Diffusion of Ellipsoids in Bacterial Suspensions

Active fluids such as swarming bacteria and motile colloids exhibit exotic properties different from conventional equilibrium materials. As a peculiar example, a spherical tracer immersed inside active fluids shows an enhanced translational diffusion, orders of magnitude stronger than its intrinsic Brownian motion. Here, rather than spherical tracers, we investigate the diffusion of isolated ellipsoids in a quasi-two-dimensional bacterial bath. Our study shows a nonlinear enhancement of both translational and rotational diffusions of ellipsoids. More importantly, we uncover an anomalous coupling between particles’ translation and rotation that is strictly prohibited in Brownian diffusion. The coupling reveals a counterintuitive anisotropic particle diffusion, where an ellipsoid diffuses fastest along its minor axis in its body frame. Combining experiments with theoretical modeling, we show that such an anomalous diffusive behavior arises from the generic straining flow of swimming bacteria. Our work illustrates an unexpected feature of active fluids and deepens our understanding of transport processes in microbiological systems.

Figure 1

Diffusion of an ellipsoid in quasi-two-dimensional bacterial bath. (a) Velocity field of swarming bacteria around an ellipsoid obtained from particle image velocimetry (PIV). Scale bar: $20\mu \mathrm{m}$. (b) Trajectory of an ellipsoid in a time interval $\mathrm{\Delta }t=0.2\mathrm{s}$. The trajectory is obtained from a custom algorithm for particle tracking velocimetry (PTV). The center of mass of the particle at different times $\mathit{x}(t)$ is indicated by black dots. The color indicates the orientation of the particle $\theta (t)$ with respect to the $x$ axis fixed in the lab frame. Bacterial concentration $n=30n_0$.

Figure 2

Enhanced translational and rotational diffusions. (a) Translational mean-squared displacements (${\mathrm{MSD}}_T$) and (b) rotational mean-squared displacements (${\mathrm{MSD}}_R$). The slopes of the lines are indicated. (c) Translational and rotational diffusion coefficients $D_T$ and $D_R$ as a function of bacterial concentrations $n$. Solid lines indicate linear and nonlinear enhancements. The vertical line indicates the onset of obvious bacterial swarming.

Figure 3

Translation-rotation coupling. (a) Cross correlations at different $n$. The dashed line indicates zero coupling. (b) The ratio of diffusion coefficients along the major and minor axes in the body frame. Solid squares are from direct measurements and blue triangles are the product of $⟨\mathrm{\Delta }{x}_{\parallel }^2⟩/⟨\mathrm{\Delta }{x}_{\perp }^2⟩$ and ${\tau }_{\parallel }/{\tau }_{\perp }$ (see text). The solid line is an empirical relation for visual guide. Inset illustrates schematically the distinct particle motions with positive (blue) and negative (red) translation-rotation couplings. (c) The extremum of $C(t)$ vs $D_{\parallel }/D_{\perp }$. The solid line is a linear fit. The “prohibited zone” of Brownian motion is indicated by the red areas in (b) and (c).

Figure 4

$D_{\parallel }/D_{\perp }$ as a function of $n$ for ellipsoids of different aspect ratios, $p$. Inset shows the linear relation between $(1-D_{\parallel }/D_{\perp })/(1+D_{\parallel }/D_{\perp })$ and $(p^2-1)/(p^2+1)$, as predicted by the hydrodynamic model [Eq. (2)], where $D_{\parallel }/D_{\perp }$ are taken at $n=40n_0$.

Figure 5

Origin of the anomalous translation-rotation coupling. (a) Ratio of short-time MSDs along the major and minor axes in the superdiffusive regime with $\mathrm{\Delta }t=0.1\mathrm{s}$. (b) Ratio of correlation times along the major and minor axes. Schematics showing the average effective straining on ellipsoids induced by the extensile dipole flow of pushers such as E. coli (c) and the contractile dipole flow of pullers such as C. reinhardtii (d).