Symmetric shear banding and swarming vortices in bacterial "superfluids"
Shuo Guo, Devranjan Samanta, Yi Peng, Xinliang Xu, Xiang Cheng
aa r X i v : . [ c ond - m a t . s o f t ] M a y D R A F T Symmetric shear banding and swarming vorticesin bacterial “superfluids”
Shuo Guo a,b , Devranjan Samanta a,1 , Yi Peng a , Xinliang Xu b,2 , and Xiang Cheng a,2 a Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA; b Beijing Computational Science Research Center,Beijing 100193, ChinaThis manuscript was compiled on May 31, 2018
Bacterial suspensions—a premier example of active fluids—show anunusual response to shear stresses. Instead of increasing the vis-cosity of the suspending fluid, the emergent collective motions ofswimming bacteria can turn a suspension into a “superfluid” withzero apparent viscosity. Although the existence of active “super-fluids” has been demonstrated in bulk rheological measurements,the microscopic origin and dynamics of such an exotic phase havenot been experimentally probed. Here, using high-speed confocalrheometry, we study the dynamics of concentrated bacterial suspen-sions under simple planar shear. We find that bacterial “superfluids”under shear exhibit unusual symmetric shear bands, defying the con-ventional wisdom on shear-banding of complex fluids, where the for-mation of steady shear bands necessarily breaks the symmetry of un-sheared samples. We propose a simple hydrodynamic model basedon the local stress balance and the ergodic sampling of nonequi-librium shear configurations, which quantitatively describes the ob-served symmetric shear-banding structure. The model also success-fully predicts various interesting features of swarming vortices in sta-tionary bacterial suspensions. Our study provides new insights intothe physical properties of collective swarming in active fluids andillustrates their profound influences on transport processes. active fluids | bacterial suspensions | shear banding A ctive fluids, suspensions of self-propelled particles, haveattracted enormous research interests in recent years (1–5). With examples across biological and physical systems ofwidely different scales, active fluids exhibit many novel prop-erties, such as the emergence of collective swarming (6–9),giant number fluctuations (10, 11) and enhanced diffusion ofpassive tracers (12–16). Among all these unusual features,the flow behavior of active fluids demonstrates the nonequi-librium nature of active systems in the most striking manner.Surprising phenomena including superfluid-like behaviors (17)and spontaneous directional flows (18, 19) have been observedin active fluids.Using a phenomenological model that couples hydrody-namic equations with active nematic order parameters, Hat-walne et al. first showed that pusher microswimmers suchas E. coli can significantly lower the bulk viscosity of activesuspensions, to such an extent that suspensions can have alower viscosity than the suspending fluids (20). Based on asimilar approach, Cates et al. further predicted that near thedisorder-to-order transition to collective motions, a pusher ac-tive fluid can enter a “superfluidic” regime where its apparentshear viscosity vanishes (21). Later theory by Giomi et al. revealed even richer dynamics and predicted the existence ofshear banding, yield stress, and “superfluidity” of active fluids(22). Unusual rheology of active fluids has also been studiedbased on the microhydrodynamics of microswimmers at lowconcentrations (23–27), swimming pressures (28) and gener- alized Navier-Stokes equations (29). Experimentally, Sokolov et al. and Gachelin et al. showed the low viscosity of bacte-rial suspensions in thin films (30, 31). Lopez et al. demon-strated the superfluid-like transition in concentrated
E. coli suspensions using a rotational rheometer (17). Under channelconfinements, this “superfluidic” behavior displays as sponta-neous directional flows (18, 19). In comparison, puller swim-mers such as swimming algae were shown to enhance, insteadof suppressing, the viscosity of suspensions (32).Although the vanishing shear viscosity of active “superflu-ids” has been demonstrated in bulk rheology studies (17), themicroscopic dynamics of such an exotic phase under simpleshear flows have not be experimentally explored. The shear-banding structure—an important prediction of hydrodynamictheories (21, 22)—has not been verified. Here, using fast con-focal rheometry, we study the dynamics of concentrated bac-terial suspensions under planar oscillatory shear. We findthat bacterial superfluids exhibit symmetric shear-bandingflows with three shear bands. We systematically investigatethe variation of the shear-banding structure with shear rates,bacterial concentrations and bacterial motility. Based on theexisting hydrodynamic theories, we construct a simple phe-nomenological model that quantitatively describes the shapeof the symmetric shear bands. The model also predicts severalnontrivial properties of swarming vortices in stationary bac-terial suspensions, including the linear relation between the..
Significance Statement
Bacterial suspensions can flow without apparent viscosity.Such a superfluid-like behavior stems from the collective mo-tions of swimming bacteria. Here, we explore the microscopicflow profile of bacterial “superfluids” under simple shear. Wefind that, instead of deforming uniformly, bacterial “superflu-ids” develop multiple shear bands, i.e., regions with differentshear deformations. We construct a simple model that quan-titatively describes the shape of the shear-banding structureand reveals important physical properties of collective bacte-rial motions. Our study sheds light on complex interactionsbetween swimming microorganisms and ambient fluid flows,crucial for the survival of microorganisms in nature and themanipulation of bacterial suspensions in engineering settings.
D.S. and X.C. conceived the project. S.G., D.S. and X.C. conducted the experiments and analyzedthe data with help from Y.P. X.X. and X.C. developed the model. S.G., X.X. and X.C. wrote themanuscript with all authors contributed to the final version.The authors declare no conflict of interest. Current address: Department of Mechanical Engineering, Indian Institute of Technology, Ropar,Nangal Road, Rupnagar, Punjab 140001, India To whom correspondence should be addressed. E-mail: [email protected] [email protected]
May 31, 2018 | vol. XXX | no. XX | R A F T Fig. 1.
Bacterial suspensions under planar oscillatory shear. (A) Bacterial swarming at a concentration n = 80 n . The scale bar is 20 µ m. The fluorescently tagged E.coli serve as tracer particles for particle imaging velocimetry (PIV). (B) Schematic showing our custom shear cell. A Cartesian coordinate system is defined, where x , y and z are the flow, shear gradient and vorticity directions, respectively. (C) Temporal variation of mean suspension velocities ˙ x ( t ) at different heights, y , above the bottom plate.Red curves are for shear-rate amplitude ˙ γ = 0 . s − . Black curves are for ˙ γ = 0 . s − . Velocities are normalized by the imposed velocity amplitudes, V . Time t isnormalized by shear period T = 1 /f . y is normalized by gap thickness H . ˙ x ( t ) at different y are shifted vertically for clarity. Dashed lines are sinusoidal fits. kinetic energy and the enstrophy of suspension flows and thesystem-size dependence of the length and strength of swarm-ing vortices. We conclude the paper by discussing the uniquefeature of the shear-banding flow of bacterial suspensions incomparison with conventional shear-banding complex fluids.Our study provides new insights into the collective swarmingof active fluids and illustrates the unexpected consequence ofcollective swarming on momentum transports of active sys-tems. Our results also help to understand complex interac-tions between bacteria and ambient shear flows encounteredin many natural and engineering settings. Results
We use fluorescently tagged
Escherichia coli
K-12 strain(BW25113). The bacteria are suspended in a mobility bufferto a concentration n . We vary n between 10 n and 100 n with n = 8 × ml − the concentration of bacteria in thestationary phase of growing. When n & n , collective bac-terial swarming can be observed (Fig. 1A and SI AppendixA).We investigate the 3D fluid flow of E. coli suspensions un-der planar oscillatory shear. A suspension of 20 µ l is con-fined between the two parallel plates of a custom shear cellwith a constant spacing H = 60 µ m unless other stated(Fig. 1B)(33, 34). A circular top plate of radius 2.5 mm is sta-tionary, whereas a much larger bottom plate driven by a piezo-electric actuator moves sinusoidally with x ( t ) = A sin(2 πft ).The shear amplitude and frequency, A and f , determine theamplitude of imposed shear rates, ˙ γ = V /H = 2 πfA /H ,where V is the applied velocity amplitude. For most exper-iments, we vary ˙ γ by changing A and fixing f = 0 . n > n (7). Bothshear boundaries yield qualitatively similar results. Symmetric shear banding.
The average velocity of a concen-trated bacterial suspension under shear at different heights y above the bottom plate, ˙ x ( y, t ), is shown in Fig. 1C. Here,the average is taken along both the flow ( x ) and the vor-ticity ( z ) directions. ˙ x ( y, t ) is sinusoidal following ˙ x ( y, t ) = V ( y ) cos(2 πft ), where V ( y ) is the velocity amplitude at y .A drastic difference in suspension dynamics can be identifiedbetween suspensions in the normal phase under strong shearand those in the “superfluidic” phase under weak shear (17).Under strong shear, V ( y ) decreases linearly with y , similar tothe response of dilute colloidal suspensions (Fig. 2A). How-ever, under weak shear, interesting nonlinear shear profilesare observed. All the applied shear concentrates near the cen-ter of the suspensions. Near the top and bottom plates, localshear gradients are small and may even vanish, resulting inapproximately symmetric shear profiles rarely seen in othercomplex fluids (Fig. 2A). A crossover from the linear to thenonlinear shear profiles is observed with decreasing ˙ γ .The shape of shear profiles also depends on the strength ofcollective bacterial swarming. We vary the swarming strengthby changing bacterial concentrations n (SI Appendix Fig. S2)(6). At large n , bacteria show strong collective motions, lead-ing to the nonlinear shear profiles at low ˙ γ (Fig. 2B). Below40 n where the collective swarming is weak, the shear profileappears to be linear even at low ˙ γ . A similar crossover tothe linear profile is also observed when bacterial swarmingweakens due to the depletion of oxygen. A concentrated sus-pension of immobile bacteria shows a linear shear profile atall ˙ γ (Fig. 2B).The competition between the shear flow and the collectivebacterial swarming dictates the microscopic suspension dy-namics. The strength of shear flows is naturally quantified bythe imposed shear rate amplitude, ˙ γ . The strength of bacte-rial swarming can be quantified by the enstrophy of bulk sta-tionary suspension flows without external shear, Ω y ≡ h ω y / i (9). Here, ω y = ∂ x v z − ∂ z v x is the in-plane vorticity, where v x and v z are local suspension velocities along the flow andvorticity directions. The average h·i is again taken over the et al. R A F T Fig. 2.
Shear profiles of bacterial suspensions. (A) Normalized shear profiles atdifferent shear rates. V is the applied shear velocity amplitude. Bacterial concen-tration is fixed at n = 50 n . The shear-rate amplitude ˙ γ = 0 . s − (blacksquares), 0.16 s − (red circles) and 0.055 s − (blue triangles). Si wafer is used asthe top plate. (B) Normalized shear profiles at different bacterial concentrations. ˙ γ is fixed at . s − . n = 10 n (black squares), n (red circles) and n (blue triangles). To maintain bacterial motility at high n , a porous membrane is usedas the top plate. The stop height, h s , of the profile at n is indicated. Emptysquares are for a suspension of immobile bacteria at n . flow–vorticity plane. We then construct a dimensionless shearrate ˙ γ / p Ω y . To characterize the shape of shear profiles,we measure the stop height, h s , above which the shear flowvanishes (Fig. 2B). h s is obtained experimentally by fittingshear profiles piecewise with three linear lines (SI AppendixFig. S4). When plotting h s as a function of ˙ γ / p Ω y , all ourdata at different imposed shear rates, bacterial activities andgap thicknesses collapse onto a master curve (Fig. 3). Above˙ γ / p Ω y ≈
2, the shear profiles are linear with h s = H . Atsmall ˙ γ / p Ω y , h s increases linearly with ˙ γ / p Ω y and ap-proaches H/ Model.
The existence of bacterial “superfluids” have been pre-dicted by hydrodynamic theories of active fluids (21, 22).These theories show that the constitutive equation of activefluids is nonmonotonic across zero (Fig. 4A). The mechani-cal instability induced by the negative slope of the constitu-tive relation then leads to a zero-stress “superfluidic” plateau(35, 36). The instability also predicts a nonmonotonic shearprofile with two shear bands of opposite shear rates (Fig. 4B).To understand the symmetric shear profiles in our experi-ments, we construct a simple phenomenological model basedon the constitutive equation of the hydrodynamic theory (21)(SI Appendix C). The local total shear stress, σ t , can be di-vided into two parts, σ t = σ s + σ a , where σ s = η ˙ γ loc is thelocal viscous shear stress with suspension viscosity η and lo-cal shear rate ˙ γ loc . σ a = −| σ a | sgn( ˙ γ loc ) is the active stress Fig. 3.
Shape of shear profiles. The stop height, h s , as a function of the dimen-sionless shear rate ˙ γ / p Ω y . h s is normalized by the gap thickness H . H = 30 µ m (squares) and 60 µ m (circles). Color symbols are obtained with the symmetricshear boundary using Si wafer at f = 0 . Hz. Gray symbols are obtained with theasymmetric shear boundary using the porous membrane. Solid gray symbols are for f = 0 . Hz and empty gray symbols are for other shear frequencies between 0.025Hz and 0.3 Hz. Inset shows the same data in a log-linear plot. The solid line is thetheoretical prediction in the “superfluidic” phase and the dashed line is the predictionin the normal phase (Eq. [2]). originated from bacterial swimming (22), where sgn is thesign function. Here, we assume that the degree of local ne-matic ordering of bacteria is determined by steric and hydro-dynamic interactions between bacteria, whereas the orienta-tion of the nematic order is selected by the local shear flow. | σ a | is a function of bacterial concentrations and motility, butis insensitive to the magnitude of local shear rates (21, 22).A shear-rate-dependent | σ a | based on detailed hydrodynamictheories does not change the predictions of our simple model(SI Appendix F). For simplicity, we also ignore the complexbacteria-boundary interaction, which may influence the av-erage bacterial orientation near walls (37). Considering thebacteria-boundary interaction should not affect the key pre-dictions of our model either (SI Appendix G).In the “superfluidic” phase, the stress balance, σ s + σ a = 0,gives rise to two solutions, i.e., ˙ γ loc = ˙ γ ∗ and ˙ γ loc = − ˙ γ ∗ ,where ˙ γ ∗ ≡ | σ a | /η is the characteristic shear rate of bacterialsuspensions. To satisfy the no-slip boundary condition, wehave the nonmonotonic shear-banding flow (Figs. 4B and C),where the width of the shear band with − ˙ γ ∗ , w , follows (SIAppendix C) wH = 12 (cid:18) − ˙ γ ˙ γ ∗ (cid:19) = 12 − ˙ γ C p y ! [1]Here, we replace ˙ γ ∗ by the experimental observable Ω y . Ina stationary sample without external applied shear, bacterialswarming is solely driven by the active stress. Thus, the activestress balances the viscous stress, | σ a | = Cηω y = Cη p y ,where C is a proportionality constant close to one. Thus,˙ γ ∗ = | σ a | /η = C p y . Since w ≥
0, ˙ γ ≤ ˙ γ ∗ , setting thenecessary condition for “superfluids”.It should be emphasized that there are two and only twoshear configurations with two shear bands satisfying the stressbalance and the no-slip boundary condition, which are shownin Figs. 4B and C, respectively. Since both shear configu-rations satisfy the local stress balance, we hypothesize they Guo et al.
PNAS |
May 31, 2018 | vol. XXX | no. XX || vol. XXX | no. XX |
May 31, 2018 | vol. XXX | no. XX || vol. XXX | no. XX | R A F T Fig. 4.
Duality of shear configurations. (A) A schematic showing the constitutiverelation of active fluids from hydrodynamic theories (21, 22). The nonmonotonictrend predicts shear-banding flows with two shear bands of opposite shear rates, ˙ γ = ˙ γ ∗ and ˙ γ = − ˙ γ ∗ . The corresponding shear profile are shown in (B) and(C). Red arrows indicate shear velocities at different heights. Gap thickness, H , andthe width of the shear band with ˙ γ , w , are indicated. (D) Symmetric shear profile(thick red line) resulting from the average of the two shear configurations in (B) and(C) (yellow and blue dashed lines). Symbols are the experimental shear profile at n = 80 n and ˙ γ = 0.26 s − . The stop height, h s , is indicated. (E) The dualityof shear profiles at zero applied shear rate ˙ γ = 0 . The mean flow is zero (thickred line), whereas the two shear-banding configurations (yellow and blue dashedlines) are symmetric with respect to the mean flow. Inset of (E): at given y , the twoconfigurations moving along and against the shear flow complete a bacterial vortexin the x - z plane. emerge in a sheared sample “ergodically” with equal proba-bility, an assumption that shall be tested a posteriori . Themeasured shear profile should then be an “ensemble” averageof the two shear configurations. A possible physical inter-pretation is as follows: a single bacterial vortex normal to theflow-vorticity plane extending across the two shear plates (seeFig. 1A) can be viewed as composed of the two shear config-urations (Fig. 4E inset). The half of the vortex moving alongthe shear direction represents the configuration of Fig. 4B,whereas the other half moving against shear gives the config-uration in Fig. 4C. Thus, the ensemble average is achievedexperimentally through a spatiotemporal average over multi-ple swarming vortices. Vortices have a characteristic diameter ∼ µ m when H = 60 µ m (Fig. 1A) and a life time of a fewseconds (7, 29), whereas the spatial and temporal scales ofour experiments are 180 µ m and 40 s, respectively.The ensemble average of the two shear configurations nat-urally leads to a symmetric shear profile (Fig. 4D), consistentwith our observations. Using Eq. [1] and a simple geometricrelation h s + w = H , we have h s H = ( (cid:16) ˙ γ C √ y (cid:17) if ˙ γ / p Ω y ≤ C √ , γ / p Ω y > C √ , [2]which successfully predicts the linear relation between h s and˙ γ / p Ω y in the “superfluidic” phase (Fig. 3). A quantitativefitting of experimental data shows C = 1 . ± . H , three or more shear Fig. 5.
Probability distribution function of local velocities along the flow direction, v x ,at different shear rates, P ( v x ) . (A) ˙ γ / p Ω y = 0 , (B) ˙ γ / p Ω y = 0 . and(C) ˙ γ / p Ω y = 2 . . Local velocities are measured when the average shearvelocity reaches maximal in each shear cycle. PIV box size is chosen at R , where R is the characteristic radius of swarming vorticies. n = 80 n and H = 60 µ m.Insets show schematically the corresponding shear profiles. The thick dashed lines(red and blue) indicate the two shear configurations. The thin horizontal dashedline indicates the position of our imaging plane. The intersections give two discretevelocities, v x,l and v x,r , corresponding to the two peaks of P ( v x ) . (D) The twopeaks of P ( v x ) , v x,l (black squares) and v x,r (red disks), and velocity variance, δv x (magenta triangles), as a function of shear rate, ˙ γ / p Ω y . Dashed lines showthe model predictions. bands may emerge, which have infinite possible shear configu-rations satisfying the stress balance and the no-slip boundarycondition. The ergodic assumption would then lead to feature-less linear shear profiles (SI Appendix D). Our experimentsare different from earlier studies on bacterial suspensions un-der channel confinement, which constrains bacterial swarmingalong both the shear gradient and vorticity directions. Such aconfinement suppresses the instability that induces swarmingvortices (4). As a result, suspensions develop directional flowsand break the hypothesized “ergodicity” (18).The model incorporates an unique feature, i.e., a dynamicalternation between the two shear configurations around themean shear profile (Fig. 4D). To verify the hypothesis, wemeasure the probability distribution function of local veloci-ties at the center of the shear cell, P ( v x ), at different shearrates (Fig. 5). At zero and low shear rates (Figs. 5A and B),bimodal distributions with two distinct peaks can be identi-fied. The peaks correspond to the velocities of the two discreteshear profiles at y = H/ v x,l and v x,r (Figs. 5A and B insets).The finite width of the distributions arises presumably fromthe variation of individual bacterial mobility, an effect thatis not included in our model. The areas underneath the twopeaks are approximately the same with difference less than 5%at ˙ γ = 0, supporting our ergodic assumption. At high shearrates in the normal phase, P ( v x ) becomes unimodal (Fig. 5C),indicating the emergence of a single linear profile (Fig. 5C in-set). Our model predicts that the left peak of P ( v x ), v x,l ,increases linearly with ˙ γ in both phases, whereas the rightpeak of P ( v x ), v x,r , is constant in the “superfluidic” phaseand merges with v x,l in the normal phase. The variance ofvelocity δv x from the model follows (SI Appendix E) δv x = p h v x i − h v x i = v x,r − v x,l H (cid:16)p y − ˙ γ (cid:17) [3]in the “superfluidic” phase and becomes zero in the normalphase. Our experiments quantitatively agree with all thesepredictions (Fig. 5D). Direct measurements on instantaneousshear profiles at local scales are certainly needed to finally ver- et al. R A F T Fig. 6.
Properties of bacterial swarming in stationary samples. (A) Kinetic energy, E xz , versus enstrophy of suspension flows, Ω y . The gap size H is indicated in theplot. Flows are measured at the midplane y = H/ . The solid line indicates thelinear relation E xz ∼ Ω y . (B) Λ extracted from the slope of E xz (Ω y ) versus H .The solid linear is a linear fit. (C) Velocity spatial correlations. The horizontal dashedline is e − . H is indicated. (D) Correlation length, l , as a function of H . Bacterialconcentrations, n , are indicated. The dashed line indicates the linear relation. (E) E xz as a function of the height y at three different H . n = 64 n . (F) The maximum E xz at y = H/ versus H . The solid line is a linear fit. ify the ergodic assumption of our model, which is constructedto rationalize the 3D experimental results using simple steady-state 1D shear profiles (SI Appendix C). Swarming vortices in stationary bacterial suspensions.
Theminimal model also predicts several nontrivial properties ofswarming vortices in stationary bacterial suspensions withoutshear. First, from Eq. [3], when ˙ γ = 0, δv x = Λ Ω y , whereΛ ≡ H /
2. Since without shear h v x i = 0, δv x = h v x i . Thekinetic energy of suspension flows E xz = h v x i = δv x = Λ Ω y .Thus, the model predicts that the kinetic energy of a bacterialswarming flow is linearly proportional to the enstrophy of theflow. The square root of the slope, Λ, is proportional to thegap size of the system. Although the linear relation between E xz and Ω y has been reported in experiments on thin bacte-rial films and in simulations using generalized Navier-Stokesequations (9), there still lacks a simple physical explanation ofits origin. Our simple model shows that such a linear relationarises from the alternation of self-organized shear profiles inunsheared samples dictated by the local stress balance. Toverify the model, we measure E xz and Ω y of stationary bacte-rial suspensions. At a fixed H , E xz indeed increases linearlywith Ω y for different bacterial motility (Fig. 6A). More impor-tantly, we measure E xz (Ω y ) at different gap sizes and extractΛ from the slope of the linear relations. Λ as a function of H shows a clear linear trend (Fig. 6B), agreeing with the model,although the slope of Λ( H ) is smaller than the predicted value.Previous studies implied that Λ is associated with thelength scale of swarming vortices (9). Since Λ changes lin-early with H (Fig. 6B), we hypothesize that the size of swarm-ing vortices should also change linearly with the gap size of the system. To test the hypothesis, we measure the velocity–velocity spatial correlation (Fig. 6C) C v ( r ) = RR d ~r i d ~r j ( ~v ( ~r i ) · ~v ( ~r j )) δ ( r ij − r ) R d ~r i ( ~v ( ~r i ) · ~v ( ~r i )) , [4]where the local suspension velocity ~v is measured at the mid-plane y = H/ r ij = | ~r i − ~r j | . The correlation length l of swarming vortices is extracted from the location where C v decreases to 1 /e . l as a function of H is shown in Fig. 6D. Alinear relation is observed when H < µ m. Our results areconsistent with previous published data using different exper-imental setups. In thin chambers with height < µ m, thevortex size is of the order of 5 − µ m (8), whereas in chambersof height ∼ µ m the vortex size increases to ∼ µ m (9).At even larger H , l shows a trend for saturation. Althoughthe working distance of the confocal microscope prevents usfrom imaging samples with very large H , a large swarmingvortex with strong nematic order is known to be unstable forpusher suspensions (1, 3, 4, 20).Lastly, the two shear configurations are symmetric with-out shear, leading to zero mean velocity (Fig. 4E). E xz = δv x shows a non-monotonic trend with y , where E xz reaches amaximum, E xz, max , at the center of the cell and approachesto zero at the top and bottom walls. Our experiments con-firm the nonmonotonic trend of E xz ( y ) (Fig. 6E). Since thelocal shear gradient ˙ γ ∗ is independent of the gap size H , aswe increase H , the velocity fluctuation δv x should increase lin-early with H . Thus, E xz, max should increases as H . Our ex-periments quantitatively agree with this prediction (Fig. 6F).Thus, in addition to the length scale of swarming vortices,the model also successfully predicts the dependence of theirstrength, characterized by E xz , on the system size. Comparison with other shear-banding complex fluids.
Ourstudy on 3D suspension dynamics shows that bacterial “su-perfluids” arise from the balance of local viscous and activestresses. Moreover, the duality of shear-banding configura-tions reveals a remarkable feature of active fluids, differentfrom the shear-banding behavior of equilibrium complex flu-ids such as worm-like micelle solutions (38), colloidal suspen-sions (39) and entangled polymeric fluids (40). Shear rates inthese complex fluids are invariably positive (35, 36). The for-mation of shear bands necessarily breaks the translational androtational symmetry of the unsheared samples (Fig. 7A). Al-though the lost symmetry can be restored theoretically whenall allowed shear-banding configurations are averaged, a shear-banding complex fluid invariantly selects one of the symmetry-broken configurations in the steady state (Fig. 7A). The choiceof the specific configuration depends on initial and/or bound-ary conditions, a process analogous to the spontaneous sym-metry breaking in phase transitions. In contrast, a sheared ac-tive fluid, instead of being trapped into one of the symmetry-broken configurations, samples all allowed shear-banding con-figurations (Fig. 7B), which leads to a symmetric yet non-linear shear profile preserving the original symmetry of theunsheared sample. Although an active fluid is intrinsicallyout of equilibrium, it appears to be more “ergodic” due to itscollective motions.
Guo et al.
PNAS |
May 31, 2018 | vol. XXX | no. XX || vol. XXX | no. XX |
May 31, 2018 | vol. XXX | no. XX || vol. XXX | no. XX | R A F T Fig. 7.
Comparison of shear banding in complex and active fluids. (A) Shear band-ing in conventional complex fluids. The shear banding flow breaks the symmetry ofunsheared samples, which can be seen from the difference in the shape of shearprofiles after two physical operations: (i) a rotational operation (R), where the sys-tem is rotated counter-clockwise by π , and (ii) a translational operation (T), wherethe lab frame is transformed into a moving frame of a linear velocity − V . Althoughthe boundary conditions of the systems after the two operations are the same, theresulting shear profiles are different. Thus, the sheared sample before the opera-tions cannot simultaneously satisfy the translational and rotational symmetry of theunsheared sample. The ensemble average of the two symmetry-broken shear con-figurations is approximately linear, restoring the original symmetry of the unshearedsample. A sheared complex fluid chooses one of the two symmetry-broken configura-tions depending on initial and/or boundary conditions. The symmetry-broken processis illustrated schematically by the location of a red disk in a split-bottom potential, inanalogy to the spontaneous symmetry breaking in equilibrium phase transitions. Thevalleys (R) and (T) indicate the two possible symmetry-broken shear-banding config-urations. (B) Shear banding in active fluids. The ensemble-averaged shear profilefrom the two symmetry-broken shear-banding configurations is symmetric and non-linear. A sheared active fluid samples both symmetry-broken configurations andpreserves the symmetry of the unsheared fluid. Conclusions
We investigated the dynamics of concentrated bacterial sus-pensions under simple oscillatory shear using fast confocalrheometry. We observed unusual symmetric shear-bandingflows in the “superfluidic” phase of bacterial suspensions,rarely seen in conventional complex fluids. A minimal phe-nomenological model was constructed based on the detailedstress balance and the ergodic sampling of different shear con-figurations, which quantitatively describes the variation of theshear-banding structure with applied shear rates and bacterialactivity. Such a simple model also successfully predicts var-ious non-trivial physical properties of collective swarming instationary bacterial suspensions. Particularly, it explains thelinear relation between the kinetic energy and the enstrophyof suspension flows and shows the dependence of the lengthand strength of swarming vortices on the system size. Ourstudy provides new insights into the emergent collective be-havior of active fluids and the resulting transport properties.It illustrates the unusual rheological response of bacterial sus-pensions induced by the complex interaction between bacteriaand ambient shear flows, which is frequently encountered innatural, biomedical and biochemical engineering settings.
ACKNOWLEDGMENTS.
We thank K. Dorfman, Y.-S. Tai andK. Zhang for helps with bacterial culturing and J. Brady and Z.Dogic for discussions. The research is supported by DARPA YFA(No. D16AP00120), the Packard Foundation and NSF-CBET (No. 1702352). X. X. acknowledge support from National Natural Sci-ence Foundation of China (No. 11575020) and (No. U1530401).
1. Koch DL, Subramanian G (2011) Collective Hydrodynamics of Swimming Microorganisms:Living Fluids,
Ann. Rev. Fluid Mech.
Proceedings of the International School of Physics “En-rico Ferm”, Course CLXXXIV “Physics of Complex Colloid” , eds. BechingerC, Sciortino F, Ziherl P(IOS, Amsterdam: SIF, Bologna), pp 317–386.3. Marchetti MC, Joanny JF, Ramaswamy S, Liverpool TB, Prost J, Rao M, Simha RA (2013)Hydrodynamics of soft active matter.
Rev. Mod. Phys.
Complex fluids in biolog-ical systems , eds. Spagnolie S(Springer-Verlag, New York), pp 319–355.5. Bechinger C, Di Leonardo R, Lowen H, Reichhardt C, Volpe G, Volpe G (2016) Active parti-cles in complex and crowded environments.
Rev. Mod. Phys.
Phys. Rev. Lett.
Phys. Rev. Lett.
Proc. Natl. Acad. Sci. USA
Phys. Rev. Lett.
Science
Proc. Natl. Acad. Sci. USA
Phys. Rev. Lett.
Soft Matter
Proc. Natl. Acad. Sci. USA
Phys. Rev. Lett.
Phys. Rev. E
Phys. Rev. Lett.
New J. Phys.
Science
Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Rev. E
Phys. Rev. E
Exp. Mech.
Phys. Rev. E
EPL
Rheol.Acta
Phys. Rev. Lett.
Phys. Rev. Fluids
Phys. Rev. Lett.
Phys. Rev. Lett.
Phys.Rev. Lett.
Science
Rev. Sci. Instrum.
Rheol. Acta
Annu. Rev. Fluid Mech.
EPL
Phys. Rev. Lett. et al. R A F T
39. Cohen I, Davidovitch B, Schofield AB, Brenner MP, Weitz DA (2006) Slip, yield, and bandsin colloidal crystals under oscillatory shear.
Phys. Rev. Lett.
Phys. Rev. E
Guo et al.
PNAS |
May 31, 2018 | vol. XXX | no. XX || vol. XXX | no. XX |