Symmetric shear banding and swarming vortices in bacterial superfluids

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.

Keywords: active fluids; bacterial suspensions; shear banding.

Fig. 1.

Bacterial suspensions under planar oscillatory shear. (A) Bacterial swarming at a concentration $n=80n_0$. (Scale bar, 20 $\mathrm{\mu }$m.) The fluorescently tagged $E.\hspace{0.17em}coli$ 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 $\dot{x}\left(t\right)$ at different heights, $y$, above the bottom plate. Red curves are for shear-rate amplitude ${\dot{\gamma }}_0=0.52$ s⁻¹. Black curves are for ${\dot{\gamma }}_0=0.21$ s⁻¹. Velocities are normalized by the imposed velocity amplitudes, $V_0$. Time $t$ is normalized by shear period $T=1/f$. $y$ is normalized by gap thickness $H$. $\dot{x}\left(t\right)$ at different $y$ are shifted vertically for clarity. Dashed lines are sinusoidal fits.

Fig. 2.

Shear profiles of bacterial suspensions. (A) Normalized shear profiles at different shear rates. $V_0$ is the applied shear velocity amplitude. Bacterial concentration is fixed at $n=50n_0$. The shear-rate amplitude ${\dot{\gamma }}_0=0.42$ s⁻¹ (black squares), 0.16 s⁻¹ (red circles), and 0.055 s⁻¹ (blue triangles). A Si wafer is used as the top plate. (B) Normalized shear profiles at different bacterial concentrations. ${\dot{\gamma }}_0$ is fixed at 0.16 s⁻¹. $n=10n_0$ (black squares), $40n_0$ (red circles), and $100n_0$ (blue triangles). To maintain bacterial motility at high $n$, a porous membrane is used as the top plate. The stop height, $h_s$, of the profile at $100n_0$ is indicated. Open squares are for a suspension of immobile bacteria at $100n_0$.

Fig. 3.

Shape of shear profiles. The stop height, $h_s$, as a function of the dimensionless shear rate ${\dot{\gamma }}_0/\sqrt{{\mathrm{\Omega }}_y}$. $h_s$ is normalized by the gap thickness $H$. $H=30\hspace{0.17em}\mathrm{\mu }$m (squares) and 60 $\mathrm{\mu }$m (circles). Colored symbols are obtained with the symmetric shear boundary using a Si wafer at $f=0.1$ Hz. Gray symbols are obtained with the asymmetric shear boundary using the porous membrane. Solid gray symbols are for $f=0.1$ Hz and open gray symbols are for other shear frequencies between 0.025 Hz and 0.3 Hz. Inset shows the same data in a log-linear plot. The solid line is the theoretical prediction in the superfluidic phase and the dashed line is the prediction in the normal phase (Eq. 2).

Fig. 4.

Duality of shear configurations. (A) A schematic showing the constitutive relation of active fluids from hydrodynamic theories (21, 22). The nonmonotonic trend predicts shear-banding flows with two shear bands of opposite shear rates, ${\dot{\gamma }}_1=\dot{\gamma }*$ and ${\dot{\gamma }}_2=-\dot{\gamma }*$. The corresponding shear profile are shown in B and C. Red arrows indicate shear velocities at different heights. Gap thickness, $H$, and the width of the shear band with ${\dot{\gamma }}_2$, $w$ 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 $n=80n_0$ and ${\dot{\gamma }}_0$ = 0.26 s⁻¹. The stop height, $h_s$, is indicated. (E) The duality of shear profiles at zero applied shear rate ${\dot{\gamma }}_0=0$. The mean flow is zero (thick red line), whereas the two shear-banding configurations (yellow and blue dashed lines) are symmetric with respect to the mean flow. (E, Inset) At given $y$, the two configurations moving along and against the shear flow complete a swarming vortex in the $x-z$ plane.

Fig. 5.

Probability distribution function of local velocities along the flow direction, $v_x$, at different shear rates, $P\left(v_x\right)$. (A–C) ${\dot{\gamma }}_0/\sqrt{{\mathrm{\Omega }}_y}=0$ (A), ${\dot{\gamma }}_0/\sqrt{{\mathrm{\Omega }}_y}=0.24$ (B), and ${\dot{\gamma }}_0/\sqrt{{\mathrm{\Omega }}_y}=2.88$ (C). Local velocities are measured when the average shear velocity reaches maximal in each shear cycle. PIV box size is chosen at $R$, where $R$ is the characteristic radius of swarming vortices. $n=80n_0$ and $H=60\hspace{0.17em}\mathrm{\mu }$m. A–C, Insets show schematically the corresponding shear profiles. The thick dashed lines (red and blue) indicate the two shear configurations. The thin horizontal dashed line indicates the position of our imaging plane. The intersections give two discrete velocities, $v_{x,l}$ and $v_{x,r}$, corresponding to the two peaks of $P\left(v_x\right)$. (D) The two peaks of $P\left(v_x\right)$, $v_{x,l}$ (black squares) and $v_{x,r}$ (red circles), and velocity variance, $\delta v_x$ (magenta triangles), as a function of shear rate, ${\dot{\gamma }}_0/\sqrt{{\mathrm{\Omega }}_y}$. Dashed lines show the model predictions.

Fig. 6.

Properties of bacterial swarming in stationary samples. (A) Kinetic energy, $E_{xz}$, vs. enstrophy of suspension flows, ${\mathrm{\Omega }}_y$. The gap size $H$ is indicated in the plot. Flows are measured at the midplane $y=H/2$. The solid line indicates the linear relation $E_{xz}\sim {\mathrm{\Omega }}_y$. (B) $\mathrm{\Lambda }$ extracted from the slope of $E_{xz}\left({\mathrm{\Omega }}_y\right)$ vs. $H$. The solid line is a linear fit. (C) Velocity spatial correlations. The horizontal dashed line is $e^{-1}$. $H$ is indicated. (D) Correlation length, $l$, as a function of $H$. Bacterial concentrations, $n$, are indicated. The dashed line indicates the linear relation. (E) $E_{xz}$ as a function of the height $y$ at three different $H$. $n=64n_0$. (F) The maximum $E_{xz}$ at $y=H/2$ vs. $H^2$. The solid line is a linear fit.

Fig. 7.

Comparison of shear banding in complex and active fluids. (A) Shear banding in conventional complex fluids. The shear-banding flow breaks the symmetry of unsheared samples, which can be seen from the difference in the shape of shear profiles after two physical operations: (i) a rotational operation (R), where the system is rotated counterclockwise by $\pi$, and (ii) a translational operation (T), where the laboratory frame is transformed into a moving frame of a linear velocity $-V$. Although the boundary conditions of the systems after the two operations are the same, the resulting shear profiles are different. Thus, the sheared sample before the operations cannot simultaneously satisfy the translational and rotational symmetry of the unsheared sample. The ensemble average of the two symmetry-broken shear configurations is approximately linear, restoring the original symmetry of the unsheared sample. A sheared complex fluid chooses one of the two symmetry-broken configurations, depending on initial and/or boundary conditions. The symmetry-broken process is illustrated schematically by the location of a red circle in a split-bottom potential, in analogy to the spontaneous symmetry breaking in equilibrium phase transitions. The valleys (R) and (T) indicate the two possible symmetry-broken shear-banding configurations. (B) Shear banding in active fluids. The ensemble-averaged shear profile from the two symmetry-broken shear-banding configurations is symmetric and nonlinear. A sheared active fluid samples both symmetry-broken configurations and preserves the symmetry of the unsheared fluid.