High-frequency acoustic droplet vaporization is initiated by resonance

Vaporization of low-boiling point droplets has numerous applications in combustion, process engineering and in recent years, in clinical medicine. However, the physical mechanisms governing the phase conversion are only partly explained. Here, we show that an acoustic resonance can arise from the large speed of sound mismatch between a perfluorocarbon microdroplet and its surroundings. The fundamental resonance mode obeys a unique relationship $kR$~$\sim$ 0.65 between droplet size and driving frequency that leads to a 3-fold pressure amplification inside the droplet. Classical nucleation theory shows that this pressure amplification increases the nucleation rate by several orders of magnitude. These findings are confirmed by high-speed imaging performed at a timescale of ten nanoseconds. The optical recordings demonstrate that droplets exposed to intense acoustic waves generated by inter-digital-transducers nucleate only if they match the theoretical resonance size.

The interaction of ultrasound with a low-boiling point droplet has been subject to extensive study aiming at understanding the underlying physical mechanisms driving nucleation.
Experimentally, it has been found that a prominent peak negative pressure (PNP) nucleation threshold exists above which the nucleation probability increases with the acoustic pressure amplitude and that this threshold, counterintuitively, lowers with an increase in ultrasound frequency and with a decrease in ambient pressure [9,12,[21][22][23][24][25][26][27][28][29][30]. These observations are in line with what is predicted from both classical nucleation theory [31] and superhamonic focusing [23,32], and by the combination of the two [33,34]. Superharmonic focusing results from the focusing of higher harmonics with wavelengths on the order of the droplet diameter that are generated through nonlinear propagation of the transmitted ultrasound wave [32]. However, the required nonlinear propagation in tissue is dramatically lower from that in water due to the two orders of magnitude lower ratio of acoustic nonlinearity to attenuation, or Gol'dberg number [35]. This severely limits the effectivity of ADV by superharmonic focusing in vivo. In this Letter, a physical ADV nucleation mechanism based on an acoustic resonance of the droplet is presented that has been overlooked until now.
The theoretical resonance behavior is experimentally validated and its role in lowering the vaporization threshold is elucidated using classical nucleation theory. Resonant ADV offers an efficient approach for in vivo applications and adds to our fundamental understanding of acoustic droplet vaporization.
Here we will give the main results of the derivation; all details can be found in the Supplementary Information SI.1. The system consists of a perfluoropentane (PFP) droplet (medium 1) immersed in water (medium 0) considered to be of infinite size. We assume a purely spherical geometry. To calculate the resonance behavior we couple the pressure on the inside of the droplet interface p(R in , t) to the external acoustic driving pressure p A (t). The droplet is assumed to be small compared to the wavelength in water λ 0 and the acoustic pressure can be considered homogeneous around the droplet. Integration of the the momentum equation in water then gives: with p(R out , t) the pressure on the outside of the droplet interface. p atm is the atmospheric pressure, R(t) is the droplet radius, and its overdots represent the interface velocity and acceleration, respectively. The pressure jump across the interface is expressed using the normal stress balance: with µ i the viscosity of medium i. While in the analogous derivation of the Rayleigh-Plesset equation for bubbles the equation is closed by the highly compressible and uniform gas pressure [36], here we need to evaluate the complete acoustic pressure distribution within the droplet. A classical acoustic derivation of the particle velocity v leads to the well-known spherical Bessel equation: Here, x = k 1 r, with k 1 = ω/c 1 the wavenumber, c 1 the speed of sound in the droplet, and r the radial coordinate. Avoiding the unphysical divergence at r = 0, the solution to Eq. (3) is a spherical Bessel function of the first kind: Writing X = k 1 R, the boundary condition at the droplet interface v(R, t) =Ṙ yields: The pressure distribution is then found by inserting Eq. (5) in the mass conservation and compressibility equation and using the Bessel function recurrence relation j 1 (x) = j 0 (x) − where β 1 = 1/ρ 1 c 2 1 is the compressibility of the droplet. Note that, owing to the j 0 (x) = sinx/x sinc term, the pressure amplitude will always be maximum in the center of the droplet. Integrating Eq. (6) for small oscillation amplitudes (R = R 0 (1 + ), 1) with R 0 the resting radius and X 0 = k 1 R 0 , with the initial boundary condition p(r, 0) = p atm +2σ/R 0 at t = 0 and evaluating in r = R yields: Note that this approximation only holds when j 1 (X 0 ) is not near its zero-crossing. Combining Eqs. (1), (2) and (7) now gives the droplet dynamics equation: The reradiated pressure scattered by the droplet induces a compression of the surrounding medium. The effect of acoustic reradiation on the droplet dynamics can be expressed by an , effectively adding a damping term to the set of equations. The above equations can also be extended to include the acoustic interaction with a rigid wall through the addition of a pressure term ρ 0 representing the reflected scattering, with d the distance to the wall [38]. For a droplet at the wall, d = R, and the left-hand side of Eq. 8 takes the form ρ 0 3 2 RR + 2Ṙ 2 , see SI.2. Equation (8) can be solved numerically to obtain the resonance behavior of the system. The pressure in the center of the droplet can be obtained by re-evaluating Eq. (6) in r = 0 instead of r = R : Linearization of Eq. (8) gives a relation between the angular eigenfrequency of the droplet ω 0 and its resting radius: Using a first-order expansion cot(z) 1/z − z/3 + O(z 3 ) and by neglecting the interfacial tension term, Eq. (10) reduces to a simple classical form: with f 0 = ω 0 /2π the eigenfrequency of the droplet. The same approach leads to an expression for the damping of the system in canonical form: As a result of the negligible contribution of viscous damping, which is three orders of magnitude smaller than the first term in Eq. (12), this then further reduces to: For a droplet at the wall the linearization simply adds a prefactor 2/3 to both Eqs. (11) and (13). Equation (13) also shows that the resonance arises from a speed of sound mismatch between the droplet and its surrounding medium, i.e. when the speed of sound ratio goes to 1, the system becomes overdamped, without a resonance effect, while the largest speed of sound mismatch produces the strongest resonance.
The resonance curves, solutions of Eq. (8) and computing the maximum pressure in the drop using Eq. (9) for frequencies of 10, 25, 50 and 100 MHz were calculated using the ODE45 solver in Matlab and are displayed in Fig. 1(a). The physical parameters for PFP and water that were used for the calculations are listed in SI.3 and were extracted from [39][40][41]. The resonance frequency f R is plotted against the droplet size in Fig. 1(b) for a droplet in free-field and for a droplet against a rigid wall. Two direct results from these plots are that (1), unlike superharmonic focusing, the acoustic resonance strength has very little dependency on droplet size (see also SI.4) and (2), the resonance is expected to have a high quality factor and a frequency about 50 times higher than that of free gas bubbles, see the corresponding Minnaert bubble resonance frequency f M = 3.3 µmMHz/R 0 plotted in Fig. 1(b) [36].
Owing to the high eigenfrequencies expected, the response of PFP droplets was mea- A perfluoropentane (PFP) droplet emulsion was prepared as in [15]. Its size distribution was measured using a Coulter counter, see Fig. 2(c). The suspension was loaded by capillary suction in a chamber that was approximately 100 µm in height. The chamber was located directly above the piezoelectric substrate, open at the front end to allow for a direct coupling between the SAW and the liquid, and closed above using a microscope cover slip (24 mm length), see details in Fig. 2(b).

Droplet vaporization was imaged using an inverted microscope (Olympus BX-FM)
equipped with a 20× magnification objective (Olympus SLMPlan N) coupled to the Brandaris 128 ultra high-speed camera [45,46]   smaller at 45.4 MHz. Note that every nucleation site produced a stable bubble at both driving frequencies, as can be observed from the images captured after 100 ms, i.e. none of the nucleated bubbles were observed to recondense [48]. The theoretical resonance peaks in Fig. 4(c,d) are corrected for the presence of a rigid wall and closely match the experimental peaks in Figs. 4(a,b).
Numerical simulations were performed on the basis of the geometry of Fig. 2 Video. The amplification factor of the first resonance, as well as its location, is in very good agreement with the theoretical model, although the simulated amplification is ∼ 15% lower than predicted by the theory. Interestingly, the presence of a second mode (see SI.5) is visible at 45.4 MHz. It appears that the experimental data for a driving frequency of 45.4 MHz ( Fig. 4(b)) indeed displays a secondary nucleation peak around a radius of 5 µm. This higher mode appears to have a smaller effect on the ADV threshold despite the large pressure amplification (factor 10) found in simulation. This is most likely a limitation of the present rigid numerical simulation since higher order non-axisymmetric modes can induce droplet deformation, which may have a significant impact on the pressure distribution within the droplet.
The statistical increase in the number of vaporization events as a function of the nega-  Fig. 3c and d, combined with the measured timing of the event, were used to determine the voltage at which each droplet vaporized. The resulting nucleation rate is displayed in Fig. 4(f) (open circles). Classical nucleation theory [50,51] dictates that the nucleation rate is a function of the surface tension of the liquid σ PFP 10 mN/m [41], of the ambient temperature T amb = 293 K and of the ambient pressure: Here, p v is the vapor pressure of the liquid and k B is the Boltzmann constant. The vapor pressure of PFP can be estimated using Antoine's law and ranges from 65 kPa at a temperature of 15 • C to 95 kPa at 25 • C. As a result, and considering the large peak negative pressures typically required to induce cavitation, (p v − p atm ) /p drop 1. With this simplification, Eq. (14) can be volume-integrated during the negative pressure phase of the ultrasound cycle to determine the average nucleation rate: see details provided in SI.6. At resonance the pressure amplification factor due to the resonance effect p drop /p A 3, see Figs. 4(c,d). When the SAW devices are driven within their linear range, the acoustic pressure is proportional to the driving voltage. Since the droplet concentration is contained in the prefactor, this expression can be used to fit the experimental data and provide an estimate of the local acoustic driving pressure, which is the only free parameter in Eq. (15). The least-squares fit is shown in Fig. 4(f) and gives p A /V 40 MPa/V, leading to an acoustic pressure at the droplet location of ∼ 20 MPa for an excitation voltage of 500 mV. Note that these numbers only constitute a rough estimate since, on the one hand, classical nucleation theory is known to overestimate the peak negative pressures required for cavitation [50] and, on the other hand, the range of data available for fitting is limited. It should also be noted that this pressure is of the same order as those typically used for ADV. Interestingly, nucleation theory as presented in Eq. (15) allows to predict the importance of the resonance effect by calculating the average nucleation rate without resonance, namely for p drop /p A = 1 (with the same prefactor), see Fig. 4(f) (blue line). Thus, it is clear that a pressure amplification factor of 3 has a dramatic effect on droplet vaporization behavior.
In conclusion, it was shown that efficient vaporization can be achieved by driving phasechange PFP droplets at their fundamental resonance frequency. Good agreement was found between the modeled size-dependent pressure amplification within the droplet and the measured size dependent vaporization probability. Resonance-induced vaporization is a new phenomenon, with important impact on potential ADV strategies and on the understanding on previous experimental observations. In addition, this work shows the potential of using monodisperse phase-change agents driven by a matching resonance frequency to boost the efficiency of ADV.
We thank James Friend for stimulating discussions. This work was funded by NanoNextNL, a micro and nanotechnology consortium of the Government of the Netherlands and 130 partners.