Mechanical Properties of Acoustically Levitated Granular Rafts

We investigate a model system for the rotational dynamics of inertial many-particle clustering, in which submillimeter objects are acoustically levitated in air. Driven by scattered sound, levitated grains self-assemble into a monolayer of particles, forming mesoscopic granular rafts with both an acoustic binding energy and a bending rigidity. Detuning the acoustic trap can give rise to stochastic forces and torques that impart angular momentum to levitated objects. As the angular momentum of a quasi-two-dimensional granular raft is increased, the raft deforms from a disk to an ellipse, eventually pinching off into multiple separate rafts, in a mechanism that resembles the breakup of a liquid drop. We extract the raft effective surface tension and elastic modulus and show that nonpairwise acoustic forces give rise to effective elastic moduli that scale with the raft size. We also show that the raft size controls the microstructural basis of plastic deformation, resulting in a transition from fracture to ductile failure.

Popular Summary

Many physical systems are composed of rapidly spinning objects, from rotating black holes to self-gravitating asteroids. The stability and shape of these rotating systems is determined by a balance between attractive forces, which bind the material together, and outward pressure from the rotation. Rotating liquid droplets are often used as experimental and mathematical models, with surface tension playing the role of the attractive force. Still, even in tiny liquid droplets, the number of molecules is so large that internal restructuring under rotation is difficult to assess. To probe the limit where individual constituents in a rapidly rotating system can be tracked with precision, we acoustically levitate a granular material.

When particles are levitated, sound scattering generates attractive forces that organize the granular material into a tightly packed single layer of submillimeter-sized particles. This levitated “raft” has an overall surface-tension-like cohesion. The acoustic trap also increases the rate of rotation of levitated objects, driving transitions in the shape of the rafts and their eventual breakup. Increasing the number of particles in a raft from 10 to 200, we find a crossover from soft solid to liquidlike behavior. In contrast to liquids, however, the effective surface tension and bulk modulus of acoustically levitated rafts increase with their size, similar to the attractive forces in spinning objects held together by gravitational forces.

These results provide insight into the emergence of liquidlike behavior in clusters of cohesive particles and might even open the door toward the systematic study of “tabletop asteroids.”

Figure 1

Self-assembly of granular rafts by acoustic levitation. (a) Three-dimensional drawing of setup and sequence of images (side view) showing the self-assembly of a granular raft from its constituent particles. Piezoelectric elements (cylinder) are attached to an aluminium horn to generate ultrasound (only the base is shown). The grounded aluminium horn is spaced over a (grounded) indium-tin-oxide glass slide. The dashed black rectangle indicates the field of view in the subsequent still images. At $t=0$, a standing wave is established between the transducer (above the top of the image) and reflector surface, which is covered with loose particles. These particles are picked up from the reflector surface by the primary acoustic force and initially form small clusters, which travel in the underdamped acoustic environment until they coalesce to form a monolayer. (b) Self-assembled rafts composed of varying numbers of particles $N_0$, viewed from below. (c) Sequence of images from below, showing two rafts approaching each other and merging into a larger raft. Variations in brightness correspond to local curling of the raft out of plane.

Figure 2

Secondary acoustic forces drive self-assembly and rotation of levitated granular rafts. (a) Secondary acoustic potential due to a particle with radius $a$, computed using a scattering expansion, normalized by the energy density of the cavity $E_0$, and the particle volume $V_p$. (b) Secondary acoustic force between a pair of particles, computed using a scattering expansion (black solid line) and using a lattice simulation (blue markers). Top panel: horizontal force $F_x$ between two particles as a function of distance $x$ between their surfaces [dashed white horizontal line in panel (a)]. Bottom panel: angular force $F_{\psi }$ on a particle displaced out of the nodal plane with angle $\psi$ [$x/2a=0.1$, dotted circular contour in (a)]. (c) Example traces of rotation rate $d\theta /dt$ as a function of time, for a two-particle dimer consisting of a pair of $700–810\mu \mathrm{m}$ polyethylene particles, at different detuning parameters $\mathrm{\Delta }f/f_0$. (d) Probability of dimer rotation $P(\mathrm{rot})$ as a function of $\mathrm{\Delta }f/f_0$. Shaded areas indicate the standard error. (e) Plot of the decorrelation time ${\tau }_d$ for the rotation rate of a pair of particles, as a function of $\mathrm{\Delta }f/f_0$. Shaded areas indicate the standard error. Data are not plotted for the smallest detuning parameters, for which there is no significant change in the rotation over time.

Figure 3

Levitated granular rafts exhibiting emergent liquidlike behavior. (a) Sequence of images from below, showing the deformation of an initially circular raft that rotates in the clockwise direction. As the raft gains angular momentum, it elongates into an ellipse (at $t=t_0$, where the kinetic energy $\rho {\omega }^{2}a{R}_{\mathrm{eff}}^4$ is at its maximal value) and then splits into two smaller rafts. The corresponding traces are shown in orange in panels (c) and (d) (second largest raft). (b) Sequence of images from below, showing the deformation of a smaller raft, at first also by elongating (at $t=t_0$) but then by shedding particles. The corresponding traces are shown in purple in panels (c) and (d) (second smallest raft), with the diamond marking the image at time $t=3\mathrm{ms}$ in (b). (c) Example time series of the absolute value of the number of rotations per second $|\omega |/2\pi$ (top panel) and the dimensionless shape parameter $R^{*}$ of the levitated raft (bottom panel), as a function of the number of particles in the raft $N_0$ [see part (d) for the color key]. Before $t_0$, the rafts spin faster without deforming; after $t_0$, the rafts change shape, shed weakly bound particles, and eventually split into multiple pieces. Raw data are filtered using a moving-average filter, with a width of 20 data points (7 ms) to obtain the data in panels (c) and (d). (d) Evolution of the dimensionless shape parameter $R^{*}$, as a function of the kinetic rotational energy of the drop $\rho {\omega }^{2}a{R}_{\mathrm{eff}}^4$, for rafts with several different $N_0$. The inset images show the rafts at various points in the spinning process. After deformation into an ellipse, rafts can shed angular momentum by losing small clusters of weakly bound particles.

Figure 4

Contactless measurement of effective surface tension, effective elastic modulus, and microstructural properties. (a) Effective surface tension $\gamma$ as a function of the initial number of particles in a raft $N_0$. The dashed line displays the prediction for $\gamma$ obtained from integrating the acoustic force [Fig. 2] over the raft body (within a multiplicative prefactor, see Appendix pp1). Data are shown for transducer peak-to-peak driving voltage $V_{\mathrm{pp}}=300\mathrm{V}$. Inset: slope of $\gamma$ with respect to $N_0$, measured for different $V_{\mathrm{pp}}$. Error bars indicate standard error. (b) Effective elastic modulus $K$, defined in analogy to the bulk modulus of a three-dimensional material, as a function of the initial number of particles in a raft $N_0$. The dashed line displays the prediction for $K$ obtained from integrating the acoustic force [Fig. 2] over the raft body, with a multiplicative prefactor (see Appendix pp1). Data are shown on a log-log plot for transducer peak-to-peak driving voltage $V_{\mathrm{pp}}=300\mathrm{V}$. Error bars indicate the error due to fitting the data. Inset: slope of $K$ with respect to $N_0$, measured for different $V_{\mathrm{pp}}$. Error bars indicate standard error. (c) Average coordination number of particles in the raft interior $⟨z⟩$, as a function of $N_0$. The dotted line indicates the coordination number for close-packed configurations in two dimensions. (d) Lost number of particles up to maximal plastic deformation, $N_{\mathrm{loss}}$, normalized by $N_0$, as a function of $N_0$. Shaded regions indicate the standard error.

Figure 5

Schematic of the coordinate system for calculating the potential on a point particle due to a disc composed of point scatterers. The point particle, which has radius $a$, is placed in plane with the disc (which has radius $R$ and thickness $2a$) and at a horizontal distance $D$ from the edge of the disc.

Figure 6

Example raw data for the calculation of $K$ (dots). The fractional change in volume of the raft $V/V_0-1$ is plotted as a function of the rotational pressure, $\frac{1}{2}\rho {\omega }^2{R}_{\mathrm{eff}}^2$. Color indicates $N_0$. The best-fit line is indicated as a solid line through each data set. Data have been smoothed with a moving average filter, with a width of 10 data points (3 ms).

Figure 7

Voronoi construction reveals the onset of plasticity during raft deformation. (a,b) Images in Fig. 3 of the main text, overlaid with the corresponding Voronoi diagram of the raft interior. Cells in the Voronoi diagram are overlaid with the corresponding number of sides of the polygon: Particles with five neighbors are shaded blue, particles with six neighbors are white, and particles with seven neighbors are red. (c) Left axis: plot of the number of particles in the raft interior with six neighbors $n_6$, divided by the total number of particles in the raft $N(t)$, as a function of time (both $n_6$ and $N$ are tracked at every frame of the drop evolution), for the raft pictured in (a). Right axis: raft shape parameter $R^{*}$ as a function of time. (d) Upper: $n_6/N(t)$ as a function of time, for several different initial raft sizes $N_0$. Lower: $R^{*}$ as a function of time.

Figure 8

Nonaffine motion of particles before breakup. (a)–(c) Comparison of rotation-corrected particle positions at the start of spin-up (dark blue circles) and just before shape change (i.e., one frame before $E_{\mathrm{rot}}=E_{\mathrm{rot},\max }$, plotted as light-blue filled circles), for three rafts of varying size. Red arrows show the local displacement field from initial to final particle position. (d) Plot of average displacement of particles (averaged over all particles in the raft) during spin-up $\mathrm{\Delta }d$ as a function of $N_0$, normalized by the particle diameter $2a$. Displacement has been resolved into the radial ($\mathrm{\Delta }{d}_r$, blue data points) and azimuthal ($r\mathrm{\Delta }{d}_{\theta }$, orange data points) directions. The black dashed line indicates the noise floor for particle tracking.

Figure 9

Drag force $F_d$ on a particle on the perimeter of a raft, estimated using a linear drag model, as a function of the total number of particles in the raft, $N_0$.

Figure 10

Rotation activating out-of-plane bending in levitated granular rafts. (a) Sequence of side-view images showing the rotation of a raft, which eventually elongates into an ellipse ($t=34\mathrm{ms}$). During the process of spinning up, the raft deforms significantly out of plane. (b) Raft concavity $\chi$ (value shown by the color bar, see text for definition), as a function of the absolute rotational frequency $|\omega |$ and raft radius $R_{\mathrm{eff}}/a$. The data represent observations of 60 (independently self-assembled) rafts. The 95% confidence intervals for $\chi$ are approximately constant as a function of $|\omega |$ and $R_{\mathrm{eff}}/a$ and do not exceed 0.02 for any measurement.