From bipedal to chaotic motion of chemically fueled partially wetting liquid drops

Abstract

We employ a thermodynamically consistent out-of-equilibrium continuum model to study the motion patterns of partially wetting liquid drops covered by autocatalytically reacting surfactants. When ambient chemostats feed a chemomechanical feedback loop involving a nonlinear reaction network, surface stresses caused by the Marangoni effect and the ensuing hydrodynamic motion, drops show a variety of increasingly complex biomimetic motility modes including shuttling, bipedal, rotational, intermittently chaotic and chaotic motion. We determine the corresponding nonequilibrium phase diagram and show that the complexity of the motion arises from competing length scales.

Figures (6)

• Figure 1: Schematics of the considered system: drops of a partially wetting liquid on a flat solid substrate are described by a height profile $h(\vec{x}, t)$. The free surface is covered by two species of surfactants with density profiles $\Gamma_1(\vec{x},t)$ and $\Gamma_2(\vec{x}, t)$. They engage in an autocatalytic reaction and are exchanged with ambient reservoirs. The open exchange of fuel and waste together with a chemomechanical feedback loop drives droplet motility. The streamlines represent the height-integrated liquid flow.
• Figure 2: Panels (a) to (e) give various motility modes of chemically fueled drops covered by reacting surfactants. Here, each top [bottom] row of the left part shows snapshots of surface tension profiles where red (blue) indicates high (low) values [the height-integrated liquid flow, namely, the flux of field $h$ in Eq. (\ref{['eq:gradient_dynamics_general']})]. The streamline thickness represents the magnitude of the flux, and the dashed drop contours correspond to $h=1.1$ ($h=1.1h_a$ in dimensional units). The right parts give associated center-of-mass trajectories of the drop between points $A$ and $B$ on the substrate. In (c), the inset shows a magnification of the trajectory. Only part of the periodic computational domain $[-50,50]\times[-50,50]$ is shown. For parameters, see Appendix A. Also see Supplemental Videos 1-5.
• Figure 3: (a) Morphological phase diagram spanned by surfactant diffusivity $D$ and drop volume $V_\text{drop}$. States are represented by colors: resting drop (dark blue), shuttling (light blue), rotational (gray), bipedal (light red) and chaotic (dark red) motion. The number of spots increases with decreasing $D$ and increasing $V_\text{drop}$. The primary instability of the resting drop state is determined by continuation (black line) and fitted with the power law $D\propto V^{0.58}_\text{drop}$ (green line). Colored triangular corners indicate multistability with the respective state. Yellow lines indicate slices analyzed in Fig. \ref{['fig:parameter_slices']}. (b)-(e) Snapshots of the surface tension profiles corresponding to the four diagram corners (style as in Fig. \ref{['fig:state_collection']}). For parameters, see Appendix A.
• Figure 4: One-parameter diagram along the lines marked in Fig. \ref{['fig:phase_diagram']}, i.e., the diffusivity $D(V_\text{drop})$ is linearly related to $V_\text{drop}$. Shown are (a) the turning angle $\langle\Delta\theta\rangle$ and (b) the step size $\langle\Delta x\rangle$, along with the deformation frequency $\langle\omega \rangle$. For drop rotation, there is a transition between modes with star-shaped (gray region) and circular trajectories (white region). Trajectories are indicative of the respective mode.
• Figure 5: (a) Partial bifurcation diagram of the originally stable resting drop (black line) with diffusivity $D$ as control parameter. The solution measure is the L$_2$-norm $\vert\vert p \vert\vert_2 = 1/L\left(\int p^2\:\text{d}^2 x\right)^{1/2}$ where $p$ is given by Eq. (\ref{['app:eq:pressure_nondim']}). The domain size is $100\times100$. Strong [weak] lines represent linearly stable [unstable] states. The stable resting drop is destabilized in a subcritical pitchfork bifurcation (PF). Note that other occurring bifurcations are not marked. Red and blue lines indicate branches of unstable states that bifurcate at the PF. The disconnected branches of the same color are related by an imperfect pitchfork bifurcation. (b) Liquid pressure profiles of the different emerging states [marked by corresponding symbols in (a)]. Red (blue) indicates high (low) pressure. For parameters, see Table \ref{['app:tab:parameter_sets']}.