may still apply.

Flow can also induce additional errors on the particlelocation due to image distortion. Since the image of aparticle is scanned either via lines in 2D or via horizontalslices in 3D, it will be distorted because the particle isdisplaced between two consecutive lines as well as slices.Such distortion can be exploited to deduce the local flowvelocity, see [68], but here we are interested in the veloc-ity range for which the distortion is sufficiently small toconsider the object as effectively spherical. The particlespeed beyond which this no longer holds can be estimatedby comparing the imaging time tim with the time tf re-quired for the flow to displace the particle over its owndiameter. For a flow velocity V (in pixels per second),tf = 2a/V . We consider the particle significantly dis-torted if tim/tf ≥ 0.1, i.e. a distortion of 1 pixel for aparticle size 2a = 10. Using Eqs. 1,2 for the acquisitiontimes, the maximum velocities are:

V max2D = 0.1nfscan, V max

3D = 0.1fa/az. (3)

For typical parameters (fscan = 90 Hz, n = 256 pix-els, 1 pixel ' 0.2 µm) the limiting velocity in 2D isV max ' 500 µm/s, while for 3D images we obtainV max ∼ 2 µm/s for an x to z pixel size ratio of 1:1.In both cases, further improvement could be achieved byremoving the distortion prior to locating the particles viaimage correlation procedures [68]. In principle, improve-ment could also be achieved by using a location algorithmwith a particle template or Sphere Spread Function [80]with a distorted shape, but since this requires a prioriknowledge of the flow field, the removal of distortion priorto particle location is more practical.

V. TRACKING ALGORITHMS

Once the coordinates have been found in each frame,they need to be merged into trajectories describing the

8

particle motion. In this ‘tracking’ procedure, each parti-cle is labelled with an identification tag and an algorithmlooks for particles in the following frame that can be as-signed the same tag. Tracking has applications in fieldsas diverse as robotics and biophysics [79, 86, 87]. In thefield of digital image processing a variety of methods hasbeen devised [88]. In each method, a specific cost func-tion is calculated based on the changes in coordinates foreach set of identifications, possibly extended with a costfunction for change in feature appearance [88]. The ‘cor-rect’ identification is then obtained as the one for whichthe cost function is minimized.

A. The classic CG algorithm

1. Tracking algorithm

The algorithm we use, devised by Crocker and Grier(CG), is based on the dynamic properties of non-interacting colloids [20]. The cost function in this case isthe mean squared frame to frame displacement (MSFD,as defined before) of particles between frames. Given theposition of a particle in a frame and all the possible newpositions, in the following frame, within a tracking rangeRT of the old position, the algorithm chooses the identi-fication which results in the minimum MSFD. Particlesmoving farther than RT between frames are unable to betracked, and are either mis-identified as other particles,or else treated by the algorithm as new particles.

Note that the original CG algorithm also includes theideas discussed in Sec. IV; here we focus on trackingparticles between successive frames, rather than locat-ing particles in a single image. These two parts of theCG algorithm are decoupled: the tracking method worksindependently of how the particles were originally iden-tified.

2. Hard particle simulations

Crocker and Grier tested their algorithm by trackingthe self-diffusion of particles in dilute colloidal suspen-sions [20]. Since then, the algorithm has been appliedin various studies of quiescent colloidal systems [19, 31].However, quantitative studies of its tracking performancein realistic concentrated systems, possibly with addi-tional motion on top of Brownian diffusion, have not beenperformed to date. In fact, the study of concentratedsystems explicitly pushes the CG tracking algorithm be-yond its design parameters. We therefore apply the clas-sic CG algorithm to computer generated data, in whichthe particle identity is known a priori, and evaluate itsperformance for quiescent systems of different densitiesand for various imposed particle motions. Quiescent datawere generated by Monte-Carlo (MC) simulation of hard-disks in two dimensions and of hard-spheres in 3D (seee.g. [89]) imposing mean squared displacements between

-0.5 0.0 0.5102

104

106

0.0 0.1

0.98

1.00

RT

(b)

PCG

(φ2D

=0.01) PMC

(φ2D

=0.01)P

CG(φ

2D=0.33) P

MC(φ

2D=0.33)

P [n

. par

ticle

s]

∆x /

(a)

φ3D

0.03 0.2

φ2D

0.01 0.04 0.15 0.33 0.58

f

εn

2 (2D), εn

2/2 (3D)

FIG. 6: (a) The fraction of correctly-tracked particles, f , ver-sus the normalized true MSD ε2n between tracked frames forMC simulations of hard-disk fluids at various densities φ2D

and 3D hard-sphere fluids at φ3D = 0.03 and φ3D = 0.2. Notethe different x-axis for the 2D and the 3D case. (b) Corre-sponding distribution of normalized displacements P (∆x/`)for two densities φ2D, taken for f ' 0.98. Solid lines: thetrue PDF over n MC steps. Symbols: PCG as obtained fromclassic CG tracking between frame i and i + n, for φ = 0.01(�) and φ = 0.33 (H).

each MC iteration chosen to obtain a sufficient successrate. We simulated N > 1200 particles and, as in ex-perimental data, particles may (dis)appear at arbitrarytimes around the edge of our simulation cell. [98]

We take the data from the simulations, and treat itas the raw data of particle positions for the classic CG-algorithm. Specifically, we use the CG-algorithm to trackparticles between MC-iterations i and i + n, for whichthe true MSFD is 〈∑j(r

ji+n − rj

i )2〉 = 〈∆r2

n〉, with j =x, y, z. The key idea of the CG-algorithm is that ideally,between each frame of the movie, the majority of particlesshould move less than the typical interparticle (center-to-center) spacing `. In other words, it is desirable for theMSFD to be less than `2. In fact, since the MSFD iscalculated by averaging over all particles, many particleswill have larger motions; likewise ` is an average overall particles, so that some particles are closer together.Thus, in practice, it is desirable that the MSFD is muchless than `2.

To quantify this last statement, we tracked simulatedparticles between frames with progressively larger nor-malized MSFD, ε2n ≡ 〈∆r2

n〉/`2, by increasing n, andtested how the CG-algorithm performed when pushedpast its original design parameters. For each ε2n, we mea-sured the fraction f of correctly tracked particles. Wechecked in all cases that the vast majority of the track-ing errors are generated in the bulk of the system ratherthan at the boundaries.[99]

3. Quiescent system

In Fig. 6(a) we show f as function of ε2n for the 2Dsystem at different φ2D and φ3D. As expected, f de-creases with increasing εn, but the performance in moreconcentrated systems is considerably better than in thedilute case. To quantify this, we impose a criterion for

9

‘successful tracking’ of f > 0.99, and we find that the al-gorithm works up to εn = 0.15 at the lowest φ2D studied,but this figure rises and essentially saturates at εn = 0.3at the highest φ2D. This behavior reflects the differencein structure between a dilute and concentrated system.While ` is the average nearest neighbor spacing, in a di-lute system, particles can approach much closer than this(although still limited to be at least 2a apart). In thiscase, two closely spaced particles could potentially swappositions and confuse the tracking algorithm. In con-trast, for a concentrated system, ` ≈ 2a, and it is muchharder for particles to swap positions. Thus in a con-centrated system there are fewer misidentifications for agiven value of ε compared to a dilute system with thesame ε. For the 3D systems, the performance at largeconcentration is even better, i.e. for φ3D = 0.2, the algo-rithm works up to εn ' 0.4.

An experimental diagnostic for correct tracking isthe distribution of particle displacements from frame toframe, P (∆x) [100]. For correct tracking it should vanishsmoothly within the tracking range. Figure 6(b) com-pares the true distribution function over n frames, PMC ,with that resulting from the classic CG tracking betweenframes i and i + n, PCG, for two densities and n suchthat f ' 0.98. In both cases PCG follows the simulationdata for ∆x < RT , beyond which is cut-off, and the dis-crepancies with PMC , due to the misidentified particles(1 − f = 2%), appear in the large ∆x/` tails (clearer inthe denser case). Note that the CG algorithm is able tofollow particles for larger tracking ranges in the case oflarger area (or volume) fractions.

4. 2D system in shear flow

To test the performance of classic CG tracking in thepresence of non-uniform motion, we superimpose affineshear and random displacements with a MSFD of (ε`)2on a single 2D MC configuration with φ2D = 0.33. Thestrain increment between frames is ∆γ, i.e. the affine x-displacement over one frame for particle k is ∆γ(yk − y)(subtracting y guarantees zero net motion), again withperiodic boundary conditions. We analyze data onlyover accumulated strains < 20%, so that shear does notbring neighboring particles in close proximity, avoiding‘artificial’ reduction in performance (see Sec. VC2 andFig. 12). The true origin for tracking errors is the in-crease in the difference in advected displacements be-tween different parts of the image, the maximum of whichis ∆Sx/` = Ly∆γ/2` in units of the average spacing, withLy the system size in the velocity gradient direction. Fig-ure 7(a) shows that f rapidly decreases for ∆Sx/` & 0.4.

From the resulting tracks, we obtain the distribution ofnon-affine frame to frame displacements after subtractingthe affine shear as evaluated from the classic CG trajec-tories. The results for x and y displacements are shownin Fig. 7(b). For ∆Sx/` = 0.25 the result is identicalto that without shear, matching the superimposed ran-

0.0 0.5 1.00.7

0.8

0.9

1.0

-1.0 -0.5 0.0 0.5 1.0101

103

105(a)

f

∆Sx /

(b)

(∆Sx/ =0.25): Px

CG Py

CG

(∆Sx/ =0.54): Px

CG Py

CG

P [n

. par

ticle

s]

∆x / , ∆y /

FIG. 7: Evaluation of classic CG tracking under shear: (a)f versus the normalized maximum difference in advected dis-placement between frames ∆Sx/` = Ly∆γ/` for φ2D = 0.33and ε2 = 0.005. (b) The distribution of normalized displace-ments P (∆x/`) and P (∆y/`) for ∆Sx/` = 0.25 (squares) and∆Sx/` = 0.54 (circles). Solid line: result without shear.

dom motion. For ∆Sx/` = 0.54, the distribution of y-displacements appears very close to the correct distribu-tion, but the deformed central peak of P x and the pres-ence of prominent side bands show that tracks have beenevaluated incorrectly. The reason for the side bands toappear is linked to the fact that, for large shear, the algo-rithm ends up picking the wrong comoving frame, due tothe fact that a large fraction of particles are misidentified.

5. Summary of evaluation of CG tracking

We have shown that the classic CG algorithm cantrack particles between consecutive frames for a maxi-mum MSFD . (0.3`)2 in quiescent concentrated hard-sphere-like systems, but considerably less in dilute sys-tems. These limits are similar to those discussed in theoriginal article by CG [20]. Simply put, for larger dis-placements, the problem of uniquely identifying parti-cles becomes ill-posed, as the possibilities of particles ex-changing places become too significant. No algorithmcan succeed in this case, and the only remedy is to ac-quire images at a faster rate to resolve the intermediatesteps.

Further, for non-uniform flow the limit is set by a max-imum difference in advected motion of ∼ 0.4` over thefull image. We also found that cut-off and distortioneffects in the distribution of particle displacements canindeed be used as diagnostic of incorrect tracking, al-though one should be cautious to interpret the absenceof such features as proof of 100% performance. The nextsubsection discusses a simple modification of the classicCG algorithm to deal better with particles in uniform ornon-uniform flow.

10

B. Iterated CG tracking algorithm

1. Description

The classic CG algorithm was designed for cases whereall particles move randomly (due to Brownian motion).However, many interesting cases have particles movingin a flow with larger coherent structures, perhaps alsowith Brownian motion superimposed, or even simplynoise. For example, the coherent motion could be dueto Poiseuille flow through a pipe, overall drift of the fieldof view, or an induced shear flow. If the magnitude ofthis motion is small, the classic CG algorithm still hassome ability to track particles. Following the logic above,tracking should work reasonably well if the distance mostparticles move between frames is moderately less than theinterparticle spacing `, whatever the origin of this motionmay be.

In cases where the motion is simple and small com-pared to `, the classic CG algorithm can be iterated toproduce better results. This “Iterated Tracking” methodis as follows. (i) First identify the particle positions ateach time, as per Sec. IV. (ii) Track the particles us-ing the classic CG algorithm. (iii) Determine the coher-ent motion from the successfully tracked particles. (iv)Remove the coherent motion from the original particlepositions. (v) Repeat steps ii-iv until most particles aresuccessfully tracked, and the residual coherent motiondetected in step iii is reduced to an acceptable level. (vi)Add back in all of the coherent motion that has beenpreviously subtracted in all iterations of step iv.

As long as the motion of most particles is less thanthe tracking limit RT and thus less than the interparti-cle spacing `, at least a few particles will be successfullytracked in the first iteration. The coherent motion ofthese few particles is then used to ‘bootstrap’ the clas-sic CG algorithm, and in the subsequent iterations, moreparticles are correctly tracked. These then refine the co-herent motion and thus the Iterated Tracking methodeventually is able to converge on the correct trajecto-ries for all the particles. In practice, this usually onlytakes 3-4 iterations to produce good results. The key toiterated tracking is that the first tracking step must cor-rectly track enough particles to start the process. We useour simulated data to study the breakdown of IteratedTracking in a test case.

2. Uniformly moving system

We first superimpose a uniform x-displacement sn overn MC steps on top of the MC dynamics. Periodic bound-ary conditions keep particles within the analysis window.We then perform Iterated Tracking on these particle po-sitions. Figure 8(a) shows f versus sn for two densi-ties (f(sn = 0) > 0.995 in both cases). For small sn,f ' f(sn = 0) ≈ 1, but for sn/` & 0.5 very few correcttracks are found. Note that Iterated Tracking still pro-

0.0 0.2 0.4 0.6 0.8

0.4

0.6

0.8

1.0

-0.5 0.0 0.5

104

105

106

(a)

φ

2D=0.01, ε

n

2=0.01

φ2D

=0.33, εn

2=0.06

f

sn/

φ2D

=0.33

(b)

PMC

Px

IT(s

n/ =0.7)

PIT(s

n/ =0.2) Py

IT(s

n/ =0.7)

P [n

. par

ticle

s]

∆x/ , ∆y/

FIG. 8: Iterated tracking results. (a) f versus the normalizedshift sn/` between frames for φ = 0.01, ε2n = 0.01 and φ =0.33, ε2n = 0.06. (b) PDF’s in the co-moving frame for φ2D =0.33 and two ‘drift’ velocities: Line: the true PDF over n MCframes. (�): PCG for sn/` = 0.2, (•): distribution function ofx-displacements P x

CG, in the co-moving frame for sn/` = 0.7,(◦) same for y-displacements.

vides a result, but mostly consisting of incorrect tracks,and yields incorrect motion; in this case, the first trackingstep has failed and subsequent tracking steps are unableto improve the results. In Fig. 8(b) we show the distri-bution of particle displacements in the co-moving framefor two values of sn/`. By construction, the true PMC isidentical to that in the quiescent system, while PIT is thedisplacement distribution in the co-moving frame, i.e. theframe where the average displacement of the ‘IT-tracks’(obtained via the Iterated Tracking algorithm) vanishes.The breakdown of iterated tracking for sn/` = 0.7 in theφ = 0.33 system is brought out by the sharp cut-off ofP (∆x > RT ), and, more prominently, by the asymme-try in the distribution of displacements P x

IT along thedirection of motion.

The results are similar for non-uniform flow. If themaximum motion in any area of the data exceeds ∼ 0.4`,that region will be poorly tracked. For example, with theshearing data discussed in Sec. VA 4, at each iterationstep the measured shear strain can be removed, but thismethod will still have difficulties when ∆Sx & 0.4. Withcare, trajectories found in regions with motion less than0.4` may be used to extrapolate the motion to the moremobile regions. Note that the classic CG algorithm wasnot intended for tracking particles in flow, so that thesuccess of Iterated Tracking in non-trivial cases revealsthe strength of that algorithm.

C. Correlated Image Tracking (CIT)

1. Description

Due to the limitations shown above, 2D or 3D imageswith large drift or non-uniform motion require a modi-fied analysis for correct tracking. Similar shortcomings ofstandard tracking for granular flows have been discussedin [71] and analysis of such data will also benefit fromthe correction method we describe in this section. Thebasic ingredients of “Correlated Image Tracking” are: (i)

11

We first obtain the particle coordinates as explained inSec. IV. (ii) We then obtain independent informationon the advected motion and its spatial and time depen-dence via PIV-type correlation analysis of the raw im-ages. (iii) Next, this (time and position dependent) ad-vected motion is subtracted from the bare particle coor-dinates. This yields the particle coordinates in a ‘locallyco-moving’ (‘CM’) frame. (iv) In the CM frame, theparticles can be tracked as in a quiescent system. Thetracking efficiency is essentially limited by the value ofthe MSFD or non-affine motion in the CM frame. (v)After tracking, the advected motion is added back to theparticle coordinates to obtain the trajectories in the lab-oratory frame.

To identify frame to frame advective motion, we usestandard PIV-type image correlation methods [41]. Con-sider a region of size n×m pixels of two consecutive 2Dimages i − 1 and i. Let Ii−1(x, y) and Ii(x, y) be theintensity patterns as function of position (x, y) of these(sections of) images. The covariance is defined as:

cov[Ii−1(x, y), Ii(x, y)] =

1[(n×m)− 1]

n∑p=1

m∑q=1

[Ipqi−1 − 〈Ii−1〉

][Ipq

i − 〈Ii〉], (4)

where Ipqi−1 and Ipq

i are the intensities of the pixel corre-sponding to position (xp, yq) in each image and 〈Ii−1〉,〈Ii〉are the respective average intensities defined as 〈I〉 =

1[n×m]

n∑p=1

m∑q=1

Ipq. Analogously, the variance of a sin-

gle image I is:

var[I(x, y)] =1

[(n×m)− 1]

1∑

i=i

m∑

j=1

(Iij − 〈I〉)2. (5)

The correlation coefficient c[Ii−1(x, y), Ii(x, y)] of the twoimages is defined as:

c[Ii−1(x, y), Ii(x, y)] =cov[Ii−1(x, y)Ii(x, y)]√

var[Ii−1(x, y)]var[Ii(x, y)].

(6)The motion is obtained by shifting image i by a certainnumber of pixels (∆X ′, ∆Y ′) and computing the correla-tion coefficient c[Ii−1(x, y), Ii(x−∆X ′, y−∆Y ′)]. This isrepeated for shifts within a desired range until c is max-imized for ∆X ′ = ∆X, ∆Y ′ = ∆Y . Repeating the pro-cedure over subsequent frames yields the displacementas function of time (∆X, ∆Y )(ti) in the region of theimage series under consideration, Fig. 9. For stronglytime-dependent flows, a scan over all possible ∆X ′, ∆Y ′

in this region is required, but for smooth flows we haveimplemented a more efficient method in which image iis scanned in a narrow range centered around the shift(∆X, ∆Y )(ti−1) found from the previous images. A keypoint to recognize is that these shifts are quantized bythe size of a pixel, so while the motion obtained from im-age correlation is a good starting point for the tracking,

Y'

y2

y1

y3

x3

x1

x2

X'

FIG. 9: Illustration of the shifting and correlation procedure.Each particle in the section of the image under considera-tion is displaced by (∆x, ∆y) between frame 1 and frame2. The entire image section is shifted over a range of val-ues (∆X ′, ∆Y ′) to find the optimum shift (∆X, ∆Y ) thatmaximizes the correlation.

the subsequent tracking is necessary to achieve subpixelresolution of particle motion.

In most applications one can identify, at least withinthe microscope field of view (see Sec. III E), a principalaxis along which the flow takes place. The entire imagecan then be rotated such that the advective motion oc-curs in only one direction, which we denote by x. Toobtain the advection profile over the entire image, thecorrelation method is applied in different ways dependingon the uniformity of the motion and image dimensional-ity, as we describe now.

For 2D images, when the motion is spatially uniform inthe xy plane, the above procedure is applied to the entireimage, Fig. 10(a), resulting in ∆R(r, ti) = ∆X(r, ti) =∆X(ti). The advective motion can also depend on theposition y transverse to the flow, see Fig. 10(b). Theimage is then decomposed into strips, which are shiftedand correlated separately, yielding an advection profile∆X(yq) discretized at the centers yq of the strips.

For 3D images, the basic manifestation of nonuniformflow is simple shear, Fig. 10(c), where the average motionis a function of z only. The sheared volume is then de-composed in xy slices at different z and the correlationprocedure is performed on each 2D slice separately. Amore complex flow is shown in Fig. 10(d), where shearoccurs both in the y and z direction, as for instance in 3Dchannel flows. Here the 3D images are first decomposedin xy slices at different z and then each slice is furtherdecomposed in y-bins for which the motion is analyzed.

In the most general form ∆R is both position and timedependent, ∆R = ∆R(r, ti), and includes shifts in allthree directions ∆X(r, ti), ∆Y (r, ti),∆Z(r, ti). For ex-ample, in experiments where a point-like force source isapplied in the medium, e.g. by dragging a magnetic ortracer bead through a colloidal suspension [91], the direc-tion and the magnitude of the ‘advected’ motion depend

12

FIG. 10: Examples of 2D (a,b) and 3D (c,d) image correlationprocedures. (a) A uniform shift ∆X across the entire field ofview maximizes the correlation. (b) The advected motion is afunction of y; the image is then decomposed in bins centeredat yq, each of which is shifted by ∆X(yq) to obtain maximumcorrelation. (c) The motion is a function of z only. The 3Dimage is decomposed in slices centered at zr, each of which isshifted by ∆X(zr) to obtain maximum correlation. (d) Theadvected motion is a function of y and z. Decomposition intoy and z bins yields the advection profile ∆X(yq, zr).

on x, y and z. In such a case the imaged area (vol-ume) must be decomposed into squares (cubes), and afull PIV analysis must be carried out to characterize themotion ∆R(xp, yq, zr, ti) in each element p, q, r. Anotherexample is sedimentation, where ∆X = 0, ∆Y = 0 but∆Z depends on z. Extensions to our algorithm dealingwith such cases are possible but we have not implementedthis. For our experiments (simple shear or channel flow)it suffices to consider shifts in one direction, which areindependent of the coordinate in that direction.

Once the advected motion has been measured, the dis-crete displacement profile ∆R(xp, yq, zr, ti) is then in-terpolated to give the continuous profile ∆R(x, y, z, ti).This is more appropriate to subtract from the particle co-ordinates, which are themselves continuously distributed.Using the continuous profile, the transformation of theposition rk(ti) = [xk, yk, zk](ti) of particle k in the lab-oratory frame to its position rk(ti) in the CM frame ofreference is:

rk(ti) = rk(ti)−i∑

j=1

∆R(rk(ti), tj), (7)

where ∆R(rk(ti), tj) is the past motion between framej and j − 1, at the current location rk(ti) of the par-ticle. This reduces to ∆R(r, tj) = ∆X(y, z, tj) for ourexperiments.

In the CM frame, the average particle motion (nearly)vanishes. The use of the classic CG algorithm at thispoint therefore allows particle tracking limited only bythe MSFD in the CM frame. Occasionally, the CM tracksshow some residual motion, in which case a modest im-provement may be obtained by using Iterated Tracking(Sec. VB). We also note that, in the event that a clearlyanisotropic non-affine motion is found after a first CM

tracking step, it could be beneficial to perform the track-ing in the CM-frame again using an ellipsoidal shape ofthe tracking area or volume rather than the standard cir-cular or spherical shape defined by the range RT . How-ever, the examples discussed in Sec. VI did not requiresuch a modification and we have not implemented this.

Once tracking is completed, the advected motion ∆Ris added back by inverting Eq. 7. This then provides thetrajectories of the particles in the laboratory frame ofreference.

2. Limitations of ’CIT’

A possible limitation to Correlated Image Tracking isa failure of the PIV-type correlation method. This couldarise when the frame to frame shifts are a significantfraction of the actual image or image-bin size. However,this method can work successfully for quite large shiftsamounting to nearly the image size. To show this, weanalyzed an experimental image series taken from plugflow. In this image series, the motion was slow enoughthat classic CG tracking worked; we use the results ofthis tracking as the “true” motion. Furthermore, therewas little relative particle motion (nearly zero MSFD).We then took pairs of images from this image series, sep-arated by n frames, to model an effectively much fasterflow rate. From the tracking, we know the shift ∆Xtra

that should best align these two images. For each pair ofimages, we calculate the shift ∆Xshift from image corre-lation, and compare that apparent shift to the true shift∆Xtra, in Fig. 11 (red symbols). The correlation coeffi-cient (thin blue line) decays roughly linearly with ∆Xtra,in line with the reduction of the correlated portion of theimages, and plateaus at a value ∼ 0.2, correspondingto the coefficient for two entirely different images of thesame system. The latter value is specific to our highdensity system, and may vary for different systems.

When comparing the image at time j and the shiftedimage at time j + 1, it is important to note that in eachcase the full image is compared. Thus the shifted imageat j+1 has some pixels wrapped around from one edge ofthe image to the other, which has no physical meaning.An alternate idea would be to crop the two images, sothat any pixels shifted outside the boundary are removed.Thus, when considering very large shift factors approach-ing the width of the image, only two narrow strips of thetwo images would be compared to determine the corre-lation coefficient. We find that this method is generallyless successful, despite its intuitive appeal. When the re-quired shifts are large, generally using the full image ismore likely to find the correct shift value. Comparing thelarge regions of the two images that could be potentiallycropped, these will be uncorrelated, and thus in generalthe correlation coefficient is dominated by the small re-gions that correctly overlap. This then yields the resultsof Fig. 11 where the correct shift value is found even for∆Xtra almost as much as the full width of the image.

13

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

300

350

400

Xsh

ift [p

ixel

s]

Xtra [pixels]

Cor

rela

tion

coef

ficie

nt

FIG. 11: Test of the shift-correlation procedure on images ofplug-flow. The shift in the flow direction ∆Xshift (�), ob-tained from the correlation method, is plotted against the ex-act accumulated displacement obtained from tracking ∆Xtra

(–), see the text. Image size: 331 × 580 pixels. Right axis:correlation coefficient (blue line) versus ∆Xtra.

A stronger limitation to the correlation procedure isan excessive amount of relative particle motion betweenframes. This obviously limits the correlation comparedto that of plug flow described above, but, more directly,large relative displacements cause failure of classic CGtracking in the CM frame, as discussed in Sec. VA 3.In practice we found that a maximum MSFD in the co-moving frame (the non-affine motion) of ∼ (0.3`)2 is thelimiting value for the method to work.

Non-affine motion during flow has an additional effectwhich may limit the tracking performance when subtract-ing shear advection accumulated over many frames. Sup-pose two particles start off as nearest neighbors, at adistance ≥ 2a, but at streamlines with different veloci-ties. After one frame, when the advected motion is sub-tracted, they remain at that distance if their motion wasfully affine. In contrast, with non-affine motion, theirseparation after advection removal may become < 2a,that is, they would apparently be in contact. Over sin-gle frames this effect is limited, i.e. particle separationsstill considerably exceed their non-affine displacementsbetween frames and tracking is not affected. However,after multiple frames, the non-affine displacements canaccumulate and the particle separation in the CM framemay have been reduced to a value comparable to thenon-affine frame to frame displacements. This gives riseto tracking errors based on the same arguments as inSec. V A3. The artificial reduction in particle separationin the CM frame is visible in the pair correlation functiong(r), Fig. 12, where we see that some particles apparentlyoverlap after the removal of large advective motion.

This problem is remedied by piecewise tracking of par-ticles in intervals over which the accumulated relativemotion is small. Each interval the particles are assignedtheir identification tag, and the full trajectory is ob-tained by matching the tags in a one-frame overlap ofthe intervals. For the case in Fig. 12, particle separa-

0 1 2 3 40

1

2

3

4

5=62%

rate=0.01 s-1lab.frame'CM' frame, =1'CM' frame, =2

g(r)

r [ m]

FIG. 12: Short distance behavior of the pair correlation func-tion g(r) from 3D coordinates of a sheared colloidal glass(φ = 0.62, a = 850 nm), in the laboratory frame (•), in the lo-cally co-moving (‘CM’) frame after subtracting advected mo-tion corresponding to an accumulated strain of ∆γ = 100%(�), and ∆γ = 200% (N). The local shear rate is γ = 0.01 s−1,strain accumulation during acquisition of a single 3D stack is∼ 1.5%, while between frames it is 4%. The data show thatparticles may come in close proximity after removal of largeadvected motion.

tions in the co-moving frame are ≥ a for accumulatedstrains ∆γ ≤ 100%, so that tracking is only affected for∆γ > 100%, but in general a different limit may applysince non-affine motion does not necessarily scale withaccumulated strain [70] and may depend on φ.

Summarizing, the main advantage of Correlated Im-age Tracking is that particles are tracked in a locallyco-moving frame of reference where limitations to track-ing are the same as in a quiescent system, as describedin Sec. V A3. In other words, our method permits thetracking of particles in flowing colloids to the same levelof accuracy as that in non-flowing systems.

VI. APPLICATIONS

In this section, we give example results from particletracking in the two flow geometries already introducedin Sec. III, viz., simple shear and capillary flow. Wealso describe the application of the confocal rheoscope toperform simultaneous rheology and velocity profiling ofsoft materials.

A. 2D tracking of channel flows

We start with a 2D example, the characterization ofpressure-driven channel flow of colloidal pastes. We useda φ ≈ 0.63 suspension of fluorescent PMMA spheres (ra-

14

dius a = 1.3 ± 0.1 µm, from microscopy), suspended ina mixture of CHB and mixed decalin for refractive indexand buoyancy matching. A pressure difference, ∆P , wasapplied to drive the suspension into the square channels(side 2b = 50 µm, smooth inner walls). The flow acrossthe full channel width was imaged in 2D at 107 framesper second (image size 44×58µm2, 256×320 pixels) at adepth of 17 µm from the lower surface. The images werecollected at a distance corresponding to ∼ 2000 particlediameters from the channel inlet where entry effects havedied out and the flow has negligible x-dependence on ascale compared to image size. The particles are locatedin 2D with accuracy ∼ 50 nm [20, 77]. According toEq. 3, distortion of the particle image becomes signif-icant only for V > V max

2D ' 600 µm/s, which exceedsthe maximum velocities at which we are able to trackthe particle. Assuming uniform motion, the results inSec. VB 2 suggest that Iterated Tracking fails for flowvelocities V > V max ' a × fscan ' 140 µm/s, but inpractice the limit is smaller due to the presence of largevelocity gradients.

For the experiments considered here, the flow profilesconsist of a central region of uniform velocity V ' Vc

and lateral zones adjacent to the channel walls wherethe shear is localized, Fig. 13. We have discussed thedetailed physics elsewhere [15]. To comment briefly, wenote that such a profile in itself, could be consistent withpredictions from the rheology of yield stress fluids [92].However, in contrast to yield stress fluid predictions, thevelocity profiles scale with flow rate and the width of theshear zones is independent of flow rate [15]. By examin-ing the microscopic dynamics of the particles we observethat they are dominated by interparticle collisions andcontacts with similarities to dense granular flows. Thesimilarities extend to the shape and scaling of the velocityprofiles, which allowed to interpret the data using a stressfluctuation model initially conceived for dry grains [15].

In these experiments, the advected motion ∆X(y) isanalyzed using Correlated Image Tracking with a schemesimilar to that in Fig. 10(b). We divide the image in hor-izontal bins with sufficiently uniform displacements andobtain ∆X(yq) from correlation in each of these. Forchannels with smooth walls, the particles near the wallare arranged in well defined layers, Fig. 13. Combinedwith the fact that velocity gradients are largest near theedge, this motivates the choice of one-particle-wide binsnear the walls and a larger bin in the center. An ex-ample of the displacement profile ∆X(yq, ti) is shown inFig. 13(b) and, combined with Eq. 7, can be used to trackthe particles in the CM frame. Figure 14 shows a parti-cle trajectory inside a shear zone both in the laboratory(left) and in the CM frame (right).

In Figs. 15(a),(b) we first show the displacement dis-tribution functions for a small velocity Vc ' 28 µm/s(1.5 pixel/frame), for which particles can be tracked di-rectly using classic CG tracking. These vanish smoothlywithin the tracking range and are consistent with theresults (in the co-moving frame) from Iterated Tracking

-1.0 -0.5 0.0 0.5 1.00

15

30

45

60

0.0

0.5

1.0

<V>

[m

/s]

y/beff a)

n/n

max

1 2 3 4 5 6 7 8 9

b)

x1

x1

X1X2X3X4X5X6X7X8X9

FIG. 13: (a) Velocity profile (◦,left axis) and histogram ofparticle positions (–,right axis) as a function of the trans-verse coordinate y normalized by effective channel half-widthbeff = b−a. The experiment refers to the flow of a 63.5% sus-pension in a 2b = 50µm wide smooth, square channel. Theparticles are arranged in well defined layers close to the wallsand the shear decays towards the channel center where thesuspension flows as a plug. The vertical lines highlight the y-bins and the numbers match those in Fig. (b). (b) Illustrationof the non-uniform shifts during 2D channel flows. The re-sulting displacements ∆X(yq) (q = 1..9) are subtracted fromthe particle coordinates. Image size: 34 µm× 50 µm.

(open squares) and with the results from tracking in theCM frame, i.e. after removing non-uniform motion (filledsquares).

For faster flow, Vc ' 223 µm/s (∼ 12 pixels/frame),Correlated Image Tracking is required. In Fig. 15(c) weshow P (∆x) from classic CG tracking (filled circles). Thestrong asymmetry and sharp cutoff show the inapplicabil-ity of this method. Results from iterative tracking, i.e. af-ter removing uniform motion obtained from a precedingdirect tracking step (open squares), still reveal asymme-try and are not reliable. Instead, the results from Cor-related Image Tracking (full squares) are symmetric andshow virtually no cut-off effects. The y-dependent mo-tion after restoring these tracks to the laboratory frameare consistent with the advection profile ∆X(y) from cor-relation, confirming the success of the method. We notethat for the different methods, the distributions P (∆y),shown in Figs. 15(d), do not show any difference despite

15

5 m

1 m

FIG. 14: Example of particle tracks in the laboratory (left)and in the co-moving frame (right). Circles indicate the startof the trajectory. The particle is situated in a shear zone(local shear rate 5 s−1) of a suspension flowing into a squarechannel with smooth inner walls. The length of the track is∼ 2.3 s.

-10-8 -6 -4 -2 0 2 4 6 8100

101

102

103

104

-8 -6 -4 -2 0 2 4 6 8 10100

101

102

103

104

P [n

. par

ticle

s]

∆x [pixels]

a)

∆y [pixels]

P [n

. par

ticle

s]

b)

-10-8 -6 -4 -2 0 2 4 6 8100

101

102

103

104

P [n

. par

ticle

s]

∆x [pixels]

c)

-8 -6 -4 -2 0 2 4 6 8 10100

101

102

103

104

P [n

. par

ticle

s]

∆y [pixels]

d)

FIG. 15: Comparison of displacement distributions along x(a),(c) and y (b),(d) for 2D channels flows. Classic CGtracking in the laboratory frame (•), Iterated CG tracking(�), Correlated Image Tracking (�). (a),(b): slow flow

(Vc = −1.5 pixel/frame); classic CG tracking is sufficient.

(c),(d): fast flow (Vc = −12 pixel/frame). Here the classicCG tracking is inapplicable in the lab frame of reference (•),but Correlated Image Tracking utilizing CG tracking in theco-moving frame of reference is successful.

the variations observed in P (∆x). This again illustratesthat care must taken in interpreting displacement distri-bution functions from particle tracking as described inSec. VA.

Finally, we also tested the method on ' 30% volumefraction suspensions in quasi-2D channels yielding the ex-pected parabolic flow profiles [77, 93]. With CorrelatedImage Tracking we have been able to successfully trackparticles in flows with velocity as high as 250 µm/s cor-responding to almost twice the limit V max mentionedabove.

B. 3D particle tracking in simple shear flow

Next, we move to full 3D imaging and consider a col-loidal glass in steady shear flow, measured in the rheome-ter or the shear cell with rough, coated surfaces. The

apparatus has already been described in Sec. III. Thecolloids (radius a = 850 nm) are suspended in a charge-screened CHB-decalin mixture for refractive index anddensity matching (η = 2.6 mPa·s). The volume fraction,measured from the average Voronoi volume determinedfrom particle coordinates, is φ = 0.62. Each 3D imageconsists of 76 slices (256×256 pixels each, 2D frame ratef = 45 s−1), imaged over a height zmax − z0 = 15 µmwith z0 either 10 µm or 15 µm (z = 0 at the coverslide) and was acquired in ∼ 1.7 s. The voxel size is0.11× 0.11× 0.20 µm3. The local shear rate γ which wemeasure (see below) may exceed the overall applied rateγa. This is due to global shear localization, which weobserved directly in velocity profiles v(z) measured fromimage series on a coarser z-scale, which we will describein detail elsewhere [94].

The intrinsic accuracy for locating particles (with therefinement in [80]), obtained from the MSD in a ran-dom close packed system (no flow) is ±30 nm in x,y and±70 nm in the z-direction. Under flow, using Eq. 3 inSec. IVB, the velocity for which distortion of the particleimage sets in is V max

3D ∼ 1 µm/s, exceeding our largestmeasured velocities V max ∼ γzmax. The error due toshort time thermal displacement is ±30 nm, not exceed-ing those mentioned above.

In Fig. 16 we show results for slow shear, where classicCG tracking is sufficient. Figure 16(a) shows the distri-bution of frame to frame displacements along x (the ve-locity direction) and y (the vorticity direction). The for-mer is shifted and slightly broader compared to P (∆y)due to the z-dependence of ∆X and the zero-velocityplane being outside the image, which is illustrated by thedisplacement profile ∆x(z) (as obtained from the particletrajectories) in Fig. 16(b). The profile is linear on aver-age on this z-scale, the slope gives the local shear rateγ = 9.3 × 10−4 s−1, and it extrapolates to zero withinexperimental uncertainties at the cover-slide (z = 0),confirming that the coating provides a stick boundarycondition.

Locally, the shear induces plastic breaking of the par-ticle cages, causing diffusive behavior at long times, asshown by the MSDs in Fig 16(c) (see also [70]). Thisdiffusion contrasts a quiescent colloidal glass where thelong time particle dynamics remains caged. Figure 16(c)includes the MSDs in the three directions x, y and z,where for the x direction we use 〈∆x2(t)〉, with

∆x(t) = x(t)− x(0)− γ

∫ t

0

z(t′)dt′, (8)

which represents only the non-affine displacement. Notethat with this definition, the usual effect of Taylor disper-sion is suppressed, see e.g. [95]. As observed, the MSDsare nearly isotropic, i.e. the (non-affine) structural relax-ation due to cage breaking is nearly the same for all di-rections [70]. The distribution of the non-affine displace-ments dx is also included in Fig. 16(a), and coincideswith P (∆y).

16

-0.5 0.0 0.5100

102

104

10.0 20.0 30.00

1

1 10 100

0.01

0.1

∆t=4s(a)

P [n

. par

ticle

s]

~∆x ∆y

∆x (non-affine)

~∆x, ∆x, ∆y [µm]

(b) ∆t=40s(10 frames)

∆x [µ

m]

z [µm]

(c)

~<∆x2> <∆y

2><∆z

2> 2 Dy ∆t

∆t [s]

<∆r j2 >

[µm

2 ]

FIG. 16: Direct tracking of a sheared glass at γ = 9.3 ×10−4 s−1 (a) Histograms of frame to frame displacements.(b) Displacements ∆x versus z for all particles over 10 frames(∆t = 40 s). The line is a linear fit, the slope of which givesthe local shear rate γ = 9.3 × 10−4 s−1. (c) Mean squareddisplacement 〈∆y2(∆t)〉 in the vorticity direction, 〈∆z2(∆t)〉in the gradient direction, and the non-affine MSD 〈∆x2(∆t)〉in the velocity direction. Line: 〈∆y2(∆t)〉 = 2Dy∆t withDy = 3.6× 10−4 µm2/s.

We now turn to faster shear, γ = 0.019 s−1. We firstconsider a 2D image series taken at z = 18 µm from the3D stacks. From correlation we find a uniform motionwith a constant velocity V = 1.6 µm/frame. To com-pare with Sec. VB 2, using ` ∼ 1.8 µm as the particlespacing, this corresponds to a reduced shift s/` ' 1 be-tween frames. We locate the particles in 2D, and trackthem both with iterated CG tracking, and with Corre-lated Image Tracking, using Eq. 7. In Fig. 17 the frame toframe displacement distributions from Iterated Trackingare shown, with P (∆x) evaluated in a co-moving frameso that 〈∆x〉 = 0. These distributions are cut-off at∆r = RT = 1 µm by definition; the tracking programdoes not consider possible displacements larger than RT

(frame-to-frame). PIT (∆x) is asymmetric, similar to theMC data in Fig. 8(b).

This changes when we use Correlated Image Track-ing and then examine the measured displacements inthe co-moving frame. Now, the displacement distribu-tion functions, Fig. 17, are no longer cut-off and coin-cide for ∆y and ∆x (in the co-moving frame), indicatingcorrect tracking. The inset shows the resulting MSDs〈∆y2(∆t)〉 and 〈∆x2(∆t)〉 (the latter again in the co-moving frame). As for slow shear, the dynamics is nearlyisotropic. A more detailed description on the shear-induced structural relaxation is given in [70]. From the

-1.0 -0.5 0.0 0.5 1.0100

101

102

103

104

1 10

0.1

1

Px

CI T

Py

CI T

Px

I T

Py

I T

P [n

. par

ticle

s]

∆x, ∆y [µm]

<∆x2 >

[µm

2 ]

∆t [s]

FIG. 17: Histograms of frame to frame displacements, in acomoving frame with 〈∆x〉 = 0, obtained from iterated CGtracking (PIT, squares) and from Correlated Image Tracking(PCIT, circles) for 2D imaging in the velocity vorticity planeof a sheared glass (V = 1.6 µm/frame, γ = 0.019 s−1). Inset:the MSD 〈∆y2(∆t)〉 (�), and 〈∆x2(∆t)〉 in the co-movingframe (�), from tracking in the CM frame. Line: 〈∆y2〉 =2Dy∆t with Dy = 5.4× 10−3 µm2/s.

frame to frame MSD 〈∆y2(∆t = 4 s)〉 ' 0.05 µm2 we ob-tain 〈∆r2〉/`2 = ε2 ' 0.03, within the limits for trackingin a concentrated quiescent system, Fig. 6.

To analyze the 3D data, we use Correlated ImageTracking where the correlation procedure is performedon image sequences at different heights zr = 10 + 2r µm(r integer), as shown in Fig. 10(c). From this, we obtainthe accumulated displacements profile ∆Xa(zr, t, ∆t) =∑t

t−∆t ∆X(zr, t), an example of which is shown inFig. 18(a) for t = 40 s and ∆t = 16 s. The profileis linear and again shows approximately stick boundaryconditions. Time averages 〈∆Xa(zr, t, ∆t = 16 s)〉t (notshown) virtually overlap these data, showing that theflow is steady. To subtract this advected motion fromthe particle coordinates, we use the linear interpolationprofile ∆Xa(z, t, ∆t) shown by the lines connecting thesymbols. Since both accumulated strain and non-affinedisplacements are large in this case, we performed boththe subtraction of ∆X(z, t) (see Eq. 7) and the “piece-wise tracking”, in intervals of 10 frames (∆γ = 80%),as described in Sec. VC2. The resulting displacementdistribution functions are shown in Fig. 18(b), both forthe non-affine and the real displacements. Note the largerange, 0 µm ≤ ∆x ≤ 3 µm, of the latter, resulting fromthe strong z gradient in advected motion. Finally, weshow the MSDs for x, y and z calculated from these 3Ddata. The results for ∆x2 and ∆y2 match those in theinset to Fig. 17 while ∆z2 shows that despite the largeshear rate, the dynamics remain nearly isotropic.

17

0 5 10 15 20 250

5

10

-1 0 1 2 3100

101

102

103

104

1 10 1000.01

0.1

1

(a) ∆t=16s

∆Xa [

µm]

z [µm]

∆x, ∆x, ∆y [µm]

(b) ∆t = 4s

~P(∆x)P(∆x) P(∆y)

P [n

. par

ticle

s]

~

~

(c)

<∆y2>

<∆x2>

<∆z2>

< ∆r j2 >

[µm

2 ]

∆t [s]

FIG. 18: 3D analysis of a sheared glass at γ = 0.019 s−1.(a) (•) accumulated displacement ∆Xa(zr, t, ∆t) from im-age correlation, for t = 40 s over ∆t = 16 s (4 frames).Connecting lines represent the interpolating profile, defin-ing ∆Xa(z, t, ∆t). Line: linear fit giving an accumulatedstrain d∆Xa/dz = 0.28. (b) Distribution of frame to framedisplacements P (∆x) and P (∆y) after tracking in the CMframe (RT = 1.1 µm) and restoring the coordinates in thelaboratory frame. Also shown is P (∆x) of the non-affine x-displacements, using Eq. 8 and γ = 0.019 s−1. (c) the (non-affine) MSD in the three directions. The data are consistentwith those in Fig. 17, inset. Line: 〈∆y2(∆t)〉 = 2Dy∆t withDy = 5.4× 10−3 µm2/s.

C. Rheology and velocity profiling

As a last example we describe the results of experi-ments on a more dilute colloidal suspension using theconfocal rheoscope in cone plate geometry. Here we mea-sure simultaneously the rheological response and map thevelocity profile during flow, see Fig. 5.

The sample consists of a φ ∼ 55% suspension of non-fluorescent PMMA-PHS colloids (radius = 150 nm) in anindex matching (decalin-tetralin) mixture, seeded with∼ 0.5% fluorescent tracers (radius = 652 nm). Both thecone and the cover slide are coated with a layer of tracerparticles. From the image series of the cone motion, wecan verify the rotation speed of the rheometer during op-eration and by focusing on the top and bottom coatingswe can map the spatial profile of the cone plate geom-etry. In Fig. 19 we show the variation of the gap sizeas function of the distance r from the center of the conemeasured using the lateral objective translation. The

0 2 4 6 80

20

40

60

80

100

120

140

z [

m]

R [mm]

Truncation gap 32 +/- 1 m (32 m)Cone angle 1:08:04 degrees (1:00:12)

FIG. 19: Gap profile of the cone plate geometry, measuredby confocal microscopy with fluorescent coating on both thebottom (glass) surface and the surface of the cone (radius20 mm). The truncation gap is clearly visible and the profileextrapolates at zero height in the center. The nominal valuesof the truncation gap and of the cone angle are in brackets inthe figure legend.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.01 0.1 1 10

1

10

100

[Pa]

[s-1]

V/V

cone

Z/Zgap

FIG. 20: Main panel: steady state stress σ versus appliedshear rate γ for a φ ∼ 55% suspension of hard-sphere colloidsof radius 150 nm. Inset: velocity profiles, normalized by thevelocity of the cone Vcone, as a function of the height, normal-ized by the local gap size Zgap = θr of the geometry. Datafor γ = 10 s−1 are taken at different positions r = 2.5 mm(•), r = 4 mm (�) and r = 5 mm (N) and for γ = 0.5 s−1

data are taken at r = 5 mm (�).

truncation gap is nicely resolved and the data show thatbending of the cover slide is negligible.

The velocity measurements were performed in twoways. One is the time-resolved mode: similarly to whatis described in Sec. VIB, we rapidly scan 3D stacks of∼ 20 to 50 slices, covering the entire gap from z = 0to z = Zgap = θr with θ the cone angle, Fig. 5(b).We use an oil-immersion objective with 200 µm workingdistance (60× magnification) and scan at equal speed

18

both up and down to avoid disturbance of the sam-ple due to large sudden displacements of the objective.By extracting the image series at each height and us-ing the correlation method (Sec. VC 1), we obtain thetime resolved displacements and shear profile. Typicallyit takes ∆t ∼ 1 s to scan the gap. For the particularcase of images containing tracers, displacements up toapproximately half the image size S can be measured.Therefore, this mode is successful for maximum veloci-ties v ∼ S/(2∆t) ∼ 50 µm/s, with S = 100 µm, whichtranslates to γar < S/(2θ∆t). In the second mode (‘step-ping’) we simply record a time series at each z and re-construct the velocity profile v(z). Here the maximumvelocity which can be measured is considerably larger:v ∼ Sfscan/2 ∼ 5 mm/s for fscan = 100 s−1. This cor-responds to a maximum shear rate of γ ∼ 30 s−1 forr = 10 mm. Note that even larger velocities can be mea-sured by using smaller magnification objectives.

Figure 20 (main panel) shows the steady state flowcurve of the sample. Since the suspension at this vol-ume fraction is in the proximity of the glass transition(φg ∼ 57%), the expected low-rate Newtonian regimeoccurs at shear rates below our experimental window. Inthe regime we measured, 0.006 s−1 < γ < 30 s−1 the sam-ple exhibits strongly nonlinear rheology with pronouncedshear thinning response (the viscosity decreases dramat-ically on increasing shear rate). In the inset we showsome velocity profiles, measured in the ‘stepping’ modeat different shear rates and various distances r from thecenter of the cone. The corresponding gap sizes rangefrom Zgap ' 50 µm at r = 2.5 mm to Zgap ' 100 µmat r = 5 mm. It is clear that the normalized profilesare linear, independent of γ and r. In the case of largervolume fractions, above the glass transition, φ > φg, weobserve either slip or shear localization, depending on theboundary conditions. These results will be discussed in

detail elsewhere [76, 94].

VII. CONCLUSION

In this paper we have described new instrumentationand analysis algorithms for 2D and 3D imaging studiesof concentrated (colloidal) suspensions during flow. Thecombination of fast confocal microscopy and controlledflow, such as in a rheometer where simultaneous rheolog-ical information is available, opens up new horizons forthe study of driven soft matter systems at high concen-trations. Our evaluation of the CG tracking algorithmestablishes the validity of previous experiments on col-loidal dynamics. Finally, our method for particle track-ing in a locally co-moving frame allows us to investigatethe affine and non-affine dynamics of colloids during flowup to relatively large velocity (gradients), limited primar-ily by the amount of non-affine motion during flow. Themethod could therefore be of use in a variety of other ap-plications, including the study of granular flow at single-particle level.

VIII. ACKNOWLEDGMENTS

We thank N. Pham, J. Arlt and K.N. Pham for ad-vice on the experiments and design of the confocal rheo-scope, A.B. Schofield for particle synthesis, A. Garrie,A. Downie, and V. Devine for technical support, and E.Kim for providing the 3D Monte Carlo simulation datafor hard-spheres. Eric R. Weeks thanks J.C. Crockerfor helpful discussions over many years. R. Besselingand W.C.K. Poon acknowledge funding through EPSRCGR/S10377/01 and EP/D067650. L. Isa was funded bythe EU network MRTN-CT-2003-504712. E.R. Weekswas funded through NSF DMR-0603055 (US).

[1] P. N. Pusey, in: J. P. Hansen, D. Levesque, J. Zinn-Justin (Eds.), Liquids, Freezing and Glass Transition,Elsevier, 1991, Ch. 10, pp. 767–942.

[2] W. C. K. Poon, P. N. Pusey, in: M. Baus, L. F. Full,J. P. Ryckaert (Eds.), Observation, Prediction and Sim-ulation of Phase Transitions in Complex Fluids, Kluwer,1995, Ch. 1, pp. 3–51.

[3] W. C. K. Poon, The physics of a model colloid–polymermixture, J. Phys.-Condens. Mat. 14 (2002) R859–R880.

[4] D. G. A. L. Aarts, M. Schmidt, H. N. W. Lekkerkerker,Interfacial dynamics in demixing systems with ultralowinterfacial tension, Science 304 (5672) (2004) 847–850.

[5] V. J. Anderson, H. N. W. Lekkerkerker, Insights intophase transition kinetics from colloid science, Nature416 (6883) (2002) 811–815.

[6] K. A. Dawson, The glass paradigm for colloidal glasses,gels, and other arrested states driven by attractive in-teractions, Curr. Opin. Colloid Interface Sci. 7 (2002)218–227.

[7] F. Sciortino, P. Tartaglia, Glassy colloidal systems, Adv.

Phys. 54 (2005) 471–524.[8] E. Zaccarelli, Colloidal gels: equilibrium and non-

equilibrium routes, J. Phys.-Condens. Mat. 19 (2007)323101.

[9] S. M. Fielding, P. Sollich, M. E. Cates, Aging and rhe-ology in soft materials, J. Rheol. 44 (2000) 323–369.

[10] M. Fuchs, M. E. Cates, Theory of nonlinear rheologyand yielding of dense colloidal suspensions, Phys. Rev.Lett. 89 (2002) 248304.

[11] V. Kobelev, K. S. Schweizer, Strain softening, yielding,and shear thinning in glassy colloidal suspensions, Phys.Rev. E 71 (2005) 021401.

[12] D. I. Wilson, S. L. Rough, Exploiting the curious char-acteristics of dense solid-liquid pastes, Chem. Eng. Sci.61 (2006) 4147–4154.

[13] J. A. Lewis, Colloidal processing of ceramics, J. Am.Ceramic Soc. 83 (2000) 2341–2359.

[14] H. M. Jaeger, S. R. Nagel, R. P. Behringer, Granularsolids, liquids, and gases, Rev. Mod. Phys. 68 (4) (1996)1259–1273.

19

[15] L. Isa, R. Besseling, W. C. K. Poon, Shear zones andwall slip in the capillary flow of concentrated colloidalsuspensions, Phys. Rev. Lett. 98 (18) (2007) 198305.

[16] P. Coussot, C. Ancey, Rheophysical classification of con-centrated suspensions and granular pastes, Phys. Rev.E 59 (1999) 4445–4457.

[17] P. Lindner, T. Zemb (Eds.), Neutron X-rays & light:scattering methods applied to soft condensed matter,North Holland, 2002.

[18] J. B. Perrin, Atoms, Constable, London, 1916.[19] V. Prasad, D. Semwogerere, E. R. Weeks, Confocal mi-

croscopy of colloids, J. Phys.: Condens. Mat. 19 (2007)113102.

[20] J. C. Crocker, D. G. Grier, Methods of digital videomicroscopy for colloidal studies, J. Colloid Interface Sci.179 (1996) 298.

[21] K. Zahn, G. Maret, Dynamic criteria for melting in twodimensions, Phys. Rev. Lett. 85 (2000) 3656–3659.

[22] K. Zahn, A. Wille, G. Maret, S. Sengupta, P. Nielaba,Elastic properties of 2D colloidal crystals from videomicroscopy, Phys. Rev. Lett. 90 (2003) 155506.

[23] F. Ebert, P. Keim, G. Maret, Local crystalline order ina 2d colloidal glass former, Eur. Phys. J. E 26 (2008)161–168.

[24] J. Baumgartl, C. Bechinger, On the limits of digitalvideo microscopy, Europhys. Lett. 71 (2005) 487–493.

[25] M. S. Elliot, W. C. K. Poon, Conventional optical mi-croscopy of colloidal suspensions, Adv. Colloid InterfaceSci. 92 (2001) 133–194.

[26] M. S. Elliot, B. T. F. Bristol, W. C. K. Poon, Directmeasurement of stacking disorder in hard sphere col-loidal crystals, Physica A 235 (1997) 216–223.

[27] T. Wilson, Confocal Microscopy, Academic Press, SanDiego, 1990.

[28] A. van Blaaderen, P. Wiltzius, Real-space structure ofcolloidal hard-sphere glasses, Science 270 (5239) (1995)1177–1179.

[29] A. D. Dinsmore, E. R. Weeks, V. Prasad, A. C. Levitt,D. A. Weitz, Three–dimensional confocal microscopy ofcolloids, App. Optics 40 (2001) 4152–4159.

[30] P. Habdas, E. R. Weeks, Video microscopy of colloidalsuspensions and colloidal crystals, Curr. Opin. ColloidIn. 7 (2002) 196–203.

[31] D. Semwogerere, E. R. Weeks, Confocal microscopy, En-cyclopedia of Biomaterials and Biomedical Enigineer-ing, Taylor & Francis (2005).

[32] U. Gasser, E. R. Weeks, A. Schofield, P. N. Pusey, D. A.Weitz, Real-space imaging of nucleation and growth incolloidal crystallization, Science 292 (2001) 258–262.

[33] W. K. Kegel, A. van Blaaderen, Direct observation ofdynamical heterogeneities in colloidal hard-spheres sus-pensions, Science 287 (2000) 290–293.

[34] E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield,D. A. Weitz, Three–dimensional direct imaging of struc-tural relaxation near the colloidal glass transition, Sci-ence 287 (2000) 627–631.

[35] W. C. K. Poon, R. Besseling, L. Isa, A. B. Schofield,Imaging of hard-sphere colloids under flow, Adv. Poly-mer Sci. to be published.

[36] K. N. Pham, G. Petekidis, D. Vlassopoulos, S. U. Egel-haaf, P. N. Pusey, W. C. K. Poon, Yielding of colloidalglasses, Europhys. Lett. 75 (2006) 624–630.

[37] K. N. Pham, G. Petekidis, D. Vlassopoulos, S. U. Egel-haaf, W. C. K. Poon, P. N. Pusey, Yielding behavior of

repulsion- and attraction-dominated colloidal glasses, J.Rheol. 52 (2008) 649–676.

[38] W. Russel, M. Grant, Distinguishing between dynamicyielding and wall slip in a weakly flocculated colloidaldispersion, Colloids Surf. A 161 (2000) 271.

[39] R. Buscall, J. I. McGowan, A. J. Morton-Jones, Therheology of concentrated dispersions of weakly attract-ing colloidal particles with and without wall slip, Col-loids Surf. A 37 (1993) 621.

[40] P. D. Olmsted, Perspectives on shear banding in com-plex fluids, Rheol. Acta 47 (2008) 283–300.

[41] M. Raffel, C. Willert, J. Kompenhans, Particle ImageVelocimetry, a Practical Guide, Springer, Berlin, 1998.

[42] S. P. Meeker, R. T. Bonnecaze, M. Cloitre, Slip and flowin soft particle pastes, Phys. Rev. Lett. 92 (19) (2004)198302.

[43] S. P. Meeker, R. T. Bonnecaze, M. Cloitre, Slip andflow in pastes of soft particles: Direct observation andrheology, J. Rheol. 48 (6) (2004) 1295–1320.

[44] J.-B. Salmon, S. Manneville, A. Colin, B. Pouligny, Anoptical fiber based interferometer to measure velocityprofiles in sheared complex fluids, Eur. Phys. J. Appl.Phys. 22 (2003) 143.

[45] S. Manneville, L. Becu, A. Colin, High-frequency ul-trasonic speckle velocimetry in sheared complex fluids,Eur. Phys. J. Appl. Phys. 28 (2004) 361.

[46] L. Becu, S. Manneville, A. Colin, Yielding and flowin adhesive and nonadhesive concentrated emulsions,Phys. Rev. Lett. 96 (13) (2006) 138302.

[47] L. Becu, D. Anache, S. Manneville, A. Colin, Evidencefor three-dimensional unstable flows in shear-bandingwormlike micelles, Phys. Rev. E 76 (1) (2007) 011503.

[48] E. Fukushima, Nuclear magnetic resonance as a toolto study flow, Annu. Rev. Fluid Mech. 31 (1999, andreferences therein) 95–123.

[49] P. T. Callaghan, Rheo-NMR: nuclear magnetic reso-nance and the rheology of complex fluids, Rep. Progr.Phys. 62 (4) (1999) 599–670.

[50] D. Bonn, S. Rodts, M. Groenink, S. Rafai,N. Shahidzadeh-Bonn, P. Coussot, Some applications ofmagnetic resonance imaging in fluid mechanics: Com-plex flows and complex fluids, Annu. Rev. Fluid Mech.40 (2008) 209–233.

[51] L. F. Gladden, P. Alexander, Applications of nuclearmagnetic resonance imaging in process engineering,Meas. Sci. Technol. 7 (3) (1996) 423–435.

[52] G. Ovarlez, F. Bertrand, S. Rodts, Local determinationof the constitutive law of a dense suspension of noncol-loidal particles through magnetic resonance imaging, J.Rheol. 50 (3) (2006) 259–292.

[53] J. S. Raynaud, P. Moucheront, J. C. Baudez,F. Bertrand, J. P. Guilbaud, P. Coussot, Direct determi-nation by nuclear magnetic resonance of the thixotropicand yielding behavior of suspensions, J. Rheol. 46 (3)(2002) 709–732.

[54] N. Huang, G. Ovarlez, F. Bertrand, S. Rodts, P. Cous-sot, D. Bonn, Flow of wet granular materials, Phys. Rev.Lett. 94 (2005) 028301.

[55] A. Fall, N. Huang, F. Bertrand, G. Ovarlez, D. Bonn,Shear thickening of cornstarch suspensions as a reen-trant jamming transition, Phys. Rev. Lett. 100 (1)(2008) 018301.

[56] V. Breedveld, D. van den Ende, A. Tripathi, A. Acrivos,The measurement of the shear-induced particle and

20

fluid tracer diffusivities in concentrated suspensions bya novel method, J. Fl. Mech. 375 (1998) 297.

[57] V. Breedveld, D. van den Ende, M. Bosscher,R. J. J. Jongschaap, J. Mellema, Measuring shear-induced self-diffusion in a counterrotating geometry,Phys. Rev. E 63 (2001) 021403.

[58] M. D. Haw, W. C. K. Poon, P. N. Pusey, Direct ob-servation of oscillatory-shear-induced order in colloidalsuspensions, Phys. Rev. E 57 (6) (1998) 6859–6864.

[59] M. D. Haw, W. C. K. Poon, P. N. Pusey, P. Hebraud,F. Lequeux, Colloidal glasses under shear strain, Phys.Rev. E 58 (4) (1998) 4673–4682.

[60] P. Smith, G. Petekidis, S. U. Egelhaaf, W. C. K. Poon,Yielding and crystallization of colloidal gels under oscil-latory shear, Phys. Rev. E 76 (2007) 041402.

[61] R. Biehl, T. Palberg, Modes of motion in a confinedcolloidal suspension under shear, Europhys. Lett. 66 (2)(2004) 291–295.

[62] V. A. Tolpekin, M. H. G. Duits, D. van den Ende,J. Mellema, Aggregation and breakup of colloidal par-ticle aggregates in shear flow, studied with video mi-croscopy, Langmuir 20 (7) (2004) 2614–2627.

[63] H. Hoekstra, J. Vermant, M. J, Flow-induced anisotropyand reversible aggregation in two-dimensional suspen-sions, Langmuir 21 (24) (2003) 9134–9141.

[64] E. J. Stancik, G. T. Gavranovic, M. J. O. Widenbrant,A. T. Laschtisch, J. Vermant, G. G. Fuller, Structureand dynamics of particle monolayers at a liquid-liquidinterface subject to shear flow, Faraday Discuss. 123(2003) 145–156.

[65] P. Varadan, M. J. Solomon, Direct visualization of flow-induced microstructure in dense colloidal gels by confo-cal laser scanning microscopy, J. Rheol. 47 (4) (2003)943–968.

[66] I. Cohen, T. Mason, D. Weitz, Shear-induced configura-tions of confined colloidal suspensions, Phys. Rev. Lett.93 (2004) 046001.

[67] T. Solomon, M. J. Solomon, Stacking fault structure inshear-induced colloidal crystallization, J. Chem. Phys.124 (13) (2006) 134905.

[68] D. Derks, H. Wisman, A. van Blaaderen, A. Imhof, Con-focal microscopy of colloidal dispersions in shear flowusing a counter-rotating cone plate shear cell, J. Phys.-Condens. Mat. 16 (2004) S3917–S3927.

[69] Y. L. Wu, J. H. J. Brand, J. L. A. van Gemert, J. Verk-erk, H. Wisman, A. van Blaaderen, A. Imhof, A newparallel plate shear cell for in situ real-space measure-ments of complex fluids under shear flow, Rev. Sci. In-strum. 78 (2007) 103902.

[70] R. Besseling, E. R. Weeks, A. B. Schofield, W. C. K.Poon, Three-dimensional imaging of colloidal glassesunder steady shear, Phys. Rev. Lett. 99 (2007) 028301.

[71] H. Xu, A. P. Reeves, M. Y. Louge, Measurement error inthe mean and fluctuation velocities of spherical grainsform a computer analysis of digital images, Rev. Sci.Instrum. 75 (4) (2004) 811–819.

[72] L. Antl, J. W. Goodwin, R. D. Hill, R. H. Ottewill, S. M.Owens, S. Papworth, J. A. Waters, The preparation ofpoly(methyl methacrylate) lattices in non-aqueous me-dia, Colloid. Surface. 17 (1) (1986) 67–78.

[73] B. J. Ackerson, P. N. Pusey, Shear–induced order insuspensions of hard–spheres, Phys. Rev. Lett. 61 (8)(1988) 1033–1036.

[74] A. Yethiraj, A. van Blaaderen, A colloidal model system

with an interaction tunable from hard sphere to soft anddipolar, Nature 421 (6922) (2003) 513–517.

[75] P. J. Lu, J. C. Conrad, H. M. Wyss, A. B. Schofield,D. A. Weitz, Fluids of clusters in attractive colloids,Phys. Rev. Lett. 96 (2006) 028306.

[76] P. Ballesta, R. Besseling, L. Isa, G. Petekidis,W. C. K. Poon, arXiv:0807.1437v1 (2008).

[77] L. Isa, R. Besseling, E. R. Weeks, W. C. K. Poon, Exper-imental studies of the flow of concentrated hard spheresuspensions into a constriction, J. Phys.–Conference Se-ries 40 (2006) 124–132.

[78] E. R. Weeks, J. C. Crocker, Particle tracking using IDL:website, http://www.physics.emory.edu/∼weeks/idl/.

[79] C. P. Brangwynne, G. H. Koenderink, E. Barry,Z. Dogic, F. C. MacKintosh, D. A. Weitz, Bending dy-namics of fluctuating biopolymers probed by automatedhigh-resolution filament tracking, Biophys. J. 93 (2007)346–359.

[80] M. Jenkins, S. U. Egelhaaf, Confocal microscopy of col-loidal particles: Towards reliable, optimum coordinates,Adv. Col. Int. Sci. 136 (1–2) (2008) 65–92.

[81] P. N. Pusey, R. J. A. Tough, Hydrodynamic interactionsand diffusion in concentrated particle suspensions, Fara-day Discuss. 76 (76) (1983) 123–136.

[82] C. Beenhakker, P. Mazur, Diffusion of spheres in a con-centrated suspension .2., Physica A 126 (1984) 349–370.

[83] M. Tokuyama, I. Oppenheim, Dynamics of hard-spheressuspensions, Phys. Rev. E. 50 (1994) R16.

[84] J. Brady, Brownian-motion, hydrodynamics, and theosmotic-pressure, J. Chem. Phys. 567 (1993) 99.

[85] W. van Megen, T. C. Mortensen, S. R. Williams,J. Muller, Measurements of the self–intermediate scat-tering function of suspensions of hard spherical particlesnear the glass transition, Phys. Rev. E 58 (5) (1998)6073–6085.

[86] E. Steager, C.-B. Kim, J. Patel, S. Bith, C. Naik, L. Re-ber, M. J. Kim, Control of microfabricated structurespowered by flagellated bacteria using phototaxis, Appl.Phys. Lett 90 (2007) 263901.

[87] S. S. Rogers, T. A. Waigh, J. R. Lu, Intracellular mi-crorheology of motile amoeba proteus, Biophys. J. 94 (8)(2008) 3313–3322.

[88] D. Chetverikov, Particle image velocimetry by fea-ture tracking, Lecture Notes in Computer Science 2124(2001) 325.

[89] B. Doliwa, A. Heuer, Cage effect, local anisotropies, anddynamic heterogeneities at the glass transition: A com-puter study of hard spheres, Phys. Rev. Lett. 80 (1998)4915.

[90] A. Jaster, Orientational order of the two-dimensionalhard-disk system, Europhys. Lett. 42 (3), 277-281(1998)

[91] P. Habdas, D. Schaar, A. C. Levitt, E. R. Weeks, Forcedmotion of a probe particle near the colloidal glass tran-sition, Europhys. Lett. 67 (3) (2004) 477–483.

[92] R. R. Huilgol, Z. You, Application of the augmented La-grangian method to steady pipe flows of Bingham, Cas-son and Herschel–Bulkley fluids, J. Non-Newton. FluidMech. 128 (2005) 126–143.

[93] M. Frank, D. Anderson, E. R. Weeks, J. F. Morris, Par-ticle migration in pressure-driven flow of a browniansuspension, J. Fluid Mech 493 (2003) 363–378.

[94] R. Besseling, P. Ballesta, L. Isa, G. Petekidis,W. C. K. Poon, Shear localization in hard-sphere col-

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loidal glasses, in preparation.[95] R. Yamamoto, A. Onuki, Dynamics of highly super-

cooled liquids: Heterogeneity, rheology, and diffusion,Phys. Rev. E 58 (3) (1998) 3515–3529.

[96] To accommodate the wavelength dependent deflectionof the AOD, the instrument uses a slit instead of a pin-hole, but in practice the resolution is very similar tothat of standard pinhole configuration.

[97] In actual fact, due to a possible different pixel size inthe x, y and z directions the images may not appearas spherical; the crucial assumption is that the imagedfeatures are spherical in reality. Any stretching of theimage can then be removed prior to the location proce-dure.

[98] Similar to Ref. [89], we observe, at sufficiently high con-centration, a transition from short (‘in-cage’) to long-time (‘cage breaking’) diffusion in the mean squared

displacement 〈∆r2(∆t)〉 (the ‘MSD’). In addition, in2D, our hard-disk system shows dislocation mediated,two stage, melting according to the Kosterlitz-Thoulessscenario, with a melting density φ2D ∼ 0.7, see [90].For 3D our densities are also below the freezing fractionφF

3D = 0.494. Our simulations are by no means exhaus-tive, they are merely performed to test the success ofthe tracking algorithm.

[99] We note the following points: (1) The particle radius ais not relevant to the tracking algorithm, and mattersin the simulation only as a hard-particle constraint anddetermines the density of the system via the area or vol-ume fraction, φ2D or φ3D, respectively. (2) The trackingrange RT used to test the CG-algorithm is the largestpossible beyond which combinatorics become excessive.

[100] This is also known as the self-part of the van Hove cor-relation function.