Numerical study of the effect of liquid compressibility on acoustic droplet vaporization

In acoustic droplet vaporization (ADV), a cavitated bubble grows and collapses depending on the pressure amplitude of the acoustic pulse. During the bubble collapse, the surrounding liquid is compressed to high pressure, and liquid compressibility can have a significant impact on bubble behavior and ADV threshold. In this work, a one-dimensional numerical model considering liquid compressibility is presented for ADV of a volatile microdroplet, extending our previous Rayleigh-Plesset based model [Ultrason. Chem. 71 (2021) 105361]. The numerical results for bubble motion and liquid energy change in ADV show that the liquid compressibility highly inhibits bubble growth during bubble collapse and rebound, especially under high acoustic frequency conditions. The liquid compressibility effect on the ADV threshold is quantified with varying acoustic frequencies and amplitudes.     

Predicting initial nucleation events occurred in a metastable nanodroplet during acoustic droplet vaporization

Acoustic droplet vaporization (ADV) capable of converting liquid perfluorocarbon (PFC) micro/nanodroplets into gaseous microbubbles has gained much attention due to its medical potentials. However, its physical mechanisms for nanodroplets have not been well understood due to the disappeared superharmonic focusing effect and the prominent Laplace pressure compared to microdroplets, especially for the initial ADV nucleation occurring in a metastable PFC nanodroplet. The classical nucleation theory (CNT) was modified to describe the ADV nucleation via combining the phase-change thermodynamics of perfluoropentane (PFP) and the Laplace pressure effect on PFP nanodroplets. The thermodynamics was exactly predicted by the Redlich–Kwong equation of state (EoS) rather than the van der Waals EoS, based on which the surface tension of the vapor nucleus as a crucial parameter in the CNT was successfully obtained to modify the CNT. Compared to the CNT, the modified CNT eliminated the intrinsic limitations of the CNT, and it predicted a larger nucleation rate and a lower ADV nucleation threshold, which agree much better with experimental results. Furthermore, it indicated that the nanodroplet properties exert very strong influences on the nucleation threshold instead of the acoustic parameters, providing a potential strategy with an appropriate droplet design to reduce the ADV nucleation threshold. This study may contribute to further understanding the ADV mechanism for PFC nanodroplets and promoting its potential theranostic applications in clinical practice.     

Computational study of state equation effect on single acoustic cavitation bubble’s phenomenon

Many models have been established to study the evolution of the bubble dynamics and chemical kinetics within a single acoustic cavitation bubble during its oscillation. The content of the bubble is a gas medium that generates the evolution of a chemical mechanism governed by the internal bubble conditions. These gases are described by a state equation, linking the pressure to the volume, temperature and species amounts, and influencing simultaneously the dynamical, the thermal and the mass variation in the cavitation bubble. The choice of the state equation to apply has then a non-neglected effect on the obtained results. In this paper, a comparative study was conducted through two numerical models based on the same assumptions and the same scheme of chemical reactions, except that the first one uses the ideal gas equation to describe the state of the species, while the second one uses the Van der Waals equation. It was found that though the dynamic of the bubble is not widely affected, the pressure and temperature range are significantly increased when passing from an ideal gas model to a real one. The amounts of chemical products are consequently raised to approximately the double. This observation was more significant for temperature and pressure at low frequency and high acoustic amplitude, while it is noticed that passing from ideal gas based approach to the Van der Waals one increases the free radicals amount mainly under high frequencies. When taking the results of the second model as reference, the relative difference between both results reaches about 60% for maximum attained temperature and 100% for both pressure and free radicals production.     

Numerical investigation of the translational motion of bubbles: The comparison of capabilities of the time-resolved and the time-averaged methods

In the present study, the accuracies of two different numerical approaches used to model the translational motion of acoustic cavitational bubble in a standing acoustic field are compared. The less accurate but less computational demanding approach is to decouple the equation of translational motion from the radial oscillation, and solve it by calculating the time-averaged forces exerted on the bubble for one acoustic cycle. The second approach is to solve the coupled ordinary differential equations directly, which provides more accurate results with higher computational effort. The investigations are carried out in the parameter space of the driving frequency, pressure amplitude and equilibrium radius. Results showed that both models are capable to reveal stable equilibrium positions; however, in the case of oscillatory solutions, the difference in terms of translational frequency may be more than three fold, and the amplitude of translational motion obtained by the time-averaged method is roughly 1.5 times higher compared to the time-resolved solution at particular sets of parameters. This observation implies that where the transient behaviour is important, the time-resolved approach is the proper choice for reliable results.     

鼓泡床中超声驻波的模拟及其对气泡的调制机理

采用计算流体动力学（CFD）的方法，数值生成了鼓泡床中一对声换能器以16kHz高频振动引发的超声场。数值计算是基于包括粘性影响的可压缩流体基本守恒方程，并耦合了水的状态方程。模拟结果表明，在本研究所用的几何布置和换能器与时间相关的速度入口边界条件下，反应器中形成了一个稳定的驻波声场；由于波的非线性以及水的粘性，压力波节点呈现出轻微的时间漂移性。模拟结果与前人的实验结果定性吻合。在模拟的声压分布的基础上，分析了驻波声场调制气泡的机理。如比较熟知，气泡在驻波声场作用下或者向压力波节点运动或者向压力波腹点运动，取决于气泡尺寸与共振尺寸的关系。     

Numerical simulation of the nonlinear ultrasonic pressure wave propagation in a cavitating bubbly liquid inside a sonochemical reactor

We investigate the acoustic wave propagation in bubbly liquid inside a pilot sonochemical reactor which aims to produce antibacterial medical textile fabrics by coating the textile with ZnO or CuO nanoparticles. Computational models on acoustic propagation are developed in order to aid the design procedures. The acoustic pressure wave propagation in the sonoreactor is simulated by solving the Helmholtz equation using a meshless numerical method. The paper implements both the state-of-the-art linear model and a nonlinear wave propagation model recently introduced by Louisnard (2012), and presents a novel iterative solution procedure for the nonlinear propagation model which can be implemented using any numerical method and/or programming tool. Comparative results regarding both the linear and the nonlinear wave propagation are shown. Effects of bubble size distribution and bubble volume fraction on the acoustic wave propagation are discussed in detail. The simulations demonstrate that the nonlinear model successfully captures the realistic spatial distribution of the cavitation zones and the associated acoustic pressure amplitudes.     

Radial oscillation and translational motion of a gas bubble in a micro-cavity

According to classical nucleation theory, a gas nucleus can grow into a cavitation bubble when the ambient pressure is negative. Here, the growth process of a gas nucleus in a micro-cavity was simplified to two “events”, and the full confinement effect of the surrounding medium of the cavity was considered by including the bulk modulus in the equation of state. The Rayleigh–Plesset-like equation of the cavitation bubble in the cavity was derived to model the radial oscillation and translational motion of the cavitation bubble in the local acoustic field. The numerical results show that the nucleation time of the cavitation bubble is sensitive to the initial position of the gas nucleus. The cavity size affects the duration of the radial oscillation of the cavitation bubble, where the duration is shorter for smaller cavities. The equilibrium radius of a cavitation bubble grown from a gas nucleus increases with increasing size of the cavity. There are two possible types of translational motion: reciprocal motion around the center of the cavity and motion toward the cavity wall. The growth process of gas nuclei into cavitation bubbles is also dependent on the compressibility of the surrounding medium and the magnitude of the negative pressure. Therefore, gas nuclei in a liquid cavity can be excited by acoustic waves to form cavitation bubbles, and the translational motion of the cavitation bubbles can be easily observed owing to the confining influence of the medium outside the cavity.     

Numerical investigation on acoustic cavitation characteristics of an air-vapor bubble: Effect of equation of state for interior gases

The cavitation dynamics of an air-vapor mixture bubble with ultrasonic excitation can be greatly affected by the equation of state (EOS) for the interior gases. To simulate the cavitation dynamics, the Gilmore-Akulichev equation was coupled with the Peng–Robinson (PR) EOS or the Van der Waals (vdW) EOS. In this study, the thermodynamic properties of air and water vapor predicted by the PR and vdW EOS were first compared, and the results showed that the PR EOS gives a more accurate estimation of the gases within the bubble due to the less deviation from the experimental values. Moreover, the acoustic cavitation characteristics predicted by the Gilmore-PR model were compared to the Gilmore-vdW model, including the bubble collapse strength, the temperature, pressure and number of water molecules within the bubble. The results indicated that a stronger bubble collapse was predicted by the Gilmore-PR model rather than the Gilmore-vdW model, with higher temperature and pressure, as well as more water molecules within the collapsing bubble. More importantly, it was found that the differences between both models increase at higher ultrasound amplitudes or lower ultrasound frequencies while decreasing as the initial bubble radius and the liquid parameters (e.g., surface tension, viscosity and temperature of the surrounding liquid) increase. This study might offer important insights into the effects of the EOS for interior gases on the cavitation bubble dynamics and the resultant acoustic cavitation-associated effects, contributing to further optimization of its applications in sonochemistry and biomedicine.     

A simple model of ultrasound propagation in a cavitating liquid. Part I: Theory, nonlinear attenuation and traveling wave generation

The bubbles involved in sonochemistry and other applications of cavitation oscillate inertially. A correct estimation of the wave attenuation in such bubbly media requires a realistic estimation of the power dissipated by the oscillation of each bubble, by thermal diffusion in the gas and viscous friction in the liquid. Both quantities and calculated numerically for a single inertial bubble driven at 20 kHz, and are found to be several orders of magnitude larger than the linear prediction. Viscous dissipation is found to be the predominant cause of energy loss for bubbles small enough. Then, the classical nonlinear Caflish equations describing the propagation of acoustic waves in a bubbly liquid are recast and simplified conveniently. The main harmonic part of the sound field is found to fulfill a nonlinear Helmholtz equation, where the imaginary part of the squared wave number is directly correlated with the energy lost by a single bubble. For low acoustic driving, linear theory is recovered, but for larger drivings, namely above the Blake threshold, the attenuation coefficient is found to be more than 3 orders of magnitude larger then the linear prediction. A huge attenuation of the wave is thus expected in regions where inertial bubbles are present, which is confirmed by numerical simulations of the nonlinear Helmholtz equation in a 1D standing wave configuration. The expected strong attenuation is not only observed but furthermore, the examination of the phase between the pressure field and its gradient clearly demonstrates that a traveling wave appears in the medium.     

Stable and transient bubble formation in acoustically-responsive scaffolds by acoustic droplet vaporization: theory and application in sequential release

Acoustically-responsive scaffolds (ARSs), which are fibrin hydrogels containing monodispersed perfluorocarbon (PFC) emulsions, respond to ultrasound in an on-demand, spatiotemporally-controlled manner via a mechanism termed acoustic droplet vaporization (ADV). Previously, ADV has been used to control the release of bioactive payloads from ARSs to stimulate regenerative processes. In this study, we used classical nucleation theory (CNT) to predict the nucleation pressure in emulsions of different PFC cores as well as the corresponding condensation pressure of the ADV-generated bubbles. According to CNT, the threshold bubble radii above which ADV-generated bubbles remain stable against condensation were 0.4 µm and 5.2 µm for perfluoropentane (PFP) and perfluorohexane (PFH) bubbles, respectively, while ADV-generated bubbles of any size in perfluorooctane (PFO) condense back to liquid at ambient condition. Additionally, consistent with the CNT findings, stable bubble formation from PFH emulsion was experimentally observed using confocal imaging while PFO emulsion likely underwent repeated vaporization and recondensation during ultrasound pulses. In further experimental studies, we utilized this unique feature of ADV in generating stable or transient bubbles, through tailoring the PFC core and ultrasound parameters (excitation frequency and pulse duration), for sequential delivery of two payloads from PFC emulsions in ARSs. ADV-generated stable bubbles from PFH correlated with complete release of the payload while transient ADV resulted in partial release, where the amount of payload release increased with the number of ultrasound exposure. Overall, these results can be used in developing drug delivery strategies using ARSs.     

Numerical modelling of acoustic cavitation threshold in water with non-condensable bubble nuclei

Numerical modelling of acoustic cavitation threshold in water is presented taking into account non-condensable bubble nuclei, which are composed of water vapor and non-condensable air. The cavitation bubble growth and collapse dynamics are modeled by solving the Rayleigh-Plesset or Keller-Miksis equation, which is combined with the energy equations for both the bubble and liquid domains, and directly evaluating the phase-change rate from the liquid and bubble side temperature gradients. The present work focuses on elucidating acoustic cavitation in water with a wide range of cavitation thresholds (0.02–30 MPa) reported in the literature. Computations for different nucleus sizes and acoustic frequencies are performed to investigate their effects on bubble growth and cavitation threshold. The numerical predictions are observed to be comparable to the experimental data in the previous works and show that the cavitation threshold in water has a wide range depending on the bubble nucleus size.     

A single oscillating bubble in liquids with high Mach number

The oscillation characteristics of a single bubble and its induced radiation pressure and the dissipated power are essential for a wide range of applications. For bubble oscillations with high Mach number, the influence of the liquid compressibility is significantly strong and should be fully considered. In the present paper, the bubble wall motion equation with the second-order Mach number is employed for investigating a free oscillating bubble in the liquid with numerical and experimental verifications. For the purpose of comparisons, the revised Keller-Miksis equation up to the first-order Mach number is solved with the same conditions (e.g. the initial conditions and the ambient pressure). Through our simulations, comparing with the predictions by the first-order equation, we find that: (1) The bubble radius, the bubble wall radial velocity and the bubble wall radial acceleration predicted by the second-order equation with high Mach number are significantly different respectively, and the dimensionless differences increase with the increase of the Mach number. (2) The valid prediction range of the second-order equation is much larger. (3) The dissipated power predicted by the second-order equation with high Mach number is smaller.     

Bubble size distribution in acoustic droplet vaporization via dissolution using an ultrasound wide-beam method

Performance and efficiency of numerous cavitation enhanced applications in a wide range of areas depend on the cavitation bubble size distribution. Therefore, cavitation bubble size estimation would be beneficial for biological and industrial applications that rely on cavitation. In this study, an acoustic method using a wide beam with low pressure is proposed to acquire the time intensity curve of the dissolution process for the cavitation bubble population and then determine the bubble size distribution. Dissolution of the cavitation bubbles in saline and in phase-shift nanodroplet emulsion diluted with undegassed or degassed saline was obtained to quantify the effects of pulse duration (PD) and acoustic power (AP) or peak negative pressure (PNP) of focused ultrasound on the size distribution of induced cavitation bubbles. It was found that an increase of PD will induce large bubbles while AP had only a little effect on the mean bubble size in saline. It was also recognized that longer PD and higher PNP increases the proportions of large and small bubbles, respectively, in suspensions of phase-shift nanodroplet emulsions. Moreover, degassing of the suspension tended to bring about smaller mean bubble size than the undegassed suspension. In addition, condensation of cavitation bubble produced in diluted suspension of phase-shift nanodroplet emulsion was involved in the calculation to discuss the effect of bubble condensation in the bubble size estimation in acoustic droplet vaporization. It was shown that calculation without considering the condensation might underestimate the mean bubble size and the calculation with considering the condensation might have more influence over the size distribution of small bubbles, but less effect on that of large bubbles. Without or with considering bubble condensation, the accessible minimum bubble radius was 0.4 or 1.7 μm and the step size was 0.3 μm. This acoustic technique provides an approach to estimate the size distribution of cavitation bubble population in opaque media and might be a promising tool for applications where it is desirable to tune the ultrasound parameters to control the size distribution of cavitation bubbles.     

Modeling acoustic cavitation with inhomogeneous polydisperse bubble population on a large scale

A model for acoustic cavitation flows able to depict large geometries and time scales is proposed. It is based on the Euler–Lagrange approach incorporating a novel Helmholtz solver with a non-linear acoustic attenuation model. The method is able to depict a polydisperse bubble population, which may vary locally. The model is verified and analyzed in a setup with a large sonotrode. Influences of the initial void fraction and the population type are studied. The results show that the velocity is strongly influenced by these parameters. Furthermore, the largest bubbles determine the highest pressure amplitude reached in the domain, which corresponds to the Blake threshold of these bubbles. Additionally, a validation is performed with a small sonotrode. The model reproduces most of the experimentally observed phenomena. In the experiments, neighboring bubbles are found which move in different directions depending on their size. The numerical results show that the responsible mechanism here is the reversal of the primary Bjerknes force at a certain pressure amplitude.     

Influence of bulk nanobubble concentration on the intensity of sonoluminescence

The present study mainly examined the effects of the volumetric concentration of nanobubbles (ultrafine bubbles) on the intensity of sonoluminescence (SL). The addition of nanobubbles at high acoustic amplitude enhanced the SL intensity for various bubble concentrations in comparison with that in pure water. This probably means that the resulting high amplitude is over the Blake threshold, and accordingly nanobubbles expand to some extent, leading to higher SL intensity. Therefore, nanobubbles have the potential to provide nucleation sites for sonochemistry. The influence of bubble size on the intensity of SL was also evaluated.     

Stable bubble oscillations beyond Blake’s critical threshold

The equilibrium radius of a single spherical bubble containing both non-condensable gas and vapor is determined by the mechanical balance at the bubble interface. This expression highlights the fact that decreasing the ambient pressure below the so called Blake’s critical threshold, the bubble has no equilibrium state at all. In the last decade many authors have tried to find evidence for the existence of stable bubble oscillation under harmonic forcing in this regime, that is, they have tried to stabilize the bubble motion applying ultrasonic radiation on the bubble. The available numerical results provide only partial proof for the existence as they are usually based on linearized or weakly nonlinear (higher order approximation) bubble models. Here, based on numerical techniques of the modern nonlinear and bifurcation theory, the existence of stable bubble motion has been proven without any restrictions in nonlinearities. Although the model, applied in this paper, is the rather simple Rayleigh–Plesset equation, the presented technique can be extended to more complex bubble models easily.     

Effects of medium viscoelasticity on bubble collapse strength of interacting polydisperse bubbles

Due to its physical and/or chemical effects, acoustic cavitation plays a crucial role in various emerging applications ranging from advanced materials to biomedicine. The cavitation bubbles usually undergo oscillatory dynamics and violent collapse within a viscoelastic medium, which are closely related to the cavitation-associated effects. However, the role of medium viscoelasticity on the cavitation dynamics has received little attention, especially for the bubble collapse strength during multi-bubble cavitation with the complex interactions between size polydisperse bubbles. In this study, modified Gilmore equations accounting for inter-bubble interactions were coupled with the Zener viscoelastic model to simulate the dynamics of multi-bubble cavitation in viscoelastic media. Results showed that the cavitation dynamics (e.g., acoustic resonant response, nonlinear oscillation behavior and bubble collapse strength) of differently-sized bubbles depend differently on the medium viscoelasticity and each bubble is affected by its neighboring bubbles to a different degree. More specifically, increasing medium viscosity drastically dampens the bubble dynamics and weakens the bubble collapse strength, while medium elasticity mainly affects the bubble resonance at which the bubble collapse strength is maximum. Differently-sized bubbles can achieve resonances and even subharmonic resonances at high driving acoustic pressures as the elasticity changes to certain values, and the resonance frequency of each bubble increases with the elasticity increasing. For the interactions between the size polydisperse bubbles, it indicated that the largest bubble generally has a dominant effect on the dynamics of smaller ones while in turn it is almost unaffected, exhibiting a pattern of destructive and constructive interactions. This study provides a valuable insight into the acoustic cavitation dynamics of multiple interacting polydisperse bubbles in viscoelastic media, which may offer a potential of controlling the medium viscoelasticity to appropriately manipulate the dynamics of multi-bubble cavitation for achieving proper cavitation effects according to the desired application.     

Multi-time scale characterization of acoustic droplet vaporization and payload release of phase-shift emulsions using high-speed microscopy

Acoustic droplet vaporization (ADV) is the phase-transitioning of perfluorocarbon emulsions, termed phase-shift emulsions, into bubbles using focused ultrasound. ADV has been utilized in many biomedical applications. For localized drug release, phase-shift emulsions with a bioactive payload can be incorporated within a hydrogel to yield an acoustically-responsive scaffold (ARS). The dynamics of ADV and associated drug release within hydrogels are not well understood. Additionally, emulsions used in ARSs often contain high molecular weight perfluorocarbons, which is unique relative to other ADV applications. In this study, we used ultra-high-speed brightfield and fluorescence microscopy, at frame rates up to 30 million and 0.5 million frames per second, respectively, to elucidate ADV dynamics and payload release kinetics in fibrin-based ARSs containing phase-shift emulsions with three different perfluorocarbons: perfluoropentane (PFP), perfluorohexane (PFH), and perfluorooctane (PFO). At an ultrasound excitation frequency of 2.5 MHz, the maximum expansion ratio, defined as the maximum bubble diameter during ADV normalized by the initial emulsion diameter, was 4.3 ± 0.8, 4.1 ± 0.6, and 3.6 ± 0.4, for PFP, PFH, PFO emulsions, respectively. ADV yielded stable bubble formation in PFP and PFH emulsions, though the bubble growth rate post-ADV was three orders of magnitudes slower in the latter emulsion. Comparatively, ADV generated bubbles in PFO emulsions underwent repeated vaporization/recondensation or fragmentation. Different ADV-generated bubble dynamics resulted in distinct release kinetics in phase-shift emulsions carrying fluorescently-labeled payloads. The results provide physical insight enabling the modulation of bubble dynamics with ADV and hence release kinetics, which can be used for both diagnostic and therapeutic applications of ultrasound.