Shape stability of a gas bubble in a soft solid

Predicting the onset of non-spherical oscillations of bubbles in soft matter is a fundamental cavitation problem with implications to sonoprocessing, polymeric materials synthesis, and biomedical ultrasound applications. The shape stability of a bubble in a Kelvin-Voigt viscoelastic medium with nonlinear elasticity, the simplest constitutive model for soft solids, is analytically investigated and compared to experiments. Using perturbation methods, we develop a model reducing the equations of motion to two sets of evolution equations: a Rayleigh-Plesset-type equation for the mean (volume-equivalent) bubble radius and an equation for the non-spherical mode amplitudes. Parametric instability is predicted by examining the natural frequency and the Mathieu equation for the non-spherical modes, which are obtained from our model. Our theoretical results show good agreement with published experiments of the shape oscillations of a bubble in a gelatin gel. We further examine the impact of viscoelasticity on the time evolution of non-spherical mode amplitudes. In particular, we find that viscosity increases the damping rate, thus suppressing the shape instability, while shear modulus increases the natural frequency, which changes the unstable mode. We also explain the contributions of rotational and irrotational fields to the viscoelastic stresses in the surroundings and at the bubble surface, as these contributions affect the damping rate and the unstable mode. Our analysis on the role of viscoelasticity is potentially useful to measure viscoelastic properties of soft materials by experimentally observing the shape oscillations of a bubble.     

Nonlinear non-spherical oscillations of gas bubble under a periodic variation of ambient liquid pressure

A mathematical model is constructed for the bubble dynamics, in which the interphase surface variation is presented in the form of a series in spherical harmonics, and the equations are written with the accuracy up to the squared amplitude of the distortion of the spherical shape of the bubble. In the oscillation regimes close to periodic sonoluminescence of a single bubble in a standing acoustic wave, the character of air bubble oscillations in water was studied depending on the bubble initial radius and the amplitude of the liquid pressure variation. It was found that non-spherical oscillations of bounded amplitude can take place outside the region of linearly stable spherical oscillations. Both the oscillations with a period equal to one or several periods of the liquid pressure variation and aperiodic oscillations are observed. It is shown that neglecting the distortions in the form of spherical harmonics with large numbers (i > 3) may lead to a change of oscillation regimes. The influence of distortions on the bubble surface shape for the harmonics with i > 8 is insignificant.     

The resonant mechanism of subdivision of a gas bubble in a fluid

Small nonlinear oscillations of a bubble in a fluid at the resonance of the frequencies of the radial mode and an arbitrary deformational mode 2 : 1 are considered. The deformational mode is determined by the associated Legendre polynomial with indices n = 2, 3, ??, m = 0, 1, ??, n. The energy transfer from the radial mode into the Legendre deformational mode is described by the method of invariant normalization. An analogy is established with oscillations of the material point on the string with the frequency ratio of the vertical mode to the horizontal one of 2. During the transfer, the amplitude of the Legendre mode with indices n, m exceeds the amplitude of radial oscillations by a factor of 3n at m = 0. As index m increases, the transfer time increases considerably and the maximal amplitude of the Legendre mode increases insignificantly in this case. From here, it is concluded that the deformational Legendre mode with indices n, m = n has the greatest probability to rise. The considered effect can serve as a mechanism of subdivision of gas bubbles under varying the external pressure in the fluid.     

Effects of argon sparging rate,ultrasonic power,and frequency on multibubble sonoluminescence spectra and bubble dynamics in NaCl aqueous solutions

The sonoluminescence spectra from acoustic cavitation in aqueous NaCl solutions are systematically studied in a large range of ultrasonic frequencies under variation of electrical power and argon sparging. At the same time, bubble dynamics are analysed by high-speed imaging. Sodium line and continuum emission are evaluated for acoustic driving at 34.5, 90, 150, 365, and 945 kHz in the same reactor vessel. The results show that the ratio of sodium line to continuum emission can be shifted by the experimental parameters: an increase in the argon flow increases the ratio, while an increase in power leads to a decrease. At 945 kHz, the sodium line is drastically reduced, while the continuum stays at elevated level. Bubble observations reveal a remarkable effect of argon in terms of bubble distribution and stability: larger bubbles of non-spherical shapes form and eject small daughter bubbles which in turn populate the whole liquid. As a consequence, the bubble interactions (splitting, merging) appear enhanced which supports a link between non-spherical bubble dynamics and sodium line emission.     

Ultrasonic scattering cross sections of shell-encapsulated gas bubbles immersed in a viscoelastic liquid: first and second harmonics

The oscillations of gas bubbles, without shell, immersed in viscoelastic liquids and driven by an acoustic wave have been the subject of several investigations. They demonstrate that the viscosity coefficient and the spring constant of the liquid have significant influence on the scattering cross section of the gas bubble. For shell-encapsulated gas bubbles, the investigations have been concentrated to bubbles immersed in a pure viscous liquid. This present work computes the ultrasonic scattering cross section, first and second harmonics, of shell-encapsulated gas bubbles immersed in a viscoelastic liquid. The theoretical model of the bubble oscillation is based on the generalized Rayleigh-Plesset equation of motion of a spherical cavity immersed in a viscoelastic liquid represented by a three-parameter linear Oldroyd model. The scattering cross section is computed for Albunex type of bubble (shell thickness=15 nm, shell shear viscosity=1.77 Pas, shell modulus of rigidity=88.8 MPa) irradiated by a 3.5 MHz ultrasonic pressure wave with an amplitude of 30 kPa. The results demonstrate that encapsulated bubbles respond independently of the surrounding liquid being pure viscous or viscoelastic as long as the surrounding liquid shear viscosity is as low as 10(-3) Pas. Nevertheless, for higher shear viscosities, the bubble responds differently if the surrounding liquid is pure viscous or viscoelastic. In general, the scattering cross sections of first and second harmonics are larger for the viscoelastic liquid.     

Weakly nonlinear focused ultrasound in viscoelastic media containing multiple bubbles

To facilitate practical medical applications such as cancer treatment utilizing focused ultrasound and bubbles, a mathematical model that can describe the soft viscoelasticity of human body, the nonlinear propagation of focused ultrasound, and the nonlinear oscillations of multiple bubbles is theoretically derived and numerically solved. The Zener viscoelastic model and Keller–Miksis bubble equation, which have been used for analyses of single or few bubbles in viscoelastic liquid, are used to model the liquid containing multiple bubbles. From the theoretical analysis based on the perturbation expansion with the multiple-scales method, the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation, which has been used as a mathematical model of weakly nonlinear propagation in single phase liquid, is extended to viscoelastic liquid containing multiple bubbles. The results show that liquid elasticity decreases the magnitudes of the nonlinearity, dissipation, and dispersion of ultrasound and increases the phase velocity of the ultrasound and linear natural frequency of the bubble oscillation. From the numerical calculation of resultant KZK equation, the spatial distribution of the liquid pressure fluctuation for the focused ultrasound is obtained for cases in which the liquid is water or liver tissue. In addition, frequency analysis is carried out using the fast Fourier transform, and the generation of higher harmonic components is compared for water and liver tissue. The elasticity suppresses the generation of higher harmonic components and promotes the remnant of the fundamental frequency components. This indicates that the elasticity of liquid suppresses shock wave formation in practical applications.     

Dynamics of three cavitation bubbles with pulsation and symmetric deformation

A new system of dynamical equations was obtained by using the perturbation and potential flow theory to couple the pulsation and surface deformation of the second-order Legendre polynomials (P₂) of three bubbles in a line. The feasibility and effectiveness of the model were verified by simulating the radial oscillations, surface deformation with P₂, and shape evolution of three bubbles. The spherical radial pulsation and surface deformation of the three bubbles exhibit periodic behavior. The maximum secondary Bjerknes forces (SBFs) on the three bubbles are found not to depend on the system’s resonance frequency. Within a stable region, the SBFs of the three bubbles increase with increasing sound pressure amplitude but decrease with increasing distance between the bubbles. The primary Bjerknes force (PBF) on a bubble is significantly higher than the SBF on it.     

Bubble oscillation and inertial cavitation in viscoelastic fluids

Non-linear acoustic oscillations of gas bubbles immersed in viscoelastic fluids are theoretically studied. The problem is formulated by considering a constitutive equation of differential type with an interpolated time derivative. With the aid of this rheological model, fluid elasticity, shear thinning viscosity and extensional viscosity effects may be taken into account. Bubble radius evolution in time is analyzed and it is found that the amplitude of the bubble oscillations grows drastically as the Deborah number (the ratio between the relaxation time of the fluid and the characteristic time of the flow) increases, so that, even for moderate values of the external pressure amplitude, the behavior may become chaotic. The quantitative influence of the rheological fluid properties on the pressure thresholds for inertial cavitation is investigated. Pressure thresholds values in terms of the Deborah number for systems of interest in ultrasonic biomedical applications, are provided. It is found that these critical pressure amplitudes are clearly reduced as the Deborah number is increased.     

Shock wave emission from laser-induced cavitation bubbles in polymer solutions

The role of extensional viscosity on the acoustic emission from laser-induced cavitation bubbles in polymer solutions and near a rigid boundary is investigated by acoustic measurements. The polymer solutions consist of a 0.5% polyacrylamide (PAM) aqueous solution with a strong elastic component and a 0.5% carboxymethylcellulose (CMC) aqueous solution with a weak elastic component. A reduction of the maximum amplitude of the shock wave pressure and a prolongation of the oscillation period of the bubble were found in the elastic PAM solution. It might be caused by an increased resistance to extensional flow which is conferred upon the liquid by the polymer additive. In both polymer solutions, however, the shock pressure decays proportionally to r⁻¹ with increasing distance r from the emission centre.     

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 $\mathrm{\mu }\text{m}$ 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 $\mathrm{\mu }\text{m}$, 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.     

Modeling non-spherical oscillations and stability of acoustically driven shelled microbubbles

The oscillation and destruction of microbubbles under ultrasound excitation form the basis of contrast enhanced ultrasound imaging and microbubble assisted drug and gene delivery. A typical microbubble has a size of a few micrometers and consists of a gas core encapsulated by a shell. These bubbles can be driven into surface mode oscillations, which not only contribute to the measured acoustic signal but can lead to bubble destruction. Existing models of surface model oscillations have not considered the effects of a bubble shell. In this study a model was developed to study the surface mode oscillations in shelled bubbles. The effects of shell viscosity and elasticity on the surface mode oscillations were modeled using a Boussinesq-Scriven approach. Simulation was conducted using the model with various bubble sizes and driving acoustic pressures. The occurrence of surface modes and the number of ultrasound cycles needed for the occurrence were calculated. The simulation results show a significant difference between shelled bubbles and shell free bubbles. The shelled bubbles have reduced surface mode amplitudes and a narrower bubble size range within which these modes develop compared to shell free bubbles. The clinical implications were also discussed.     

A model for the dynamics of gas bubbles in soft tissue

Understanding the behavior of cavitation bubbles driven by ultrasonic fields is an important problem in biomedical acoustics. Keller-Miksis equation, which can account for the large amplitude oscillations of bubbles, is rederived in this paper and combined with a viscoelastic model to account for the strain-stress relation. The viscoelastic model used in this study is the Voigt model. It is shown that only the viscous damping term in the original equation needs to be modified to account for the effect of elasticity. With experiment determined viscoelastic properties, the effects of elasticity on bubble oscillations are studied. Specifically, the inertial cavitation thresholds are determined using R(max)/R(0), and subharmonic signals from the emission of an oscillating bubble are estimated. The results show that the presence of the elasticity increases the threshold pressure for a bubble to oscillate inertially, and subharmonic signals may only be detectable in certain ranges of radius and pressure amplitude. These results should be easy to verify experimentally, and they may also be useful in cavitation detection and bubble-enhanced imaging.     

Flow of an electrically conducting bubble liquid in the field of an electromagnetic force

Physical processes accompanying the flow of a conducting bubble liquid in crossed electric and magnetic fields are considered. Based on the general equations of mechanics of multiphase media, we develop a one-dimensional model of the flow of and heat exchange in a compressible bubble liquid when the phases are not in thermal and velocity equilibrium. The model is numerically investigated. It is demonstrated that, when the bubble liquid flows along the electromagnetic force vector, the bubbles lag behind the carrying flow and are compressed and warmed up. This causes oscillations of the bubble volume, as well as oscillations of the parameters of both the disperse and carrying phase. In particular, the compression of the bubbles reduces the volumetric gas content, as well as increases the effective conductivity of the flow and the electromagnetic force in the downstream direction. This sets conditions for crisis of the bubble flow when the electromagnetic force expels the bubbles against the main stream. On the basis of the solutions obtained, the efficiency of a gas compressor is calculated.     

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.     

Microbubble spectroscopy of ultrasound contrast agents

A new optical characterization of the behavior of single ultrasound contrast bubbles is presented. The method consists of insonifying individual bubbles several times successively sweeping the applied frequency, and to record movies of the bubble response up to 25 million frames/s with an ultrahigh speed camera operated in a segmented mode. The method, termed microbubble spectroscopy, enables to reconstruct a resonance curve in a single run. The data is analyzed through a linearized model for coated bubbles. The results confirm the significant influence of the shell on the bubble dynamics: shell elasticity increases the resonance frequency by about 50%, and shell viscosity is responsible for about 70% of the total damping. The obtained value for shell elasticity is in quantative agreement with previously reported values. The shell viscosity increases significantly with the radius, revealing a new nonlinear behavior of the phospholipid coating.     

Numerical modeling of ultrasonic cavitation by dividing coated microbubbles into groups

Homogeneous cavitation models usually use an average radius to predict the dynamics of all bubbles. However, bubbles with different sizes may have quite different dynamic characteristics. In this study, the bubbles are divided into several groups by size, and the volume-weighted average radius is used to separately calculate the dynamics of each group using a modified bubble dynamics equation. In the validation part, the oscillations of bubbles with two sizes are simulated by dividing them into 2 groups. Comparing with the predictions by the Volume of Fluid (VOF) method, the bubble dynamics of each size are precisely predicted by the proposed model. Then coated microbubbles with numerous sizes are divided into several groups in equal quantity, and the influence of the group number is analyzed. For bubble oscillations at f = 0.1 MHz and 1 MHz without ruptures, the oscillation amplitude is obviously under-estimated by the 1-group model, while they are close to each other after the group number increases to 9. For bubble ruptures triggered by Gaussian pulses, the predictions are close to each other when more than 5 groups are used.     

Energy analysis during acoustic bubble oscillations: Relationship between bubble energy and sonochemical parameters

In this work, energy analysis of an oscillating isolated spherical bubble in water irradiated by an ultrasonic wave has been theoretically studied for various conditions of acoustic amplitude, ultrasound frequency, static pressure and liquid temperature in order to explain the effects of these key parameters on both sonochemistry and sonoluminescence. The Keller–Miksis equation for the temporal variation of the bubble radius in compressible and viscous medium has been employed as a dynamics model. The numerical calculations showed that the rate of energy accumulation, dE/dt, increased linearly with increasing acoustic amplitude in the range of 1.5–3.0 atm and decreased sharply with increasing frequency in the range 200–1000 kHz. There exists an optimal static pressure at which the power w is highest. This optimum shifts toward a higher value as the acoustic amplitude increases. The energy of the bubble slightly increases with the increase in liquid temperature from 10 to 60 °C. The results of this study should be a helpful means to explain a variety of experimental observations conducted in the field of sonochemistry and sonoluminescence concerning the effects of operational parameters.     

Detonation waves in bubble-drop media

Wave processes in chemically active multicomponent media: liquid — gas bubbles — liquid drops have been studied experimentally. Existence of detonation waves in multicomponent (bubble-drop) media has been proved. Structure of detonation waves in bubble-drop and bubble media is qualitatively identical: detonation waves are solitary waves with pulsation profile the pressure behind which is close in value to the one in unperturbed medium. Propagation velocity of detonation waves in bubble and bubble-drop media drops with the increase in medium gas phase concentration and with the decrease in carrier liquid viscosity. Presence of liquid drops decreases detonation wave velocity compared with bubble medium that does not contain liquid drops. Detonation wave propagation in multicomponent media causes gas bubbles fragmentation as well as fragmentation of individual liquid drops. The work was financially supported by the Russian Foundation for Basic Research (Grant No. 04-03-33106).     

Influence of orifice spacing on twin bubbles formation in shear-thinning fluids: Laser image measurement

A laser image system for investigating the influence of orifice spacing on twin bubbles formation in shear-thinning fluid was established. The bubbles formation process at two orifices could be directly visualized and real-time recorded through computer by means of He–Ne laser as light source using the beam expanding and light amplification technology. The shape and size of bubbles generating in carboxymethylcellulose (CMC) aqueous solutions were studied experimentally at the orifices spacing 1D_o, 2D_o and 3D_o (D_o orifice diameter). The results reveal that the minute bubbles can be imaged clearly and amplified without distortion in the higher concentration solutions, and therefore the shape and size of bubbles are obtained accurately. With the increase of orifice spacing, bubble instantaneous volume in shear-thinning fluid decreases at the radial expansion stage but then increases at vertical elongation period, and the deformation and deviation of bubbles forming goes down due to the reduced interaction of adjacent bubble.     

Enhancement of oscillation amplitude of cavitation bubble due to acoustic wake effect in multibubble environment

Acoustic cavitation occurs in ultrasonic treatment causing various phenomena such as chemical synthesis, chemical decomposition, and emulsification. Nonlinear oscillations of cavitation bubbles are assumed to be responsible for these phenomena, and the neighboring bubbles may interact each other. In the present study, we numerically investigated the dynamic behavior of cavitation bubbles in multi-bubble systems. The results reveal that the oscillation amplitude of a cavitation bubble surrounded by other bubbles in a multi-bubble system becomes larger compared with that in the single-bubble case. It is found that this is caused by an acoustic wake effect, which reduces the pressure near a bubble surrounded by other bubbles and increases the time delay between the bubble contraction/expansion cycles and sound pressure oscillations. A new parameter, called “cover ratio” is introduced to quantitatively evaluate the variation in the bubble oscillation amplitude, the time delay, and the maximum bubble radius.