Confinement Stabilizes a Bacterial Suspension into a Spiral Vortex

Confining surfaces play crucial roles in dynamics, transport, and order in many physical systems, but their effects on active matter, a broad class of dynamically self-organizing systems, are poorly understood. We investigate here the influence of global confinement and surface curvature on collective motion by studying the flow and orientational order within small droplets of a dense bacterial suspension. The competition between radial confinement, self-propulsion, steric interactions, and hydrodynamics robustly induces an intriguing steady single-vortex state, in which cells align in inward spiraling patterns accompanied by a thin counterrotating boundary layer. A minimal continuum model is shown to be in good agreement with these observations.

Figure 1

Overview. (a) Experimental setup. (b) Bright field image of a $40\mu \mathrm{m}$ drop, and definition of cell orientation angle relative to main circulation direction.

Figure 2

Steady-state circulation in highly concentrated B. subtilis droplets. (a) PIV flow field for a droplet with a volume filling fraction $\phi \sim 0.4$. For clarity, not all PIV vectors are shown. (b) Enlarged region reveals the counterrotating boundary layer. All PIV vectors are shown. (c),(d) Vortex order parameter $\Phi$ for varying diameter $d$. (c) Drops of constant height $h\sim 25\mu \mathrm{m}$. Dashed lines denote the highly ordered single-vortex regime. (d) Averaged vortex order parameter $\Phi$ ($5\mu \mathrm{m}$ bins) for $h\sim 15\mu \mathrm{m}$ (red dashed line) and $h\sim 25\mu \mathrm{m}$ (blue solid line). Error bars indicate the standard deviation. (e) Azimuthal flow $v_t(r)=⟨\mathbf{v}·\mathbf{t}{⟩}_{\theta }$ profile for three different experiments (blue solid lines), compared with continuum bulk flow model results (red dashed lines). Negative flow indicates the counterrotating boundary layer.

Figure 3

Schematic cell organization in droplets. (a) Dashed line indicates continuum model boundary, where bulk flow begins. (b)–(d) Physical mechanisms driving boundary layer formation. (b) Shear flow reorients cells to face upstream. (c) Contact angle ${\theta }_m$ decreases with the drop diameter, restricted by steric interactions. (d) Ratchetlike steric repulsion and inward flow (red arrows) created by boundary cells force the next layer to move in the opposite azimuthal direction, thereby setting the bulk flow direction.

Figure 4

Bacterial orientation. (a) Local orientation, averaged over 2 s. External ring lies at the water-oil interface and shows local azimuthal direction $\mathbf{t}$, and cellular orientation appears in the central disc. Discontinuity in color between ring and disc indicates the angle between cells and the azimuthal direction. (b) Boundary angle [Fig. 3c] as a function of drop diameter. Symbols denote different bacterial concentrations. Dashed black lines indicate geometric estimates of minimum packing angle $\Theta$ for different cell lengths.