Experimental Realization of the Green-Kubo Relation in Colloidal SuspensionsEnabled by Image-based Stress Measurements

Neil Y.C. Lin, Matthew Bierbaum, and Itai CohenDepartment of Physics, Cornell University, Ithaca, New York 14853

By combining confocal microscopy and Stress Assessment from Local Structural Anisotropy(SALSA), we directly measure stresses in 3D quiescent colloidal liquids. Our non-invasive andnon-perturbative method allows us to measure forces . 50 fN with a small and tunable probingvolume, enabling us to resolve the stress fluctuations arising from particle thermal motions. Weuse the Green-Kubo relation to relate these measured stress fluctuations to the bulk Brownian vis-cosity at different volume fractions and comparing against simulations and conventional rheometrymeasurements. We demonstrate that the Green-Kubo analysis gives excellent agreement with theseprior results. This agreement provides a strong demonstration of the applicability of the Green-Kubo relation in nearly hard-sphere suspensions and opens the door to investigations of local flowproperties in many poorly understood far-from-equilibrium systems, including suspensions that areglassy, strongly-sheared, or highly-confined.

PACS numbers: 05.40.-a, 05.60.-k, 82.70.Dd, 83.85.Cg

All quiescent thermal systems may seem static macro-scopically, but microscopically they fluctuate strongly.By observing the system’s response to these thermal fluc-tuations, a material’s linear transport coefficients canbe predicted using the Green-Kubo relation [1–4]. Thisfoundational relation – a central achievement of nonequi-librium statistical mechanics – has enabled numerousdiverse theoretical calculations ranging from electricaland magnetic susceptibilities in quantum systems [5, 6]to thermal conductivities in nanotubes [7–10]. In par-ticular, it has been widely used to theoretically deter-mine the viscosities in bulk [11], confined [12, 13], super-cooled [14, 15], and quantum [16] liquids, where exter-nal load is problematic or heterogeneities play a crucialrole. Unfortunately, these applications have remainedstrictly theoretical due to the difficulties in experimen-tally observing fluctuations in atomic systems, which aretoo rapid (∼ ps) and weak (∼ µN) to mechanically re-solve in experiments.

Here, by using high-speed confocal microscopy in con-junction with Stress Assessment from Local StructuralAnisotropy (SALSA) [17], we directly measure the stressfluctuations in nearly hard-sphere colloidal liquids. Col-loidal suspensions are comprised of particles that aresmall enough to demonstrate Brownian motions, whilelarge enough to be optically imaged, providing length-and time-scales that are associated with system relax-ation [18]. To measure a suspension’s stress fluctuations,we use a confocal microscope to image the 3D microstruc-ture of the sample, then use SALSA to determine itsBrownian stress arising from interparticle thermal colli-sions. Since SALSA is image-based, non-invasive, non-perturbative, and able to measure the suspension stresswith a tunable probing volume, it can resolve the weakstress fluctuations that are usually averaged out in con-ventional bulk measurements due to the requisite largeprobing volume.

The suspension samples are comprised of silica sphereswith a radius a = 490 nm in a water-glycerine mixture

0 50 100 150 200 250 300Time t (s)

-0.5

0

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yx

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FIG. 1: (a) Schematic of the experiment setup and axis. Thesuspension sample is hermetically sealed in a sample cell, andplaced on a high-speed confocal microscope to image its mi-crostructure. The measurement window (dashed box) is ∼ 5µm above the coverslip avoiding boundary effects. (b) Thefeatured particle positions are used to calculate the stress us-ing SALSA. The selected particle’s (orange) local structuralanisotropy is calculated based on the configuration of its col-liding neighbors (blue) that lie within a thin shell ∆ ≈ 106nm (green). This process is done for each snapshot giving theinstantaneous Brownian stresses σxz (orange line) and σxy

(blue line) fluctuating within ∼ ± 0.5 mPa.

that has a matched refractive index and viscosity η0 = 60mPa·s. We add 1.25 mg/ml of fluorescein sodium salt tothe solvent to shorten screening length (≤10 nm) andobtain nearly hard sphere interactions. The added flu-orescein also makes the solvent fluorescent, so the sol-vent appears bright and the particles appear dark. Wethen image the particle configuration using a high-speedconfocal microscope with a hyper-fine scanner that max-imizes the stability in the vertical (z-axis) scanning posi-tion (schematic in Fig. 1(a)). To ensure that the suspen-sion structure remains homogenous throughout the ex-periment, we image the sample within a minute after thesample cell is made. We capture 216 frames per secondand acquire stacks of 100 images within 0.5 s ∼ 0.02τB,

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FIG. 2: (a) Time-time autocorrelation functions of σxz (orange) and σxy (blue) are calculated from the time series of stress.The green line shows an exponential fit to the data to extract the time scale of the stress autocorrelation functions. Thecross-correlation 〈σxz(t+ ∆t)σxy(t)〉 (gray) is consistent with zero showing low coupling between components. For clarity, theautocorrelation is normalized by its corresponding fluctuation’s variance, and the cross-correlation is normalized by the meanvariance of all autocorrelations. (b) The correlation time τ of the stress fluctuation varies weakly with the volume fraction φ,a trend that is consistent with the variation of the inverse self-diffusivity D0/Dss(φ) found in Accelerated Stokesian Dynamicssimulations (ASD) [19]. Here, D0 is the diffusivity in ultra-dilute limit and a2/Dss roughly determines the relaxation time-scaleof the suspension. (c) Mean variance, Cij , of all shear stress components is plotted versus the normalized probing volume V/Vp,where Vp is particle volume. The gray line denotes an inverse proportionality between Cij . The inset shows that the measuredviscosity η ≈ 50.5 mPa.s is roughly constant when V/Vp ≥ 200 and starts to decay slightly at smaller probing volumes. We setthe measurement window V = 61µm×15µm×12µm, V/V p ∼ 22, 280 (orange line) throughout all measurements.

where τB = 6πa3η0/kBT is the self-diffusion time of thesample.

By implementing the previously developed SALSAmethod [17], we determine the stress in our 3D suspen-sions. SALSA uses the featured particle positions tocalculate the local structural anisotropy or fabric tensor

ψαij(∆) =∑β∈nn r

αβi rαβj of particle α, where nn is the

set of colliding neighbors that lie within a distance 2a+∆from particle α (∆ = 106 nm in the current work), i,j arespatial indices, and rij is the unit vector between parti-cles (see Fig. 1(b) and SI). Scaling the ensemble-averagedψαij(∆) by ∆ enables us to estimate the probability ofthermal collisions between particles. Consequently, theinstantaneous Brownian stress of the sample can be ap-proximated: σij(V,∆) = kBT

Va∆

∑α∈V ψ

αij(∆)+nkBTδij ,

where V is the averaging window volume, kBT is thermalenergy, n is number density, and δij is Kronecker deltafunction. Here, nkBT is simply the ideal gas term.

The typical volume of our probed region V =61µm×15µm×12µm ∼ 10 pL contains approximately6,000 particles at a volume fraction φ ∼ 0.27. This smallvolume ensures that the stress fluctuations are not sup-pressed by the volume averaging, ∝ 1/V , while preserv-ing bulk behavior. We plot the instantaneous stress σxzand σxy in Fig. 1(b), where z is the gravitational axis andx and y are horizontal. In contrast to a flat line at zerolevel anticipated in a macroscopic measurement, we findthat both σxz and σxy fluctuate up to ± 0.5 mPa. Wenote that the force fluctuations corresponding to thesestresses are less than 50 fN, difficult to resolve using me-chanical methods.

We calculate the time-time autocorrelation function

〈σij(t+∆t)σij(t)〉 for the stress components σxz and σxy,and show the correlation decay in a log-linear plot, seeFig. 2 (a). Despite the slight sedimentation due to thedensity mismatch between the particle and solvent, bothautocorrelation functions decay in the same fashion indi-cating an isotropic viscosity of the sample (see SI). Wefurther examine the cross-correlation 〈σxz(t+ ∆t)σxy(t)〉and find it negligibly small, which is consistent with thesystem symmetry. While the exact function form of theautocorrelation decay cannot be determined from thecurrent data due to the limited measurement time span,we use an exponential decay (∼ e−∆t/τ ) to quantify thecorrelation time. In doing this, we find that the corre-lation time τ varies weakly with the suspension volumefraction φ (see Fig. 2 (b)). We compare our observedtrend with previous simulations of short-time diffusivityDss where a2/Dss roughly sets the relaxation time-scaleof the system [19,20]. In simulations, Dss decays ap-proximately as Dss ∼ D0(1 − bφ) (red line, Fig. 2 (b))with b on the order of 1.5 at intermediate volume frac-tions. Here, we find a weaker trend b ∼ 0.60± 0.23 (bluedashed line) indicating either our measurements are notsufficiently precise to determine b accurately or that thefunctional form changes at volume fractions approachingclose-packing.

With the measured stress fluctuations, we can directlycalculate the shear viscosity of our sample via the Green-Kubo formula ηB = 〈 V

kBT

∫〈σij(t + ∆t)σij(t)〉d∆t〉i 6=j ,

where ηB is the Brownian contribution to the total shearviscosity ηtot. Since our suspension systems are nearlyhard-sphere, we anticipate that the stresses are weaklycorrelated in space, and thus the sample viscosity is

3

roughly independent of probe window size. To verify this,we change our probing (averaging) volume V , and investi-gate how the stress fluctuations vary. In Fig. 2(c), we plotthe mean variance of shear stress Cij = 〈σij(t)σij(t)〉t,i6=jas a function of V/Vp where Vp is the particle volume(4/3)πa3. We find that Cij is inversely proportional toV/Vp when V/Vp ≥ 200 corresponding to a cubic volumethat is approximately six particles across. This inverseproportionality and constant viscosity shown in the insetof Fig. 2(c) are consistent with the Green-Kubo formula.When V/Vp ≤ 200, we find that the viscosity slightlydeviates from its bulk value. The viscosity reduction isaround 20% of the mean for the smallest probing volumeexplored – a three-particle wide cube. While this reduc-tion is reminiscent of the system size-dependent viscosityassociated with long-ranged stress correlations in atomicsimulations [20–26], in our nearly hard-sphere liquid sys-tem we do not anticipate such long-ranged correlationsthat lead to nonlocal viscosities. Instead, at small vol-umes, the stress fluctuations are strongly influenced bychanges in particle number as particles pass into and outof the constrained field of view.

→→

SALSA

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FIG. 3: Relative Brownian viscosity ηB/η0 calculated fromGrenn-Kubo relation (red circles) is plotted versus volumefraction φ, where η0 is the solvent viscosity. The errorbars denote the standard errors over 14 runs of measure-ments. The experimental results are quantitatively consistentwith accelerated Stokesian dynamics simulations, ASD (bluesquares) [19]. Furthermore, we find that the measured Brow-nian viscosity of our suspension is also consistent with ourBrownian dynamics simulation results determined with di-rect stress calculation ~F~x (purple diamonds) and the SALSAmethod (green diamonds). Finally, we find our results arein excellent agreement with rheometry measurements (purplecrosses) [27]. In particular, we subtract the hydrodynamiccontribution ηH from the total viscosity ηtot measured us-ing bulk rheometry to determine the Brownian component,where ηH is obtained from analytical calculations reported inprevious work [27–30].

To compare our results with macroscopic flow measure-

ments and simulations, we use the measured stress auto-correlation in conjunction with the Green-Kubo relationto determine the Brownian viscosity ηB of suspensionsat eight different volume fractions 0.12 ≤ φ ≤ 0.45 (seeFig. 3). The resulting viscosities (red circles) show ex-cellent agreement with previous hydrodynamic Stokesiansimulations (blue squares) [19]. To further confirm theaccuracy of our SALSA stress measurement, we also useBrownian Dynamics simulations to generate sets of par-ticle configurations matching the experimental parame-ters (e.g. particle size, solvent viscosity, and tempera-ture), and compare the stresses calculated from actualvirials FijXij (purple diamonds) with those calculatedon the same data set with SALSA (green diamonds) [31].Both results again show a quantitative agreement withthe experimental measurements. Finally, the measuredBrownian viscosities are compared with conventional me-chanical measurements by subtracting the hydrodynamiccontribution ηH from the total viscosity ηtot determinedusing rheometry (purple crosses) [27, 32]. The rheologydata points (colloidal PMMA and silica systems) are ob-tained from previous experiments [27] and the hydrody-namic contribution is calculated from previous analyti-cal approximation for the high frequency viscosity [27–30]. We find good agreement between the viscositiesdetermined by our stress fluctuation measurements andconventional rheometry at all volume fractions explored.Collectively, the agreements between our results, simula-tions, and bulk measurements provide a clear demonstra-tion of the Green-Kubo formula in hard-sphere systems.

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FIG. 4: (a) Osmotic pressure Π (red disks) averaged over timeis plotted versus φ. We find that the measured trend Π(φ) isconsistent with the Carnahan-Starling equation of state (CSEoS, gray line). (b) Log-log plot of the normalized Brown-ian contributions to the bulk ηbulkB /ηo (dark red points) andshear ηB/ηo (light red points) viscosities versus Π showing aΠ2 scaling (dashed black line), where η0 is the solvent viscos-ity. The atomic liquid viscosity η/η′0 (blue curve, reproducedfrom [24]) exhibits a similar scaling behavior only at highpressures, where η′0 is the ideal gas viscosity derived from theBoltzmann equation (see SI).

In contrast to conventional mechanical measurements,which can only measure the flow-gradient stress and thedifference between normal stresses, SALSA measures allstress components simultaneously. In Fig. 4(a) we re-port the pressure of the suspension at the eight volume

4

fractions explored in Fig. 3. We find that the measuredosmotic pressure arising from Brownian collisions (reddisks) is well described by the Carnahan-Starling equa-tion of state (gray line) [54]. In addition, we find thatboth the shear (ηB , light red points) and bulk (ηbulkB , darkred points) viscosities roughly exhibit Π2 scaling (dashedblack line), as shown in Fig. 4(b). While the underly-ing mechanism of such an empirical scaling remains anopen question, we can qualitatively understand this scal-ing for the bulk viscosity using a dimensional analysis.Since the correlations in the Green-Kubo formula decayapproximately exponentially in time, and the relaxationtime τ does not increases significantly with increasingpressure over the range measured, we have

ηbulkB ∼∫ ∞

0

〈(Π(t+ ∆t)− Π)(Π(t)− Π)〉d∆t

∼∫ ∞

0

CΠe−∆t/τd∆t ∼ 〈Π2〉 − 〈Π〉2

where CΠ is the variance of pressure [33, 34].While many previous studies have made analogies be-

tween the transport phenomena of colloidal systems andsimple liquids [32, 35–37], we find that the observed Π2

scaling is actually absent in atomic systems. Specifically,the atomic viscosity (blue curve in Fig. 4(b)) exhibits asimilar scaling behavior, but only at high pressures corre-sponding to large φ. At low pressures, the viscosity trenddeviates from the Π2 scaling. We conjecture that this de-viation is associated with the kinetic contribution to theviscosity [4, 38], which is associated with atom velocity,insensitive to Π, and dominates in the dilute limit (seeSI). Collectively, our findings, which are made possibleby SALSA, suggest that even the Brownian contributionto the colloidal viscosity can have a distinct transportmechanism than that in simple liquids.

In conclusion, we measure the stress fluctuation in col-

loidal liquids with SALSA, and experimentally demon-strate the well-known Green-Kubo relation [1, 2]. Ourmeasurements essentially show that “as far as linear re-sponses are concerned, the admittance is reduced to thecalculation of time-fluctuations in equilibrium” [1]. Pre-vious pioneering experiments were able to combine theGreen-Kubo relation with numerical simulations to ex-tract the viscosity of a 2D dusty plasma [39]. These mea-surements, however, relied on assumptions for the inter-particle potentials and ignored power-law decays in thestress correlation characteristic of 2D systems, which areknown to lead to diverging integrals [40–42]. The analy-sis presented here avoids many of these complications andopens the door to further investigations of stress distri-butions in liquids under shear, confinement, and at highdensities where the suspension becomes glassy [43–45].In such situations SALSA is still applicable since the sol-vent remains in equilibrium. More importantly, since theSALSA measurement is non-invasive, it also allows forprobing the mechanical heterogeneity in a 3D colloidalglass [46–52], in which we can perform a time-average forparticle-scale stress calculation. Measuring the tempo-ral and spatial stress fluctuations in such a system willshed light on the generalization of the Green-Kubo re-lation in far-from-equilibrium systems and elucidate themechanisms that underly the flow behaviors of disorderedsystems.

The authors thank James Sethna, Brian Leahy, JamesSwan, John Brady, and Wilson Poon for helpful discus-sions. I.C. and N.Y.C.L. gratefully acknowledge the PoonLaboratory at School of Physics & Astronomy, Univer-sity of Edinburgh for generous use of their PMMA sus-pensions. I.C. and N.Y.C.L. acknowledge funding fromNational Science Foundation (NSF) NSF CBET-PMPAward 1509308. M. B. was supported by Department ofEnergy DOE-DE-FG02-07ER46393 and continued sup-port from NSF DMR-1507607.

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[54] We note that in previous confocal measurements wherethe pressure is determined by calculating the availablevolume to insert an additional sphere into a system, andits surface area [53], the volume and corresponding sur-face area become exceedingly small and difficult to mea-sure when suspensions are dense. This uncertainty re-sults in a mismatch between data and theory. In our ex-periment, SALSA accurately captures the particle colli-sion probabilities and correctly reports the pressure atall tested volume fractions.