Ultrasonic Atomization—From Onset of Protruding Free Surface to Emanating Beads Fountain—Leading to Mist Spreading

Highlights

What are the main finding?

• As a prerequisite to effective ultrasonic atomization, the process of liquid fountaining is detailed.
• Its whole process spans a series of structural variations of free-surface morphology in four phases.
• Size specificity and periodicity exhibited by undulating and beading fountains are model- predicted.

What is the implication of the main finding?

• Critical values of ultrasound excitation frequency are evaluated/proposed for possible bifurcation.

Abstract

The process of ultrasonic atomization involves a series of dynamic/topological deformations of free surface, though not always, of a bulk liquid (initially) below the air. This study focuses on such dynamic interfacial alterations realized by changing some acousto-related operating conditions, including ultrasound excitation frequency, acoustic strength or input power density, and the presence/absence of a “stabilizing” nozzle. High-speed, high-resolution imaging made it possible to qualitatively identify four representative transitions/demarcations: (1) the onset of a protrusion on otherwise flat free surface; (2) the appearance of undulation along the growing protuberance; (3) the triggering of emanating beads fountain out of this foundation-like region; and (4) the induction of droplets bursting and/or mist spreading. Quantitatively examined were the two-parameters specifications—on the degrees as well as induction—of the periodicity in the protrusion-surface and beads-fountain oscillations, detected over wider ranges of driving/excitation frequency (0.43–3.0 MHz) and input power density (0.5–10 W/${\mathrm{c}\mathrm{m}}^2$) applied to the ultrasound transducer of flat surface on which the nozzle was either mounted or not. The resulting time sequence of images processed for the extended operating ranges, regarding the fountain structure pertaining, in particular, to recurring beads, confirms the wave-associated nature, i.e., their size “scalability” to the ultrasound wavelength, predictable from the traveling wave relationship. The thresholds in acoustic conditions for each of the four transition states of the fountain structure have been identified—notably, the onset of plausible “bifurcation” in the chain-beads’ diameter below a critical excitation frequency.

1. Introduction

Ultrasonic atomization (UsA), with the fundamental and paramount task of generating a swarm of small droplets (mist) in uniform distribution, requires the prior formation of a fountain or jet of liquid from an initially induced protrusion of the free surface [1,2]. The most extensively studied in the literature, thus, involve the droplet size distribution (DSD) of emerging mist [3,4,5], followed by the extent of separation/concentration of a solute (or in some cases “targeted” suspension) into mist, often along with its mechanisms described [6,7,8]. While the dynamic structure of the (general) liquid fountain could be examined visually [1,9], more systematic visual elucidation has been conducted on the beads-regulated fountain in association with internal cavity and/or (external) droplets bursting [2,10,11,12,13]. Some efforts have been directed towards quantifying the periodic nature, specifically dominant frequencies, exclusive to fountain-beads —modeled using a series of isolated drops—oscillations [2,14,15,16].

Among those aspects listed above, the UsA’s structural aspect is of the primary concern here; more specifically, it is the present study’s intent to focus on four representative transitions/demarcations in the acoustic fountain structure and dynamics: (1) the onset of a protrusion on an otherwise flat free surface; (2) the appearance of undulation along the growing protuberance; (3) the triggering of an emanating beads fountain out of this foundation-like region; and (4) the induction of droplets bursting and/or mist spreading. Prior to providing the relevant literature findings specific to each transitional “state/phase”, the UsA mechanisms—i.e., how the mist generation would occur in association with selective separation into the mist—are reviewed first.

Mechanistic descriptions of UsA: A couple of mechanisms have been proposed in the literature, in particular, for selective separation of solutes including surfactants or suspended particles from a solution: the capillary-wave hypothesis supported by, e.g., Qi et al. (2008), Collins et al. (2012), and Blamey et al. (2013) [17,18,19]; and the cavitation hypothesis supported by, e.g., Neppiras and Noltingk (1951), Kojima et al. (2010), Ramisetty et al. (2013), and Inui et al. (2021) [20,21,22,23]. Besides these, Antonevich (1959) [24] claimed for the first time that UsA would be an outcome of a combination of cavitation-bubble collapsing and capillary-wave instability, with the size of emitted droplets determined by their release mechanism; since then, this conjunction hypothesis has been supported by Boguslavskii and Eknadiosyants (1969), Rozenberg (1973), Barreras et al. (2002), Kirpalani and Toll (2002), Simon et al. (2012, 2015), Tomita (2014), and Zhang et al. (2020) [11,12,13,25,26,27,28,29].

Some visual evidence supports the capillary-wave hypothesis, suggesting that μm-size droplets are pinched off from the wave crests during surface oscillations prevailing along a perturbed protuberance or a conical fountain [1,27]. There is little direct (visual) evidence, however, that mist generation is related to the occurrence of cavitating bubbles within the fountain [10,11,13]. In some limited circumstances, the extent of cavitation yield is represented by that of sono[chemical] luminescence [30,31,32]. As demonstrated by Kojima et al. (2010) [21], sonochemical luminescence could be observed not only in the bulk liquid near the transducer and liquid surface but also in the fountain when UsA occurs.

The following four subsections with paragraph headers are to be described in terms of “stability of the liquid-surface protrusion” or its dynamic “structural variation” by specifying its controlling factors: radiation pressure (or traveling wave) field, anti-resonant pressure (or standing wave) field, induced flow field, and the factors’ thresholds levels.

Protrusion on free surface: The radiation force resulting from the traveling wave will induce a “mound”—as claimed by Simon et al. (2015) [13] for a focused ultrasound wave —on the free surface. This mound of “parabolic” curvature on its tip would, in turn, provide the interactive field between the “waves incident on and reflected from” the pressure-releasing (i.e., wave-absorbing) interface [13], causing the generation of cavitation bubbles “inside” the protuberance. Alternative insights into the contribution of cavitation to driving the UsA would be gained [30,31,32] for a rather flat free surface—in the absence of any UsA liquid fountain triggered.

With a properly selected set of three different driving frequencies, 0.17, 0.45, and 0.73 MHz, for a given applied power of 20 W (1.1 W/${\mathrm{c}\mathrm{m}}^2$), Lee et al. (2011) [31] proposed the following distinctive—and possibly inherent—shift in mechanistic trends: The lower frequency (in the order of ${10}^{-1}$ MHz) would induce a standing wave field as the wave starts propagating upwards. The intermediate (∼${10}^0/2$-MHz) frequency would result in an appreciable attenuation in the acoustic pressure amplitude, developing a traveling wave field above the transducer vs. the standing wave still prevailing below the free surface. At the higher (∼${10}^0$-MHz) frequency, with this acoustic energy attenuation becoming significant enough to intensify the energy gradient along the propagating acoustic wave, the predominant mechanism would shift from the radiation pressure associated with the traveling wave to acoustic streaming.

Orisaki and Kajishima (2022) [33] conducted a direct numerical simulation of the very beginning phase of water surface rising/protruding under 0.5-MHz ultrasound irradiation. Depending on the rising rate of the water surface, its height-increasing process was divided into three phases/stages: increasing slightly, with acceleration, and at constant speed. Separate explanations of the protruding mechanisms for the individual stages were provided: (1) The acoustic radiation pressure equals the acoustic kinetic energy density, resulting in a slight rise in the water surface. (2) As the region between the sound source and the free surface turns to resonance, the acoustic radiation pressure acting on the surface will increase profoundly, causing the surface to rise rapidly. Through the entrainment of fluid via rapid rising of the water surface, acoustic streaming is induced towards the risen region. (3) Eventually, the water surface will shift to non-resonance, leading to the acoustic radiation pressure decreasing significantly. In this last phase, the velocity of the acoustic streaming towards the rising is reported to become nearly constant.

Stability of growing protuberance: Realizing the shape and evaluating the height of the acoustic fountain through numerical simulation under different input pressure levels, Xu et al. (2016) [34] claimed that, while the ultrasonic field was an anti-resonant pressure (i.e., standing wave) field, the radiation pressure was minimum and the fountain surface tended to be stable. Their findings, however, are limited to just a mound-like precursory structure. Still confining to an essentially protuberant water fountain, Kim et al. (2021) [35] conducted an experimental study, by applying focused ultrasound at two frequencies in a proper range (see above), 0.55 and 1.1 MHz, and using particle image velocimetry (PIV) to map the flow field induced about the “ultrasonic focal spot”. They noted three different regimes—through increasing the pressure level of the transducers—all of which were claimed to have an acoustic radiation force as the predominant driving force.

Aikawa and Kudo (2021) [36] rendered useful insights, in terms of the thresholds for triggering free radical generation and atomization, into a demarcation in the structural variation of the free surface. The former phenomenon is an outcome of the cavitation collapsing process; the latter should require the shape transition of free surface [9] from a protrusion to a “slender” fountain (of aspect ratio exceeding unity; [36]). Upon increasing the transducer input power density, they detected an essentially identical threshold level, 2.5 W/${\mathrm{c}\mathrm{m}}^2$, for all four [cavitation collapsing, free radical(s) generation, free- surface shape transition: from protuberance to prolate fountain, and atomization] phenomena. They further recognized, through focused shadowgraphy, the so-called “spotty-shaped high-intensity nodes”—originating from the pressure-releasing “parabolic mirror” interface [13]. It is important to note that its converging effect would promote cavitation inside the protrusion and that the prevalence of such cavitation could, in turn, trigger the “abruptly shape-transitioning” fountain, which would lead to the atomization.

Emergence of beadsfountain: When the ultrasonic wave has sufficient intensity or acoustic radiation pressure to overcome some threshold limit, a beads fountain—a fountain consisting of a chain (sometimes limited to a doublet or triplet) of beads in contact—will emerge from the foundation-like region, often accompanied by bursting droplets [2,10,11,12,13]. Fujita and Tsuchiya (2013) [10], applying ultrahigh-speed imaging (at 0.25 megaframes/s) with a set of parameters (a flat ultrasound of 2.4 MHz with 2.0-W/${\mathrm{c}\mathrm{m}}^2$ intensity), managed to capture a series of evidential images (reproduced in [2]) of a cavity—either a single (irregular-shaped) void or tightly clustered (tiny cavitating) bubbles in a group—within a fountain bead. This cavity was observed to move laterally towards the gas–liquid interface, leading to droplet bursting.

Tomita (2014) [11], using a focused ultrasound transducer (of 1 MHz), provided visual evidence (at 50 kfps) for cavitation inside the primary bead of a double-beads fountain/jet to recognize the surface elevation and jet breakup, with their threshold conditions specified, and fine droplets sprayed out of the neck—claimed to be induced by the collapse of capillary waves—between the primary and secondary beads. Simon et al. (2015) [13] employed also focused transducers over rather wide ranges of ultrasonic frequencies (2.165 mainly, as well as 1.04 and 0.155 MHz) and sound speeds (1.14–1.90 km/s), but with moderate focal acoustic intensities in the order of several ${10}^2$ but requiring at least 180 W/${\mathrm{c}\mathrm{m}}^2$. They provided similar visual evidence (at 5–30 kfps) within the threshold of acoustic intensity for atomization being liquid-dependent.

Dropletsbursting and/or mist spreading: In addition to the (rather stable) mound- like structure described so far, the present study concerns the acoustic fountains that are not only unstable but are also associated with eventual mist spreading [out of the Finely-Structured Surface Region (FSSR)] as well as liquid ligaments formed due to tip breakup [the Lumped-Crest Region (LCR)] [1,9].

Over a wider input-power range of 5–50 W (≅0.25–2.5 W/${\mathrm{c}\mathrm{m}}^2$), but at a fixed driving frequency of 0.49 MHz, Kojima et al. (2010) [21] conducted measurements, through a laser Doppler velocimetry (LDV) and sonochemical luminescence, on the spatial distributions of liquid flow and acoustic pressure, respectively. They confirmed—with increasing input power beyond 1.5 W/${\mathrm{c}\mathrm{m}}^2$—that the fountain indeed prevailed with appreciable atomization, along with sonochemical luminescence detected within both the bulk liquid and the fountain.

Simon et al. (2015) [13] reported a non-spherical (triangular) deformation of the fountain beads prior to atomization, demonstrating that cavitation bubbles should be a significant driving force for atomization in beads fountains. Wang et al. (2022) [2] observed the surface oscillations of a beads fountain, before atomization set in, from ellipsoid-shaped to diamond-shaped and finally to a hexagonal shape. They suggested, in addition, that a minimum driving frequency for the beads to be formed was roughly identified to be ≈0.8 MHz or higher; however, a systematic evaluation of this lower bound has not been performed.

In the present study, high-speed, high-resolution visualization was utilized to identify the surface dynamics of different states of acoustic fountains—ranging from a mere protruding state to a mist-associated state with a higher aspect ratio—over wider ranges of acoustic parameters (mainly, the excitation frequency and the input power density). The objectives of the study are mainly threefold: to find the relevant acoustic thresholds for the demarcations in the structural variations of free-surface morphology in the fountain; to confirm and extend (to wider acoustic conditions) the size specificity and periodicity exhibited by the beads fountain through both experimental (including fast Fourier transform) and model-based approaches; and to attempt to physically interpret visually observed, additional phenomena such as possible bifurcation.

2. Experimental

Figure 1 provides a schematic of the experimental setup used for visualizing the UsA process. An ultrasonic transducer with a wide range of driving/excitation frequencies [KAIJO QT-011 with frequency shifting/tuning capability: $f_{ex}=$ 0.43, adjustable between 0.60 and 0.95 (with an increment of 0.05), 1.0, 1.6, 2.0, and 3.0 MHz] was placed on the bottom of a square vessel with its dimensions 200 × 200 × 185 (height) mm. For selected runs, on top of the transducer’s oscillating disk (20.0 mm in diameter with an effective diameter of oscillation, 16 mm), a Teflon^® nozzle [2] was installed to help stabilize the liquid column. Its installation angle was fixed at 0° (viz., the ultrasonic irradiation was directed vertical; [9]). The input power applied to the transducer was changed over 1–20 W (its examined density then ranged $I_0=$ 0.5–10 W/${\mathrm{c}\mathrm{m}}^2$).

The other conditions as well as procedure are essentially identical to those in [2]. They are listed here for the sake of completeness: The effective liquid depth (i.e., the vertical distance from the center of the oscillator to the free surface of the liquid) was kept at 25 mm or 32.5 mm above the vessel bottom. The liquid used was an aqueous ethanol solution with an initial concentration of 50 wt% (28 mol%) at 25 °C. The UsA operation used was, at most, 10 s.

High-speed visualization: A digital camera (Photron FASTCAM MINI AX100) with a macro-lens (Nikon Micro-Nikkor 105 mm f/2.8) was used, as in [2], for imaging the dynamics of a liquid protrusion or fountain, as well as their associated phenomena. The image-capturing parameters were as follows: frame rate(s) up to 5000 fps (with the shortest exposure time set as 197 µs) and a resolution of at least 512 × 512 pixels; in most runs, 5 s after the transducer turned on, a 0.2-s period of image data was acquired. A metal halide lamp (Lighterrace MID-25FC, Takatsuki, Japan) provided backlighting, whose non-uniformity was reduced through a light-diffuser sheet (see Figure 1) between the protrusion/fountain and the lamp.

Image processing: Two types of image analyses were conducted: dynamic tracking (including time-series data) of the free-surface protrusion and static outlining (time- averaged, instantaneous shape) of the fountain beads. The former (see Section 3.1 for the resulting outcome) was realized using an image analysis software (DItect Dipp Motion Ver. 224d) and the latter (see Section 3.2) using another software (DItect Dipp Macro Ver. I). Regarding the latter in particular, for details of the image processing and data acquisition procedure, refer to [2]. The original images of the liquid surface (both the protrusion and the column) in the air were binarized; including this binarization, the procedure for obtaining the image-evaluated feature of an ellipse representing each bead in contact along the liquid column, via a video analysis software (Keyence Movie Editor Ver. 1), can be found also in [2].

Time-series data analysis: The same procedure as in [2,9] was used to analyze a time sequence of images of an instantaneous state or phase —more specifically, the protruding height $f\left(t\right)$. Provided that it exhibits fluctuations, we attempted to apply fast Fourier transform (FFT) to the resulting time-series data/signals $f\left(t\right)$ to extract the dominant frequency(ies), if any. As detailed by Wang et al. (2022) [2], a given time- domain signal $f\left(t\right)$ was decomposed into a frequency-domain spectrum $F\left(\omega \right)$, employing a fast, efficient algorithm (MathWorks MATLAB^® 2022b). The acquired signal corresponding to the former sum of discrete elements, $f_{j+1}\left(t\right)$ $(j=0,1,\dots ,N-1)$, sampled at equal intervals, was processed to obtain the latter spectrum:

$$F_{k+1}(\omega )={\sum }_{j=0}^{N-1}{\Psi }^{jk}{f}_{j+1}\left(t\right)$$

(1)

where the term specifying the inner product is given by the $jk$-th power of $\Psi =e^{-2\pi i/N}$ [37].

3. Results and Discussion

The UsA sequence of the initiating free-surface protuberance or “mound”, the resulting liquid fountain, followed by its eventual association with mist emergence is captured in images over wide ranges of experimental conditions. It is, thus, characterized in a stagewise manner and discussed from mechanistic viewpoints in both static/time-averaged and dynamic natures, over selected ranges of driving frequencies as broad as $f_{ex}=$ 0.43–3.0 MHz and input power densities of $I_0=$ 0.5–10 W/${\mathrm{c}\mathrm{m}}^2$ (see Section 2). Figure 2 shows typical images of the four states/phases signifying the transitions described in Section 1, along with the relevant acoustic conditions summarized in

Table 1. Set of acoustic conditions, excitation frequency and applied power, required for attaining each of the four transition states.

Driving Frequency (MHz)	$\mathbf{Input}\mathbf{Power}\mathbf{Density}(\mathbf{W}/{\mathbf{c}\mathbf{m}}^2$)
	State 1	State 2	State 3	State 4
0.43	6, 7, 8	9, 10	–– b	–– b
0.80	6, 7	8	9, 10	–– b
1.0	–– a	0.5	1, 1.5, 2	2.5
1.6	–– a	0.5	1	1.5

^a Not detected within the lowest operational limit (0.5 W/${\mathrm{c}\mathrm{m}}^2$); ^b Not detected within the highest operational limit (10 W/${\mathrm{c}\mathrm{m}}^2$).

.

The first state has a rather shallow, smooth mound kept stationary [see Figure 2a] under the sets of acoustic conditions (driving frequencies and input power densities) as given (in the first and second columns, respectively, in Table 1). As the combination of these two parameters is properly adjusted (third column vs. first in Table 1), the protrusion becomes sharper with the appearance of characteristic waves “dissecting” its surface; its shape as a whole still maintains the bell-like mound [Figure 2b]. With the combination of the parameters in different settings (fourth column as well as first in Table 1)—requiring a minimum/threshold driving frequency exceeding 0.43 (as high as 0.80) MHz—a chain of liquid beads will emanate out of the mound [Figure 2c]. As the minimum driving frequency is further raised, this beading structure would become destabilized, exhibiting a complex structure [Figure 2d]. A similar set of transition states, reported (for water fountaining) by Kim et al. (2021) [35] who applied focused ultrasound at two distinct frequencies, 0.55 and 1.1 MHz, were termed as “weak, intermediate (stable) and highly forced (explosive)” fountains (see Section 1). The detailed description of each transition state is provided below in a separate section.

3.1. Free-Surface Protrusion and Its Growth

The geometric characterization of the bell-shaped mound of the liquid free surface can be obtained by tracing its outline, as shown in Figure 3. As an outcome of the image processing described in Section 2, the interface boundary/outline extracted, by confining it down to 1 pixel, is provided for a specific combination of the two operating parameters: (a) for the lowest driving frequency of 0.43 MHz with varying (over a widest range of) input power density from the minimum of 6 W/${\mathrm{c}\mathrm{m}}^2$—below which no appreciable protrusion was visually recognized in this study—to 10 W/${\mathrm{c}\mathrm{m}}^2$ and (b) for the proper combinations (listed in Table 1) to realize the second state [see Figure 2b].

Geometric universality: An important finding to be stressed here [shown in Figure 3a] is that—while the mound exhibits an essentially identical height as well as shape for the input power range of 6–8 W/${\mathrm{c}\mathrm{m}}^2$—the height will increase stepwise as the power is raised from 8 to 9 W/${\mathrm{c}\mathrm{m}}^2$ and again maintains the same yet new level at 9 and 10 W/${\mathrm{c}\mathrm{m}}^2$. Similar observations, with quantitative evaluations of the steady, attained height of the acoustic fountain, were made numerically by Xu et al. (2016) [34] and experimentally by Kim et al. (2021) [35], who both reported a step increment—with an increase in the acoustic pressure—of half the ultrasound wavelength. Such a stepwise transition from the first to second state identified in this study would correspond to that from the weak to the stable fountain reported by Kim et al. (2021) [35]; this transitional regime, spanning the states of topologically identical geometries, should be a crucial “stage” covered from induction to the establishment of the Foundation Region (FR; [1]) of the general configuration of the UsA fountain. It is to be added here that the bottom diameter, or width, of the protrusion appears to be preserved, due probably to the driving frequency fixed at 0.43 MHz, i.e., under the same directivity of ultrasound irradiated.

Another point to be noted in regard to the mound geometry is shown in Figure 3b: its outline for the second state, i.e., the being-established rather steady/stable FR of the fountain, extracted via image processing shares essentially the same “universal” shape and—more importantly—height for all the combinations of the two acoustic conditions as listed on the first and third columns in Table 1. In this set of combinations, however, the width of protrusion differs slightly, depending on the driving frequency used; in general, the higher the frequency is, the narrower—with some exception—the width would be. Further investigation is needed to confirm the reliability as well as reproducibility of the present findings regarding the data given in Figure 3.

Protrusion of concentric circles induced on free surface: As can be seen in Figure 2b, the mound will “grow”—become higher—with either increasing input power density (at a fixed driving frequency) or vice versa; this process is often observed to concur with the appearance of “regular undulation” (a series of concentric circles) superimposed on the general surface outline of the protrusion, some indication of which could be already recognizable at lower input powers in the first state. As claimed by Tsuchiya et al. (2011) [1], who proposed that the vertical distance spanning each “local peak” of the undulation (undulation “pitch”) should correspond to half the wavelength, a unique feature of such undulation lies in its regularity being controlled by the frequency—thus, the wavelength—of the propagating ultrasound.

Figure 4 depicts such undulation along the mound surface observed at different driving frequencies. For the driving, or excitation, frequencies of $f_{ex}=$ 0.80, 1.0, and 2.0 MHz, the undulation pitches are measured to be 0.90, 0.79, and 0.34 mm, respectively, corresponding to roughly half the ultrasonic wavelength. Note that the pertaining wavelengths (${\lambda }_{wave}=v_{wave}/f_{ex}$—as excited) are evaluated to be 1.84, 1.47, and 0.74 mm, respectively, where the sound speed $v_{wave}$ of 1470 m/s in 50-wt% ethanol aqueous solution at 25 °C [38] is adopted. Tsuchiya et al. (2011) [1] used an ultrasound transducer (Honda Electronics HM-2412) with a driving frequency of 2.4 MHz in water to provide—though limited—the detected undulation, with a vertical distance of 0.32 mm for each local peak.

The wave-specific nature—i.e., the characteristic length being strongly dependent on the acoustic wavelength—in the phenomenon of concern, as above, appears to have a role in physically describing (almost) every state of the UsA process. In such a description, it is important to realize that the frequency of the ultrasonic wave $f_{wave}$ may not always equal $f_{ex}$: it has been claimed in the literature that once sufficient instabilities are set in to induce the liquid-column fluctuations observed (in the 3rd state to be described in detail below), there is a tendency to have $f_{wave}=f_{ex}/2$ [2,17,39]—otherwise, $f_{wave}=f_{ex}$ [11,13]. In describing the above undulation prevailing in the second state of negligible fluctuating instability, which can be characterized by the vertical span of peak-to-peak distance of the undulation or its pitch, two pitches were found and are presumed to correspond to ${\lambda }_{wave}=v_{wave}/f_{wave}$ where $f_{wave}=f_{ex}$.

Height fluctuations: While a first-sight outline of the second-state protrusion appears stable/steady, detailed observations reveal that its height will start fluctuating with small amplitudes—(indication at least of) an onset of instability, in addition to its surface undulation. Figure 5 demonstrates typical oscillation patterns exhibited in terms of time- series data/signals for the protrusion/fountain altitude/height and their corresponding FFT outcomes. The periodicity in the height fluctuations appears to be of “quasi-nature”. At 0.43 MHz, the signals inherently contain a dominant frequency of 6 Hz (in a rather wider range spanning 6–24 Hz) and 12 Hz (in a much narrower range) for 9 and 10 W/${\mathrm{c}\mathrm{m}}^2$ [Figure 5a,b], respectively; the effects of input power density are inconclusive, though some critical shift in periodic trend appears to exist between these two values. As the driving frequency is raised from 0.43 to 0.80, 1.0, and 1.6 MHz, the dominant frequency of height fluctuations tends to increase from 6 to 10, 22, and 54 Hz; the assigned values of the input power density here are irrelevant in discussing the trend for this set of data attaining the second state (see Table 1). The rather distinctive trends exhibited above are believed to provide mechanistically useful information; however, as in the case of the data in Figure 3, this series of data analyses would demand confirmation in both reliability (reducing uncertainties) and reproducibility (evaluating errors), warranting further investigation.

Bulk-liquid cavitation: As stated repeatedly in Section 1, obtaining information regarding the concurrent formation of cavitation—along with the structural transitioning of the free surface from flat to protruding to jetting—should be quite helpful [13,36]. While it is extremely difficult in the present study to capture any images of cavitation bubbles inside the protuberance, doing so through the bulk liquid above the UsA transducer up to the free surface has been attempted.

Figure 6 shows the spatial distribution of bubbles, which should have originated from cavitation, generated within the bulk liquid and captured via side-lighting. Note that two sets of comparisons are made in terms of two-parameter ($f_{ex},I_0$) combinations: different $f_{ex}$’s of 0.43 and 0.80 MHz for a given $I_0$ of 8 W/${\mathrm{c}\mathrm{m}}^2$; and different $I_0$’s of 8 and 9 W/${\mathrm{c}\mathrm{m}}^2$ for a fixed $f_{ex}$ of 0.80 MHz. In the first series of images (top row), the bubbles appear encircling around the vertical axis of acoustic wave propagation in a standing-wave pattern (for $t\ge$ 0.93 ms with $t$ being the time elapsed after the UsA transducer turned on). As the time exceeded 1.8 ms, some ripples began to appear on the liquid surface (not recognizable in the figure); the bubbles then spread out appreciably towards the vessel wall as they were approaching the free surface ($t\ge$ 5.0 ms); prior to the free surface deforming to a mound (appreciation of the 1st state, 6.3 ms).

For the same $I_0$ (8 W/${\mathrm{c}\mathrm{m}}^2$) but with $f_{ex}$ raised to 0.80 MHz (middle row), the free- surface rippling was recognizable at 0.4 ms; a bell-shaped mound with some undulation was observed to start developing at 0.93 ms (2nd state). In comparison to the first case, some additional information has been obtained: The size of the visible bubbles was significantly reduced at the higher $f_{ex}$. Despite faint streaks between nodes being identified, the standing-wave structure was clearly disrupted. The bubbles were mainly localized near the transducer with no obvious tendency to spread. Maintaining $f_{ex}$ (0.80 MHz) while increasing $I_0$ to 9 W/${\mathrm{c}\mathrm{m}}^2$ (bottom row), a developing protrusion began to appear as early as at 0.12 ms, resulting in the solid formation of a beads fountain at 1.92 ms. The apparent structure of the standing wave was no longer identifiable in the images; only randomly dispersed/migrating bubbles of a very limited number could be identified closer to the liquid surface. Note that such a series of cavitation-associated behaviors in the bulk liquid would resemble those reported by Lee et al. (2011) [31], who investigated the mechanism of liquid flow by monitoring the spatial distribution of visible bubbles, for a similar range of $f_{ex}$ but different $I_0$ values (see Section 1).

Further discussion regarding the dynamic behavior of the bulk-phase cavitation bubbles are possible, such that those visible bubbles would be expelled from the pressure antinodes and become trapped at adjacent pressure nodes due to active bubbles becoming merged by the actions of the primary and secondary Bjerknes forces [31]. The larger bubbles or bubble clusters/clouds generated via coalescence can scatter and attenuate the ultrasound wave [40]. It is generally known that the size of bubbles will decrease with increasing ultrasound frequency [41,42,43,44,45]; in the present study, however, no further attempts in this regard have been made to extend the ranges of the acoustic operating conditions. To be somewhat conclusive, however, in light of the difference in bubble flow between 8 and 9 W/${\mathrm{c}\mathrm{m}}^2$ at 0.80 MHz, it could be presumed under the present conditions that acoustic streaming is most likely responsible for achieving instability-associated fountain formation. The large “gap” in the applied power density detected between 0.80 MHz (9 W/${\mathrm{c}\mathrm{m}}^2$) and 1.0 MHz (1 W/${\mathrm{c}\mathrm{m}}^2$) further implies that, behind the mechanism of acoustic fountain formation, there could be some “unforeseen” thresholding factor(s), which should be clarified in an upcoming work.

3.2. Characterization of Emanating Beads Fountain

In regard to a chain of beads emanating out of the foundation-like region [FR alone when no appreciable fluctuating instabilities are apparent yet, or Bumpy-Surface Region (BSR) on top of it; [1,9]], some experimental and model-based findings were made in our previous work [2]. The acoustic pair of conditions was confined to $f_{ex}=$ 1–3 MHz mostly and $I_0=$ 4–6 W/${\mathrm{c}\mathrm{m}}^2$, in the presence of the stabilizing nozzle, to obtain two quantitative/inherent relationships for (1) the beads size ($d_{bead}$) vs. $f_{ex}$ (size specificity) and (2) the beads resonance/natural frequency ($f_{bead}$ or $f_n$) vs. $f_{ex}$ (periodicity). The present study was conducted to extend the same line of analyses to a wider range of the paired parameters with or without the nozzle equipped.

Minimum driving frequency for triggering beadsfountain: One specific aspect to be clarified in this extended work lies in the uncertainty of minimum driving frequency for realizing the beads fountain (3rd state; see Figure 2 and Table 1). Figure 7 provides a series of images implying the critical role played by the UsA-driving frequency—over the range $f_{ex}=$ 0.65–3.0 MHz—in the formation of a beads fountain. The specificity as well as uncertainty (associated with utilizing the “frequency shifting/tuning capability” provided on the ultrasonic transducer system; see Section 2) in $I_0$, on the other hand, should be noted to be rather insignificant or even irrelevant over a typical range examined, as reported by Wang et al. (2022) [2]. As the driving frequency was increased, except $f_{ex}=$ 0.65 MHz, the beads’ diameter tended to decrease; the extent of reduction can be clearly seen for $f_{ex}\ge$ 1.0 MHz. It is noteworthy that, for this lowest 0.65 MHz tested, no triggering of beads fountain has been detected (on all the images obtained under the same repeated conditions) even at the highest operational limit tested ($I_0=$ 10 W/${\mathrm{c}\mathrm{m}}^2$).

It is to be noted here that Wang et al. (2022) [2] presumed the required frequency should exceed at least 0.8 MHz, based on their argument that “the pertaining bead diameter $d_{bead}$ should not exceed the capillary length ($l_c$)”:

$$l_c=\sqrt{\sigma /{\rho }_{l}g},$$

(2)

which depends on the surface tension $\sigma$, density ${\rho }_l$ of the liquid, and the gravitational acceleration $g$. As Wang et al. (2022) [2] further argued, having met the corresponding (counter) condition ($d_{bead}>l_c$), the prevalent surface tension/capillary wave would become less dominant than the gravity wave. Substituting the liquid properties, $\sigma =$ 28.8 mN/m and ${\rho }_l=$ 902 kg/${\mathrm{m}}^3$, for the present 50-wt% ethanol aqueous solution at 25 °C [46], $l_c$ would be estimated to be 1.80 mm, slightly less than the characteristic bead diameter—confirmed (see Figure 7) to be almost identical to an estimate—of $d_{bead}\cong {\lambda }_{wave}/2=\left({\displaystyle \frac{1}{2}}\right)1470/(800\times {10}^3/2)=$ 1.84 mm; thus the above-stated condition was met for $f_{ex}$ of 0.80 MHz.

Phase-averaged size specificity of beadsfountain: The fountain-beads’ diameter averaged under each specific condition tested is shown in Figure 8 as a function of the driving frequency; for ease of specific and detailed comparison between each of the data points, all the obtained values (of standard deviation as well as average) are summarized in

Table 2. Statistically evaluated (using 60 images of beads under each condition) beads diameters for all the excitation frequencies tested.

Driving Frequency (MHz)	Beads Diameter without Nozzle (μm)	Beads Diameter with Nozzle (μm)
0.65	— a	— a
0.70	2065 ± 37	— a
0.75	1902 ± 23	— a
0.80	$\left\{\begin{array}{c}1810\pm 76\\ 610 \pm 20\end{array}\right.$	— a
0.85 (0.80)	$\left\{\begin{array}{c}{—}^{\mathrm{b}}\\ 551 \pm 15\end{array}\right.$	$\left.{\displaystyle \genfrac{}{}{0pt}{}{ 1731 \pm 34}{ 547 \pm 19}}\right\}$
0.85 (1.0)	1792 ± 41	1803 ± 29
0.90 (0.80)	$\left\{\begin{array}{c}{—}^{\mathrm{b}}\\ 533 \pm 26\end{array}\right.$	$\left.{\displaystyle \genfrac{}{}{0pt}{}{ 1653 \pm 28}{ 516 \pm 19}}\right\}$
0.90 (1.0)	1662 ± 36	1641 ± 35
0.95 (1.0)	1554 ± 22	1461 ± 50
1.0	1394 ± 25	1420 ± 80
1.6	859 ± 18	850  ± 37
2.0	573 ± 14	670 ± 40
3.0	— b	440 ± 25

^a Not detected below the highest operational limit (10 W/${\mathrm{c}\mathrm{m}}^2$); ^b Not detected.

. Plotted in log-log scale (to “expand” the lower range of the frequency), the trends exhibited by the experimental data can be well represented by straight lines of slope $-1$. Such trends, i.e., $d_{bead}$ increases—in line with the hyperbolic variation—with decreasing $f_{ex}$, can be represented by the physical principle—signifying the inherent traveling wave relationship:

$$k{d}_{bead}\cong {\lambda }_{wave}=v_{wave}/f_{wave}=k{v}_{wave}/b{f}_{ex}$$

(3)

where two parameters $k$ and $b$ are introduced: the former signifies, as stated above (see Section 3.1 concerning the protrusion of concentric circles induced on the free surface), whether the frequency of the ultrasonic wave $f_{wave}$ equals $f_{ex}$ or not; the latter concerns the possible “bifurcation” in $d_{bead}$ whose onset may occur below some critical excitation frequency. Depending on the fluctuating instabilities setting in along the protuberance, $k=2$, if they are sufficient (States 3 and 4); otherwise, $k=1$ (States 1 and 2).

It is to be noted in the figure that the beads diameter does not appear to be influenced by the presence/absence of the regulating nozzle. In principle, its value is argued to not exceed the capillary length—in our present ethanol solution, $d_{bead}\le l_c=$ 1.80 mm (as detailed above). The experimental results indicate that $d_{bead}$, when carefully evaluated (refer to [2] for its detailed measurement procedure), could take a value as high as 2.07 mm at $f_{ex}=0.70 \mathrm{M}\mathrm{H}\mathrm{z}$. While this largest value of $d_{bead}$ might be in the gravity-wave (rather than capillary-wave) domain, it could be inferred that the minimum driving frequency discussed in the previous “subsection” would be as low as 0.7 MHz—in comparison to the preliminary value 0.8 MHz proposed by Wang et al. (2022) [2].

It is also noteworthy, in Figure 8, that a separate set of data points shown for the frequencies lower than 0.9–1.0 MHz could be an indication of the bifurcation candidate—though not provided with solid evidence at this stage of data collection—requiring further verification. As a piece of partial evidence, shown in Figure 9, beads with diameters of about one-third the “normal” ones (see the data along the solid line in Figure 8) could be detected concurrently between 0.80 and 0.90 MHz—plausible bifurcation into normal-sized beads [primary beads; $b=1$ in Equation (3)] and smaller ones (secondary beads; $b=3$), observed experimentally as in a sequence of images provided subsequently.

The appearance of secondary beads is often accompanied by—emanating out of—a single primary bead formed right above the FR (or BSR if present); as mentioned above, the occurrence of such secondary beads will be realized alternately with the primary ones, leading us to a speculative, mechanistic description, as follows. As the driving frequency is reduced below a threshold—possibly the above-discussed “bifurcation- inducing” frequency of $f_{ex}\simeq$ 0.9 MHz—the “highly excited radial oscillations” would become unstable [13]. The acoustic radiation force could then be insufficient to support the stable prevalence of a fountain solely consisting of a chain of primary beads. In a sense, a solitary “base” primary bead acts as an “energy concentrator” [13]; the acoustic radiation pressure is effectively trapped inside the base bead and focused towards the tip, producing a chain of recurring secondary beads of roughly one-third the diameter of the primary ones. It can be hypothesized, then, that the relevant threshold for instability signifies a shift in the dominant mechanism, leading to the beads-emanating state— from acoustic radiation pressure to streaming.

4. Concluding Remarks

In irradiating ultrasound under wide ranges of operating conditions—specifically, the driving frequency $f_{ex}=$ 0.43–3.0 MHz and the input power density $I_0=$ 0.5–10 W/${\mathrm{c}\mathrm{m}}^2$—the UsA process is confirmed to go through a sequence of four states of transition: the onset of protrusion, the appearance of undulation, the triggering of beads fountain emergence, and the induction of droplets bursting. In this study, the specific findings are given as follows.

The threshold (lower-bound) values for the set of acoustic operating parameters ($f_{ex},$ $I_0$) are determined visually to be (0.43 MHz, 6 W/${\mathrm{c}\mathrm{m}}^2$) or (0.80 MHz, 6 W/${\mathrm{c}\mathrm{m}}^2$) for the protrusion onset—State 1; (0.43 MHz, 9 W/${\mathrm{c}\mathrm{m}}^2$), (0.80 MHz, 8 W/${\mathrm{c}\mathrm{m}}^2$), (1.0 MHz, 0.5 W/${\mathrm{c}\mathrm{m}}^2$), or (1.6 MHz, 0.5 W/${\mathrm{c}\mathrm{m}}^2$) for the undulation appearance—State 2; (0.80 MHz, 9 W/${\mathrm{c}\mathrm{m}}^2$), (1.0 MHz, 1 W/${\mathrm{c}\mathrm{m}}^2$), or (1.6 MHz, 1 W/${\mathrm{c}\mathrm{m}}^2$) for the beads-fountain triggering—State 3; and (1.0 MHz, 2.5 W/${\mathrm{c}\mathrm{m}}^2$) or (1.6 MHz, 1.5 W/${\mathrm{c}\mathrm{m}}^2$) for the droplets-bursting induction—State 4. Note that there is a large “gap” in the applied power density detected in State 2 as well as State 3 between 0.80 and 1.0 MHz, implying that there could be some drastically different thresholding factor(s) responsible for describing the mechanism of acoustic fountain formation, warranting further research.

Three peculiar characteristics to be pointed out in State 2 are as follows: first, the wave-specific nature of the stable-mound undulation pitch, i.e., the vertical span of peak-to-peak distance of the undulated protuberance, quantified by every two pitches corresponding to ${\lambda }_{wave}=v_{wave}/f_{wave}$ where $f_{wave}=f_{ex}$; second, the stepwise increase in the steady-attained height of the mound with negligible fluctuating instability as the input power is raised critically from the upper bound of State 1 to the lower one of State 2; and third, the mound height fluctuating with small amplitudes, in addition to its surface undulation. The last characteristic, having a quasi-periodic nature (as evaluated through the FFT analysis), possesses a dominant frequency that increases with the driving frequency, while the input power density exhibited inconclusive effects.

The third phase, or State 3, is associated with the fountain structure comprising a chain of recurring beads whose size should be scalable to the UsA wavelength. The average beads diameter $d_{bead}$ evaluated experimentally, over the present extended range of the acoustic parameters set, exhibits a general trend that $d_{bead}$ will increase—in line with the hyperbolic variation—with decreasing $f_{ex}$. This trend has been confirmed to coincide with the wave nature predicted using the simple physical principle over the entire range examined in this study—provided $d_{bead}$ would not exceed the capillary length $l_c$ appreciably— corresponding, here, to a minimum driving frequency as low as 0.7 MHz; an additional characteristic is found concerning possible “bifurcation” in $d_{bead},$ whose onset may occur below some critical $f_{ex}$.

As an indication of bifurcation, beads with diameters of about one-third the “normal” ones (or the primary beads, as predicted above) have been detected intermittently between 0.80 and 0.90 MHz. Such secondary beads, realized alternately with the primary ones, tend to emanate out of a single primary bead—a solitary “round base” acting as an “energy concentrator“—formed right above the Foundation Region.