Quantized orbital-chasing liquid metal heterodimers directed by an integrated pilot-wave field

A millimetric bouncing droplet sustained on a vibrating bath becomes a moving wave source (particle) through periodically interacting with the local wave field it generates during the droplet-bath impact. By virtue of such particle-wave duality, the macroscopic hydrodynamic system imitates enigmatic behaviors of the quantum realm. Here we show that it is possible to create an integrated pilot-wave field to better prescribe the droplet trajectories, via amplified bath capillarity. This is demonstrated with a liquid metal droplet-bath system in which the local wave field generated by droplet bouncing is superposed by the global wave field induced by bath meniscus oscillation. The resulting dual pilot-wave configuration enables a class of directional chasing motions of two bound dissimilar droplets (heterodimers) in multilevel hydrodynamic traps (orbits), featuring two quantized regime parameters, namely, the interdroplet binding level and the orbit level. We investigate the dynamics of the vibrating liquid metal bath, with its level-split ring-wave field and its peculiar vortex field being highlighted. We also rationalize the exotic droplet motions by considering the interdroplet particle-wave interactions mediated by the integrated pilot-wave field. It is revealed that a temporal bouncing phase shift between the two droplets in the heterodimers, due to size mismatch, gives rise to their horizontal propulsion, while their spatial binding regime exclusively determines the collective chasing direction. It is further evidenced that the horizontal in-orbit chasing motion is directly related to vertical droplet bouncing. Our findings unveil the integrated pilot-wave field as a trail towards improved droplet guiding, thereby extending the hydrodynamic particle-wave analogy to optical systems and beyond.

Figure 1

(a) Schematic side view of the liquid metal droplet-bath system. The bath is filled with a bottom liquid metal layer and a top alkaline electrolyte layer. The vibrating bath and the bouncing droplet induce a global wave field ${\mathcal{W}}^G$ and a local wave field ${\mathcal{W}}^D$, respectively. (b) Schematic top view of the directional orbital-chasing motion of the liquid metal heterodimers and the definition of the regime parameters. ${\lambda }_G$ and ${\lambda }_D$ represent the wavelength of ${\mathcal{W}}^G$ and ${\mathcal{W}}^D$, respectively. (c) The regime diagram of the orbital-chasing motion. ($\circ$) The large droplet ${\mathcal{D}}_L$ chases the small droplet ${\mathcal{D}}_S$; ($\square$) ${\mathcal{D}}_S$ chases ${\mathcal{D}}_L$; ($\diamond$) ${\mathcal{D}}_S$ chases ${\mathcal{D}}_L$ but the heterodimers either cease or derail before a full-circle orbiting is completed (partial orbiting regimes); ($▵$) high-orbit levels in which the confinement becomes too weak to sustain the heterodimers. Regimes requiring $d_l>2r_n$ are geometrically invalid and left blank. (d) Top-view snapshots of the orbiting heterodimers with different binding levels $l$ ($n$ = 2 fixed). (e) Top-view snapshots of the orbiting heterodimers in different orbit levels $n$ ($l$ = 1 fixed). The arrows indicate the direction of orbiting and chasing.

Figure 2

(a) The orbiting trajectories of the large droplet ${\mathcal{D}}_L$(red) and the small droplet ${\mathcal{D}}_S$ (blue) featuring different orbit levels $n$ ($l$ = 1 fixed). (b) The magnitudes of the instant velocity $v$ and the average velocity $\overline{v}$ of the heterodimers in (a) as a function of time. The errors in the measurement of $v$ and $\overline{v}$ are $<$ 0.2 mm ${\mathrm{s}}^{-1}$ and $<$ 0.04 mm ${\mathrm{s}}^{-1}$, respectively.

Figure 3

(a) Binding distance $d{}_l$ variations of the heterodimers at different binding levels $l$ ($n$ = 2 fixed). (b) Cycle-averaged binding distance ${\overline{d}}_l$ in different orbiting regimes as a function of the normalized driving acceleration ${\gamma }_0$/g. (c) Quantization of the orbital motion of the liquid metal heterodimers. In all figures, the solid lines and scatters represent experimental data, and the dashed lines are their linear fittings.

Figure 4

(a) Time-lapse side-view snapshots (slightly oblique) of the liquid metal bath at the vibrating onset. From top to bottom, $t$ = 0.00 ms, 22.23 ms, 41.67 ms, and 86.12 ms. (b) Side-view snapshots of a single droplet bouncing on the vibrating bath (with no horizontal motion). The superposition of the droplet-induced local wave field ${\mathcal{W}}^D$ and the global wave field ${\mathcal{W}}^G$ can be seen from the bottom panel. (c) Bath vibrating amplitude $\mathrm{\Delta }z$ measured along the radial direction $r$ at different driving accelerations as indicated. (d) Surface profile of the quiescent liquid metal surface. (e), (f) Top-view snapshots featuring the maximum bath deformation at the bath center due to meniscus oscillating (e) and the maximum localized bath deformation induced by droplet impact (f). The maximum size of the dark region (deformed liquid metal surface) at the bath center (e) and the droplet impact site (f) is indicated. $\gamma {}_0$ = 2.2 g.

Figure 5

(a) Top-view of the vibrating liquid metal bath showing the global annular wave field ${\mathcal{W}}^G$. (b) Visualized vortex field at the liquid metal-alkaline interface and (c) the corresponding vorticity color map (${\lambda }_0=1.6$ g).

Figure 6

(a), (b) Oblique side views of the orbiting heterodimers in the short-range binding regime $l$ = 0 (a) and the long-range binding regime $l$ = 1 (b). (c), (d) Vertical bouncing trajectories of the small droplet ${\mathcal{D}}_S$ (upper panel) and the large droplet ${\mathcal{D}}_L$ (lower panel) as a function of time for five consecutive bouncing periods $T$ of a (0, 3) heterodimer (c) and a (1, 3) heterodimer (d). (e) The spatialtemporal trajectories of the same heterodimer in (e) obtained by visual tracking. (f) The spatialtemporal trajectories of the same heterodimer in (d) obtained by visual tracking. The impact time difference $\mathrm{\Delta }t$ of the two droplets of the heterodimers are indicated.

Figure 7

(a) Schematics of the temporal phase shift between ${\mathcal{W}}^D$ induced by the impact of ${\mathcal{D}}_S$ (black) and ${\mathcal{D}}_L$ (red). (b), (c) Directional propulsion resulted from the interaction between individual droplets and the wave field generated by their partners in the $l$ = 0 regime (b) and the $l$ = 1 regime (c). The characteristic time of impact for cases i–iv is indicated in (a). The arrows indicate the direction of the propulsion that the local wave field exerts on its partner droplet during impact. Here (b) and (c) are in the one-dimensional spatial reference frame along the orbit.

Figure 8

(a) A survey of droplet size distribution. (b) Relative droplet size and its acceleration dependence (inset).

Figure 9

(a), (b) Acceleration dependence of $\theta /2\pi$ for the (0, 2), (0, 3) regimes (a) and the (1, 2), (1, 3) regimes (b). The error in the measurement of $\theta /2\pi$ is 0.02. The pentagrams show the acceleration-dependent phase shift of the same heterodimer. (c), (d) Acceleration dependence of $\overline{v}$ for the (0, 2), (0, 3) regimes (c), and the (1, 2), (1, 3) regimes (d). Error in $\overline{v}$ measurement $<$ 0.04 mm ${\mathrm{s}}^{-1}$. The scatters represent the measured values, based on which the filled regions are drawn as guides for the eyes. The inset of (d) depicts the geometrical constraints, which shows how the tangent projection of the droplet momentum is influenced by the binding distance ${\mathit{d}}_l$ and orbit radius ${\mathit{r}}_n$.