Acoustically propelled winged macroparticles

Active or self-propelled particles locally harvest and harness energy from their environment, converting it into mechanical force that propels their motion.^1–4 The assembly of such particles, active matter, reproduces collective behavior observed in nature (flocks of birds,⁵ fishes,⁶ and bacteria⁷) and human-made inventions in science^8,9 and technology.^10,11 Active particles recently demonstrated a wide range of applications,^1,12 among which, targeted drug delivery,¹³ recovery of precious metals,¹⁴ waste water treatment,¹⁵ and energy production.¹⁶ 

A single particle is said to be active if the direction of its motion depends on the characteristics of the particle itself (e.g., its shape), rather than being entirely controlled by the energy source.^17,18 Yet, each particle needs energy to pursue its motion, which can be delivered, for example, through an electric field,¹⁸ light,¹⁹ magnetic field,²⁰ temperature,²¹ viscosity,²² or concentration gradient.²³ 

Acoustic fields have been used to propel nano- and microparticles through resonance,^24–26 notably in liquids,^27–30 where it is easier to manipulate environmental parameters than in air. Macroscopic particles (typical length scale $∼1 cm$⁠) are propelled by acoustic fields in the range 10–200 Hz, known as low frequency noise (LFN). Small particles (order of $1μ$m and below) resonate in the ultrasound range and are generally excited in the range of 1–4 MHz.^24–30 While larger particles would facilitate the experimental study of their shape and collective motion and open new applications in ambient energy harvesting,³¹ limited investigations have been conducted on the propulsion of active macroparticles.^32,33 Most previous studies used standing waves^24,34–39 to excite the particles, which forces a certain position and orientation upon them.⁴⁰ Besides, homogeneous acoustic fields are prevalent.^41–44 While simplifying experimental studies, these characteristics further the system from real-life applications.

The macroparticles studied in this research represent a variation of the vibrots proposed by Altshuler et al.³³ Vibrots are macroparticles equipped with elastic cantilever legs that transform oscillations from a vibrating platform into controlled motion. Their individual moving mechanism⁴⁵ and collective motion^8,9,46,47 have been studied.

We propose a method for propelling macroparticles, similar to vibrots, but using an acoustic field. We induce the acoustic resonance of thin polymer plates, referred to as wings, where the propulsion mechanism is not based on bending elastic legs, but on the displacement of air produced by the wings’ oscillations. We demonstrate that tweaking the wings’ shape allows to control the direction and magnitude of the force produced and elucidate the mechanisms at the origin of motion, and the dependence of motion speed and direction on wings’ parameters. Conflicting findings exist regarding the influence of particle size on propulsion velocity. Existing models^48,49 predict that larger particles move faster, which is corroborated by computer simulation,⁵⁰ but experimental work by Soto et al.³⁷ shows the inverse trend. We find a non-monotonic evolution of the force produced by wings of increasing size, highlighting two competing effects: higher frequency needed to excite smaller particles results in a higher excitation speed, while larger particles move more fluid, but slower.

Large active particles, akin to vibrots and our winged particles, offer numerous advantages over smaller systems. They are cost-effectively manufacturable, even in large quantities and without high-end fabrication tools; their individual and collective behavior is easily observable. In particular, the winged particles can be propelled by traveling waves, largely simplifying experimental procedures. The utilization of environmental LFN, often considered a disturbance,⁵¹ presents a potential green energy source for propelling winged macroparticles.^52,53 LFN constitutes a significant portion of ambient or environmental noise,⁵⁴ sources including wind turbines,⁵⁵ urban traffic,⁵⁶ and industrial machinery.⁵⁷ 

As asymmetric winged particles respond to frequency variations by changing rotation speed and direction, they could be used in frequency-guided navigation systems, or adaptive materials that change properties based on acoustic field variation. We provide a proof of concept of this principle: a macroparticle with bidirectional rotation, whose rotation direction varies with the acoustic field frequency.

After introducing our experimental methods, we analyze the effect of wing size (length, width, and thickness) on the force produced, finding that the force evolves non-monotonically with increasing wing size. We then tweak the force direction by creating asymmetric wings. We apply these findings to create rotating macroparticles and analyze the change in angular acceleration for prolonged times. Finally, we design a macroparticle with bidirectional rotation: inverting its rotation direction when excited at a different frequency.

We study two types of macroparticles: with rectangular and circular wings. Rectangular wings are used for force measurements, while circular ones are used to probe the propelling-ability of asymmetrical wings in rotation. The wings are cut from polylactic acid (PLA) sheets of thickness $k∈[0.2,0.3,0.4]$ mm, at varying length, $l$⁠, and width, $w$⁠.

The force produced by a rectangular wing is measured by attaching the bottom side of the wing to a load sensor, its top edges freely moving, as shown in Fig. 1. The force generated by acoustic excitation is measured with an Arduino UNO, XH711 amplifier and XIN NUO QI force sensor. The later has a maximum load of 0.02 N and accuracy $10 − 5$ N. The measurement system has a sampling rate of 11 Hz.

The acoustic excitation is produced by a transducer (Monacor SP-300PA) controlled by a function generator, which provides waveform and amplitude. We generate sinusoidal functions at frequencies corresponding to the natural frequencies of the wings. The amplitude is controlled by modifying the input voltage of the function generator and measured in decibels using a Risepro HT-80A decibel-meter (accuracy $1.5± 0.1 dB$⁠), placed at the wing’s position. Measurements of sound intensity are performed without a wing on the scale; the decibel-meter is placed at the position where wings are located during a force measurement.

The first two natural frequencies of the plates are determined experimentally as the frequencies at which a wing experiences a peak in oscillation amplitude (see Fig. 2).

At its first natural frequency, the wing’s oscillation results in low force. For example, the wing shown in Fig. 4(b) produces a force $F f 1= 10 − 2$ mN at its first natural frequency, $f 1$⁠, while at its second natural frequency, $f 2$⁠, it is one order of magnitude higher, reaching $F f 2=2× 10 − 1$ mN (with a constant acoustic field amplitude of $90 dB$⁠). Unless specified otherwise, we use their second natural frequency to excite all wings.

The acoustic propulsion of winged macroparticles relies on different local vibrations of the polymer wing(s), although the whole wing is submitted to the same acoustic field. To validate this physical mechanism, the measurements must be made in homogeneous acoustic. We verify the field’s homogeneity by measuring the sound intensity in the plane corresponding to the wing’s position, shown in Fig. 3. The amplitude of the acoustic field or noise intensity is measured in dB at a frequency of 47 Hz (the transducer’s input voltage is 1.4 V). The distance between the transducer and the wing along the $x$-axis is kept at 20 cm [see Fig. 1(a)], while $y$ and $z$ positions are varied. Figure 3 shows that the acoustic field variations are within maximum 2 dB, which we consider homogeneous.

In all measurements presented, except when specified otherwise, the noise level used is 90 dB.

The second possible source of error is artifacts due to interaction between the load sensor and the acoustic field. The force recorded by the load sensor without wing attached is measured [Fig. 4(a)] and compared to the force when wings of different dimensions are attached [Figs. 4(b) and 4(c)]. The load recorded without wing is $O( 10 − 3 mN)$⁠, two orders of magnitude smaller than the values for wings excited at their second natural frequency (for sound higher than 80 dB), and independent of the excitation frequency [Fig. 4(a)]. The effect of the acoustic field on the load sensor can hence be neglected in our range of study.

Note that a smaller wing [Fig. 4(b)] can produce higher force than a larger wing [Fig. 4(c)] in the region of interest (80–100 dB). While this result might seem counter-intuitive at first, comparing the natural frequencies, one can see that the small wing has higher natural frequencies, resulting in oscillations at a higher speed. This effect can dominate over larger wing’s dimensions, in which case, a larger vertical force $F z$ is produced by a smaller wing, exemplifying that a higher excitation frequency can overwrite smaller dimensions in the final force produced. This point will be addressed in more details in Sec. III B, focusing on the influence of the wings’ dimensions on $F z$⁠.

A threshold in acoustic field amplitude, below which no force is produced, is visible in Figs. 4(b) and 4(c). The transition from immobile (⁠ $F z≈0$⁠) to oscillating (⁠ $F z>0$⁠) happens around 70 dB for the small wing [Fig. 4(b)], and 60 dB for the large one [Fig. 4(c)]. The large wing begins earlier to move enough air to produce a force, due to its larger area. But its lower natural frequency results in a slower oscillation. The effect of frequency soon dominates over amplitude, producing the steeper $F z$ increase for the small over the large wing.

Rotary macroparticles are manufactured by attaching four asymmetric wings to a 3D printed square base of edges 4 cm. The base contains a bearing that blocks all degrees of freedom except rotation along its axis of symmetry. The experimental setup is shown in Figs. 5(a) and 5(b) which illustrates a rotatory winged particle of radius $R=9 cm$⁠. The $z$-distance between the transducer and the macro-particle is 20 cm. A camera records the particle’s rotation at 30 fps from below [see Fig. 5(a)]. Automated tracking is implemented using colored dots placed below the base.

To understand the mechanism by which the macroparticle is activated by an acoustic field and how to control its motion, we first focus on the propulsion mechanism, and the influence of wing’s geometry on the resulting force. Then, we apply the propelling mechanism to control macroparticles’ motion.

Winged macroparticles harness energy from the environment through the wings’ oscillation. By acoustically exciting the wings, they displace a mass of air that creates a force in the opposite direction of the air stream.

The motion of air is captured using smoke in Fig. 6, where the bottom edge of a rectangular wing is clamped, while all other edges are free to move. When the smoke source is outside the wing’s oscillation range, no deviation is observed on the string of smoke [Fig. 6(a)]; once it reaches the wing’s vicinity, the smoke highlights the change in air-stream close to the wing, particularly at its free extremity, where the motion amplitude is maximum [Fig. 6(b)].

A quantitative estimate of the force produced from a simple force balance is provided in the supplementary material. The combination of parameters like the wing’s stiffness, the area left free to oscillate, or frequency and amplitude of excitation, determine the amount of air displaced, and in turn the force created. Furthermore, the direction of the force, trivial for the rectangular wing, changes with its geometry: the direction in which the resulting force is produced depends on which edge is displacing a larger amount of air. We proceed to study the relation of magnitude and direction of the force produced with the wings’ characteristics.

A wing’s transverse area, $A$⁠, is the product of its width, $w$⁠, by its length, $l$⁠, as defined in Fig. 1(b). While intuition might dictate that larger wings produce larger force, we observed the opposite behavior in Fig. 4 (as reported by Soto et al.³⁷), prompting the following analysis of the relationship between wing’s area, $A$⁠, and the force it produces, $F z$⁠.

The force produced by rectangular wings of increasing transverse area is studied, maintaining $w$ constant and varying only $l$ (Fig. 7). This is repeated for three widths, $w∈[2,3,4]$ cm.

Excited at their respective $f 2$⁠, the force $F z$ produced by the wings [Fig. 7(b)] first increases with increasing $A$⁠, but then decreases. (Note that we ignore the initial plateau, where wings are too small to harvest any force.) This non-monotonic behavior, and the counter-intuitive decrease of $F z$ at high $A$⁠, can be explained by comparing the natural frequencies: at constant $w$⁠, wings of smaller area have higher natural frequencies, resulting in the displacement of a mass of air at a higher speed. The quantity of air displaced by wings’ motion depends on both their oscillation frequency and transverse area. As $f 2∼1/ l 2$ [Eq. (2)], while $A∝l$⁠, at low $A$⁠, increasing wings size dominates, while at high $A$⁠, the decrease in oscillation frequency results in a decrease in $F z$⁠. These counteracting effects lead to the maximums in Fig. 7(b) for the intermediate area of each width and might be a route to explore to understand contradictory results found in previous studies^37,48–50 regarding the relation between the size and speed of active particles.

The transition point between the two regimes, plotted as an inset in Fig. 7(b), increases with $w$⁠.

Despite being thin plates, wings have a non-zero thickness (shortest length), which influences the force generated. We study its influence on the force’s magnitude.

Since all wings’ are made of the same material, increasing their thickness also increases their effective stiffness. As a result, a thicker wing oscillates less, as observed in Fig. 8(a): the oscillations’ amplitude, which represents the distance between the two extremes of the oscillating wing, is consistently lower for thicker wings. Inversely, motion amplitude increases for increasing wings’ area; we do not find a visible difference in the rate of increase for the different thicknesses studied, within experimental error.

In Fig. 8(b), we observe that larger motion amplitude results in higher force: the upward orientation of the cloud of points shows a clear correlation between these two variables, irrespective of the specific geometry of each wing. This is confirmed by a Pearson correlation coefficient^60,61 of 0.67.

The propulsion mechanism proposed depends on the amount and direction of air displaced by the wing’s motion. In the case of a simple rectangular wing [as the one shown in Fig. 1(b)], the air is expelled by the wing’s oscillation in the $+z$-direction, but also incidentally in the $y$-direction. Thus, the wing’s motion also creates a force in the $y$-direction. In the case of a rectangular wing, both lateral sides (edges along $z$⁠) have the same length and mobility, hence producing forces of similar magnitude in opposite directions, which cancels each other out. But if one lateral side of the wing is prolonged, the force produced by the wing’s oscillation will have a different magnitude along the positive and negative $y$-direction. Such asymmetric wings allow to control the direction of the force produced, which can translate into the macroparticle’s direction of motion.

Figure 9 exhibits the force produced in $z$-direction (measured by the setup in Fig. 1), by two types of asymmetric wings: angular and rounded geometries, represented, respectively, in Figs. 9(a) and 9(b). Both geometries share a base side of equal length, $l$⁠, and are distinguished by the length of the prolonged side, $l ′$ (angular) and $l ″$ (rounded). In the angular case, the prolonged side is always longer than the base, while in the rounded one, it is shorter, because of the center of rotation used (represented by a red dot). These two wing shapes were selected to produce forces in opposite directions.

Figure 9(c) illustrates the force generated by the wings when placed horizontally on the scale. The solid/dashed lines represent a rotation of 180°, as represented in Figs. 9(a) and 9(b).

For both geometries, we see a symmetric behavior between the forces created in each position. This result supports our claim that the position of the wing’s longer side defines the direction of the force generated. The larger side mobilizes a greater amount of air, which creates a total force in the direction of the smaller side. This effect can be seen in Fig. 10, where the longer sides (b) and (c) have a bigger amplitude of oscillation than the shorter ones (a) and (d), respectively.

Comparing the angular and rounded geometries, the forces produced follow different trends. In particular, for larger areas, the rounded wings produce larger force than the angular ones. This can be understood through the ratio of lengths of base (⁠ $l$⁠) and prolonged (⁠ $l ′$⁠, $l ″$⁠) sides: for the rounded wings, the prolonged side represents a fraction of the base (i.e., $l ″/l<1$⁠) so that the prolonged side gains greater freedom of motion when the wings area is increased. On the contrary, for the angular wings, $l$ and $l ″$ have a fixed ratio so that increasing the area results in larger force produced on both sides of the wing, conducting eventually to the stagnation of total force produced, $F z$⁠.

We showed in Sec. III C that asymmetric wings produce a force in the lateral direction. We use this to create rotating macroparticles, by combining four asymmetric wings placed horizontally on a square base (Fig. 5), to produce a force perpendicular to the rotation axis, which induces rotation of the system.

To maximize the force generated, we select the asymmetric shape and size that produced the highest forces among that previously tested: rounded edge, with 90° inclination angle. Figure 11 illustrates the two geometries used in this section. The first one is created from the rounded wings described in Sec. III C [Fig. 11(a)]; the second one is a new geometry, named circular [Fig. 11(b)], where both sides have the same length (i.e., $l= l ″$⁠), the four wings forming a circle.

In Fig. 12, the angular speed is plotted vs acoustic field amplitude. For the circular geometry [illustrated by Fig. 11(b)], we observe a sharp increase in rotation speed around 80 dB. In this macroparticle geometry, the prolonged side $l ′$ has the same length as the base $l$ (i.e., longer than for the rounded geometry); therefore, the part of the wing that is on the prolonged side is always larger than the part on the base side, hence having more freedom of motion. In turn, it produces more force and, therefore, leads to rotation. On the contrary, in the rounded geometry [Fig. 11(a)], both sides of the wing have equal areas, hence similar freedom of motion. The macroparticle being pulled to rotate in both directions equally ends up remaining static.

We study the evolution of the angular speed over a longer time period, by recording the circular particle while it rotates for one full hour. Figure 13 shows the angular speed as a function of time (left axis) and its angular acceleration (right axis). Videos of the rotation are available as the supplementary material.

The acceleration of the macroparticle depends on the force produced by its wings, and since the energy input from the loudspeaker is constant, the acceleration is also constant, thus producing a linear increase in the angular speed. However, the variability in angular speed is high throughout the experiment; for higher angular speeds, it is reflected in the acceleration (calculated on intervals on 60 s), which becomes relatively unstable, albeit the rotation never stopping.

Each wing geometry has its own natural frequencies, different wings can be distinctively activated by changing the acoustic field frequency. A macroparticle endowed with two different wings’ geometry allows to control of the motion direction through the acoustic field.

To create such bidirectional particle, we combine the geometries from Figs. 9(a) and 9(b) (rounded and angular asymmetric wings) and place two similar wings opposite to each other, in a configuration where wings of the same geometry produce force in the same direction, but each geometry produces force in the opposite direction. Figure 14(a) shows the resulting configuration.

Figure 15 exhibits the angular speed of this bidirectional macroparticle, when each natural frequency is activated consecutively. Calculating the average rotation speed of each time interval where one or the other sound frequency is activated, the macroparticle clearly changes direction with the change in acoustic field frequency, exhibiting the desired bidirectionality. The rounded geometry (which creates negative angular speed in our coordinate system) produces higher absolute values of speed. That confirms results shown in Fig. 12: the rounded wings, at area 92 cm, produced higher force than the corresponding angular wing. Note that we also observe large variations in angular speed within one interval, and regardless of the rotation direction. They can be related to our current experimental setup, but while the inhomogeneity of the acoustic field used to produce motion may result in such large speed variations, the macroparticle never completely stops rotating.

With this bidirectional winged macroparticle, we demonstrate the control of propulsion direction through varying sound frequency, offering potential applications in directing the path of particles using adjustable acoustic fields.

We introduce the acoustic propulsion of macroscopic particles, where the propulsion force and direction can be controlled by the particles’ geometry. The macroparticles are composed of a 3D printed base endowed with four wings: thin polymer sheets placed horizontally around the base. The wings enter in resonance with the acoustic field and displace a mass of air by oscillating, thus generating a force in the opposite direction, which can enable the macroparticle’s motion.

By altering the dimensions of simple rectangular wings, we find that their oscillation speed (related to their resonance frequency), and the wing’s transverse area, have counteracting effects in the force produced. Because small wings are excited with high frequencies, they oscillate at higher speed; larger wings, on the other hand, are slower, but move more fluid (air) with every oscillation. This leads to a non-monotonic force magnitude with respect to wings’ length or area, as observed in previous findings.^37,48,49 The maximum value of force is obtained for wings of intermediate area. We also find that the force produced is correlated to the wings’ oscillation amplitude, regardless of their specific geometrical and mechanical characteristics (within the range of parameters tested).

For rectangular wings, the force is perpendicular to the clamped side (⁠ $z$-direction): their symmetry results in zero effective force in the horizontal direction (⁠ $xy$-plane). Thus, when a rectangular wing is attached to a freely moving base, it translates following a linear path. To achieve rotational motion of the macroparticle, we use asymmetrical wings, in which the lateral free sides have different lengths. The side oscillating with a larger motion amplitude displaces more air, thus creating a higher force in the opposite direction, which leads to rotation. This rotation can be kept at constant acceleration by maintaining the acoustic field.

By placing wings of different geometries on the same macroparticle, we showed that bidirectional rotation can be achieved: wings of different natural frequency allow to selectively excite each rotation direction.

Active macroparticles, propelled without direct contact and with controllable paths through adjustment of the acoustic field frequency, hold significant potential for applications requiring remote control. The collective motion of a large number of these particles, under heterogeneous acoustic fields with region-specific frequency variations, can provide insights into segregation behaviors in mixtures of macroparticles with different natural frequencies. The airflow generated by one particle when excited, and its impact on neighboring particles, might provide a model system to reproduce exclusion zones observed in system more complex and less amendable to empirical testing. Examples of such systems include bacterial colonies,^62,63 charged particles in tribocharged granular media^64–66 or complex plasmas,^67,68 and colloidal suspensions.⁶⁹ 

Future research will explore the use of these particles for precise 3D trajectory control. In robotics, winged micro-robots, excited using acoustic fields and/or piezoelectric oscillating cantilevers,⁷⁰ could be used as frequency-guided navigation systems, including in higher-viscosity media like water.

The systems proposed offer a number of interesting avenues for further studies, which we hope will stimulate interest in acoustically propelled winged macroparticles.

1.A simple model for the resulting force F z is proposed. The model results in a value of F z = 0.1868 mN, consistent with the experimental result of 0.1877 mN (standard deviation 0.05 mN).2.Tables S.I and S.II contain the parameters used for fitting the data in Figs. 7(b) and 8, respectively.3.Some of the videos used to extract the data presented are provided: ∙Video 1 shows an excerpt from the rotation of the winged macroparticle over 1 h, studied in Fig. 13. The full original recording is publicly available with the rest of the data (Figshare repository 25265860). ∙Video 2 shows the bidirectional rotation of particle illustrated in Fig. 14.

We thank Thorsten Pöschel and Ernesto Altshuler for their support, Thomas Voigtmann for insightful comments, and Walter Pucheanu for assistance with the experimental setups.

Funding for this work was provided by the Bayerischen Hochschulzentrums für Lateinamerika (BAYLAT) through the Scholarship for Latinoamerican Students and the Institute of Multiscale Simulation of Particulate Systems (MSS) through the MSS Scholarship 2023.

The authors have no conflicts to disclose.

A.E. conceived the initial idea, fabricated the winged particles, performed the experiments, and analyzed the data. A.E. and A.S. manufactured the experimental setups. A.E. and O.D’A. wrote the manuscript. O.D’A. provided overall guidance. All authors contributed to the experiments’ design and scientific discussion.

Adriana Enriquez: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Achim Sack: Conceptualization (supporting); Investigation (supporting); Supervision (supporting). Olfa D’Angelo: Conceptualization (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.