Wetting dynamics by mixtures of fast and slow self-propelled particles

We study active surface wetting using a minimal model of bacteria that takes into account the intrinsic motility diversity of living matter. A mixture of “fast” and “slow” self-propelled Brownian particles is considered in the presence of a wall. The evolution of the wetting layer thickness shows an overshoot before stationarity and its composition evolves in two stages, equilibrating after a slow elimination of excess particles. Nonmonotonic evolutions are shown to arise from delayed avalanches towards the dilute phase combined with the emergence of a transient particle front.

Figure 1

(a) Snapshot for speed diversity $\delta =0.5$ and $\varphi =0.18$ in the SS. Fast (slow) particles are in red (blue). (b) Temporal evolution of the mean wetting layer thickness for $\varphi =0.18$ and $\delta$ from 0 to 1 every 0.1. Stars indicate SS averages ($t>100\tau$). Insets: $\delta =0$ and $\delta =1$ on the left and $\delta =0.5$ on the right. Solid lines are the theory.

Figure 2

Top: Wetting layer composition evolution for $\delta$ from 0 to 1 every 0.1 and $\varphi =0.18$ (6000 particles in total), measured by the amount of fast and slow particles in it. The diagonal corresponds to the same amount of fast and slow particles. Star symbols show results for the SS. The color bar shows time (in a nonlinear scale, to focus on the final stage). Bottom: Theory

Figure 3

Spatiotemporal diagram of the self-propulsion orientation parameter $\alpha \left(x\right)=\langle \widehat{n}·\widehat{\nu }\rangle$, for fast (left) and slow (right) particles, with $\delta =0.5$ and $\varphi =0.18$. The black dashed lines show the mean thickness data shown in Fig. 1. The dashed red and blue lines are the “cleaning signals” with velocities $\langle v_x^{(\mathrm{f}/\mathrm{s})}\rangle$ (see main text).

Figure 4

Stationary concentration profiles for slow [${\varphi }_{\mathrm{s}}\left(x\right)$] (top), fast [${\varphi }_{\mathrm{s}}\left(x\right)$] (medium), and all [${\varphi }_{\mathrm{f}}\left(x\right)+{\varphi }_{\mathrm{s}}\left(x\right)$] (bottom) particles for various values of speed diversity degree $\delta$. The global area fraction is $\varphi =0.18$.

Figure 5

Sequence of snapshots (from left to right and top to bottom), showing an avalanche event for $\delta =0$. The first escaping particle is shown in red, followed by a small avalanche of one particle (in light blue).

Figure 6

Ratio between the number of slow particles and the total number of particles in the layer for $\varphi =0.18$ as a function of $\delta$. Dashed and solid lines show the theory for the transient and for the SS, respectively. Triangles and stars show simulation results for the transient and for the SS, respectively. Inset: Stationary mean layer thickness. Stars are simulation and the solid line is the theory.