Validation of an approximate model for the thermal behavior in acoustically driven bubbles

Effect of tissue viscoelasticity and adjacent phase-changed microbubbles on vaporization process and direct growth threshold of nanodroplet in an ultrasonic field

Understanding the behavior of nanodroplets converted into microbubbles with applied ultrasound is an important problem in tumor therapeutical and diagnostic applications. In this study, a comprehensive model is proposed to investigate the vaporization process and the direct growth threshold of the nanodroplet by following the vapor bubble growth, especially attention devoted to the effect of tissue viscoelasticity and adjacent phase-changed microbubbles (PCMBs). It is shown that the ultrasonic energy must be sufficiently strong to counterbalance the natural condensation of the vapor bubble and the tissue stiffness-inhibitory effect. The softer tissue with a lower shear modulus favors the vaporization process, and the nanodroplet has a lower direct growth threshold in the softer tissue. Moreover, the adjacent PCMBs show a suppression effect on the vaporization process due to the negative value of the secondary Bjerknes force, implying an attractive force, preventing the nanodroplet from escaping from the constraint of the adjacent PCMBs. However, according to the linear scattering theory, the attractive force signifies that the constraint is weak, causing the direct growth threshold to increase in the range of 0.09-0.24 MPa. The weak increase in threshold demonstrates that the direct growth threshold is relatively unaffected by the adjacent PCMBs. The prediction results of our model are in good agreement with the experiment results obtained by the echo enhancement method, in which the threshold is relatively independent of the intermediate concentration. The findings presented here provide physical insight that will be further helpful in understanding the complex behavior of the nanodroplet responses to ultrasound in practical medical applications.

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.

Probing the pressure dependence of sound speed and attenuation in bubbly media: Experimental observations, a theoretical model and numerical calculations

The problem of attenuation and sound speed of bubbly media has remained partially unsolved. Comprehensive data regarding pressure-dependent changes of the attenuation and sound speed of a bubbly medium are not available. Our theoretical understanding of the problem is limited to linear or semi-linear theoretical models, which are not accurate in the regime of large amplitude bubble oscillations. Here, by controlling the size of the lipid coated bubbles (mean diameter of ≈5.4μm), we report the first time observation and characterization of the simultaneous pressure dependence of sound speed and attenuation in bubbly water below, at and above microbubbles resonance (frequency range between 1-3 MHz). With increasing acoustic pressure (between 12.5-100 kPa), the frequency of the peak attenuation and sound speed decreases while maximum and minimum amplitudes of the sound speed increase. We propose a nonlinear model for the estimation of the pressure dependent sound speed and attenuation with good agreement with the experiments. The model calculations are validated by comparing with the linear and semi-linear models predictions. One of the major challenges of the previously developed models is the significant overestimation of the attenuation at the bubble resonance at higher void fractions (e.g. 0.005). We addressed this problem by incorporating bubble-bubble interactions and comparing the results to experiments. Influence of the bubble-bubble interactions increases with increasing pressure. Within the examined exposure parameters, we numerically show that, even for low void fractions (e.g. 5.1×10-6) with increasing pressure the sound speed may become 4 times higher than the sound speed in the non-bubbly medium.

Penetration of hydroxyl radicals in the aqueous phase surrounding a cavitation bubble

In the sonochemical degradation of nonvolatile compounds, the free radicals must be delivered into the aqueous solution from the cavitation bubble to initiate reduction-oxidation reactions. The penetration depth in the liquid becomes an important parameter that influences the radical delivery efficiency and eventual treatment performance. However, the transport of radicals in the liquid phase is not well understood yet. In this paper, we focus on the most reactive OH radical and numerically simulate its penetration behavior. This is realized by solving the coupled equations of bubble dynamics, intracavity chemistry, and radical dispersion in the aqueous phase. The results present both the local and global penetration patterns for the OH radicals. By performing simulations over a wide range of acoustic parameters, we find an undesirable phenomenon that the penetration can be adversely suppressed when strengthening the radical production. A mechanistic analysis attributes this to the excessively vigorous recombination reactions associated with high radical concentrations near the bubble interface. In this circumstance, the radicals are massively consumed and converted into molecular species before they can appreciably diffuse away. Our study sheds light on the interplay between radical production inside the bubble and dispersion in the outside liquid. The derived conclusions provide guides for sonochemical applications from a new perspective.

A simplified model for the gas-vapor bubble dynamics

This paper presents a full numerical model accounting for the heat transfer and phase-change by combining the modified Keller-Miksis equation with the second order term of compressibility of liquid, partial differential equations (PDEs), and Hertz-Knudsen-Langmuir equation. Then, a simplified model for studying the dynamics of the cavitation bubble or bubble excited by the acoustic waves is proposed. The major contribution is to simplify the full model with PDEs to a set of coupled ordinary differential equations (ODEs). Specifically, two energy PDEs are converted to three ODEs by coupling the boundary conditions. The comparison among the full model and other simplified models is used to validate the accuracy and superiority of the simplified model, from which the application range of the proposed simplified model can be determined.

Production and dispersion of free radicals from transient cavitation Bubbles: An integrated numerical scheme and applications

As an advanced oxidation process with a wide range of applications, sonochemistry relies on acoustic cavitation to induce free radicals for degrading chemical contaminants. The complete process includes two critical steps: the radical production inside the cavitation bubble, and the ensuing dispersion of these radicals into the bulk solution. To grasp the physicochemical details in this process, we developed an integrated numerical scheme with the ability to quantitatively describe the radical production-dispersion behavior. It employs coupled simulations of bubble dynamics, intracavity chemical reactions, and diffusion-reaction-dominated mass transport in aqueous solutions. Applying this method to the typical case of argon and oxygen bubbles, the production mechanism for the main radicals is revealed. Moreover, the temporal-spatial distribution of the radicals in the liquid phase is presented. The results demonstrate that the enhanced radical production observed in oxygen bubbles can be traced to the initiation reaction O₂ + H₂O → OH+HO₂, which requires relatively low activation energy. In the outside liquid region, the dispersion of radicals is limited by robust recombination reactions. The simulated penetration depth of OH is around 0.2 μm and agrees with reported experimental measurements. The proposed numerical approach can be employed to better capture the radical activity and is instrumental in optimizing the engineering application of sonochemistry.

The importance of chemical mechanisms in sonochemical modelling

A state-of-the-art chemical mechanism is introduced to properly describe chemical processes inside a harmonically excited spherical bubble placed in water and saturated with oxygen. The model uses up-to-date Arrhenius-constants, collision efficiency factors and takes into account the pressure-dependency of the reactions. Duplicated reactions are also applied, and the backward reactions rates are calculated via suitable thermodynamic equilibrium conditions. Our proposed reaction mechanism is compared to three other chemical models that are widely applied in sonochemistry and lack most of the aforementioned modelling issues. In the governing equations, only the reaction mechanisms are compared, all other parts of the models are identical. The chemical yields obtained by the different modelling techniques are taken at the maximum expansion of the bubble. A brief parameter study is made with different pressure amplitudes and driving frequencies at two equilibrium bubble sizes. The results show that due to the deficiencies of the former reaction mechanisms employed in the sonochemical literature, several orders of magnitude differences of the chemical yields can be observed. In addition, the trends along a control parameter can also have dissimilar characteristics that might lead to false optimal operating conditions. Consequently, an up-to-date and accurate chemical model is crucial to make qualitatively and quantitatively correct conclusions in sonochemistry.

On the threshold of 1/2 order subharmonic emissions in the oscillations of ultrasonically excited bubbles

The pressure threshold for 1/2 order subharmonic (SH) emissions and period doubling during the oscillations of ultrasonically excited bubbles is thought to be minimum when the bubble is sonicated with twice its resonance frequency (f_r). This estimate is based on studies that simplified or neglected the effects of thermal damping. In this work, the nonlinear dynamics of ultrasonically excited bubbles is investigated accounting for the thermal dissipation. Results are visualized using bifurcation diagrams as a function of pressure. Here we show that, and depending on the gas, the pressure threshold for 1/2 order SHs can be minimum at a frequency between 0.5f_r≤f≤0.6f_r. In this frequency range, the generation of 1/2 order SHs are due to the occurrence of 5/2 order ultra-harmonic resonance. The stability of such oscillations is size dependent. For an air bubble immersed in water, only bubbles bigger than 1 μm in diameter are able to emit non-destructive SHs in these frequency ranges.

Modeling the thermal behavior of an acoustically driven gas bubble

Numerical simulation of an acoustically driven gas bubble is usually achieved by solving a Rayleigh-Plesset-type equation, in which the time-dependent pressure of the gas inside the bubble needs to be appropriately modeled. This is done in most existing methods by assuming a polytropic relation between the gas pressure and the bubble volume, which sometimes oversimplifies the thermal interaction between the bubble and the ambient liquid. In this paper, a model is developed aiming to perform an accurate and efficient calculation of the pressure variation in the bubble. The approach is different from that in the recent paper by the author and his collaborator which used a combination of an integral and a collocation method to solve the energy equation in the gas [Zhou and Prosperetti (2020). J. Fluid Mech. 901, R3]. The starting point of the proposed method in this paper is the gas continuity equation which is manipulated to lead to three ordinary differential equations. In this way, the thermal behavior of an oscillating gas bubble is captured at a modest coding and computational cost.

Hybrid sideways/longitudinal swimming in the monoflagellate Shewanella oneidensis: from aerotactic band to biofilm

Shewanella oneidensis MR-1 are facultative aerobic electroactive bacteria with an appealing potential for sustainable energy production and bioremediation. They gather around air sources, forming aerotactic bands and biofilms. Here, we experimentally follow the evolution of the band around an air bubble, and we find good agreement with the numerical solutions of the pertinent transport equations. Video microscopy reveals a transition between motile and non-motile MR-1 upon oxygen depletion, preventing further development of the biofilm. We discover that MR-1 can alternate between longitudinal fast and sideways slow swimming. The resulting bimodal velocity distributions change in response to different oxygen concentrations and gradients, supporting the biological functions of aerotaxis and confinement.

Nonlinear power loss in the oscillations of coated and uncoated bubbles: Role of thermal, radiation and encapsulating shell damping at various excitation pressures

This study presents the fundamental equations governing the pressure dependent disipation mechanisms in the oscillations of coated bubbles. A simple generalized model (GM) for coated bubbles accounting for the effect of compressibility of the liquid is presented. The GM was then coupled with nonlinear ODEs that account for the thermal effects. Starting with mass and momentum conservation equations for a bubbly liquid and using the GM, nonlinear pressure dependent terms were derived for power dissipation due to thermal damping (Td), radiation damping (Rd) and dissipation due to the viscosity of liquid (Ld) and coating (Cd). The pressure dependence of the dissipation mechanisms of the coated bubble have been analyzed. The dissipated energies were solved for uncoated and coated 2-20 μm in bubbles over a frequency range of 0.25f_r-2.5f_r (f_r is the bubble resonance) and for various acoustic pressures (1 kPa-300 kPa). Thermal effects were examined for air and C3F8 gas cores. In the case of air bubbles, as pressure increases, the linear thermal model looses accuracy and accurate modeling requires inclusion of the full thermal model. However, for coated C3F8 bubbles of diameter 1-8 μm, which are typically used in medical ultrasound, thermal effects maybe neglected even at higher pressures. For uncoated bubbles, when pressure increases, the contributions of Rd grow faster and become the dominant damping mechanism for pressure dependent resonance frequencies (e.g. fundamental and super harmonic resonances). For coated bubbles, Cd is the strongest damping mechanism. As pressure increases, Rd contributes more to damping compared to Ld and Td. For coated bubbles, the often neglected compressibility of the liquid has a strong effect on the oscillations and should be incorporated in models. We show that the scattering to damping ratio (STDR), a measure of the effectiveness of the bubble as contrast agent, is pressure dependent and can be maximized for specific frequency ranges and pressures.

A strict formulation of a nonlinear Helmholtz equation for the propagation of sound in bubbly liquids. Part II: Application to ultrasonic cavitation

This paper addresses one of the greatest challenges in sonochemistry that has impaired scaling up ultrasonic processes, which is the lack of models capable of predicting the pressure distribution in sonoreactors. This work studies the effect of acoustic pressure on the transmission of sound thought cavitating bubbly liquids by utilizing the nonlinear Helmholtz equation that was demonstrated on the paper part I. The model showed that the wave number and the attenuation can be estimated from the bubble dynamics of inertial bubbles and the local bubble density. The linear model of Commander and Prosperetti is encompassed by the nonlinear model. The model was employed to predict the pressure distribution below an ultrasonic horn tip achieving a relatively close prediction of the experimental data and certainly an accurate qualitative description of the distribution of the pressure field in spite of the simplifications of the model and the assumptions of unknown variables such as the bubble density, bubble distribution and the vessel boundary conditions.

Simulation of cavitation enhanced temperature elevation in a soft tissue during high-intensity focused ultrasound thermal therapy

The present study aims to investigate temperature distribution caused by bubble oscillations in a soft tissue during focused ultrasound therapy by introducing a coupled temperature-cavitation model. The proposed model is capable of describing bubble dynamics, viscoelastic properties of surrounding tissue-like medium, temperature distribution inside and outside the bubble, vapor diffusion within the bubble and vapor flux through the bubble wall to the exterior. The continuous temperature distribution inside and outside the oscillating bubble in soft tissue subject to ultrasound wave with high acoustic pressure is presented. The temperature close to the bubble wall can reach the value of about 10³ K. The elasticity of soft tissue reduces temperature values. The relaxation time effect strongly depends on the period of the ultrasound wave. If the vapor mass flux effect is taken into account in the simulations, the rectified growth effect can be observed, which can lead to the decrease of the temperature values. Due to the growth of the bubble, the effects of elasticity and relaxation time on the temperature become less prominent during several bubble oscillation cycles. The impact of cavitation heat source terms on the exterior temperature was examined and led us to draw conclusion that, even though these heat sources can increase the outside temperature values, they can not be treated as main mechanisms for the temperature elevation during a few microseconds. The performed comparison with uncoupled conventional model for the outside temperature calculation revealed that coupling with inside temperature model delivers incomparably higher values to the bubble's exterior and, therefore, it is essential for the accurate description of the treatment process.

Mechanical damage induced by the appearance of rectified bubble growth in a viscoelastic medium during boiling histotripsy exposure

In boiling histotripsy, the presence of a boiling vapour bubble and understanding of its dynamic behaviour are crucially important for the initiation of the tissue fractionation process and for the control of the size of a lesion produced. Whilst many in vivo studies have shown the feasibility of using boiling histotripsy in mechanical fractionation of solid tumours, not much is known about the evolution of a boiling vapour bubble in soft tissue induced by boiling histotripsy. The main objective of this present study is therefore to investigate the formation and dynamic behaviour of a boiling vapour bubble which occurs under boiling histotripsy insonation. Numerical and experimental studies on the bubble dynamics induced in optically transparent tissue-mimicking gel phantoms exposed to the field of a 2.0 MHz High Intensity Focused Ultrasound (HIFU) transducer were performed with a high speed camera. The Gilmore-Zener bubble model coupled with the Khokhlov-Zabolotskaya-Kuznetsov and the Bio-heat Transfer equations was used to simulate bubble dynamics driven by boiling histotripsy waveforms (nonlinear-shocked wave excitation) in a viscoelastic medium as functions of surrounding temperature and of tissue elasticity variations. In vivo animal experiments were also conducted to examine cellular structures around a freshly created lesion in the liver resulting from boiling histotripsy. To the best of our knowledge, this is the first study reporting the numerical and experimental evidence of the appearance of rectified bubble growth in a viscoelastic medium. Accounting for tissue phantom elasticity adds a mechanical constraint on vapour bubble growth, which improves the agreement between the simulation and the experimental results. In addition the numerical calculations showed that the asymmetry in a shockwave and water vapour transport can result in rectified bubble growth which could be responsible for HIFU-induced tissue decellularisation. Strain on liver tissue induced by this radial motion can damage liver tissue while preserving blood vessels.

Bubble dynamics in boiling histotripsy

Boiling histotripsy is a non-invasive, cavitation-based ultrasonic technique which uses a number of millisecond pulses to mechanically fractionate tissue. Though a number of studies have demonstrated the efficacy of boiling histotripsy for fractionation of solid tumours, treatment monitoring by cavitation measurement is not well studied because of the limited understanding of the dynamics of bubbles induced by boiling histotripsy. The main objectives of this work are to (a) extract qualitative and quantitative features of bubbles excited by shockwaves and (b) distinguish between the different types of cavitation activity for either a thermally or a mechanically induced lesion in the liver. A numerical bubble model based on the Gilmore equation accounting for heat and mass transfer (gas and water vapour) was developed to investigate the dynamics of a single bubble in tissue exposed to different High Intensity Focused Ultrasound fields as a function of temperature variation in the fluid. Furthermore, ex vivo liver experiments were performed with a passive cavitation detection system to obtain acoustic emissions. The numerical simulations showed that the asymmetry in a shockwave and water vapour transport are the key parameters which lead the bubble to undergo rectified growth at a boiling temperature of 100°C. The onset of rectified radial bubble motion manifested itself as (a) an increase in the radiated pressure and (b) the sudden appearance of higher order multiple harmonics in the corresponding spectrogram. Examining the frequency spectra produced by the thermal ablation and the boiling histotripsy exposures, it was observed that higher order multiple harmonics as well as higher levels of broadband emissions occurred during the boiling histotripsy insonation. These unique features in the emitted acoustic signals were consistent with the experimental measurements. These features can, therefore, be used to monitor (a) the different types of acoustic cavitation activity for either a thermal ablation or a mechanical fractionation process and (b) the onset of the formation of a boiling bubble at the focus in the course of HIFU exposure.

Depth effect on the inertial collapse of cavitation bubble under ultrasound: Special emphasis on the role of the wave attenuation

Acoustic cavitation concentrates and releases a very large amount of energy in localized areas, which can be used for many physical and chemical processes. Even though acoustic cavitation has been studied widely for decades in lab-scale sonoreactors, only few studies have been devoted to characterize this event in big-scale sonoreactors, where the liquid depth may have a critical influence on the bubble collapse. The present computational study furnished numerical data about the effect of depth (z = 0-10 m) on acoustic cavitation with special focus on the role of attenuation of the ultrasound wave on the dramatic conditions developed within bubbles at collapse. The used mathematical model takes into account the liquid compressibility, surface tension and viscosity, depth as well as the attenuation of the ultrasound wave with depth. It was found that the maximum bubble temperature (T_max) and pressure (p_max) at the collapse diminished considerably with deepening into water up to 10 m with a considerable contribution of the ultrasound wave attenuation in the overall reduction event. The reduction in T_max and p_max with depth was more pronounced at higher frequency (1000 kHz) and lower temperature (10 °C) in which losses of about up to 72% in T_max and till 94% in p_max (as compared with values at z = 0) were obtained at z = 10 m. Depending on operating conditions, i.e. frequency, acoustic intensity or liquid temperature, the ultrasound wave attenuation may contribute with up to 47% and 79% in the overall reductive effect of depth toward T_max and p_max, respectively. These results were discussed, interpreted and used to support some available experimental observations. Finally, the results of the present study may help in designing large-scale sonoreactors through providing data about the effect of one of the missing links between lab-scale sonoreactors and industrial large-scale sonoreactors.

The effects of heat and mass diffusion on freely oscillating bubbles in a viscoelastic, tissue-like medium

In certain cavitation-based ultrasound techniques, the relative importance of thermally vs mechanically induced damage is unclear. As a first step to investigate this matter, a numerical model for bubble dynamics in tissue-like, viscoelastic media is presented in which full thermal effects are included inside and outside the bubble, as well as interdiffusion of vapor and non-condensible gas inside the bubble. Soft tissue is assumed to behave according to a Kelvin-Voigt model in which viscous and elastic contributions are additive. A neo-Hookean formulation, appropriate for finite-strain elasticity, accounts for the large deformations produced by cavitation. Numerical solutions to problems of relevance to therapeutic ultrasound are examined, and linear analysis is used to explain the underlying mechanisms. The dependence between the surrounding medium's elasticity (shear modulus) and the extent to which the effects of heat and mass transfer influence bubble dynamics is quantified. In particular, the oscillation properties are related to the eigenvalues determined from linear theory. Regimes under which a polytropic relation describes the heat transfer to sufficient accuracy are identified, for which the complexity and computational expense associated with solving full partial differential equations can be avoided.

A new pressure formulation for gas-compressibility dampening in bubble dynamics models

We formulated a pressure equation for bubbles performing nonlinear radial oscillations under ultrasonic high pressure amplitudes. The proposed equation corrects the gas pressure at the gas-liquid interface on inertial bubbles. This pressure formulation, expressed in terms of gas-Mach number, accounts for dampening due to gas compressibility during the violent collapse of cavitation bubbles and during subsequent rebounds. We refer to this as inhomogeneous pressure, where the gas pressure at the gas-liquid interface can differ to the pressure at the centre of the bubble, in contrast to homogenous pressure formulations that consider that pressure inside the bubble is spatially uniform from the wall to the centre. The pressure correction was applied to two bubble dynamic models: the incompressible Rayleigh-Plesset equation and the compressible Keller and Miksis equation. This improved the predictions of the nonlinear radial motion of the bubble vs time obtained with both models. Those simulations were also compared with other bubble dynamics models that account for liquid and gas compressibility effects. It was found that our corrected models are in closer agreement with experimental data than alternative models. It was concluded that the Rayleigh-Plesset family of equations improve accuracy by using our proposed pressure correction.

Numerical modeling of bubble dynamics in viscoelastic media with relaxation

Cavitation occurs in a variety of non-Newtonian fluids and viscoelastic materials. The large-amplitude volumetric oscillations of cavitation bubbles give rise to high temperatures and pressures at collapse, as well as induce large and rapid deformation of the surroundings. In this work, we develop a comprehensive numerical framework for spherical bubble dynamics in isotropic media obeying a wide range of viscoelastic constitutive relationships. Our numerical approach solves the compressible Keller-Miksis equation with full thermal effects (inside and outside the bubble) when coupled to a highly generalized constitutive relationship (which allows Newtonian, Kelvin-Voigt, Zener, linear Maxwell, upper-convected Maxwell, Jeffreys, Oldroyd-B, Giesekus, and Phan-Thien-Tanner models). For the latter two models, partial differential equations (PDEs) must be solved in the surrounding medium; for the remaining models, we show that the PDEs can be reduced to ordinary differential equations. To solve the general constitutive PDEs, we present a Chebyshev spectral collocation method, which is robust even for violent collapse. Combining this numerical approach with theoretical analysis, we simulate bubble dynamics in various viscoelastic media to determine the impact of relaxation time, a constitutive parameter, on the associated physics. Relaxation time is found to increase bubble growth and permit rebounds driven purely by residual stresses in the surroundings. Different regimes of oscillations occur depending on the relaxation time.

Vaporization dynamics of volatile perfluorocarbon droplets: a theoretical model and in vitro validation

PURPOSE

Perfluorocarbon (PFC) microdroplets, called phase-change contrast agents (PCCAs), are a promising tool in ultrasound imaging and therapy. Interest in PCCAs is motivated by the fact that they can be triggered to transition from the liquid state to the gas state by an externally applied acoustic pulse. This property opens up new approaches to applications in ultrasound medicine. Insight into the physics of vaporization of PFC droplets is vital for effective use of PCCAs and for anticipating bioeffects. PCCAs composed of volatile PFCs (with low boiling point) exhibit complex dynamic behavior: after vaporization by a short acoustic pulse, a PFC droplet turns into a vapor bubble which undergoes overexpansion and damped radial oscillation until settling to a final diameter. This behavior has not been well described theoretically so far. The purpose of our study is to develop an improved theoretical model that describes the vaporization dynamics of volatile PFC droplets and to validate this model by comparison with in vitro experimental data.

METHODS

The derivation of the model is based on applying the mathematical methods of fluid dynamics and thermodynamics to the process of the acoustic vaporization of PFC droplets. The used approach corrects shortcomings of the existing models. The validation of the model is carried out by comparing simulated results with in vitro experimental data acquired by ultrahigh speed video microscopy for octafluoropropane (OFP) and decafluorobutane (DFB) microdroplets of different sizes.

RESULTS

The developed theory allows one to simulate the growth of a vapor bubble inside a PFC droplet until the liquid PFC is completely converted into vapor, and the subsequent overexpansion and damped oscillations of the vapor bubble, including the influence of an externally applied acoustic pulse. To evaluate quantitatively the difference between simulated and experimental results, the L2-norm errors were calculated for all cases where the simulated and experimental results are compared. These errors were found to be in the ranges of 0.043-0.067 and 0.037-0.088 for OFP and DFB droplets, respectively. These values allow one to consider agreement between the simulated and experimental results as good. This agreement is attained by varying only 2 of 16 model parameters which describe the material properties of gaseous and liquid PFCs and the liquid surrounding the PFC droplet. The fitting parameters are the viscosity and the surface tension of the surrounding liquid. All other model parameters are kept invariable.

CONCLUSIONS

The good agreement between the theoretical and experimental results suggests that the developed model is able to correctly describe the key physical processes underlying the vaporization dynamics of volatile PFC droplets. The necessity of varying the parameters of the surrounding liquid for fitting the experimental curves can be explained by the fact that the parts of the initial phospholipid shell of PFC droplets remain on the surface of vapor bubbles at the oscillatory stage and their presence affects the bubble dynamics.

Effect of surfactants on single bubble sonoluminescence behavior and bubble surface stability

The effect of surfactants on the radial dynamics of a single sonoluminescing bubble has been investigated. Experimentally, it is observed that an increase in the surfactant concentration leads to a decline in the oscillation amplitude and hence light emission intensity. Numerical simulations support this result, showing that under the driving pressures required to achieve single bubble sonoluminescence (SBSL), the surface properties, namely, the surface elasticity and dilatational viscosity, contribute to the damping of the radial amplitude in the bubble oscillation. In most cases this stabilizes the bubble surface, and contributes to a decreased light intensity. A stronger driving pressure is necessary to achieve equivalent light emission to a surfactant-free bubble. However, as the driving pressure is increased, the surface stability also decreases, making it practically very difficult for a bubble to achieve high SBSL intensities in concentrated surfactant solutions. Although more stable owing to more mild pulsations, the instability mechanism for a surfactant-coated bubble at higher ambient radii is more likely to be of the Rayleigh-Taylor type than that of a clean bubble at the same given acoustic parameters, which can lead to bubble disintegration before correcting mechanisms can bring the bubble back into the stable sonoluminescence regime.

Radical production inside an acoustically driven microbubble

The chemical production of radicals inside acoustically driven bubbles is determined by the local temperature inside the bubbles and by their composition at collapse. By means of a previously validated ordinary differential equations (ODE) model [L. Stricker, A. Prosperetti, D. Lohse, Validation of an approximate model for the thermal behavior in acoustically driven bubbles, J. Acoust. Soc. Am. 130 (5) (2011) 3243-3251], based on boundary layer assumption for mass and heat transport, we study the influence of different parameters on the radical production. We perform different simulations by changing the driving frequency and pressure, the temperature of the surrounding liquid and the composition of the gas inside the bubbles. In agreement with the experimental conditions of new generation sonochemical reactors, where the bubbles undergo transient cavitation oscillations [D. F. Rivas, L. Stricker, A. Zijlstra, H. Gardeniers, D. Lohse, A. Prosperetti, Ultrasound artificially nucleated bubbles and their sonochemical radical production, Ultrason. Sonochem. 20 (1) (2013) 510-524], we mainly concentrate on the initial chemical transient and we suggest optimal working ranges for technological applications. The importance of the chemical composition at collapse is reflected in the model, including the role of entrapped water vapor. We in particular study the exothermal reactions taking place in H2 and O2 mixtures. At the exact stoichiometric mixture 2:1 the highest internal bubble temperatures are achieved.

Interacting bubble clouds and their sonochemical production

An acoustically driven air pocket trapped in a pit etched on a surface can emit a bubble cluster. When several pits are present, the resulting bubble clusters interact in a nontrivial way. Fernández Rivas et al. [Angew. Chem. Int. Ed. 49, 9699-9701 (2010)] observed three different behaviors at increasing driving power: clusters close to their "mother" pits, clusters attracting each other but still well separated, and merging clusters. The last is highly undesirable for technological purposes as it is associated with a reduction of the radical production and an enhancement of the erosion of the reactor walls. In this paper, the conditions for merging to occur are quantified in the case of two clusters, as a function of the following control parameters: driving pressure, distance between the two pits, cluster radius, and number of bubbles within each cluster. The underlying mechanism, governed by the secondary Bjerknes forces, is strongly influenced by the nonlinearity of the bubble oscillations and not directly by the number of nucleated bubbles. The Bjerknes forces are found to dampen the bubble oscillations, thus reducing the radical production. Therefore, the increased number of bubbles at high power could be the key to understanding the experimental observation that, above a certain power threshold, any further increase of the driving does not improve the sonochemical efficiency.

The role of gas in ultrasonically driven vapor bubble growth

In this paper we study both experimentally and theoretically the dynamics of an ultrasound-driven vapor bubble of perfluoropentane (PFP) inside a droplet of the same liquid, immersed in a water medium superheated with respect to the PFP boiling point. We determine the temporal evolution of the bubble radius with ultra-high speed imaging at 20 million frames per second. In addition, we model the vapor-gas bubble dynamics, based on a Rayleigh-Plesset-type equation, including thermal and gas diffusion inside the liquid. We compare the numerical results with the experimental data and find good agreement. We underline the fundamental role of gas diffusion in order to prevent total recondensation of the bubble at collapse.

Ultrasound artificially nucleated bubbles and their sonochemical radical production

We describe the ejection of bubbles from air-filled pits micromachined on a silicon surface when exposed to ultrasound at a frequency of approximately 200 kHz. As the pressure amplitude is increased the bubbles ejected from the micropits tend to be larger and they interact in complex ways. With more than one pit, there is a threshold pressure beyond which the bubbles follow a trajectory parallel to the substrate surface and converge at the center point of the pit array. We have determined the size distribution of bubbles ejected from one, two and three pits, for three different pressure amplitudes and correlated them with sonochemical OH· radical production. Experimental evidence of shock wave emission from the bubble clusters, deformed bubble shapes and jetting events that might lead to surface erosion are presented. We describe numerical simulations of sonochemical conversion using the empirical bubble size distributions, and compare the calculated values with experimental results.