# Symmetric shear banding and swarming vortices in bacterial superfluids

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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 29, 2018 (received for review December 26, 2017)

## Significance

Bacterial suspensions can flow without apparent viscosity. Such a superfluid-like behavior stems from the collective motions of swimming bacteria. Here, we explore the microscopic flow profile of bacterial “superfluids” under simple shear. We find that, instead of deforming uniformly, bacterial superfluids develop multiple shear bands, i.e., regions with different shear deformations. We construct a simple model that quantitatively describes the shape of the shear-banding structure and reveals important physical properties of collective bacterial motions. Our study sheds light on complex interactions between swimming microorganisms and ambient fluid flows, crucial for the survival of microorganisms in nature and the manipulation of bacterial suspensions in engineering settings.

## Abstract

Bacterial suspensions—a premier example of active fluids—show an unusual response to shear stresses. Instead of increasing the viscosity of the suspending fluid, the emergent collective motions of swimming bacteria can turn a suspension into a superfluid with zero apparent viscosity. Although the existence of active superfluids has been demonstrated in bulk rheological measurements, the microscopic origin and dynamics of such an exotic phase have not been experimentally probed. Here, using high-speed confocal rheometry, we study the dynamics of concentrated bacterial suspensions under simple planar shear. We find that bacterial superfluids under shear exhibit unusual symmetric shear bands, defying the conventional wisdom on shear banding of complex fluids, where the formation of steady shear bands necessarily breaks the symmetry of unsheared samples. We propose a simple hydrodynamic model based on the local stress balance and the ergodic sampling of nonequilibrium shear configurations, which quantitatively describes the observed symmetric shear-banding structure. The model also successfully predicts various interesting features of swarming vortices in stationary bacterial suspensions. Our study provides insights into the physical properties of collective swarming in active fluids and illustrates their profound influences on transport processes.

Active fluids, suspensions of self-propelled particles, have attracted enormous research interest in recent years (1⇓⇓⇓–5). With examples across biological and physical systems of widely different scales, active fluids exhibit many novel properties, such as the emergence of collective swarming (6⇓⇓–9), giant number fluctuations (10, 11), and enhanced diffusion of passive tracers (12⇓⇓⇓–16). Among all these unusual features, the flow behavior of active fluids demonstrates the nonequilibrium nature of active systems in the most striking manner. Surprising phenomena including superfluid-like behaviors (17) and spontaneous directional flows (18, 19) have been observed in active fluids.

Using a phenomenological model that couples hydrodynamic equations with active nematic order parameters, Hatwalne et al. (20) first showed that pusher microswimmers such as *Escherichia coli* can significantly lower the bulk viscosity of active suspensions, to such an extent that suspensions can have a lower viscosity than the suspending fluids. Based on a similar approach, Cates et al. (21) further predicted that near the disorder-to-order transition to collective motions, a pusher active fluid can enter a “superfluidic” regime where its apparent shear viscosity vanishes. Later theory by Giomi et al. (22) revealed even richer dynamics and predicted the existence of shear banding, yield stress, and “superfluidity” of active fluids. Unusual rheology of active fluids has also been studied based on the microhydrodynamics of microswimmers at low concentrations (23⇓⇓⇓–27), swimming pressures (28), and generalized Navier–Stokes equations (29). Experimentally, Sokolov et al. (30) and Gachelin et al. (31) showed the low viscosity of bacterial suspensions in thin films. Lopez et al. (17) demonstrated the superfluid-like transition in concentrated *E. coli* suspensions using a rotational rheometer. Under channel confinements, this “superfluidic” behavior displays as spontaneous directional flows (18, 19). In comparison, puller swimmers such as swimming algae were shown to enhance, instead of suppress, the viscosity of suspensions (32).

Although the vanishing shear viscosity of active superfluids has been demonstrated in bulk rheology studies (17), the microscopic dynamics of such an exotic phase under simple shear flows have not be experimentally explored. The shear-banding structure—an important prediction of hydrodynamic theories (21, 22)—has not been verified. Here, using fast confocal rheometry, we study the dynamics of concentrated bacterial suspensions under planar oscillatory shear. We find that bacterial superfluids exhibit symmetric shear-banding flows with three shear bands. We systematically investigate the variation of the shear-banding structure with shear rates, bacterial concentrations, and bacterial motility. Based on the existing hydrodynamic theories, we construct a simple phenomenological model that quantitatively describes the shape of the symmetric shear bands. The model also predicts several nontrivial properties of swarming vortices in stationary bacterial suspensions, including the linear relation between the kinetic energy and the enstrophy of suspension flows and the system-size dependence of the length and strength of swarming vortices. We conclude this paper by discussing the unique feature of the shear-banding flow of bacterial suspensions in comparison with conventional shear-banding complex fluids. Our study provides insights into the collective swarming of active fluids and illustrates the unexpected consequence of collective swarming on momentum transports of active systems. Our results also help to understand complex interactions between bacteria and ambient shear flows encountered in many natural and engineering settings.

## Results

We use a fluorescently tagged *E. coli* K-12 strain (BW25113). The bacteria are suspended in a mobility buffer to a concentration n. We vary n between ^{−1} the concentration of bacteria in the stationary phase of growing. When *A* and *SI Appendix*, section A).

We investigate the 3D fluid flow of *E. coli* suspensions under planar oscillatory shear. A suspension of 20 μL is confined between the two parallel plates of a custom shear cell with a constant spacing *B*) (33, 34). A circular top plate of radius 2.5 mm is stationary, whereas a much larger bottom plate driven by a piezo-electric actuator moves sinusoidally with *SI Appendix*, section B). The bottom plate is made of a smooth glass coverslip, enabling us to image 3D suspension dynamics via an inverted confocal microscope. The top plate is made of either a smooth silicon wafer or a rough porous membrane that allows for the influx of oxygen (*SI Appendix*, section A). While the symmetric smooth shear boundary with the Si wafer eliminates the biased influence of the boundary on shear profiles, the porous membrane allows us to maintain high bacterial activities for

### Symmetric Shear Banding.

The average velocity of a concentrated bacterial suspension under shear at different heights y above the bottom plate, *C*. Here, the average is taken along both the flow (x) and the vorticity (z) directions. *A*). However, under weak shear, interesting nonlinear shear profiles are observed. All of the applied shear concentrates near the center of the suspensions. Near the top and bottom plates, local shear gradients are small and may even vanish, resulting in approximately symmetric shear profiles rarely seen in other complex fluids (Fig. 2*A*). A crossover from the linear to the nonlinear shear profiles is observed with decreasing

The shape of shear profiles also depends on the strength of collective bacterial swarming. We vary the swarming strength by changing bacterial concentrations n (*SI Appendix*, Fig. S2) (6). At large n, bacteria show strong collective motions, leading to the nonlinear shear profiles at low *B*). Below *B*).

The competition between the shear flow and the collective bacterial swarming dictates the microscopic suspension dynamics. The strength of shear flows is naturally quantified by the imposed shear rate amplitude, *B*). *SI Appendix*, Fig. S4). When plotting

### Model.

The existence of bacterial superfluids has been predicted by hydrodynamic theories of active fluids (21, 22). These theories show that the constitutive equation of active fluids is nonmonotonic across zero (Fig. 4*A*). The mechanical instability induced by the negative slope of the constitutive relation then leads to a zero-stress superfluidic plateau (35, 36). The instability also predicts a nonmonotonic shear profile with two shear bands of opposite shear rates (Fig. 4*B*).

To understand the symmetric shear profiles in our experiments, we construct a simple phenomenological model based on the constitutive equation of the hydrodynamic theory (21) (*SI Appendix*, section C). The local total shear stress, *SI Appendix*, section F). For simplicity, we also ignore the complex bacteria–boundary interaction, which may influence the average bacterial orientation near walls (37). Considering the bacteria–boundary interaction should not affect the key predictions of our model either (*SI Appendix*, section G).

In the superfluidic phase, the stress balance, *B* and *C*), where the width of the shear band with *SI Appendix*, section C)

It should be emphasized that there are two and only two shear configurations with two shear bands satisfying the stress balance and the no-slip boundary condition, which are shown in Fig. 4 *B* and *C*, respectively. Since both shear configurations satisfy the local stress balance, we hypothesize they emerge in a sheared sample “ergodically” with equal probability, an assumption that shall be tested *a posteriori*. The measured shear profile should then be an “ensemble” average of the two shear configurations. A possible physical interpretation of the ensemble average is as follows: A single swarming vortex normal to the flow–vorticity plane extending across the two shear plates (Fig. 1*A*) can be viewed as composed of the two shear configurations (Fig. 4*E*, *Inset*). The half of the vortex moving along the shear direction represents the configuration of Fig. 4*B*, whereas the other half moving against shear gives the configuration in Fig. 4*C*. Thus, the ensemble average is achieved experimentally through a spatiotemporal average over multiple swarming vortices. Vortices have a characteristic diameter *A*) and a lifetime of a few seconds (7, 29), whereas the spatial and temporal scales of our experiments are 180 μm and 40 s, respectively.

The ensemble average of the two shear configurations naturally leads to a symmetric shear profile (Fig. 4*D*), consistent with our observations. Using Eq. **1** and a simple geometric relation *B* and *C* are achieved in our experiments via the 1D confinement imposed by our shear cell along the shear gradient direction. At sufficiently large H, three or more shear bands may emerge, which have infinite possible shear configurations satisfying the stress balance and the no-slip boundary condition. The ergodic assumption would then lead to featureless linear shear profiles (*SI Appendix*, section D). Our experiments are different from earlier studies on bacterial suspensions under channel confinement, which constrains bacterial swarming along both the shear gradient and vorticity directions. Such a confinement suppresses the instability that induces swarming vortices (4). As a result, suspensions develop directional flows and break the hypothesized “ergodicity” (18).

The model incorporates a unique feature, i.e., a dynamic alternation between the two shear configurations around the mean shear profile (Fig. 4*D*). To verify the hypothesis, we measure the probability distribution function of local velocities at the center of the shear cell, *A* and *B*), bimodal distributions with two distinct peaks can be identified. The peaks correspond to the velocities of the two discrete shear profiles at *A* and *B*, *Insets*). The finite width of the distributions arises presumably from the variation of individual bacterial mobility, an effect that is not included in our model. The areas underneath the two peaks are approximately the same with difference less than *C*), indicating the emergence of a single linear profile (Fig. 5*C*, *Inset*). Our model predicts that the left peak of *SI Appendix*, section E)*D*). Direct measurements on instantaneous shear profiles at local scales are certainly needed to finally verify the ergodic assumption of our model, which is constructed to rationalize the 3D experimental results using simple steady-state 1D shear profiles (*SI Appendix*, section C).

### Swarming Vortices in Stationary Bacterial Suspensions.

The simple model also predicts several nontrivial properties of swarming vortices in stationary bacterial suspensions without shear. First, from Eq. **3**, when *A*). More importantly, we measure *B*), agreeing with the model, although the slope of

Previous studies implied that Λ is associated with the length scale of swarming vortices (9). Since Λ changes linearly with H (Fig. 6*B*), we hypothesize that the size of swarming vortices should also change linearly with the gap size of the system. To test the hypothesis, we measure the velocity–velocity spatial correlation (Fig. 6*C*)*D*. A linear relation is observed when

Finally, the two shear configurations are symmetric without shear, leading to zero mean velocity (Fig. 4*E*). *E*). Since the local shear gradient *F*). Thus, in addition to the length scale of swarming vortices, the model also successfully predicts the dependence of their strength, characterized by

### Comparison with Other Shear-Banding Complex Fluids.

Our study on 3D suspension dynamics shows that bacterial superfluids arise from the balance of local viscous and active stresses. Moreover, the duality of shear-banding configurations reveals a remarkable feature of active fluids, different from the shear-banding behavior of equilibrium complex fluids such as worm-like micelle solutions (38), colloidal suspensions (39), and entangled polymeric fluids (40). Shear rates in these complex fluids are invariably positive (35, 36). The formation of shear bands necessarily breaks the translational and rotational symmetry of the unsheared samples (Fig. 7*A*). Although the lost symmetry can be restored theoretically when all allowed shear-banding configurations are averaged, a shear-banding complex fluid invariantly selects one of the symmetry-broken configurations in the steady state (Fig. 7*A*). The choice of the specific configuration depends on initial and/or boundary conditions, a process analogous to the spontaneous symmetry breaking in phase transitions. In contrast, a sheared active fluid, instead of being trapped into one of the symmetry-broken configurations, samples all allowed shear-banding configurations (Fig. 7*B*), which leads to a symmetric yet nonlinear shear profile preserving the original symmetry of the unsheared sample. Although an active fluid is intrinsically out of equilibrium, it appears to be more “ergodic” due to its collective motions.

## Conclusions

Using fast confocal rheometry, we investigated the dynamics of concentrated bacterial suspensions under simple oscillatory shear. We observed unusual symmetric shear-banding flows in the superfluidic phase of bacterial suspensions, rarely seen in conventional complex fluids. A minimal phenomenological model was constructed based on the detailed stress balance and the ergodic sampling of different shear configurations, which quantitatively describes the variation of the shear-banding structure with applied shear rates and bacterial activity. Such a simple model also successfully predicts nontrivial physical properties of collective swarming in stationary bacterial suspensions. Particularly, it explains the linear relation between the kinetic energy and the enstrophy of suspension flows and shows the dependence of the length and strength of swarming vortices on the system size. Our study provides insights into the emergent collective behavior of active fluids and the resulting transport properties. Our work illustrates the unusual rheological response of bacterial suspensions induced by the complex interaction between bacteria and ambient shear flows, which is frequently encountered in natural, biomedical, and biochemical engineering settings.

## Acknowledgments

We thank K. Dorfman, Y.-S. Tai, and K. Zhang for help with bacterial culturing and J. Brady and Z. Dogic for discussions. This research is supported by Defense Advanced Research Projects Agency (DARPA) Young Faculty Award D16AP00120, the Packard Foundation, and National Science Foundation Chemical, Bioengineering, Environmental, and Transport Systems Award 1702352. X.X. acknowledges support from the National Natural Science Foundation of China (Grants 11575020 and U1530401).

## Footnotes

↵

^{1}Present address: Department of Mechanical Engineering, Indian Institute of Technology, Ropar, Rupnagar, Punjab 140001, India.- ↵
^{2}To whom correspondence may be addressed. Email: xcheng{at}umn.edu or xinliang{at}csrc.ac.cn.

Author contributions: S.G., D.S., X.X., and X.C. designed research; S.G., D.S., X.X., and X.C. performed research; S.G., D.S., Y.P., X.X., and X.C. analyzed data; and S.G., X.X., and X.C. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1722505115/-/DCSupplemental.

Published under the PNAS license.

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