Dynamic clustering of passive colloids in dense suspensions of motile bacteria

Mixtures of active and passive particles are predicted to exhibit a variety of nonequilibrium phases. Here we report a dynamic clustering phase in mixtures of colloids and motile bacteria. We show that colloidal clustering results from a balance between bond breaking due to persistent active motion and bond stabilization due to torques that align active particle velocity tangentially to the passive particle surface. Furthermore, dynamic clustering spans a broad regime between diffusivity-based and motility-induced phase separation that subsumes typical bacterial motility parameters.

Figure 1

Dynamic clustering of passive colloids by motile bacteria. (a) Schematic of the experimental setup. (b), (c) Representative snapshots of colloids at ${\rho }_{\mathrm{b}}=0$ (b) and ${\rho }_{\mathrm{b}}=50{\rho }_0$ (c). Monomers are shown in black and clusters in blue, with lighter shades indicating larger clusters. The scale bar is $10\mu \mathrm{m}$. (d) $g\left(r\right)$ for ${\rho }_{\mathrm{b}}=0$ (black) and ${\rho }_{\mathrm{b}}=50{\rho }_0$ (blue). (e) $P\left(n\right)$ for ${\rho }_{\mathrm{b}}=0$ (black) and ${\rho }_{\mathrm{b}}=50{\rho }_0$ (blue). Inset shows the mean cluster size computed from data in the main plot. Shaded regions in (d) and (e) and error bars in inset to (e) are standard errors of the mean across three independent experiments.

Figure 2

Torque-induced stabilization of clusters in simulations. (a) Schematic showing the orientation of active particles parametrized by ${\theta }_i$, and ${\theta }_{ij}$ is the orientation of the relative position vector between active particle $i$ and passive particle $j$. (b), (c) No clustering is observed for $\mathrm{\Gamma }=0/\mathrm{s}$ (b) but substantial clustering is observed at $\mathrm{\Gamma }=25/\mathrm{s}$ (c). Monomers are shown in black and clusters in blue, with lighter shades indicating larger clusters. Insets in (b) and (c) reveal the difference in particle trajectories during a collision, with (c) and without (b) torque. The scale bar is $10\mu \mathrm{m}$.(d) Mean cluster size as a function of ${\varphi }_{\mathrm{a}}$ for $\mathrm{\Gamma }=0/\mathrm{s}$ (dotted line) and $\mathrm{\Gamma }=25/\mathrm{s}$ (solid line). In (b)–(d), error bars are standard errors of the mean across three independent simulations.

Figure 3

Effective attraction between passive particles leads to clustering. (a) Effective diffusivity (open green diamonds) and bond lifetime (open orange squares) of passive particles vs ${\varphi }_{\mathrm{a}}$ in simulations. (b) Mean cluster size vs $\delta =D\tau /\left(D_0{\tau }_0\right)$ in simulations (open symbols) and experiments (filled symbols). (c) Effective diffusivity (filled green diamonds) and bond lifetime (filled orange squares) of colloids vs ${\rho }_{\mathrm{b}}/{\rho }_0$ in experiments. In (a)–(c), black symbols correspond to ${\varphi }_{\mathrm{a}}=0$ in simulations and ${\rho }_{\mathrm{b}}=0$ in experiments. Error bars in (a)–(c) are standard errors of the mean from three independent replicates.

Figure 4

Numerical phase diagram of mixtures of active and passive particles. (a) Phase diagram in the Péclet number (Pe) - alignment rate ($\mathrm{\Gamma }$) plane showing diffusivity-based phase separation at low Pe (purple region), dynamic clustering at intermediate Pe (white region), and MIPS at high Pe (yellow region). The purple color map (left) corresponds to the demixed phase order parameter ($O_{\mathrm{DM}}$), and yellow color map (right) corresponds to the MIPS order parameter ($O_{\mathrm{MIPS}}$). The stripes in the purple region result from sparse sampling of the parameter space and incomplete domain coarsening due to the finite duration of simulations. The red arrow corresponds to $\text{Pe}=55$, the critical Péclet number for the onset of MIPS in 2D [31]. Blue stars correspond to experimental data points for P. aurantiaca and E. coli bacteria. (b)–(d) Representative simulation snapshots of the demixed phase (b), clustered phase (c), and MIPS phase (d). Triangles in (a) denote the values of Pe and $\mathrm{\Gamma }$ used for the simulations in (b)–(d). The scale bar is $10\mu \mathrm{m}$.