Motion of Heavy Particles on a Submerged Chladni Plate

Heavy particles are traditionally believed to gather at the nodes of a resonating plate, forming standard Chladni patterns. Here, for the first time, we experimentally show that heavy particles, i.e., sub-mm particles, can move towards the antinodes of a resonating plate. By submerging the resonating plate inside a fluidic medium, the acoustic radiation force and the lateral effective weight become dominant at the sub-mm scale. Those forces, averaged over a vibration cycle, move the particles towards the antinodes and generate sophisticated patterns. We create a statistical model that relates the complex motion of particles to their locations and plate vibration frequencies in a wide spectrum of both resonant and nonresonant frequencies. Additionally, we employ our model to control the motion of single particles and a swarm of particles on the submerged plate. Our device can move particles with sufficient power at an exceptionally wide frequency range, potentially opening a path to new particle manipulation techniques at sub-mm scale in fluidic media.

Figure 1

(a) Schematic of the experimental device: a silicon plate is mounted on a piezoelectric actuator and submerged in a water tank. A computer generates a signal that excites the submerged plate and moves the particles on the plate towards the antinodes. The white dashed lines represent the nodal lines. (b) and (c) Acting forces on a particle on the submerged Chladni plate at two moments during a vibration cycle where in (b) the lateral effective weight ($⟨{\mathbf{W}}_{||}⟩$) is larger than in (c). The acoustic radiation force ${\mathbf{F}}_r$ points towards the antinodes, and the drag force ${\mathbf{F}}_d$ acts in a direction that is opposite to the motion of the particle during the whole vibration cycle.

Figure 2

(a) Experimental and theoretical resonant patterns: in the experimental figures (first row), the black dashed lines are overlaid for better visualization of the nodal lines. In the theoretical figures (second row), the time-averaged horizontal effective weight $⟨{\mathbf{W}}_{||}⟩$ (arrows) is overlaid on the theoretical vibration amplitude $\xi$, where the force points towards the antinodes. (b) Experimental and theoretical nonresonant patterns: in the experimental figure, arrows show the vortexlike motion path. It follows that the adjacent resonant patterns, which are underlaid in the experimental figure, affect the path of the vortexlike motion. Scale bar, 10 mm.

Figure 3

Modeled displacement fields: first column shows the evenly spaced streamlines of the displacement field ${\mathbf{d}}_n(x,y)$ at 4978 and 14 917 Hz. Second column shows the divergence of the displacement field underlaid below the modeled displacement vector field, whereby the red and blue regions represent the nodal and antinodal regions of vibration, respectively. Third column shows the signed magnitude of curl, overlaid on the corresponding displacement field. Inset $X$ indicates a spot with a clockwise vortical motion.

Figure 4

Manipulation of single particles and a swarm of particles: (a) Manipulation of a $750\mu \mathrm{m}$ glass bead on a maze-shaped trajectory (Video SM3). (b) Simultaneous and independent manipulation of two $750\mu \mathrm{m}$ glass beads on two predefined trajectories resembling the letters of $L$ and $C$ (Video SM4). (c) Manipulation of a swarm of particles (Video SM5): placing multiple particles on the plate, aggregating them into a bundle, transporting the swarm of particles towards the center and then to the splitting location and splitting them into two clusters of particles. Scale bar, 10 mm.