Oscillations of polymeric microbubbles: effect of the encapsulating shell

A model for the oscillation of gas bubbles encapsulated in a thin shell has been developed. The model depends on viscous and elastic properties of the shell, described by thickness, shear modulus, and shear viscosity. This theory was used to describe an experimental ultrasound contrast agent from Nycomed, composed of air bubbles encapsulated in a polymer shell. Theoretical calculations were compared with measurements of acoustic attenuation at amplitudes where bubble oscillations are linear. A good fit between measured and calculated results was obtained. The results were used to estimate the viscoelastic properties of the shell material. The shell shear modulus was estimated to between 10.6 and 12.9 MPa, the shell viscosity was estimated to between 0.39 and 0.49 Pas. The shell thickness was 5% of the particle radius. These results imply that the particles are around 20 times more rigid than free air bubbles, and that the oscillations are heavily damped, corresponding to Q-values around 1. We conclude that the shell strongly alters the acoustic behavior of the bubbles: The stiffness and viscosity of the particles are mainly determined by the encapsulating shell, not by the air inside.     

I. Formation and rise of a bubble stream in a viscous liquid

The continuous emission of gas bubbles from a single ejection orifice immersed in a viscous fluid is considered. We first present a semi empirical model of spherical bubble growth under constant flow conditions to predict the bubble volume at the detachment stage. In a second part, we propose a physical model to describe the rise velocity of in-line interacting bubbles and we derive an expression for the net viscous force acting on the surrounding fluid. Experimental results for air/water-glycerol systems are presented for a wide range of fluid viscosity and compared with theoretical predictions. An imagery technique was used to determine the bubble size and rise velocity. The effects of fluid viscosity, gas flow rate, orifice diameter and liquid depth on the bubble stream dynamic were analyzed. We have further studied the effect of large scale recirculation flow and the influence of a neighbouring bubble stream on the bubble growth and rising velocity. Received: 23 July 1997 / Revised: 16 December 1997 / Accepted: 11 May 1998     

Maxwell rheological model for lipid-shelled ultrasound microbubble contrast agents

The present paper proposes a model that describes the encapsulation of microbubble contrast agents by the linear Maxwell constitutive equation. The model also incorporates the translational motion of contrast agent microbubbles and takes into account radiation losses due to the compressibility of the surrounding liquid. To establish physical features of the proposed model, comparative analysis is performed between this model and two existing models, one of which treats the encapsulation as a viscoelastic solid following the Kelvin-Voigt constitutive equation and the other assumes that the encapsulating layer behaves as a viscous Newtonian fluid. Resonance frequencies, damping coefficients, and scattering cross sections for the three shell models are compared in the regime of linear oscillation. Translational displacements predicted by the three shell models are examined by numerically calculating the general, nonlinearized equations of motion for weakly nonlinear excitation. Analogous results for free bubbles are also presented as a basis to which calculations made for encapsulated bubbles can be related. It is shown that the Maxwell shell model possesses specific physical features that are unavailable in the two other models.     

Simulation of a bubble chain in a container of high aspect ratio exposed to a magnetic field

The influence of a homogeneous magnetic field acting in the direction of gravity on a bubble chain is studied with phase-resolving numerical simulations. The bubbles rise in a narrow container filled with liquid metal. Individual bubbles are represented by an immersed boundary method with the bubble shape being described by spherical harmonics and deformed by the surrounding liquid metal. A Gaussian bubble size distribution is realized as suggested by corresponding experiments. Bubble-bubble and bubble-wall interactions are modelled based on a repelling potential. With a magnetic field, the averaged trajectory of the bubble chain becomes more rectilinear, and the transverse dispersion is reduced. The average rise velocity decreases under the impact of the field.     

II. Recirculation flow induced by a bubble stream rising in a viscous liquid

The recirculation flow induced by the rising motion of a bubble stream in a viscous fluid within an open-top rectangular enclosure is studied. The three-dimensional volume averaged conservation equations are solved by a control-volume method using a hybrid finite differencing scheme to describe the liquid phase hydrodynamics. The momentum exhange between the bubbles and the liquid phase is modeled with a source term equals to the volumetric buoyancy force acting on the gas in the bubble stream. The volumetric buoyancy force accounts for in line interactions between bubbles through the average gas volume fraction in the gas liquid column which depends on the size and the rising velocity of bubbles. The fluid flow within an open-top rectangular enclosure is further investigated by particle image velocimetry for a bubble stream rising in a water-glycerol solution. The measured fluid velocities in a vertical plane are compared with the predictions of the numerical model over a wide range of fluid viscosity (43 mPa s-800 mPa s) and gas flow rates. Finally, the recirculation flows resulting from the interaction of two neighbouring vertical bubble streams are studied. Received: 23 July 1997 / Revised: 19 December 1997 / Accepted: 11 May 1998     

Acoustic manifestations of gas hydrate shelled bubbles

The hydrocarbon seeps emitting buoyant bubble plumes from seafloor vents—gas flares have been actively investigated in different regions of the World Ocean, in particular, on the Sakhalin slope in the Sea of Okhotsk. The gas flares can be easily detected by regular echo sounders, because the scattering cross section of a gas bubble is large. Within the gas-hydrate stability zone—for high hydrostatic pressures and low temperatures, methane-hydrate ice skins are formed on rising seep bubbles which are typically methane. The objective of the present study was to develop a suitable model describing rheological characteristics of gas-hydrate shell and to analyze acoustic manifestations of such bubbles for the frequency range used in marine field experiments.     

Bubble acoustical scattering cross section under multi-frequency acoustic excitation

The acoustical scattering cross section is usually employed to evaluate the scattering ability of the bubbles when they are excited by the incident acoustic waves. This parameter is strongly related to many important applications of performance prediction for search sonar or underwater telemetry, acoustical oceanography, acoustic cavitation, volcanology, and medical and industrial ultrasound. In the present paper, both the analytical and numerical analysis results of the acoustical scattering cross section of a single bubble under multi-frequency excitation are obtained. The nonlinear characteristics(e.g.,harmonics, subharmonics, and ultraharmonics) of the scattering cross section curve under multi-frequency excitation are investigated compared with single-frequency excitation. The influence of several paramount parameters(e.g., bubble equilibrium radius, acoustic pressure amplitude, and acoustic frequencies) in the multi-frequency system on the predictions of scattering cross section is discussed. It is shown that the combination resonances become significant in the multi-frequency system when the acoustic power is big enough, and the acoustical scattering cross section is promoted significantly within a much broader range of bubble sizes and acoustic frequencies due to the generation of more resonances.     

Numerical simulation of bubble dynamics in a Phan-Thien–Tanner liquid: Non-linear shape and size oscillatory response under periodic pressure

We study non-linear bubble oscillations driven by an acoustic pressure with the bubble being immersed in a viscoelastic, Phan-Thien–Tanner liquid. Solution is provided numerically through a method which is based on a finite element discretization of the Navier–Stokes flow equations. The proposed computational approach does not rely on the solution of the simplified Rayleigh–Plesset equation, is not limited in studying only spherically symmetric bubbles and provides coupled solutions for the velocity, stress fields and bubble interface. We present solutions for non-spherical bubbles, with asphericity being addressed by means of Legendre polynomials or associated Legendre functions. A parametric investigation of the bubble dynamical oscillatory response as a function of the fluid rheological properties shows that the amplitude of bubble oscillations drastically increases as liquid elasticity (quantified by the Deborah number) increases or as liquid viscosity decreases (quantified by the Reynolds number). Extensive numerical calculations demonstrate that increasing elasticity and/or viscosity of the surrounding liquid tend to stabilize the shape anisotropy of an initially non-spherical bubble. Results are shown for pressure amplitudes 0.2–2 MPa and Deborah, Reynolds numbers in the intervals of 1–8 and 0.094–1.256, respectively.     

Sound scattering and localized heat deposition of pulse-driven microbubbles

The sound scattering of free microbubbles released from strongly driven ultrasound contrast agents with brittle shell (e.g., Sonovist) is studied numerically. At high peak pressure of the driving pulses, the bubbles respond nonlinearly with cross sections pronouncedly larger than in the linear case; a large portion of the energy is radiated into high frequency ultrasound. Subsequent absorption of these high frequencies in the surrounding liquid (blood) diminishes the effective scattering cross section drastically. The absorption results in highly localized heating, with a substantial temperature rise within the first few microm from the bubble surface. The maximum heating in 1 microm distance is strongly dependent on driving pressure. Temperature elevations of more than 100 K can be achieved for amplitudes of Pa approximately 30 atm, which coincides with the highest pressures used in ultrasound diagnostics. The perfectly spherical collapses assumed here occur rarely, and the heating is highly localized and transient (approximately 10 micros). Therefore, a thermal hazard would only be expected at driving pressures beyond the diagnostic range.     

Acoustic manifestation of bubbles frozen in an ice sheet

Hydrocarbon sources on the ocean floor produce buoyant bubble plumes, i.e., gas flares. In winter, bubbles reaching the surface freeze in an ice sheet. Such clouds of frozen bubbles are observable in Arctic seas and are usual elements of ice sheets of lakes, e.g., Lake Baikal. Based on the general solution of the problem of scattering by a sphere in an isotropic elastic medium, the frozen bubble scattering cross section is found. The theory of multiple scattering by frozen bubble plume is derived. The structure of low-frequency resonances corresponding to collective oscillations of a bubble cloud is described.     

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).     

Model for the dynamics of two interacting axisymmetric spherical bubbles undergoing small shape oscillations

Interaction between acoustically driven or laser-generated bubbles causes the bubble surfaces to deform. Dynamical equations describing the motion of two translating, nominally spherical bubbles undergoing small shape oscillations in a viscous liquid are derived using Lagrangian mechanics. Deformation of the bubble surfaces is taken into account by including quadrupole and octupole perturbations in the spherical-harmonic expansion of the boundary conditions on the bubbles. Quadratic terms in the quadrupole and octupole amplitudes are retained, and surface tension and shear viscosity are included in a consistent manner. A set of eight coupled second-order ordinary differential equations is obtained. Simulation results, obtained by numerical integration of the model equations, exhibit qualitative agreement with experimental observations by predicting the formation of liquid jets. Simulations also suggest that bubble-bubble interactions act to enhance surface mode instability.     

Nonlinear response to ultrasound of encapsulated microbubbles

The acoustic backscatter of encapsulated gas-filled microbubbles immersed in a weak compressible liquid and irradiated by ultrasound fields of moderate to high pressure amplitudes is investigated theoretically. The problem is formulated by considering, for the viscoelastic shell of finite thickness, an isotropic hyperelastic neo-Hookean model for the elastic contribution in addition to a Newtonian viscous component. First and second harmonic scattering cross-sections have been evaluated and the quantitative influence of the driving pressure amplitude on the harmonic resonance frequencies for different initial equilibrium bubble sizes and for different encapsulating physical properties has been determined. Conditions for optimal second harmonic imaging have been also investigated and some regions in the parameters space where the second harmonic intensity is dominant over the fundamental have been identified. Results have been obtained for albumin, lipid and polymer encapsulating shells, respectively.     

Spatio-temporal dynamics of an encapsulated gas bubble in an ultrasound field

Coupled equations describing the radial and translational dynamics of an encapsulated gas bubble in an ultrasound field are derived by using the Lagrangian formalism. The equations generalize Church's theory by allowing for the translation motion of the bubble and radiation losses due to the compressibility of the surrounding liquid. The expression given by Church for the inner bubble radius corresponding to the unstrained state of the bubble shell is also refined, assuming that the shell can be of arbitrary thickness and impermeable to gas. Comparative linear analysis of the radial equation is carried out relative to Church's theory. It is shown that there are substantial departures from predictions of Church's theory. The proposed model is applied to evaluate radiation forces exerted on encapsulated bubbles and their translational displacements. It is shown that in the range of relatively high frequencies encapsulated bubbles are able to translate more efficiently than free bubbles of the equivalent size.     

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.     

Nonlinear dynamics and acoustic emissions of interacting cavitation bubbles in viscoelastic tissues

The cavitation-mediated bioeffects are primarily associated with the dynamic behaviors of bubbles in viscoelastic tissues, which involves complex interactions of cavitation bubbles with surrounding bubbles and tissues. The radial and translational motions, as well as the resultant acoustic emissions of two interacting cavitation bubbles in viscoelastic tissues were numerically investigated. Due to the bubble–bubble interactions, a remarkable suppression effect on the small bubble, whereas a slight enhancement effect on the large one were observed within the acoustic exposure parameters and the initial radii of the bubbles examined in this paper. Moreover, as the initial distance between bubbles increases, the strong suppression effect is reduced gradually and it could effectively enhance the nonlinear dynamics of bubbles, exactly as the bifurcation diagrams exhibit a similar mode of successive period doubling to chaos. Correspondingly, the resultant acoustic emissions present a progressive evolution of harmonics, subharmonics, ultraharmonics and broadband components in the frequency spectra. In addition, with the elasticity and/or viscosity of the surrounding medium increasing, both the nonlinear dynamics and translational motions of bubbles were reduced prominently. This study provides a comprehensive insight into the nonlinear behaviors and acoustic emissions of two interacting cavitation bubbles in viscoelastic media, it may contribute to optimizing and monitoring the cavitation-mediated biomedical applications.     

Real-Time Measurements and Modelling on Dynamic Behaviour of SonoVue Bubbles Based on Light Scattering Technology

The dynamic behaviour of SonoVue microbubbles, a new generation ultrasound contrast agent, is investigated in real time with light scattering method. Highly diluted SonoVue microbubbles are injected into a diluted gel made of xanthan gum and water. The responses of individual SonoVue bubbles to driven ultrasound pulses are measured. Both linear and nonlinear bubble oscillations are observed and the results suggest that SonoVue microbubbles can generate strong nonlinear responses. By fitting the experimental data of individual bubble responses with Sarkar＇s model, the shell coating parameter of the bubbles and dilatational viscosity is estimated to be 7.0 nm-s-Pa.     

Nonlinear oscillation and acoustic scattering of bubbles

The scattered acoustic pressure and scattered cross section of bubbles is studied using the scattered theory of bubbles. The nonlinear oscillations of bubbles and the scattering acoustic fields of a spherical bubble cluster are numerically simulated based on the bubble dynamic and fluid dynamic. The influences of the interaction between bubbles on scattering acoustic field of bubbles are researched. The results of numerical simulation show that the oscillation phases of bubbles are delayed to a certain extent at different positions in the bubble cluster, but the radii of bubbles during oscillation do not differ too much at different positions. Furthermore, directivity of the acoustic scattering of bubbles is obvious. The scattered acoustic pressures of bubbles are different at the different positions inside and outside of the bubble cluster. The scattering acoustic fields of a spherical bubble cluster depend on the driving pressure amplitude, driving frequency, the equilibrium radii of bubbles, bubble number and the radius of the spherical bubble cluster. These theoretical predictions provide a further understanding of physics behind ultrasonic technique and should be useful for guiding ultrasonic application.     

Numerical simulation of 3D bubbles rising in viscous liquids using a front tracking method

The rise of bubbles in viscous liquids is not only a very common process in many industrial applications, but also an important fundamental problem in fluid physics. An improved numerical algorithm based on the front tracking method, originally proposed by Tryggvason and his co-workers, has been validated against experiments over a wide range of intermediate Reynolds and Bond numbers using an axisymmetric model [J. Hua, J. Lou, Numerical simulation of bubble rising in viscous liquid, J. Comput. Phys. 22 (2007) 769–795]. In the current paper, this numerical algorithm is further extended to simulate 3D bubbles rising in viscous liquids with high Reynolds and Bond numbers and with large density and viscosity ratios representative of the common air–water two-phase flow system. To facilitate the 3D front tracking simulation, mesh adaptation is implemented for both the front mesh on the bubble surface and the background mesh. On the latter mesh, the governing Navier–Stokes equations for incompressible, Newtonian flow are solved in a moving reference frame attached to the rising bubble. Specifically, the equations are solved using a finite volume scheme based on the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm, and it appears to be robust even for high Reynolds numbers and high density and viscosity ratios. The 3D bubble surface is tracked explicitly using an adaptive, unstructured triangular mesh. The numerical model is integrated with the software package PARAMESH, a block-based adaptive mesh refinement (AMR) tool developed for parallel computing. PARAMESH allows background mesh adaptation as well as the solution of the governing equations in parallel on a supercomputer. Further, Peskin distribution function is applied to interpolate the variable values between the front and the background meshes. Detailed sensitivity analysis about the numerical modeling algorithm has been performed. The current model has also been applied to simulate a number of cases of 3D gas bubbles rising in viscous liquids, e.g. air bubbles rising in water. Simulation results are compared with experimental observations both in aspect of terminal bubble shapes and terminal bubble velocities. In addition, we applied this model to simulate the interaction between two bubbles rising in a liquid, which illustrated the model’s capability in predicting the interaction dynamics of rising bubbles.