Size and shape dependent rotation characteristics of thin film ultrasonic rotors

The controlled rotation of acoustically levitated samples is beneﬁcial for analyzing sample properties, e.g., in a recently reported room temperature x-ray diffraction experiment, wherein thin ﬁlm sample holders comprising thin ﬁlm disks with short blades attached around their circumference were utilized. However, the mechanism of producing the torque and the determinant factor of the rotation direction for these planar ultrasonic rotors have been elusive. We, therefore, study the impact of the size and shape on the rotation characteristics of these ultrasonic rotors in air and further study the inﬂuence of the viscosity of ﬂuid. Theory and experiment demonstrate the essential role of the short blades in producing the acoustic torque both in air and water. In the airborne case, the shape and arrangement of the blades are found to determine the rotation direction. In water, with a dynamic viscosity 55 times higher than that of air, we demonstrate that ultrasonic rotors down to 25-l m-disk-diameter function in an optimized experimental geometry with approximately the same actuation efﬁciency as in air. Our results will be beneﬁcial to further improve the applicability of the ultrasonic rotors as sample holders for

When a sample is exposed to an acoustic radiation or levitated in an acoustic standing wave, torque is produced by the coupling of the acoustic radiation pressure with the sample shape. This has been also utilized to measure the acoustic pressure by the so-called Rayleigh disk, wherein the torque around the axis in the disk plane and perpendicular to the acoustic radiation gives the acoustic pressure. 1,2 In acoustic levitation experiments, the acoustic torque may cause not only random rotation but also positional instability and ejection from the levitator. To circumvent this obstacle, samples with symmetric shapes are normally adopted. The sample rotation can be also suppressed by modulating the acoustic field [3][4][5][6][7][8][9][10][11] but at the cost of increasing complexity of drivers and transducers, limiting their use for special purposes.
The sample rotation is favorable for applications such as the x-ray 12,13 or neutron diffraction experiments 14 and active mixing in microfluidic channels. 15,16 However, in comparison to the Rayleigh disk, the acoustic torque exerted on a thin disk around the rotation axis perpendicular to the disk plane has been hitherto less studied.
Recently, polymer thin film disks with short blades on their circumference have been reported as holders for protein crystal samples for conducting x-ray diffraction experiments. 13 The controllable rotation and the high positional stability of these ultrasonic rotors were the key to the dataset collection without using goniometers. However, the mechanisms to generate the torque and to determine the rotation direction have been elusive. Since the acoustic torque violates the axi-symmetry of the thin-films around the axis of the acoustic levitator, the residual and chiral asymmetry of the film 17 and the resultant coupling of the acoustic radiation pressure to the blades were considered as the possible origin. Given the absence of the circulation of the acoustic field in our setup, 13 viscous torque [18][19][20][21] or acoustic streaming were not thought to be responsible.
In this work, we, therefore, experimentally study the rotation characteristics of these polymer thin-film ultrasonic rotors as functions of the rotor size and blade shape. The results show that the short blades convert the acoustic pressure gradient exerted on their periphery to the torque. Another aim of the present study is to explore the miniaturization of ultrasonic rotors and their application in highly viscous fluid. For this, we fabricated ultrasonic rotors down to 25 lmdiameter and measured their rotation characteristics in water. We found that in an in-plane standing wave geometry, the samples can be rotated but via a different acoustic torque mechanism. Our findings will contribute to widening the scope of the applications of acoustic levitation as well as ultrasonic rotational manipulation for physical, chemical, and biochemical analyses. 16,[22][23][24] The ultrasonic rotors were fabricated by UV photolithography using a direct laser writer (DWL66þ, Heidelberg Instruments). The fabrication procedure is as follows. On a Si-wafer, we first spin-coat a 1.3lm-thick sacrificial layer (LOR-10B, MicroChem Corp.) and a negative resist layer, mr-DWL 40 or mr-DWL 5 (Micro Resist Technology, GmbH). Next, the ultrasonic rotor patterns were exposed by scanning a UV laser. Finally, the development in mr-Dev 600 (Micro Resist Technology, GmbH) follows. To harvest the samples, we dissolve the sacrificial layer in MF-26 (MicroChem Corp.) and wash the samples in de-ionized water. The thickness and the type of the negative resist are chosen depending on the sizes of rotors. R1-R7 were fabricated using mr-DWL 40 to achieve a nominal thickness of 22 lm. R8-R10 with a nominal thickness of 5 lm were made from mr-DWL 5. We remove residues of the sacrificial layer by gentle sonication in MF-26. Table I summarizes parameters of all the fabricated ultrasonic rotors.
In Figs. 1(a)-1(c), we show the snapshots of the acoustically levitated rotors R1-R3, respectively, with the disk diameter of 4, 0.4, and 0.2 mm, with proportionally scaled blades [see Figs. S1(a) and S1(b) of the supplementary material for their optical microscope pictures]. The dimension of R1 is the same as what was previously studied in Ref. 13. Their edges are sharp, and the residual surface roughness is at most $0.1 lm in height with a footprint of a few micrometers, negligible with regard to the acoustic wavelength of the levitator (see the following paragraph). R4-R6 share the same disk diameter of 4 mm, but with different blade shapes and arrangement as shown in Figs. 2(b)-2(d) (also in Fig.  S2 of the supplementary material). R7-R10, which have the same shape as R1 but smaller than 100 lm, were studied only in water.
For the airborne acoustic rotation experiment, we used a singleaxis levitator with the ultrasound frequency of 40.6 kHz [see Fig. 1(e), the acoustic wavelength k ¼ 8.45 mm in air]. The acoustic cavity of the levitator comprises a horn with the diameter of 20 mm attached to the bolt-clamped Langevin-type transducer and a concave spherical mirror reflector with a 20-mm focal distance. We adjusted the cavity to the fifth resonance, with the horn-to-mirror separation being approximately 5k=2. The ultrasound pressure was applied continuously while maintaining the transducer resonance by frequency feedback. We monitored the levitation and the rotation of the ultrasonic rotors by a high-speed camera (Photron mini AX100) with a zoom lens overlooking the samples with a viewing angle h ¼ 15 . The average acoustic pressure in the acoustic cavity was monitored by a sensor fixed at the mirror reflector (see supplementary material). The measured acoustic pressure was normalized by the levitation threshold pressure (corresponding to 1.35 kPa rms) for small (<0.5 mm) water droplets measured with the same levitator setup. 25 For the experiments in water, we immersed the ultrasonic rotors in a water bath fabricated on a standard glass slide and observed those by a microscope from beneath the glass slide. The magnified images were recorded by a CCD camera with the frame rate in the range of 10-130 Hz. Ultrasound at the frequencies of 2.3, 4.7, 8.5, and 13 MHz  . UV and white LED were used to illuminate fluorescent particles and ultrasonic rotors, respectively. We also evaluate the acoustic pressure P ac and the acoustic energy density E ac ¼ P 2 ac =ð4qc 2 Þ (q and c are the density and the speed of sound of the fluid) from the pulsed displacement of tracer particles upon applying pulsed ultrasound and its comparison with standard theory 28,29 as described in the supplementary material.
In Fig. 1(d), we show the relationships between the rotation speed f s in rotation per second (rps) and the normalized acoustic pressureP of R1-R3. The levitation threshold pressureP th for each sample increases with the decrease in the rotor size. BeyondP th , f s increased rapidly for all rotors with the increase inP as expected from the quadratic dependence of the acoustic radiation pressure on P ac . The small P th of R1 equal to 0.19 is attributed to the large disturbance of the acoustic field and the five-time enhancement of P ac on its surface. 13 In turn,P th of 1.05 for R3 shows that this P ac enhancement is negligible for R3 since the disk-diameter-to-k ratio (2%) is small. However, whenP is scaled byP th , their rotation characteristics were qualitatively similar, with the incidental rotational direction despite their apparently symmetric shapes. We note that f s of R3 reaches 1041 rps atP of 1.69 [see Fig. S1(c) of the supplementary material]. Such a high rotation of airborne levitated sample was recently reported using a phased-array levitator producing a rotating acoustic field. 11 Next, we study the influence of the blades on the rotation characteristics by comparing R1, R4, R5, and R6 [see the sketches in Figs. 2(a)-2(d) and snapshots in Fig. S3 of the supplementary material]. In Fig. 2(e), we compare the rotation characteristic of the blade-less rotor R4 with that of R1. We find that theP th value of R4 is approximately same as that of R1, implying the similar acoustic pressure distribution on the surface of these ultrasonic rotors. However, the observation that R4 does not rotate atP below $0.4 shows that the blades are the key to producing the acoustic torque. Furthermore, we found that the blade shape can determine the rotation direction; R5 rotates from the rounded to the straight edge as marked by the counter-clockwise (CCW) arrow in Fig. 2(c) (see Video 1 of the supplementary material]. The rotation direction of R5 is flipped to clockwise (CW) when it is flipped (see Video 2 of the supplementary material] with approximately the same rotation characteristics as shown in Fig. S3(a) of the supplementary material. Interestingly, the rotation characteristics of R5 is approximately the same as that of R1, as shown in Fig. 2(f). When the two of the rounded-edge blades were flipped as in R6, the rotation direction became again incidental as R1, even reverted upon switching the pressure (when the pressure was below approximately 0.6). In the rangeP < 0:23, R6 did not rotate (see Video 3 of the supplementary material). We note that the ultrasonic rotors studied in the present work can be rotated at f s in the range of 0-1 rps fromP near the levitation threshold. This is in contrast to the thin films reported previously. 13 As shown in Fig. S4 of the supplementary material, to initiate the rotation for the rotors with higher roughness, we need to applyP much higher thanP th and in between only oscillation but no rotation is observed.
For comparison, we calculated the acoustic pressure distribution around R5 by solving the transient Navier-Stokes equations numerically, the computational fluid dynamics (CFD) code ANSYS Fluent 2021R2 being used for this. The output of the ultrasound transducer was assumed to be 59.4 Pa rms, corresponding toP of 0.35 in our acoustic levitator. For simplicity, the ultrasonic rotor was assumed to be a rigid body and fixed to the space at the center of the middle node. Further details of the simulation are described in the supplementary material. Figure 3(a) shows the iso-surfaces of the time averaged

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scitation.org/journal/apl pressure, in which blue and yellow surfaces display 1 Pa below and above the atmospheric pressure, respectively. The top-view of the instantaneous surface pressure distribution relative to the ambient pressure is shown in Fig. 3(b) together with the stream lines around the blade. The timing was when the pressure integrated over the disk surface became maximum. We noticed a higher pressure on the rounded side of the blade than the straight side at the same lateral coordinate, as indicated by the dashed circles in Fig. 3(b). The pressure on the side of the blade is similar to that on the top surface, which results in the acoustic torque in CCW. The higher pressure on the rounded side (the lower dashed circle) is considered to be caused by the gathering of the flow in the downstream, which is depicted by the blue arrows in Fig. 3(b). From the CFD simulation, we calculated the time-averaged acoustic torque T ðCFDÞ exerted on R5 scaled forP ¼ 0:25 and found a value equal to 2:9 Â 10 À10 N m. For comparison, we evaluate the acoustic torque T ðexpÞ under the experimental conditions by noting that at the steady state, T ðexpÞ is equal to 2 pf s C, where C is the rotational friction. To calculate C, we consider two approximations for our ultrasonic rotor by a thin disk 30 and by a crossed-cylinder 31,32 given the key role of the short blades in producing the acoustic torque. We found that the former results in T ðexpÞ an order of magnitude smaller than T ðCFDÞ . In the latter case, we found that T ðexpÞ was equal to (1.34 -4.02)Â10 -10 N m for f s of 1-3 rps, in reasonable agreement with T ðCFDÞ .
From the close examination of the CFD result, we found that 76% of T ðCFDÞ is given by the acoustic radiation pressure and the remaining 24% by the viscous shear stress. 33 However, the contribution of the viscous torque [18][19][20][21] is negligible in our experiment due to the absence of the circulation of the air as shown by the flow visualization experiment reported previously. 13 We now discuss the behavior of the ultrasonic rotors in water. As depicted in Figs. 4(a) and 4(b), an in-plane standing wave is created by placing the emitting face of the transducer close to the bottom of the water bath (glass slide) We found that when the transducer-glass distance was 1.9 mm and the ultrasonic rotor size is approximately equal to k=2 (the pitch of the standing wave nodes) as shown in Fig. 4(b), the acoustic torque was enhanced. We were able to rotate R9 and R10 with the diameters of 50 and 25 lm, respectively, by aligning those in the same way (see Videos 4 and 5 of the supplementary material). Since the acoustic radiation force varies sinusoidally in the radial direction, the blades on the opposite side can pick up the acoustic torque in the same direction. Each blade is exerted by an azimuthal force, which is produced by the difference of the acoustic radiation pressure between the two sides of a blade (see the supplementary material for the detection of the sinusoidal variation of the acoustic radiation force and the evaluation of P ac by the pulsed motion and the Brownian motion of tracer particles). We found that, although the rotation direction depends on the incidental initial azimuthal angle of the blades relative to the node lines, full turns of rotation were observed even with the rotors down to the disk diameter of 25 lm (R10) with its size approaching that of biological cells. We were also able to rotate the R2 and R3 which have been studied in air (Fig. 1) at the lowest frequency of 2.3 MHz.
We note that in the airborne experiment, the E ac required to rotate the ultrasonic rotors (R1 with the disk diameter of 4 mm in 40 kHz ultrasound) at f s of 1 rps was equal to 8.5 J/m 3 . In comparison, we found that for rotating R8 at the same f s by 8.5 MHz ultrasound, E ac is equal to 5.3 6 0.5 J/m 3 . Therefore, the rotation efficiency as measured by the ratio of f s to E ac is of the same order of magnitude for the two cases despite the difference in the ultrasound frequency by a factor of 212 and the difference in the viscosity by a factor of 55.
In summary, we elucidate the mechanism to generate the acoustic torque exerted on our airborne ultrasonic rotors by experiment and numerical simulation. This is a step forward to further optimize those as sample holders for protein crystallography applications. In addition, we demonstrate miniaturized ultrasonic rotors down to 25 lm-disk- diameter in highly viscous fluid (water). The acoustic energy density required to rotate the ultrasonic rotors at a same rotation speed is found to be approximately the same in water and in air. Different from imposing the angular momentum of the acoustic field via friction by using vortex beams [34][35][36] or viscous torque, 11,24 our result introduces a unique way of rotational manipulation of planar samples by the acoustic radiation pressure without resorting to three-dimensional fabrication. 37,38 The experimentally observed relationship between the rotor size and the ultrasound wavelength suggests a possibility to realize ultrasonic rotors that are a few micrometers or even smaller in diameter. The generation of sub-micrometer wavelength ultrasound is also feasible by surface acoustic wave transducers. [39][40][41] Nevertheless, the acoustic torque mechanism of the in-water ultrasonic rotors is yet to be fully elucidated; from the coupling mechanism indicated from the experiment, the direction of the acoustic torque is reverted at every quarter turn. As such, the fact that full turns were observed for these rotors suggests that the rotational inertia plays an important role, which may substantially diminish for the rotors with a few micrometer diameters. The experimental exploration of such few micrometer ultrasonic rotors is under way.
See the supplementary material for the description of the acoustic mirror of the levitator, the high speed rotation of the ultrasonic rotors R2 and R3; the snapshots of the ultrasonic rotors R1, R4, R5, and R6; the influence of flipping on the rotational characteristics of the ultrasonic rotor R5; the CFD simulation of acoustically levitated ultrasonic rotors in air; the measurement of the acoustic pressure in the ultrasonic rotor experiment in water; and video recordings of the rotation of the ultrasonic rotors, R5, flipped R5, and R6 in air, as well as R9 and R10 in water (supplementary material Videos 1-5).
The authors thank Roderick Y. H. Lim for thoughtful comments regarding the applications in nanobiology and G. V. Shivashankar for stimulating discussions on the applications in mechanobiology. S.J. was supported by the Swiss Nanoscience Institute Ph.D. School (Project No. P2007). This work was partially supported by the Swiss National Science Foundation No. 200021_192772.

AUTHOR DECLARATIONS Conflict of Interest
The authors have no conflicts to disclose.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.