Towards active microfluidics: Interface turbulence in thin liquid films with floating molecular machines

Thin liquid films with floating active protein machines are considered. Cyclic mechanical motions within the machines, representing microscopic swimmers, lead to molecular propulsion forces applied to the air-liquid interface. We show that when the rate of energy supply to the machines exceeds a threshold, the flat interface becomes linearly unstable. As a result of this instability, the regime of interface turbulence, characterized by irregular traveling waves and propagating machine clusters, is established. Numerical investigations of this nonlinear regime are performed. Conditions for the experimental observation of the instability are discussed.

Figure 1

(Color online) Sketch of a thin film with floating molecular machines generating propulsion forces.

Figure 2

(Color online) Mechanisms of (a) instability development for upward directed propulsion forces and of (b) instability damping for downward directed propulsion forces.

Figure 3

(Color online) Real (solid line) and imaginary (light line) parts of the growth rate $S$ as functions of the wave number $K$ of the perturbation for (a) $B=0.38$, (b) $B=0.4$, and (c) $B=0.42$. Other parameters are $A=0.2$ and $D=0$.

Figure 4

(Color online) Interface turbulence. Space-time diagrams display the local film thickness $H$ (thicker regions correspond to bright and thinner regions to dark colors) depending on the spatial coordinate and time for (a) $B=0.42$, (b) $B=0.44$, (c) $B=0.46$, and (d) $B=0.48$. Other system parameters are $A=0.2$ and $D=0$. Numerical integrations for a one-dimensional system of length $L_0=512$ (with $\Delta x=1$) and the total time interval of $T_0=5\times {10}^5$ (with $\Delta t=0.01$).

Figure 5

(Color online) Snapshots of the concentration distributions (light lines) and of the respective interface profiles (black lines) for (a) $B=0.43$, (b) $B=0.46$, and (c) $B=0.49$. Other parameters are $A=0.2$ and $D=0$. Insets on the bottom show the collision of two waves (d) and the motion of a single wave (e). Arrows in the insets indicate the directions of motion of the distribution maxima.

Figure 6

Initial evolution of the pattern amplitudes $\Delta H\left(T\right)$ and of the characteristic wave numbers $K_0$ of the developing patterns for two different values $B=0.43$ and $B=0.46$ of the control parameter. Other parameters are $A=0.2$ and $D=0$. The system size is $L_0=512.$

Figure 7

Time dependences of the film height $H\left(T\right)$ and concentration $C\left(T\right)$ at a fixed spatial point for two different values of the control parameter $B$. Other parameters and the system length are the same as in Fig. 4.

Figure 8

Statistical properties of the interface turbulence at different deviations from the instability threshold: (a) characteristic wave number, (b) characteristic frequency, (c) characteristic inverse transient time, and (d) mean amplitude of the patterns. Dots show mean values; bars indicate statistical dispersion of numerical data. Solid curves show analytical results Eqs. (13, 14, 15) corresponding to the linear stability analysis predictions (see the text). Dashed curve shows a fitting power law (see the text). The same system and simulation parameters are as in Fig. 6.