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Cite this:DOI: 10.1039/c5sm02800k

Particle diffusion in active fluids is non-monotonicin size†

Alison E. Patteson,a Arvind Gopinath,ab Prashant K. Purohita and Paulo E. Arratia*a

We experimentally investigate the effect of particle size on the motion of passive polystyrene spheres in

suspensions of Escherichia coli. Using particles covering a range of sizes from 0.6 to 39 microns, we

probe particle dynamics at both short and long time scales. In all cases, the particles exhibit super-

diffusive ballistic behavior at short times before eventually transitioning to diffusive behavior. Surprisingly,

we find a regime in which larger particles can diffuse faster than smaller particles: the particle long-time

effective diffusivity exhibits a peak in particle size, which is a deviation from classical thermal diffusion.

We also find that the active contribution to particle diffusion is controlled by a dimensionless parameter,

the Peclet number. A minimal model qualitatively explains the existence of the effective diffusivity peak

and its dependence on bacterial concentration. Our results have broad implications on characterizing

active fluids using concepts drawn from classical thermodynamics.

1 Introduction

The diffusion of molecules and particles in a fluid is a processthat permeates many aspects of our lives including fog formationin rain or snow,1 cellular respiration,2 and chemical distillationprocesses.3 At equilibrium, the diffusion of colloidal particles ina fluid is driven by thermal motion and damped by viscousresistance.4 In non-equilibrium systems, fluctuations are no longeronly thermal and the link between these fluctuations and particledynamics remain elusive.5 Much effort has been devoted to under-standing particle dynamics in non-equilibrium systems, such asglassy materials and sheared granular matter.6 A non-equilibriumsystem of emerging interest is active matter. Active matter includesactive fluids, that is, fluids that contain self-propelling particles, suchas motile microorganisms,7,8 catalytic colloids9,10 and molecularmotors.11 These particles inject energy, generate mechanical stresses,and create flows within the fluid medium even in the absence ofexternal forcing.12,13 Consequently, active fluids display fascinatingphenomena not seen in passive fluids, such as spontaneous flows,8

anomalous shear viscosities,7,14 unusual polymer swelling,15,16 andenhanced fluid mixing.17–20 Active fluids also play importantroles in varied biological and ecological settings, which includeformation of biofilms,21,22 biofouling of water-treatment systems,23

and biodegradation of environmental pollutants.24

The motion of passive particles in active fluids (e.g. suspensionof swimming microorganisms) can be used to investigate the

non-equilibrium properties of such fluids. For example, atshort time, particle displacement distributions can exhibitextended non-Gaussian tails, as seen in experiments with algaeC. reinhardtii.17,18 At long times, particles exhibit enhanceddiffusivities Deff greater than their thermal (Brownian) diffusivityD0.17–20,25–27 These traits are a signature of the non-equilibriumnature of active fluids, and the deviation from equilibrium alsomanifests in violations of the fluctuation dissipation theorem.28

In bacterial suspensions, the enhanced diffusivity Deff dependson the concentration c of bacteria. In their seminal work, Wu andLibchaber25 experimentally found that Deff increased linearly with cfor suspensions of Escherichia coli. Subsequent studies26,27,29–32

have observed that this scaling holds at low concentrations and inthe absence of collective motion. In this regime, Deff can bedecomposed into additive components as Deff = D0 + DA

26,27,29–32

where D0 and DA are the thermal and active diffusivities, respectively.It has been proposed that the active diffusivity DA is a consequenceof advection due to far-field interactions with bacteria26 andmay even be higher near walls.26,27

While a majority of studies have focused on the role of bacterialconcentration c on particle diffusion, the role of particle diameterd remains unclear. In the absence of bacteria, the diffusivity of asphere follows the Stokes–Einstein relation, D0 = kBT/f0, where kB isthe Boltzmann constant, T is the temperature, and f0 = 3pmd is theStokes friction factor4 in a fluid of viscosity m. In a bacterialsuspension, this relation is no longer expected to be valid.Surprisingly, for large particles (4.5 and 10 mm), Wu andLibchaber25 suggested that Deff scales as 1/d, as in passivefluids. Recent theory and simulation by Kasyap et al.32 howeverdo not support the 1/d scaling and instead predict a non-trivialdependence of Deff on particle size, including a peak in Deff.

a Department of Mechanical Engineering & Applied Mechanics, University of

Pennsylvania, Philadelphia, PA 19104, USA. E-mail: parratia@seas.upenn.edub School of Engineering, University of California Merced, Merced, CA 95343, USA

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5sm02800k

Received 14th November 2015,Accepted 14th January 2016

DOI: 10.1039/c5sm02800k

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This non-monotonic dependence of Deff on particle size impliesthat measures of effective diffusivities,25 effective temperatures,12,33

and momentum flux26,27 intimately depend on the probe sizeand thus are not universal measures of activity. This hasimportant implications for the common use of colloidal probesin gauging and characterizing the activity of living materials,such as suspensions of bacteria,26,27 biofilms,34,35 and thecytoskeletal network inside cells,36 as well as in understandingtransport in these biophysical setting. Despite the ubiquity ofpassive particles in active environments, the effects of size onparticle dynamics in active fluids has yet to be systematicallyinvestigated in experiments.

In this manuscript, we experimentally investigate the effectsof particle size d on the dynamics of passive particles insuspensions of Escherichia coli. Escherichia coli37 are modelorganisms for bacterial studies and are rod-shaped cells with3 to 4 flagella that bundle together as the cell swims forward atspeed U of approximately 10 mm s�1. We change the particlesize d from 0.6 mm to 39 mm, above and below the effective totallength (L E 7.6 mm) of the E. coli body and flagellar bundle.We find that Deff is non-monotonic in d, with a peak at 2 o d o10 mm; this non-monotonicity is unlike the previously found 1/dscaling25 and suggests that larger particles can diffuse fasterthan smaller particles in active fluids. Furthermore, the existenceand position of the peak can be tuned by varying the bacterialconcentration c. The active diffusion DA = Deff � D0 is also a non-monotonic function of d and can be collapsed into a master curvewhen rescaled by the quantity cUL4 and plotted as a function ofthe Peclet number Pe = UL/D0 (cf. Fig. 5(b)). This result suggeststhat the active contribution to particle diffusion can be encap-sulated by a universal dimensionless dispersivity %DA that is setby the ratio of times for the particle to thermally diffuse adistance L and a bacterium to swim a distance L.

2 Experimental methods

Active fluids are prepared by suspending spherical polystyreneparticles and swimming E. coli (wild-type K12 MG1655) in abuffer solution (67 mM NaCl in water). The E. coli are preparedby growing the cells to saturation (109 cells per mL) in culturemedia (LB broth, Sigma-Aldrich). The saturated culture is gentlycleaned by centrifugation and resuspended in the buffer. Thepolystyrene particles (density r = 1.05 g cm�3) are cleaned bycentrifugation and then suspended in the buffer-bacterialsuspension, with a small amount of surfactant (Tween 20, 0.03%by volume). The particle volume fractions f are below 0.1% andthus considered dilute. The E. coli concentration c ranges from0.75 to 7.5 � 109 cells per mL. These concentrations are alsoconsidered dilute, corresponding to volume fractions f = cvb o1%, where vb is the volume26 of an E. coli cell body (1.4 mm3).

A 2 ml drop of the bacteria–particle suspension is stretchedinto a fluid film using an adjustable wire frame7,18,38 to ameasured thickness of approximately 100 mm. The film interfacesare stress-free, which minimizes velocity gradients transverse tothe film. We do not observe any large scale collective behavior in

these films; the E. coli concentrations we use are below the values forwhich collective motion is typically observed32 (E1010 cells per ml).Particles of different diameters (0.6 mm o d o 39 mm) are imaged ina quasi two-dimensional slice (10 mm depth of focus). We considerthe effects of particle sedimentation, interface deformation, andconfinement on particle diffusion and find that they do notsignificantly affect our measurements of effective diffusivity in thepresence of bacteria (for more details, please see ESI,† Section I).Images are taken with a 10� objective (NA 0.45) and a CCD camera(Sony XCDSX90) at 30 fps, which is high enough to observecorrelated motion of the particles in the presence of bacteria(Fig. 2) but small enough to resolve spatial displacements. Particlesless than 2 mm in diameter are imaged with fluorescence microscopy(red, 589 nm) to clearly visualize particles distinct from E. coli(2 mm long). We obtain the particle positions in two dimensionsover time using particle tracking methods.38,39 All experimentsare performed at temperature T0 = 22 1C.

3 Results and discussion3.1 Mean square displacements

Representative trajectories of passive particles in the absenceand presence of E. coli are shown in Fig. 1 for a time intervalof 8 s. By comparing Fig. 1(a) (no E. coli) to Fig. 1(b) (c = 3 �109 cells per mL), we readily observe that the presence ofbacteria enhances the magnitude of particle displacementscompared to thermal equilibrium. Next, we compare sampletrajectories of passive particles of different sizes d, below andabove the E. coli total length L E 7.6 mm. Fig. 1(c) and (d) showthe magnitude of particle displacement for d = 0.6 mm and

Fig. 1 Trajectories of 2 mm particles (a) without bacteria and (b) withbacteria (c = 3 � 109 cells per mL) for time interval 8 s. Trajectories of (c)0.6 and (d) 16 mm particles (c = 3 � 109 cells per mL). Scale bar is 20 mm.

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d = 16 mm, respectively. Surprisingly, we find that the particlemean square displacements in the E. coli suspension arerelatively similar for the two particle sizes even though thethermal diffusivity D0 of the 0.6 mm particle is 35 times largerthan that of the 16 mm particle. The 16 mm particles also appearto be correlated for longer times than the 0.6 mm particles.These observations point to a non-trivial dependence of particlediffusivity on d and are illustrated in sample videos includedin the ESI.†

To quantify the above observations, we measure the mean-squared displacement (MSD) of the passive particles for varying

E. coli concentration c (Fig. 2(a)) and particle size d (Fig. 2(b)).Here, we define the mean-squared particle displacement asMSD(Dt) = h|r(tR + Dt) � r(tR)|2i, where the brackets denote anensemble average over particles and reference times tR. For aparticle executing a random walk in two dimensions, the MSDexhibits a characteristic cross-over time t, corresponding to thetransition from an initially ballistic regime for Dt { t to adiffusive regime with MSD B 4DeffDt for Dt c t.

Fig. 2(a) shows the MSD data for 2 mm particles at varyingbacterial concentrations c. In the absence of bacteria (c = 0 cellsper mL), the fluid is at equilibrium and Deff = D0. For this case,we are unable to capture the crossover from ballistic to diffusivedynamics due to the lack of resolution: for colloidal particlesin water, for example, cross-over times are on the order ofnanoseconds and challenging to measure.40 Experimentally,the dynamics of passive particles at equilibrium are thusgenerally diffusive at all observable time scales. We fit the MSDdata for the d = 2 mm case (with no bacteria) to the expressionMSD = 3D0Dt and find that D0 E 0.2 mm2 s�1. This matches thetheoretically predicted value from the Stokes–Einstein relation;the agreement (,) can be visually inspected in Fig. 3(a).

In the presence of E. coli, the MSD curves exhibit a ballisticto diffusive transition, and we find that the cross-over time tincreases with c. For Dt { t, the MSD B Dt with a long-timeslope that increases with bacteria concentration c. Additionally,the distribution of particle displacements follows a Gaussiandistribution (see ESI,† Section IIA for details and measures ofthe non-Gaussian parameter). These features, MSD B Dt andGaussian displacements, indicate that the long-time dynamicsof the particles in the presence of E. coli is diffusive and can becaptured by a physically meaningful effective diffusion coeffi-cient Deff.

We next turn our attention to the effects of particle size. Forvarying particle diameter d at a fixed bacterial concentration(c = 3� 109 cells ml�1), the MSD curves also exhibit a ballistic todiffusive transition, as shown in Fig. 2(b). Surprisingly, we finda non-monotonic behavior with d. For example, the MSD curvefor the 2 mm case sits higher than the 39 mm case and the0.6 mm case. This trend is not consistent with classical diffusion

Fig. 2 (a) Mean-square displacements (MSD) of 2 mm particles versus timeDt for varying bacterial concentration c. Dashed line is a fit to Langevindynamics (eqn (1)). (b) MSD for varying particle diameter d versus time forbacterial concentration c = 3.0� 109 cells per mL. The MSD peaks at d = 2 mm.The cross-over time t (arrows) increases with d. Solid lines are Langevindynamics fits (eqn (1)).

Fig. 3 (a) Effective particle diffusivities Deff versus particle diameter d at varying c. The dashed line is particle thermal diffusivity D0. (b) The crossover-timet increases with d, scaling as approximately dn, where 1/2 t n t 1. This is not the scaling in passive fluids41 where t p d2.

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in which MSD curves are expected to decrease monotonicallywith d (D0 p 1/d). We also observe that the cross-over time tincreases monotonically with d. As we will discuss later in themanuscript, the cross-over time scaling with d also deviatesfrom classical diffusion.

3.2 Diffusivity and cross-over times

We now estimate the effective diffusivities Deff and cross-overtimes t of the passive particles in our bacterial suspensions. Toobtain Deff and t, we fit the MSD data shown in Fig. 2 to the MSDexpression attained from the generalized Langevin equation,41

that is

MSDðDtÞ ¼ 4DeffDt 1� tDt

1� e�Dtt

� �� �: (1)

Eqn (1) has been used previously to interpret the diffusion ofbacteria16 as well as the diffusion of particles in films withbacteria.35 In the limit of zero bacterial concentration, DA = 0and eqn (1) reduces to the formal solution to the Langevin equation

for passive fluids, MSDðDtÞ ¼ 4D0Dt 1� t0Dt

1� e�Dtt0

� �� �with

t0 = t(c = 0). For more details on the choice of our model, seeESI,† Section IIIA.

Fig. 3 shows the long-time particle diffusivity Deff (Fig. 3(a))and the cross-over time t (Fig. 3(b)) as a function of d forbacterial concentrations c = 0.75, 1.5, 3.0 and 7.5 � 109 cells permL. We find that for all values of d and c considered here, Deff islarger than the Stokes–Einstein values D0 at equilibrium (dashedline). For the smallest particle diameter case (d = 0.6 mm), Deff

nearly matches D0. This suggests that activity-enhanced trans-port of small (d t 0.6 mm) particles or molecules such asoxygen, a nutrient for E. coli, may be entirely negligible.32 Formore information, including figures illustrating the dependenceof Deff on c and comparisons between our measured effectivediffusivities and previous experimental work, see Sections IIB andIIC in the ESI.†

Fig. 3(a) also reveals a striking feature, a peak Deff in d. Ourdata demonstrates that, remarkably, larger particles can diffusefaster than smaller particles in suspensions of bacteria. Forexample, at c = 7.5 � 109 cells per mL ( ), the 2 mm particle hasan effective diffusivity of approximately 2.0 mm2 s�1, which isnearly twice as high as the effective diffusivity of the 0.86 mmparticle, Deff = 1.3 mm2 s�1. We also observe that the peakvanishes as c decreases. In fact, for the lowest bacterial concen-tration (c = 0.75 � 109 cells per mL), there is no peak: Deff

decreases monotonically with d. Clearly, Deff does not scale as 1/d.Fig. 3(b) shows the cross-over times t characterizing the

transition from ballistic to diffusive regimes as a function ofparticle size d for varying c. We find that the values of t increasewith d and c. We note that the variation of t with c (ESI,†Fig. S3(b)) does not follow a linear form. Instead, the datasuggests possible saturation of t for suspensions of higher – butstill dilute – concentrations. The cross-over time t scales withparticle diameter approximately as t B dn, where 1

2 t n t 1.This cross-over time does not correspond to the inertial relaxationof the particle. Therefore, this scaling does not follow the trend

seen for passive particles at thermal equilibrium,41 where t, beingthe Stokes relaxation time (order 1 ns), scales as m/f0 p d2 withparticle mass m p d3. Our data (Fig. 3(b)) highlights that in activefluids the super-diffusive motion of the passive particles cannot beignored – even for time scales as large as a second – and that thetime scales over which diffusive motion is valid (Dt 4 t) dependson the size d of the particle. Further implications of a particlesize-dependent cross-over time will be discussed below.

3.3 Effective temperature

Our data so far suggests that particle dynamics in bacterialsuspensions, while having an anomalous size-dependence(Fig. 1–3), maintain the characteristic super-diffusive to diffusivedynamics for passive fluids.41 The long-time diffusive behavior(Fig. 2) and enhancement in Deff (Fig. 3(a)), which is rooted inparticle–bacteria encounters, suggest that the particles behaveas if they are suspended in a fluid with an effectively highertemperature.

In order to further explore the concept of effective temperaturein bacterial suspensions, we measure the distribution of particlespeeds p(v) as a function of bacterial concentration c. The particlespeed distributions determine the mean kinetic energy of theparticles. If the distribution follows a Maxwell–Boltzmann form –as is always the case in fluids at equilibrium – the mean kineticenergy is related to the thermodynamic temperature via theequipartition theorem. Such a relationship may not always existfor out-of-equilibrium fluids.

Fig. 4(a) shows p(v) for d = 2 mm case for a range of c. Wedefine the particle speeds v over a time interval of 0.5 s. Thistime interval is greater than the crossover times for the 2 mmparticles (cf. ESI,† Fig. S3) to ensure that a particle samplesmultiple interactions with bacteria and exhibits diffusive behavior.In the absence of bacteria, the system is in thermal equilibriumand p(v) follows the two dimensional Maxwell–Boltzmanndistribution,

pðvÞ ¼ vm kBTeffð Þ�1e�mv2

2kBTeff ; (2)

with peak speeds vmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kBTeff=m

p, where m is the mass of

the polystyrene particle. Fitting the p(v) data in the absence ofbacteria (, in Fig. 4(a)) to eqn (2) yields Teff E T0, as expected.

In the presence of bacteria, the particle speed distributionsalso follow a Maxwell–Boltzmann form, eqn (2), with peaks thatshift toward higher values of v as the E. coli concentration cincreases. Because our data follows the Maxwell–Boltzmannform (Fig. 4(a)) for all c, this indicates that there is no correlatedmotion at long times as is the case in swarming bacterialsuspensions, for which the particle speed distributions exhibitsan exponential decay.42 We also note that the power spectra ofparticle speeds (ESI,† Section IID) are reasonably flat, consistentwith white-noise forcing and an absence of correlated motion.Fig. 4(a) thus indicates that an effective temperature Teff canbe defined from p(v) and is increasing with the bacterialconcentration.

To quantify this ‘enhanced’ temperature and the deviation fromequilibrium behavior, we fit the p(v) data (Fig. 4(a)) to eqn (2).

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These fits allow us to obtain Teff as a function of c, as shownin Fig. 4(b) ( ). Also shown in Fig. 4(b) are the values extractedof Teff ( ) from an extended Stokes–Einstein relation, Teff =3pmdDeff/kB, where Deff are the values from the MSD fits shownin Fig. 3(a) and m is the viscosity of the solvent (m E 1 mPa s). Wefind that both estimates of Teff are higher than room tempera-ture T0 and increase linearly with c.

For a fluid at equilibrium, the temperatures from these twomethods, MSD and p(v), are expected to be the same but not forsystems that are out of equilibrium,5,41,43,44 such as the bacterialsuspensions investigated here. The good agreement in Fig. 4(b)suggests that Teff may be a useful signature of bacterial activitywith some analogies to equilibrium systems. Note that in definingthe effective temperature via the generalized Stokes–Einsteinrelation, we have assumed the viscosity to be a constant andindependent of bacterial concentration, which may not be truewhen the bacterial concentrations are sufficiently high.45 Fig. 4(b)suggests that an unchanging viscosity is a valid assumption forour system.

As shown in Fig. 4(b), the estimates of Teff are higher thanroom temperature T0 and increase linearly with c at least for c o7.5 � 109 cells per mL. In this regime (c o 7.5 � 109 cells per mL),the bacterial suspensions can be consider dilute and homogenouswith particle–bacteria interactions being binary to leading order.Additionally, in the absence of collective behavior, fluctuations inbacterial concentration scale as

ffiffifficp

. These features imply that theeffective diffusivity Deff has a linear dependence on c, consistentwith our measurements (see ESI,† SII for figure and more details).Therefore, the extended Stokes–Einstein relationship suggestseffective temperature also scales linearly with concentration, asshown in Fig. 4(b).

At the highest concentration (c = 7.5 � 109 cells per mL), theslope in the Teff versus c curve decreases. This decreasing slopemay be understood using a momentum flux argument forpurely steric interactions that include the bacteria and particlesize. In over-damped systems, as in our bacteria suspensions,the momentum flux to the particle due to bacteria–particleinteractions results in excess kinetic energy, which is eventuallydissipated away viscously. At low concentrations, the flux (andtherefore the active temperature) is proportional to c. If interactions

are not purely binary, as at high concentrations, the bacteria size aswell as particle size may limit the number of bacteria interactingwith the particle at any given time and Teff may saturate due to finitesize effects.

Finally, we investigate the role of particle diameter in thesuspension effective temperature. Fig. 4(c) show the values ofTeff estimated from an extended Stokes–Einstein relation as afunction of d for different values of c. Surprisingly, we find thatTeff increases with particle size d, a behavior different fromthermally equilibrated systems where temperature does notdepend on the probe size. We note that for the largest particlediameter, d = 39 mm, estimated Teff values are approximately100 times greater than room temperature T0 = 295 K, consistentwith previous reports.25

The dependence of Teff on particle size d (Fig. 5(a)) may beunderstood through the previously introduced extendedStokes–Einstein relation, Teff = Defff0/kB = T0 + (3pmDA/kB)d,where T0 = (3pmD0/kB)d. If DA were independent of d, then Teff

would be linear in d. However, a linear fit does not adequatelycapture the trend over the full range of particle diameters. This is aconsequence of the particle size dependent Deff shown in Fig. 3(b)and hints at a particle-size dependent active diffusivity DA. Thisvariation of Teff with d highlights the interplay between particlesize and the properties of the self-propelling particles (E. coli) aswell as challenges in gauging activity using passive particles.

3.4 Active diffusivity of passive particles in bacterialsuspensions

To explore the aforementioned dependence of DA on particlesize, we plot DA = Deff � D0 with d. Here, D0 is the particlethermal diffusivity (in the absence of bacteria‡) and is obtained

Fig. 4 (a) Distribution of 2 mm particle speeds p(v) follow a Maxwell–Boltzmann distribution (solid curves) with clear peaks that shift right as c increases.(b) The effective temperature Teff extracted by fitting p(v) data to eqn (2) ( ) matches those obtained from an extended Stokes–Einstein relation ( ).(c) The effective temperature Teff increases with d for varying c.

‡ For a suspension of passive particles of volume fraction f, the Einsteinviscosity46 is m = m0(1 + 5f/2), where m0 is the viscosity of the suspending medium.In the absence of collective motion, the maximum change in the viscosity due tobacteria (since they are force free and exert no drag on the fluid) in our bacterialsuspensions (with volume fractions of f = 1%) is thus 2.5%. Therefore, m and D0 isnot expected to change significantly with the bacterial concentrations used here.This is consistent with the observation that the effective temperature estimatesfrom both an extended Stokes–Einstein relation and particle speed distribution(Fig. 4b) agree.

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from the Stokes–Einstein relation D0 ¼kBT0

3pmd. Indeed, as shown

in Fig. 5(a), DA exhibits a non-monotonic dependence on dfor all c.

To understand the observed dependence of the active diffusivityon the particle size, we consider the relevant time scales in ourparticle/bacteria suspensions, namely: (i) the time for theparticle to thermally diffuse a distance L equal to the totalbacterial length L2/D0, (ii) the time for a bacterium to swim at aspeed U for a distance L given by L/U, and (iii) the mean runtime which is the inverse of the tumbling frequency oT

�1.Dimensionless analysis then suggests there are two indepen-dent time parameters: (i) the ratio of the first two above, whichis the Peclet number, Pe � UL/D0, and (ii) t* = U/oTL, which isthe ratio of the run length U/oT to the bacterial length.

It is these two parameters, Pe and t*, that govern the particledynamics in our experiments. When the Peclet number is much lessthan one, then the thermal particle diffusion dominates andtransport by the bacteria is ineffective. When the Peclet number ismuch larger than one, then thermal diffusion is negligible and thetransport is due to the convection from bacteria. The swimmingspeed U, tumble frequencies oT

�1, and combined length of thecell body and flagellar bundle L are estimated from priorexperiments16,47 with E. coli as approximately 10 mm s�1,1 s�1, and L E 7.6 mm, respectively. Thus, in our experiments,the Peclet number varies from approximately 130 to 8600, viathe particle bare diffusivity D0 (through the particle diameter).

We note that one stain of bacteria is used – thus, the run lengthand bacteria size do not change in our experiments, andconsequently, t* E 1.8 is a constant.

In order to gain insight into the non-trivial dependence ofDA on d, we scale out the concentration dependence by introducingthe dimensionless active diffusivity %DA = DA/cUL4 and plot it againstthe Peclet number, Pe. Fig. 5(b) shows that all the active diffusionDA versus d data shown in Fig. 5(a) collapses into a single mastercurve, thereby indicating that %DA is independent of c, at least fordilute suspensions investigated here.

We find that for Pe t 103, the values of %DA initially increaseswith increasing Pe (or particle size) and follows a scaling %DA B Pe2.The observed increase of %DA with Pe may be due to the decreasingparticle Brownian motion, which allows the particle’s motion to becorrelated with the bacterial velocity disturbances for longer times.

In the limit of Pe - N, the particle’s Brownian motionvanishes and the particle displacements are dominated by theconvective transport via bacteria–particle interactions. Thus %DA

is expected to be independent of Pe and depend only on theparameter t*, defined here as the ratio of the run length U/oT tothe bacterial length L. Our experimentally measured %DA, whichcorrespond to t* E 1.8, exhibit a slight decline with Pe at highPe and have magnitudes that agree well with recent theoreticalpredictions32 (dashed lines in Fig. 5(b)) for very large Peclet att* = 1 and t* = 4. An increase in the run time (or t*) wouldincrease the asymptotic value of the scaled active diffusivity %DA

(see ESI,† Section IV).An important feature of the data shown in Fig. 5(b) is the

peak in %DA at PeA E 103. The appearance of the peak in %DA inour data may be due to the weak but non-zero effects ofBrownian motion, which allows particles to sample the bacterialvelocity field in such a way that the mean square particledisplacements and correlation times are higher compared toboth the very high Pe (negligible Brownian motion) as well asthe very low Pe (Brownian dominated).32 This feature (i.e. peakin %DA) is surprising because it suggests an optimum particle sizefor maximum particle diffusivity that is coupled to the activity ofthe bacteria. The existence of such a peak has been predicted ina recent theory/simulation investigation,32 and our data agreesat least qualitatively with such predictions. We note that thepredicted peak in %DA happens at a Pe E O(10), which is smallerthan our experimental values of (Pe E 103). One reason why maybe due to variations in bacterial concentration within the film,especially near the film’s surfaces where bacteria may cluster.

4 Maximum particle effectivediffusivity Deff

Our data (Fig. 5(b)) shows that the dimensionless active diffusion%DA collapses unto a universal curve with a peak in Pe for allbacterial concentrations. Our data also shows that, with theexception of the lowest bacterial concentration c, the particleeffective diffusivity Deff exhibits a peak in d, which varies withc (Fig. 3(a)). For instance, for c = 1.5 � 109 cells per mL, thepeak is at approximately 2 mm, while for larger concentrations

Fig. 5 (a) Active diffusivities DA = Deff � D0 are non-monotonic withparticle size for varying concentrations of bacteria. (b) Scaled hydrodynamicdiffusivity %DA = DA/cUL4 collapses with Peclet number Pe = UL/D0. Themaximum %DA occurs at PeA between 450 and 4500. For 100 o Pe { PeA,%DA scales as Pea, where a E 2.

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(c = 7.5 � 109 cells per mL), the peak (as obtained by fitting thedata to a continuous function) shifts to higher values of d. Thissuggests that one can select the particle size which diffuses themost by tuning the bacteria concentration.

In what follows (see also ESI,† SV), we provide a prediction,based on our experimentally-measured universal curve of %DA

with Pe (Fig. 5(b)) for the existence as well as the location of thepeak of Deff in d. As noted before, the particle effective diffusivityDeff can be described as the linear sum of the particle thermaldiffusivity D0, which is independent of c and decreases with d,and the active diffusivity DA, which is linear in c and non-monotonicin d, through the particle-size dependent %DA. Therefore, we canrecast the effective diffusivity as

Deff = D0 + (cL3)(UL) %DA. (3)

The criterion for the existence of a maximum Deff is obtained bytaking the derivative of eqn (3) with respect to the Pecletnumber and setting the derivative to zero. In order to estimate%DA (and its slope), we fit the data in Fig. 5(b) near the peak inthe range 200 o Pe o 4000 with a second order polynomialequation. We find that a peak exists in Deff if

cL3\ 0.4. (4)

For the bacterial length used here L = 7.6 mm, this yields c E 0.9�109 cells per mL, which is in quantitative agreement with theconcentration range (0.75 � 109 cells per mL o c o 1.5 �109 cells per mL) in which the peak in Deff emerges in our data(Fig. 3(a)). As described in ESI,† SV, we find that the location ofthe Deff peak in d here defined as dmax

eff is given by

dmaxeff � dA 1� 1

5cL3 � 2

� �� �(5)

where dA corresponds to the Peclet number PeA E 1000 at which%DA is maximum. For the E. coli used here, dA = kTPeA/3pmUL E 6 mm.Note that dmax

eff is always less than or equal to dA and increases with c,consistent with our experimental observations (Fig. 3(a)). In generalthe criterion, eqn (4), and dmax

eff , eqn (5), will depend on t*. Forsuspensions of bacteria, the universal curve of %DA informs whenand where a peak in the particle diffusivity occurs.

5 Conclusions

In summary, we find that the effective particle diffusivity Deff

and temperature Teff in suspensions of E. coli show strongdeviations from classical Brownian motion in the way theydepend on particle size d. For example, Fig. 3(a) shows thatDeff depends non-monotonically in d and includes a regime inwhich larger particles can diffuse faster than smaller particles.The existence as well as the position of a Deff peak in d can betuned by varying the bacterial concentration c. We also find thatthe cross-over time t increases with particle size and scales asapproximately dn, where 1/2 t n t 1, as shown in Fig. 3(b).

Measures of Teff obtained from either an extended Stokes–Einstein relation or particle speed distributions seem to agreequite well (Fig. 4(b)). This is surprising since this kind of agreementis only expected for systems at equilibrium. The good agreement

between the measurements suggests that Teff may be a usefulsignature of bacterial activity. However, unlike thermally equili-brated systems, Teff varies with size d (Fig. 4(c)). This non-trivialdependence of both Deff and Teff on particle size d implies thatthese common gauges of activity are not universal measures.Nevertheless, our data suggest that one can define optimalcolloidal probes of activity for suspensions of bacteria, whichcorrespond to Pe = UL/D0 E 103. At these Pe values, %DA ismaximized, which provides ample dynamical range (magnitudeof the signal). Also at these Pe values, the cross-over time t isstill relatively small, which allows for adequate temporal resolution.Both of these features are important in using passive particlesto characterize a spatially and temporally varying level of activityin materials.

Our anomalous particle-size dependent results in activefluids has important implications for particle sorting in micro-fluidic devices, drug delivery to combat microbial infections,resuspension of impurities and the carbon cycle in geophysicalsettings populated by microorganisms. A natural next step wouldbe to study the role of external fields such as gravity or shear ininfluencing particle transport in these active environments.

Acknowledgements

We would like to thank J. Crocker, S. Farhadi, N. Keim, D. Wong,E. Hunter, and E. Steager for fruitful discussions and B. Qin forhelp in the spectral analysis of data. This work was supported byNSF-DMR-1104705 and NSF-CBET-1437482.

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