Collective Surfing of Chemically Active Particles

We study theoretically the collective dynamics of immotile particles bound to a 2D surface atop a 3D fluid layer. These particles are chemically active and produce a chemical concentration field that creates surface-tension gradients along the surface. The resultant Marangoni stresses create flows that carry the particles, possibly concentrating them. For a 3D diffusion-dominated concentration field and Stokesian fluid we show that the surface dynamics of active particle density can be determined using nonlocal 2D surface operators. Remarkably, we also show that for both deep or shallow fluid layers this surface dynamics reduces to the 2D Keller-Segel model for the collective chemotactic aggregation of slime mold colonies. Mathematical analysis has established that the Keller-Segel model can yield finite-time, finite-mass concentration singularities. We show that such singular behavior occurs in our finite-depth system, and study the associated 3D flow structures.

Figure 1

(a) Schematic illustrating chemically active particles (dark circles) bound to a flat fluid surface sitting above a fluid layer of depth $H$. The gray-scale map represents the concentration field of the chemical species produced by the active particles. (b) Variation of $\mathrm{\Omega }$ as a function of $\lambda$.

Figure 2

For $\delta =1/2$, contours of (a) particle density $\psi$, (b) chemical surface concentration $C$, and (c) the surface velocity field from Eq. (8). (d) The in-plane surface vorticity field $\omega$ overlaid with a color map of vorticity magnitude $\omega$. The data correspond to $t/t_c\approx 0.9$ where $t_c$ is the estimated collapse time. Simulation parameters are $\mathrm{\Delta }t=1/400$, $\mathrm{\Delta }x=\mathrm{\Delta }y=1/256$, $P{e}_c=0.1$, $P{e}_p=400$, and $\psi (\mathbf{x},0)=1+0.1\cos (2\pi x)\cos (2\pi y)$. Vorticity is scaled by $1/\tau$.

Figure 3

The 3D velocity field overlaid with a color map of the chemical concentration $C$, at the time when ${\psi }_{\max }=\psi (L/2,L/2)\approx 10$. $\delta =1/2$, $1/10$, $1/20$ in (a), (b), and (c), respectively. Simulation parameters are the same as for Fig. 2. The $z$ axis is scaled by $\delta$. The black arcs in (a) highlight vortical flows in the bulk.

Figure 4

Time variation of ${\psi }_{\max }=\psi (1/2,1/2)$ for different values of $\delta$ and asymptotic cases of $\mathrm{\Omega }=1/2$, $1/4$ for which the evolution of $\psi$ obeys the 2D KS model. The inset shows the variation of the estimated collapse time $t_c$ as a function of $\delta$. Crosses show the asymptotic values of $t_c$ for $\mathrm{\Omega }=1/2$, $1/4$. Simulation parameters are the same as for Fig. 2.