Structural Responses of Quasi-2D Colloid Fluids to Excitations Elicited byNonequilibrium Perturbations

Jelena Pesic,∗ Joseph Zsolt Terdik,∗ Xinliang Xu, Ye Tian, Alejandro

Lopez, Stuart A. Rice, Aaron R. Dinner, and Norbert F. SchererDepartment of Chemistry and James Franck Institute,

The University of Chicago, 929 E. 57th St., Chicago, IL 60637(Dated: October 25, 2018)

We investigate the response of a dense monodisperse quasi-two-dimensional (q2D) colloid suspen-sion when a particle is dragged by a constant velocity optical trap. Consistent with microrheologicalstudies of other geometries, the perturbation induces a leading density wave and trailing wake. Wealso use a hybrid version of Stokesian Dynamics (SD) simulations to parse direct colloid-colloid andhydrodynamic interactions. We go on to analyze the underlying individual particle-particle collisionsin the experimental images. The displacements of particles form chains reminiscent of stress propa-gation in sheared granular materials. From these data, we can reconstruct steady-state dipolar-likeflow patterns that were predicted for dilute suspensions and previously observed in granular analogsto our system. The decay of this field differs, however, from point Stokeslet calculations, indicat-ing that the non-zero size of the colloids is important. Moreover, there is a pronounced angulardependence that corresponds to the surrounding colloid structure, which develops in response tothe perturbation. Put together, our results show that the response of the complex fluid is highlyanisotropic owing to the fact that the effects of the perturbation propagate through the structuredmedium via chains of colloid-colloid collisions.

PACS numbers: 83.50.-v, 82.70.-y, 83.10.-y

INTRODUCTION

Provided that one neglects inelastic collisions betweenparticles, all fluids should develop hydrodynamic-like dy-namics on a sufficiently large time scale. Hence, it isreasonable to expect that the same, or at least analo-gous, principles will govern all forms of complex fluids.To the extent that this is the case, it is important to de-termine how the average material response emerges fromthe underlying mechanisms of force transmission and dis-sipation. The extreme cases are simple fluids, for whichthe hydrodynamic description is well established [1], andathermal granular bead packs, for which structural char-acterizations exist [2–5] but descriptions of the dynamicsremain limited [6–9].

Recent theoretical [10, 11] and experimental [12, 13]advances now allow the study of microscopic force prop-agation in complex fluids, such as glasses, colloid suspen-sions [10], polymers [14], and granular materials [15]. Insuch experiments [12, 16, 17], optical or magnetic tweez-ers coupled with optical microscopy are used to micro-manipulate a probe particle to deliver a precise and lo-calized mechanical perturbation to a system while simul-taneously detecting the system response. In this fashion,microviscosities in shear thinning and thickening regimeshave been determined as functions of the speed of theprobe particle and related to the average structure of thefluid around it [13, 18, 19].

Dense colloid fluids, in which thermal motion and hy-drodynamic interactions both play significant roles, liebetween the liquid and granular regimes. Very recentwork on sheared colloid fluids [20–22] draws parallels

to experiments on sheared granular packs [2] while alsohighlighting the complex interplay between granular andliquid components. Sheared dense colloid fluids exhibita non-monotonic response with applied shear force, asexemplified by shear thinning and shear thickening. Therecent advances of microrheological methods now makeit possible to investigate the relationship between thesenon-Newtonian responses and changes in the structure ofthe sheared fluid. We expect changes in structure to playan important role in strongly confined fluids because theresponse of a confined fluid to mechanical perturbationcombines the effects of excluded volume and wall con-straints. This reasoning leads to the question of whetherthe viscosity inferred from the force required to push abead inside a microscopic cell via Stokes’ Law is appro-priate if, for example, the motion of the bead alters thelocal structure of the cell.

As a step towards establishing the structure and dy-namics during the response of a complex fluid to strongmechanical perturbation, and examining the role of indi-vidual particle-particle collisions, we used localized per-turbations to probe the complex properties associatedwith dense colloid fluids. The system that we study isa quasi-two dimensional (q2D) monodisperse colloid sus-pension. We term it q2D for two reasons. First, by con-struction the experimental chamber has a thickness onlyslightly greater than one colloid particle diameter [23],thereby restricting the centers of the colloid particles tolie close to a plane but allowing small amplitude devi-ations from that plane. Second, the supporting liquidin the colloid suspension moves in three dimensions, andthe functional form, sign, temporal decay, and density de-

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pendence of the effective colloid-colloid interactions areaffected by the boundary mediated transfer of momen-tum and mass in the liquid [24–26]. We note that at thedensities of our experiments, the system is in the hexaticphase.

First we show that the structure of the fluid aroundone particle driven steadily by an optical trap reprisesprevious observations and calculations [27–29]. Specifi-cally, we observe a density wave in front of, and a trailingwake following, the probe particle. Surprisingly, our sim-ulations suggest that hydrodynamic interactions are notrequired to obtain the density wave and trailing wake,although the fluid does influence the size of the wakeby setting the self-diffusion constant of the colloid parti-cles. Second, the angle-resolved pair correlation functionexhibits a prominent, and heretofore not observed, band-ing structure trailing the probe particle. The inter-bandspacing is consistent with a partial structural rearrange-ment of the colloid fluid from hexagonal toward squarepacking. Third, we characterize structurally anisotropicdisplacement chains (excitations of the system) that pre-cede the probe particle. We show that the orientation ofthe hexatic order in our dense q2D systems influences theformation of these displacement chains. Finally, analysisof individual colloid particle displacements reveals a flowfield around the driven particle with closed circulationloops up to 10 particle diameters from the probe parti-cle. The high particle density used in our experimentinduces long range spatial correlations that are evidentin the observed flow field. Hence, the decay of this fielddeviates from the r−2 behavior predicted for dipolar flowusing the Stokeslet approximation in q2D systems [24–26]. Furthermore, the average displacement field showswave-like modulation preceding the probe particle, withpeaks occurring at integral multiples of the particle di-ameter, consistent with the formation of displacementchains. Clearly, the fluid is profoundly affected by thestructure that develops from individual particle-particlecollisions.

METHODS

Experimental System

The experimental system that we have studied is amonodisperse colloid suspension confined between hardwalls (Fig. 1a)[30]. The colloid fluid consists of uniformsize silica spheres (diameter σ= 2.56 ± 0.04 µm; BangsLaboratories) suspended in nanopure water and confinedto a chamber with 3.2 µm wall separation maintained at23◦C. The chamber fabrication was adopted from [23].The perpendicular confinement is achieved by modulat-ing the hydrostatic pressure inside the sealed chamberusing a hand pump. The height of the sample cell is de-termined by measuring the distance traversed by a closed

FIG. 1. (Color online) Experimental setup. (a) Typical con-figuration of q2D monolayer of 2.56 µm diameter silica beadswith the probe particle at the center in red with a whitecross. The field of view is 82µm × 82µm and contains ap-proximately 1000 particles. (b) Optical trapping setup andthickness-controlled sample chamber.

loop piezo-electric stage between the epi-reflections of areference laser (532 nm) from the top and bottom glasssurfaces of the sample cell. The chamber walls are coatedwith a non-wetting hydrophobic layer to prevent adsorp-tion of colloid particles. Due to the effective repulsionbetween a colloid particle and the walls, the colloid par-ticles are confined to the center plane of the chamber andform a monolayer with a height distribution of width ±50nm. A 810 nm laser beam that is tightly focused witha high N.A. objective (Olympus SAPO 100x oil immer-sion) forms an optical trap (Fig. 1b) with a force con-stant κ = 0.2 pN/nm in the sample plane. This trapstrength is sufficient to overcome the force threshold (ap-proximately 40kBT/σ) associated with breaking throughthe cage of first neighbors at close packing [10]. All ofthe experiments were carried out in dense colloid suspen-sions; the sample was translated at constant velocity pastthe trapped probe particle (i.e., constant dragging). Thedriving speed ranged from 4.9 to 12.6 µm/s. The totalfield of view (L × L) was 82 × 82 µm2 and the packingfraction, ρ ≡ Nπσ2/4L2 ≈ 0.76. Examination of thebond orientational order and the translational order re-veals that at this density the system is in the hexaticstate [30, 31].

Analysis

We characterized the strength of the perturbation gen-erated by dragging a particle by the Peclet number,Pe = σV/D, where D is the single particle diffusion co-efficient at infinite dilution (which we estimate from theStokes-Einstein equation for spheres in 3D), V is the ve-locity of the dragged particle and σ is the particle diam-eter. Physically, Pe is the ratio of amplitudes of drivenmotion to thermal induced Brownian motion for an iso-lated particle. Here we reference the Peclet number withrespect to the diffusion coefficient for an isolated particle

3

TABLE I. Summary of experimental parameters

Drag Speed, V , (µm/s) Peclet Number, Pe12.6 1908.3 1256.5 1004.9 74

Particle Diameter (σ) 2.56 ± 0.04 µmAverage Packing Fraction (ρ) ≈ 0.76

Field of View (L2) 82 × 82 µm2

Video Frame Rate 67 Hz

in order to present a consistent measure of the strengthof the perturbation which is independent of local den-sity fluctuations around the probe particle. The particlepositions and trajectories are extracted from the experi-mental data using commercial particle tracking software(DiaTrack). The centers of the particles are determinedto ∼10 nm spatial resolution. Table I summarizes theimportant experimental parameters.

Stokesian Dynamics simulations

A key characteristic of a colloid suspension is the dis-crepancy in size and mass between the colloid particlesand the molecules of the supporting liquid. The natu-ral separation of length and time scales in such a systemimplies that the momentum relaxation of the supportingliquid is complete on the time scale of the colloid parti-cle motion. As a result, the distribution of particles canbe described by the Smoluchowski equation. A secondimplication of the natural separation of length and timescales is that the coupling between colloid particles canbe decomposed as (i) a perturbation of the hydrodynamicvelocity field generated by a single particle, and (ii) thecorresponding back action of the perturbed velocity fieldon the generating particle.

Analytical calculation of the effect of the hydrody-namic coupling on the colloid diffusive motions in a densesystem is very complicated [32, 33]. Consequently, weuse a hybrid version of Stokesian Dynamics (SD) sim-ulations to evaluate the contributions of hydrodynamicinteractions. In such a simulation, the colloid particleconfiguration space trajectories are composed of succes-sive displacements, each for a short time step ∆t. Theinfluences of particle-particle and particle-wall hydrody-namic interactions are incorporated via the values of thediffusion tensor D with elements Dij . Specifically, thedisplacement of particle i, ∆xi(t), is determined fromthe Langevin equation:

∆xi(∆t) =∑j

∂Dij

∂xj∆t+

∑j

DijFjkT

∆t+Ri(∆t). (1)

Ri(t) is a random displacement of particle i satisfying amultivariate Gaussian distribution with zero average andvariance-covariance

〈Ri(∆t)Rj(∆t)〉 = 2Dij∆t. (2)

Fi is the sum of inter-particle and external forces act-ing on particle i and all variables are evaluated at thebeginning of the time step.

In principle, for a particular colloid particle configu-ration the diffusion coefficient, Dij , at time t can be ob-tained by solving the Navier-Stokes equation with bound-ary conditions supplied by the particle spatial configura-tion at the beginning of the time step. We used, in-stead, a surrogate procedure that is equivalent. Thisprocedure, pioneered by Brady and coworkers [34, 35],captures the effect of the hydrodynamic coupling on thediffusive motions of the colloid particles yet greatly sim-plifies simulation. In this approach the hydrodynamicsare captured in the presumed known diagonal and off-diagonal diffusion coefficients Dii and Dij that appearin the Smoluchowski equation. We term our approachhybrid Stokesian Dynamics in that the diffusion coeffi-cients are derived from experimental data. In an earlierstudy of diffusion in a dense q2D colloid suspension [25]we evaluated the center of mass and relative pair diffu-sion coefficients, from which the density and separationdependences of the diagonal and off-diagonal diffusioncoefficients can be determined. A fit to those data showsthat the diagonal and off-diagonal diffusion coefficientscan be well approximated by Dii = α and Dij = β/r2

for particles with center of mass separation r, and thatboth α and β are constants depending only on the wallseparation and the colloid concentration. In our hybridStokesian Dynamics simulations, in accord with the con-ditions appropriate to the experiments reported in thispaper, these constants are taken to be α = 1 and β = 0.3.

At the beginning of each time step, the diffusion coef-ficients Dii and Dij were evaluated for the existing par-ticle distribution. With these diffusion coefficients, theN displacements {Ri} were calculated using a multivari-ate normal generator applying the Rotational Methoddescribed in reference [36]. This method utilizes the factthat the k × k variance-covariance matrix Σ is positivedefinite and symmetric. Then, if we generate k indepen-dent univariate random variables Y = (Y1, Y2, · · · , Yk),we obtain a random sample of X = YP−1, whichhas a k-dimensional normal distribution with zero meanand variance-covariance matrix Σ satisfying P

′ΣP = I,

where P′

is the transpose of P.

It is important to make two further comments concern-ing the procedure used to calculate Dii and Dij . First,because they are drawn from experimental data [26, 37],the values of Dii and Dij , calculated in the fashion de-scribed above, incorporate the effects of both far-fieldhydrodynamic and lubrication forces. We note, however,

4

that at the densities used in the earlier experiments lu-brication forces were found to be insignificant [26, 37].Second, we hypothesize that the particular forms quotedfor Dii and Dij , which are valid at packing fraction upto 0.49, remain valid at the higher packing fraction ofthe experiments reported in this paper. This idea is con-sistent with a result obtained from the point Stokesletrepresentation of the hydrodynamics. Namely, becauseof the no-slip condition at the boundaries that define theq2D chamber thickness, the three particle hydrodynamicinteraction vanishes, although higher order interactionsdo not [25]. We discuss this issue further when comparingwith the experimental results.

Our Stokesian Dynamics simulation sample consistsof N = 85 disks of diameter σ in a 10σ × 10σ two-dimensional box with periodic boundary conditions. Theparticle-particle interactions were taken to have the con-tinuous but nearly hard disk form:

U(r∗)

kBT= C(r∗ − 0.5)−γ , (3)

with r∗ = r/σ, C = 2 × 10−19, and γ = 64. Thisform for the particle-particle interaction has continuousderivatives, which simplifies the treatment of forces; ithas been used previously in simulations of dense colloidsystems [38]. The system is first brought to equilibriumusing a Monte Carlo procedure. Then, to simulate theexperimental arrangement in which a moving laser trapis used to exert force on a particle, we drag one of theparticles in the initially equilibrated sample with a har-monic potential whose center moves at a constant ve-locity V . The interaction between the dragged parti-cle and the center of the moving potential is describedby Hooke’s law, F = −κr, where r is the separationof the center of the dragged particle from the center ofthe moving potential and κ = 5000kBT/σ

2. This corre-sponds to a trap constant approximately 70 times softerthan the trap strength used in the experimental system.This choice facilitates numerical integration by allowinga larger time step but does not significantly impact thecomparison to experiment because the deviations fromthe trap center are small relative to the particle diame-ter in both cases. For the simulations, we use units suchthat σ = 1 and Dii = 1; time is then measured in terms ofσ2/Dii. The particle was dragged for 200,000 time stepsof ∆t = 5 × 10−5, after which 100,000 time steps weretaken to allow the system to relax back to equilibrium.

RESULTS AND DISCUSSION

Average structure around the probe particle

We first consider the change in average structure in-duced by a probe particle moving at constant velocity

FIG. 2. (Color online) Pair correlation function g2(r, θ) forPe = 125 centered on the probe particle, where r is the inter-particle distance and θ is the angle between the flow direction(i.e., dragged particle moving to the left) and the line con-necting the particle centers. The grayscale has been chosento highlight the average structure outside the first shell. Theextreme high (5.5-10) and low (0-2.5) values of the pair cor-relation function occur only in the first shell and the wakerespectively.

FIG. 3. Pair correlation function g2(r, θ) centered on thetrapped particle, where r is the inter-particle distance and θis the angle between the flow direction and the line connectingthe two particles. The experimental result for Pe = 125 (a) isclosely reproduced by SD simulations with full hydrodynamicinteractions (HI). (b). If the HI are turned off and the dragspeed kept the same (c), the wake diminishes. If the HI areturned off and the velocity is increased in order to restoreσV/Dii (d), the amplitude of the wake is comparable to (b).

relative to the bath. Fig. 2 displays the 2D pair correla-tion function, g2(r, θ), defined with respect to the drivenparticle at the origin. The angle defines the direction

5

of the projected line between particle centers relative tothe dragging direction. We use polar binning with ra-dial resolution δr = σ/4 and angular resolution δθ = 1◦.The definition of the pair correlation function in both theexperiment and the simulation is:

g(r) ≡ 1

A(r)

T∑t=1

N∑i=1

δ[r− ri(t)],

where ri(t) is the position of particle i at time t and A(r)is the area of the bin at position r ≡ (r, θ) relative to theprobe particle. In other words, for all T frames we bin thepositions of all N particles using polar coordinates anddivide the number of counts in each bin by its area. Theprobe, located by definition at the origin, is not counted.

The trapped particle motion generates a density wavethat precedes it and a rarefaction in its wake. These fea-tures have been previously observed in experiments inunconstrained (3D) colloid systems [13, 27] and in 2Dsimulations [11]. All experimental results shown are forPe = 125, but we observe similar features for 50 < Pe <200 (data not shown). In addition to the features al-ready discussed (i.e., preceding density wave and trail-ing wake), the experimental data shows the emergence ofclear banding structure trailing the particle. The aver-age distance between the peaks of neighboring bands is0.88-0.92 σ. The absence of comparably prominent andsharp banding preceding the particle suggests that themotion of the probe induces a partial rearrangement ofthe hexatic fluid. Furthermore, the inter band spacing(measured at 3σ behind the probe particle) is interme-diate between the spacing expected for a square lattice(1σ) and that for a triangular lattice (

√3σ/2 ≈ 0.866σ).

To parse the roles of contributing interactions, we per-formed simulations for three different constant probe ve-locities: σV/Dii = 20, 30, and 50. This ratio differs fromthe Peclet number in that Dii is not the diffusion co-efficient at infinite dilution. Empirically, we found thatthe simulations with σV/Dii = 20 gave the best cor-respondence to the experiments at Pe = 125, and weshow results from those simulations unless otherwise in-dicated. We focus on this drag speed because the slowerdragging minimizes the artifacts associated with periodicboundary conditions by allowing the system to equili-brate within a single periodic unit of the simulation. Theneed to rescale the probe speed likely arises from a num-ber of factors: the simulation is truly 2D, the simulationbox is relatively small in size, and the form of the diffu-sion tensor is extrapolated from lower densities.

Fig. 3 shows the pair correlation function for distanceranges well within the periodic boundary conditions. Wesee that the simulations reproduce the leading densitywave and trailing wake, which suggests that they cap-ture the essential physics of the system. To assess theimportance of the hydrodynamic interactions (HI), wecompare simulation results with and without HI (Fig.

3c). Without HI we see that the wake still exists, con-sistent with the observation of similar patterns in latticeMonte Carlo simulations that do not explicitly considerfrictional forces [39]. Quantitatively, the wake is dramat-ically reduced in extent without HI. However, if we in-crease the speed of the driven particle to restore σV/Dii

(Fig. 3d), the pair distribution again resembles closelythat with full HI. This comparison between simulationsperformed in a consistent fashion clarifies the role of thefluid despite potential artifacts arising from the boundaryconditions.

Individual displacement chains and hexatic order

As discussed in the Introduction, our goal is to un-derstand how the collective response of the colloid fluidarises from the dynamics of individual particle motions.In Fig. 4 we plot the displacements of individual colloidsthat move more than a threshold amount. Superimpos-ing many such displacements from different times, withthe probe particle moving over the same range, revealsa structure consistent with the density wave and wake(black) up to 2 particle diameters from the probe. Arepresentative chain of displacements from a small timeinterval of the experiment is highlighted in red. We seethat it is quite long, spanning 7 particles. Clearly, theperturbation propagates via colloid-colloid interactionsin a highly anisotropic fashion. Displacements chains arealso observed in our simulations extending up to 4σ. Sim-ilar behavior has been previously reported in granularsimulations [40]

At the densities of our experiments the colloid sus-pension is in the hexatic phase. The extent and ori-entation of the structural order around a particle canbe quantified by the 6-fold bond orientation parameterΨ6 = Re{

∑6j=1 exp(i · 6θj)}. Here, θj is the angle be-

tween a fixed axis and the line connecting the centersof the particle of interest and its j-th nearest neighbor.Within regular hexagonal domains, the value of Ψ6 isconstant. Changes in the value of Ψ6 correspond to grainboundaries and transitions to disordered domains. Fig.5 highlights typical bond orientational configurations ob-tained from simulation data under equilibrium and con-stant dragging conditions. As expected, in equilibriumwe observe bond orientational order on length scales com-parable to the simulation cell size. For the constant drag-ging perturbation, the forced motion of the probe particledistorts the hexatic order through repeated cage break-ing ultimately leading to the formation of extended grainboundaries and disordered regions.

We would like to determine if the size and orienta-tion of the displacement chains is determined by the lo-cal structure. Fig. 6 highlights typical situations. Thesign of Ψ6 characterizes the orientation of the hexaticstructure relative to the direction of motion of the probe

6

-4

-2

0

2

4

-8 -6 -4 -2 0

r/σ

r/σ

FIG. 4. (Color online) Displacement chains of the bath in-duced by a probe particle moving at a constant relative veloc-ity (V ) in experiment (top) and SD simulations (bottom). Acollection of particles forms a displacement chain if each par-ticle’s successive displacements are greater than 0.5d, where dis the displacement of the probe particle in the dragged frameof reference. The figure depicts a superposition of all displace-ment chains observed (black) and a sample of representativedisplacement chains (gray in print or red online).

particle (compare the orientation of the white hexagonswith the color scale in Fig. 6). In each case, the dis-placement chain propagates according to the orientationof the local structure; displacement chains form inlinewith the probe particle when Ψ6 > 0 and bifurcate whenΨ6 < 0. However, since the size of ordered domains istypically less than 10σ across, the displacement chainscan span multiple domains and hence are sensitive tograin boundaries. The upper displacement chain in Fig.6a follows a grain boundary, indicating that both localorder and defects influence the bath response. The ap-

FIG. 5. (Color online) Bond orientational order parame-ter, Ψ6, for simulations under equilibrium (left) and constantdragging steady state conditions (right). The drag directionis to the left and the probe is indicated in red with a whitecross at (0,0).

FIG. 6. (Color online) Relation between displacement chainsand hexatic order. Representative bath particle displacementsover a time interval of approximately 10s are shown in blackwith the probe particle in red with a white cross at (0,0).The color scale indicates the bond orientational order param-eter, Ψ6, calculated for each particle indicated in gray. Whitehexagons illustrate the fact that the sign of Ψ6 reports on theorientation of the hexatic structure relative to the directionof motion of the probe particle.

proximate angular spread (∼ 60◦) of the superposition inFig. 4 is consistent with the largest angular spread oc-curring when the probe bisects the hexatic ordering, i.e.,when Ψ6 = −6 as shown in Fig.6a. Orientational effectsin driving particles through larger ordered domains havebeen investigated previously [41].

Dipolar-like flow

In the Stokeslet approximation, the perturbationcaused by the probe particle induces a dipole-like flowin the supporting liquid. This prediction is valid for thefar-field flow in the limit of a dilute suspension of pointparticles. In contrast to these assumptions, in our systemthe particle density is high, the experimental ratio of par-ticle diameter to confinement thickness is 0.8, and the far-

7

field flow limit is not experimentally accessible owing tospatial correlations present in the hexatic phase that arecomparable to the longest measurable length scales. Fur-thermore, given the qualitative connections of the densecolloid packing to granular systems [2, 42, 43], we expectthe observed displacement chains (Fig. 4) to introducean anisotropic component to the flow field. Nevertheless,one might expect that qualitative features of the dipolarflow of the liquid to persist. Specifically, we expect theclosed loop circulation pattern to persist, but not the al-gebraic form of the fall off of velocity with the distancefrom the source. We attribute the change in spatial falloff of the velocity field to interferences generated by scat-tering from the densely packed colloid particles and thewalls, drawing on our experience with calculations usingthe method of reflections [44]. This method representsthe influence of the boundaries on the velocity field gen-erated by the moving particle as the superposition of thevelocity components reflected from each boundary en-countered by the initially generated velocity field. Thereflections are considered to be coherent, so the superpo-sition of the velocity components has both positive andnegative interferences. In a system with many reflect-ing boundaries, as in the case of densely packed colloidsstudied here, these interferences can greatly change thesymmetry and spatial falloff of the velocity field.

We recover the dipolar-like flow shown in Fig. 7through a combination of spatial binning and time av-eraging. To construct the field from experimentally mea-sured particle trajectories, the frame-to-frame displace-ment (∆x,∆y) for each particle was computed in thedragged frame of reference:

(∆x,∆y) ≡(x(t+ δt)− x(t), y(t+ δt)− y(t)

), (4)

where δt is the time between frames, and x(t) is the posi-tion of particle i at time t. The precise components of thedrag speed depend on the angular position of the cam-era relative to the stage. This introduces an additionalsource of uncertainty, estimated at < 3% in each compo-nent, in determining the displacement field componentsin the dragged frame of reference. We choose the dragvelocity within the uncertainty range that maximizes thespatial extent of the dipolar-like flow. This procedure isjustified by the consistency of the optimization process inobtaining far field dipolar-like flow, and the reproducibil-ity of the structural form of the angle dependent decayin Fig. 8.

The displacements were spatially binned based on theangle and radial distance relative to the probe particle.This was done for every frame of the video, after whicheach bin was time averaged. Only spatial bins that hadmore than 30 samples were used to generate an interpo-lated vector field. The threshold of 30 samples was suf-ficiently high to ensure that incoherent thermal motionswere averaged out while ensuring that the whole vector

field, including regions of vanishing flow, was adequatelysampled. Fig. 7a shows the interpolated vector field, de-picted by green arrows, superimposed on a line integralconvolution (LIC) flow visualization [45] which empha-sizes the weak but coherent dipolar-like flow observedout to ten diameters away from the probe particle. Fig.7b shows streamlines (green arrows) and the magnitudeof the average displacement field (gradient). To the bestof our knowledge, this is the first experimental observa-tion of dipolar-like flow in q2D colloid suspensions, and,more generally, the only quantitative analysis of such afeature in a complex fluid.

FIG. 7. (Color online) Dipolar-like flow fields around aprobe particle moving at constant velocity for Pe = 190. Thedragging direction is to the left and the field of view has adiameter of 20σ. The interpolated vector field is obtainedfrom the frame to frame displacements of every particle inthe field of view using a combination of spatial binning andtime averaging (see text). Fig. 7a shows a sample of the inter-polated vectors (green arrows) of the averaged displacementfield and the associated line integral convolution (gray) indi-cating the extent of weak but coherent dipolar flow. Fig. 7bshows streamlines (green arrows) obtained from the interpo-lated vector field and the magnitude of the average displace-ment field (gradient).

As already noted, the dipolar-like flow that is gener-ated in the supporting liquid by the driven particle is

8

expected from a simplified analysis of the hydrodynamicbehavior of a q2D colloid system wherein the driven par-ticle is treated as a force monopole in an incompressibleliquid confined between two plates [46, 47]. As each col-loid particle moves in the plane parallel to the walls itchanges the momentum and mass density of both thecolloid particle and the supporting liquid [24]. The mo-mentum of the supporting liquid is transferred to theconfining walls, and is not conserved. The mass of thecolloid particle and that of the supporting liquid are con-served, and the mass perturbation associated with parti-cle motion dominates the flow pattern at distances largerthan the thickness of the sample. To leading order in theratio of particle radius to distance from the particle, themass perturbation can be represented as a mass dipole,which in q2D creates a flow velocity proportional to r−2

at large distances from the driven particle. It is thisfunctional form that defines the long-range flow patternin q2D confinement. On a time scale short relative to thetime between colloid-colloid collisions, the colloid parti-cles act as reporters of the local fluid velocity, renderingthe dipolar-like flow observable on a length scale compa-rable to multiple colloid-colloid separations. As alreadynoted, in a dense colloid suspension we expect deviationfrom the r−2 behavior because of interference with thevelocity field generated by the zero slip boundary condi-tion on each colloid particle with nonzero volume.

Given the anisotropic structure of both the averagecolloid packing (Fig. 2) and the intermittent colloidstructure (Fig. 4), one should expect that colloid dis-placements reflect not only the average displacement ofthe underlying fluid, but also the observed structuralanisotropy. Keeping in mind the roles of (i) colloid-colloidinteractions, (ii) hydrodynamic screening due to the highdensity, and (iii) nonzero size of the probe particle, wedetermine the decay parameter p in 1/rp as the slopeof a linear fit of log10[dθ(r)] as a function of log10(r)where dθ(r) is the magnitude of the displacement vectorof the dipolar-like flow field at a radial distance r from theprobe particle and fixed angle θ. The fitting region wasr ∈ (3σ, 10σ). As shown in Fig. 8a, the decay parameterp exhibits a strong dependence on θ. The observed angledependence may be qualitatively understood as a man-ifestation of the nontrivial angle dependent structure ofthe hexatic fluid, as characterized by the pair correlationfunction, and the corresponding colloid-colloid interac-tions. We note that some quantitive aspects of Fig. 7 andFig. 8a (e.g., spatial extent of coherent dipolar-like flowbeyond 5σ in Fig. 7, and angular location of minimumand quantitative values of the weak decay parameters inFig. 8a) are sensitive to the components of the drag speedwithin our uncertainty range but the qualitative featuresdiscussed above are not.

The modulation of the displacement field precedingthe probe particle, shown in Fig. 7, exhibits peaks corre-sponding to integer particle diameters and a cone consis-

-2.7

-2.6

-2.5

-2.4

-2.3

-2.2

-2.1

-2

-1.9

-1.8

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

log(d

135(r

/))

log(r/ )

Pe=189, =135Fit

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 30 60 90 120 150 180

Dec

ay p

aram

eter

p i

n 1

/rp

Angle (Degrees)

Pe=74Pe=100Pe=125Pe=190

a)

b)

FIG. 8. (Color online) Decay parameter of the dipolar-likeflow field. (a) Angle dependence. The angle correspondingto motion of the probe particle is 180 degrees. (b) Typicallinear fit of log10-log10 data for determining the angle de-pendent decay parameter. The log10-log10 data were fitted tof(x) = b−mx and the fitted parameters were m = 1.17±0.09and b = −1.40 ± 0.07. The fitted negative slope, m, corre-sponds to a single decay parameter quoted in (a). The angleshown corresponds to 135 degrees. The uncertainties quotedare asymptotic errors in a least squares fit.

tent with the angular distribution of displacement chainsshown in Fig. 4. The anisotropic displacement chains,which are a direct result of correlated colloid-colloid col-lisions, perturb the long range dipolar flow field. Qualita-tively similar flow fields were observed in driven granularsystems [42, 43]. Our simulations, regardless of whetherHI are included, show dipolar-like flow fields (Fig. 9)that are similar to the experimental results (Fig. 7). Weobtain the displacement fields shown in Fig. 9 analo-gously to the experimental system. In the experiment,the displacements (∆x,∆y) were computed using adja-cent frames; in the simulation, the displacements werecomputed over the number of simulation time steps re-quired for the probe particle to move a distance of σ/10.

9

FIG. 9. (Color online) Dipolar-like flow fields observed insimulations. The field of view has a diameter 10σ which spansa single simulation cell. The color scale corresponds to themagnitude of the displacement field and the streamlines aredepicted by green arrows.

In addition to the dipolar-like flow fields, the displace-ment field also shows velocity modulation, preceding theprobe, occurring at integer particle diameters as in theexperiment. However, the decay is clearly faster thanin Figs. 7 and 8. Since we obtain similar results evenfor very large systems without HI (data not shown), wedo not believe this is a boundary artifact. Rather, thesimulation results suggest that the assumed form for theHI, while adequate for elucidating qualitative features ofthe observed dipolar-like flow fields, becomes inappro-priate for quantitative analysis of the decay parameters.At high densities, an alternative scaling and parameteri-zation may be required to capture the interplay betweengranular and liquid components in the colloid fluid, whichis dominated by near-field effects.

While a first-principles analysis of experiment and sim-ulation is not possible, we can relate the qualitative fea-tures observed in Fig. 8 to the phenomenology observedin the dipolar-like displacement fields in both experiment(Fig. 7) and simulation (Fig. 9). As in Fig. 8, we adoptthe convention that the driving direction is 180◦ and thesystem is approximately symmetric with respect to thedrag direction. Consider the following regions:

• Region I, |θ| ∈ (0◦, 20◦): This region correspondsto the wake. Slow decay is observed suggesting thatthe dipolar flow is dominated by the slow structuralrelaxation of the wake. Although the fitting region(> 3σ) lies outside the wake (Fig. 2 and Fig. 3) thesampling is poorer in this region than in others.

• Region II, |θ| ∈ (20◦, 60◦): This region correspondsto dipolar-like circulation currents that flow into

the wake. Decay parameters with typical valuesp = 1.5 are observed which are between those ex-pected for a point force in 3D (p = 1) and the pointstokeslet limit in q2D (p = 2).

• Region III, |θ| ∈ (60◦, 150◦): This region spans thelocation of the vortex. In each of the drag speedsconsider in the experiment, the minima in angledependent decay at ∼ 100◦ corresponds to the an-gular position of the vortex. Comparing the pointstokeslet approximation and the experimental con-ditions, we hypothesize that the non-zero size ofthe probe particle plays a significant role in form-ing the vortex. Additionally, we note that bothsimulation and experiment show characteristic an-gle dependent anisotropy in the displacement field,with the location of the vortex corresponding to asteep decline in the displacement magnitude.

• Region IV, |θ| ∈ (150◦, 180◦): This region spansthe velocity modulation of the dipolar-like field di-rectly caused by the colloid-colloid collisions. Thefaster decay parameters (p > 2) show that mo-mentum transfer through colloid-colloid collisionsdecays more quickly than hydrodynamic interac-tions, further suggesting that the long range, co-herent dipolar-like flow fields arise from HI.

CONCLUSIONS

Our study establishes the phenomenology of responseto a localized perturbation in a dense q2D colloid sus-pension and allows it to be related to individual particle-particle collisions. The trapped particle motion generatesa density wave that precedes it and a void in its wake.These ensemble-averaged features have been widely ob-served in experiments in unconstrained (3D) colloid sys-tems [13, 27] and in 2D simulations [11]. We have demon-strated that these features are preserved in a q2D geome-try as well and have shown how they arise from individualdisplacement chains.

Our study provides insight into the range of validity ofthe Smoluchowski equation treatment of the response ofa colloid suspension to the driven motion of one particle[1]. This approach neglects multiparticle (more than two)hydrodynamic interactions and is thereby limited to lowdensity when used to describe a three-dimensional sus-pension. It should remain valid to higher densities in ourcase because the three particle hydrodynamic interactionvanishes in a q2D colloid suspension at the Stokeslet levelof approximation, and our simulations did indeed capturethe leading density wave and trailing wake well.

That said, the overall deviation of the decay of thedipolar-like field from the point Stokeslet approxima-tion makes clear the importance of the fact that the

10

colloids are comparable in size to the separation be-tween the boundaries. The striking angular dependenceof the decay reflects the structure of the surroundingcolloids, which evolves through individual displacementchains arising from the perturbation. Incorporating theanisotropic propagation of forces and these memory ef-fects will be important for theories and simulations thatseek to describe the response of such complex fluids fully.Although doing so will be challenging, success would haverepercussions not only for colloid suspensions but for amuch broader range of fluids, including granular ones.

We will report complementary results from studies ofrelaxation dynamics associated with nonequilibrium ex-citations that create density waves, wakes, and persistentvoid areas in a separate publication.

ACKNOWLEDGEMENTS

We thank Binhua Lin for assistance in the design andfabrication of the sample chamber. JZT acknowledges asummer research fellowship from the James Franck Insti-tute. We acknowledge financial and central facilities as-sistance of the University of Chicago Materials ResearchScience and Engineering Center, supported by the Na-tional Science Foundation (NSF dmr-mrsec 0820054).

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