Acoustically induced breakup of levitated droplets

Acoustic levitation attracts great attention of practical applications in analytical chemistry,^1–3 biology,^4–6 and materials science^7–12 because a container-free sample manipulation can avoid the wall effect, such as an unexpected nucleation and contamination by the container wall. Acoustic levitation, which is one of the promising contactless fluid manipulation techniques, enables a sample (solid and liquid) to levitate in midair by forming an acoustic standing wave between the horn and the reflector.^13–15 The dynamic and nonlinear behaviors on a levitated droplet, such as interfacial deformation and atomization,^16–24 can be triggered due to the nonlinear acoustic field,^25–28 flow fields,^29–33 and heat/mass transfer.^34–38 Lee et al.²⁰ studied the static and equilibrium shape flattened by acoustic radiation pressure by modeling the sound field and the droplet shape to determine levitation stability. Danilov and Mironov²¹ theoretically investigated the droplet atomization mechanism from the interfacial deformation to disintegration of the droplet in a strong acoustic field. Marzo et al.³⁹ demonstrated that a solid sample can be three-dimensionally manipulated with multiple small transducers without a reflector. Hasegawa and co-workers^40,41 demonstrated the contactless coalescence and mixing of droplets with ultrasonic phased arrays. Although recent studies developed and improved the acoustic manipulation system, levitation stability for better acoustic manipulation of a sample is only partially understood. The droplets must stably levitate without any interfacial instability or atomization after levitation and coalescence in midair.⁴² 

The purpose of our research is to experimentally investigate the deformation and atomization process to understand the droplet dynamics and improve stability of a levitated droplet through its instability mechanism. We show how surface tension affects the atomization behavior of levitated droplets and compare the results to existing theories. Our experimental findings provide deeper insights into contactless liquid manipulation for prospective lab-in-a-drop applications.

In this study, a single acoustic levitator generated the acoustic field to levitate the droplet. Figure 1(a) shows a schematic of the experimental setup. A function generator (Agilent Technologies Japan, 33511B) produced the sine-wave signal input into an ultrasonic transducer (NGK SPARK PLUG CO., D4520PC) via a power amplifier (NF CORPORATION, 4502) with a power meter (Yokogawa Test and Measurement Corporation, WT310-D-C1). The sound wave was emitted from the bottom horn connected with the transducer and reflected on the top reflector. An acoustic standing wave can be formed between the horn and the reflector through precise tuning of the horn–reflector distance. The test liquid was manually injected with a syringe near the third pressure node (∼22 mm from the bottom horn) of the acoustic standing wave, as shown in Fig. 1(b). The pushed liquid from the syringe was released within a few seconds, and the droplet was formed. The motion of the levitated droplet was observed with a high-speed video camera (Photron, FASTCAM Mini AX50) that featured backlight illumination. The behavior of the droplet was computer analyzed with an in-house MATLAB code. We used two types of refractors, R∞ and R36. R indicates the curvature of the reflector. For more stable levitation, R36 was mainly selected for our experiments; however, the R∞ reflector attached with a thin glass surface was used to observe the droplets from the top through the refractor for minimizing the effect of refraction by the reflector surface. For droplet atomization, the sound pressure in the test section with stable levitation of a droplet was suddenly increased in order to lead to droplet disintegration. The droplet can be atomized within a sufficiently short time [∼O(10⁻³) s], as compared to the characteristic time of the evaporation of a droplet [d²/D ∼ O(10³) s, D (∼10⁻⁹ m²/s) is the diffusion coefficient of droplet],³⁶ so that it is not necessary to take into account the heat and mass transfer of the droplet with the surrounding air.

Table I lists the experimental conditions used in the present study. The function generator produced a frequency of ∼19.3 kHz at room temperature conditions of about 25 °C and a relative humidity of 50%. The sound pressure measured with the probe microphone (Bryel and Kjaer, Type 4182) was 1.1–3.3 kPa in the test section. Water, ethanol, and dilatant fluid (potato starch–water mixture of 2 to 1) were used to prepare the test sample to determine the effect of surface tension. The volume equivalent diameter of a droplet ranged from 0.5 mm to 3.1 mm. The aspect ratio (AR) is defined as the equatorial-to-polar ratio of the major diameter b to the minor diameter a.

The uncertainty in the droplet diameter was <4%. When d = 0.5 mm, the smallest diameter in this study, the standard deviation of the average from three measurements was less than 1 pixel, with a spatial resolution of ∼20 µm/pixel. The sound pressure was measured thrice, and the error was at most <5%.

Stable droplet levitation exists when the acoustic radiation force exceeds the gravitational force on the droplet (≧ρVg) and is lower than the surface force to maintain the droplet interface. Here, ρ, V, and g are the density and volume of the droplet, and gravitational acceleration, respectively. Precise tuning of droplet size and sound pressure is necessary to ensure stable levitation. Figure 2 shows the levitation map of (a) water and (b) ethanol droplets. The solid line representing the upper limit of sound pressure is described as⁴³ 

where σ is the surface tension of the droplet, ρ is the density, the subscript G represents the gas phase, c is the speed of sound, d is the diameter of the droplet, and λ is the wavelength of the sound wave. We confirmed the upper limit of stable levitation at 3.14 mm and 1.55 mm for the water and ethanol droplets, respectively. Above the upper limit condition, we confirmed that the droplets were atomized by the increase of the sound pressure. Both the stable and unstable (atomization) conditions of the levitated droplet were well-predicted by Eq. (1). According to the assumption of Eq. (1),⁴³ droplet atomization can be pushed into interfacial instability of the droplet surface with a higher sound pressure.

It was confirmed that the droplets could be stably levitated or become unstable and atomized in an acoustic field. In order to stably levitate and keep the droplet interface in acoustic levitation, it is necessary to first understand the instability process of the droplet and its mechanism. Figure 3 shows the typical atomization process of levitated droplets after the increase in the sound pressure exerted on the droplet.

The droplet can be levitated with almost a spherical shape [Fig. 3(a)]. By increasing the input sound pressure in the test section, the droplet interface transformed into a flattened shape [Fig. 3(b)]. The boundary of the droplet fluctuated, and the interface became unstable. Eventually, the droplet interface could no longer be maintained. It atomized from the edge of the droplet [Fig. 3(c)]. The atomized small daughter droplets aggregated due to the restoring force in the acoustic field in a planetary formation [Fig. 3(d)]. After aggregation, the smaller droplet stably levitated [Fig. 3(a′)].

Figure 4 shows the general observation of the atomization process of (a) water and (b) ethanol with time t. The droplet became stable after a maximum of 30 seconds from the droplet injection [the initial shape in Fig. 4(a)]. The sound pressure was increased by ∼30 ms before droplet atomization. Then, each droplet deformed from its initial shape and spread horizontally, and AR increased (t ≤ −3 ms). The droplet interface drastically deformed (t = −1 ms) and eventually atomized (t = 0 ms) thereafter. The daughter droplets of ethanol were finer than those of water. This was caused by the lower surface tension of the ethanol droplet than the water droplet. This confirmed the different atomization behavior due to different fluid properties. Droplets can keep their interface if the fluid property changes with strong sound pressure exerted at the droplet interface. Figure 4(c) shows a dilatant droplet. The viscosity can be increased nonlinearly, depending on the shear stress on the droplet surface. Even after the sudden increase in the sound pressure, as shown in Figs. 4(a) and 4(b), the dilatant droplet did not atomize but behaved like a solid. The aspect ratio was almost constant at 1.6. This sudden increase in shear stress at the droplet interface through the increase in the sound pressure can cause an increase in the viscosity of the dilatant droplet, preventing atomization.

Figure 5 shows the temporal evolution of the aspect ratio of water and ethanol droplets, based on Fig. 4. The initiation of atomization occurs at 0 ms. As the sound pressure increased, the aspect ratio of both droplets linearly increased (t ≤ −3 ms). Immediately before atomization (−3 ≤ t ≤ 0 ms), the aspect ratio increased nonlinearly. In the present study, the sound pressure was not gradually but suddenly increased due to the specification of the function generator. The effect of how to increase the sound pressure remains a challenge in our future work.

Figure 6 presents the time evolution of the major axis b (mm), interfacial velocity u (m/s), and acceleration Δu/Δt (m/s²) during the last 2 ms shown in Fig. 4. As the levitated droplets spread horizontally, the edge of the droplet interface was protruded from the equatorial plane, and the droplet atomized from the rapidly developed liquid film on its edge (−1 ms ≤ t ≤ 0 ms).

Figure 7 shows the time evolution of the estimated pressure difference in the droplet. The pressure difference can be roughly obtained by the following equation:

where ΔP_d is the pressure difference between the inside and outside of the droplet, and Δb is the displacement of the major axis of the droplet obtained from Fig. 6. The pressure difference between the inside and the outside of the droplet increased immediately before atomization (−1 ms ≤ t ≤ 0 ms). That pressure difference between the inside and outside of the droplets increased in both water and ethanol droplets. The liquid film was present at the droplet interface. From the measured sound pressure distribution in the test section, the sound pressure near the possible droplet levitation area was 0.4 kPa ≤ P ≤ 1.0 kPa. The estimated pressure difference obtained by Eq. (2) was in good agreement with our sound pressure measurement.⁴⁴ 

Droplets were atomized at a higher sound pressure (−1 ms ≤ t ≤ 0 ms) beyond the upper critical pressure difference to keep the droplet interface (the blue region in Fig. 7). It was technically difficult to directly measure the pressure field in the droplet under the unsteady atomization process. Contactless sound pressure measurement is a challenge left for future research.

Interfacial instability at the droplet surface can play an important role in atomization in acoustic levitation. In order to demonstrate the atomization mechanism of an acoustically levitated droplet, a clear observation of the atomization process on the droplet is required with high spatiotemporal resolution. Figure 8 shows the observation of the water droplet atomization process from the top of the droplet. The sound pressure was increased from −16 25 ms. Then, the droplet deformed and spread radially (t = −0.28 ms). After that, the capillary wave was initiated from the edge of the droplet (t = −0.14 ms), and the droplet started atomizing (t = 0 ms). The capillary wave propagated from the edge to the center of the droplet. Eventually, atomized daughter droplets were dispersed from capillary waves generated radially, and droplets were fully atomized (t = 5.00 ms).

The capillary wave on the droplet surface is of importance for the atomization mechanism. Figure 9 compares the measured capillary wavelength with the theoretical one. The black broken line represents the theoretical capillary wavelength, obtained by the following equation:⁴⁵ 

where λ is the capillary wavelength, and f is the input frequency of the sound wave. The theoretical capillary wavelength is ∼0.17 mm. The measured capillary wavelength varied from 0.11 mm to 0.22 mm over time. Our measurement result was in good agreement with the theory. Note that the capillary wave that depends on the input frequency of the sound wave was generated on the acoustically levitated droplet in the atomization process. The capillary wave affected the atomized daughter droplet size.

Figure 10 exhibits the size distribution of the atomized daughter droplets for (a) water and (b) ethanol. The blue broken line represents the theoretical diameter, described by the following equation:⁴⁵ 

where d_th is the theoretical median diameter of the droplet.

Table II summarizes the median diameter d_m of water and ethanol compared with the theoretical values calculated by Eq. (4). The theoretical value (d_th) and the experimental value (d_m) were similar. The droplet size larger than the theoretical one was observed in both the water droplet and the ethanol droplet due to the coalescence and agglomeration of droplets after atomization. Equation (4) assumed an atomization rate of ∼0.01 cm³/s so that higher atomization rates produce larger droplets than the theoretical prediction.⁴⁵ The deviation in the experimental data from the theoretical value can be attributed to the atomization rate. The median diameter of levitated droplets that are atomized acoustically can be estimated with Eq. (4). Since atomization occurs in a truly short time (within ∼5 ms), a higher spatiotemporal resolution is required to capture the fast dynamics of droplet atomization in our future work.

We investigated the breakup process of droplets using acoustic levitation and compared our experimentally obtained results with theoretical results. The unstable region of levitated droplets can be predicted by the theoretical upper limit sound pressure. Droplets were atomized in an unstable condition with a higher sound pressure. The atomization process of the droplets was affected by the difference in fluid properties. Immediately before droplet atomization, AR increased nonlinearly. The pressure difference between the inside and outside of the droplet was estimated. This estimated pressure difference was in good agreement with our sound pressure measurement. Droplets were atomized at a higher sound pressure beyond the critical pressure difference (0.4 kPa ≤ P ≤ 1.0 kPa) to maintain the droplet interface. We demonstrated that the capillary wave that depends on the input frequency of the sound wave was generated in the atomization process of the acoustically levitated droplet. The capillary wave triggered the atomization and determined the size distribution of the atomized daughter droplets. Our findings describe the atomization process and mechanism in acoustic levitation. However, direct pressure measurement of the pressure field near the droplet interface remains a challenge for future studies. Our demonstration may stimulate further research and offer a description of droplet dynamics during acoustic levitation for developing practical applications.

This work was financially supported by the Research Institute for Science and Technology, Kogakuin University, and a research grant from the Mazda Foundation. We would like to thank Editage (www.editage.com) for English language editing.