Stability of liquid films covered by a carpet of self-propelled surfactant particles

We consider a carpet of self-propelled particles at the liquid-gas interface of a liquid film on a solid substrate. The particles exert an excess pressure on the interface and also move along the interface while the swimming direction changes due to rotational diffusion. We study the intricate influence of these self-propelled insoluble surfactants on the stability of the film surface and show that depending on the strength of in-surface rotational diffusion and the absolute value of the in-surface swimming velocity, several characteristic instability modes can occur. In particular, rotational diffusion can either stabilize the film or induce instabilities of different character.

Figure 1

(a) Liquid-gas interface of a liquid film, loaded with self-propelled surfactant particles. Their swimming orientations are indicated by the unit vectors $\mathit{p}$, shown by red arrows. (b) Influence of the in-line motion of self-propelled particles on film stability. Largest growth rate of a perturbation around a flat film plotted versus wave number for different values of the in-plane self-propulsion velocity $V$ as indicated. Other parameters are $D=1$ and $d=0$. Solid (dashed) lines correspond to oscillatory (steady) dynamics.

Figure 2

(a) Stability diagram in the plane $(d_{\mathrm{eff}}=V^2/2D,V)$ at fixed $\beta =3$ and $d=0$. The inset shows the time evolution of the minimum and maximum of the local film thickness in the smooth particle dynamics simulations for $V=0$ (dashed lines) and $V=5$ (solid lines) at $D=1$. (b) and (c) The dispersion relations (maximal $\mathrm{Re}\left[\gamma \right(k\left)\right])$ for $V=1$ and different $D$ as indicated by numbers near each curve: 1, $D=0.001$; 2, $D=0.1$; 3, $D=0.3$; 4, $D=0.5$; 5, $D=0.7$; 6, $D=1$; 7, $D=2$; and 8, $D=10$. Solid (dashed) lines correspond to oscillatory (steady) perturbations. The heavy solid line represents $D\to \infty$ and coincides with the curve for $V=0$ in Fig. 1. (d) Stability diagram in the plane $(D,k)$. The solid line represents the level line of the dispersion relation at $\mathrm{Re}\left(\gamma \right)=0.03$ to illustrate its bimodal character.