Encapsulated Bubbles Research Articles

Scenarios for the appearance of strange attractors in a model of three interacting microbubble contrast agents

We study nonlinear dynamics in a model of three interacting encapsulated gas bubbles in a liquid. The model is a system of three coupled nonlinear oscillators with an external periodic force. Such bubbles have numerous applications, for instance, they are used as contrast agents in ultrasound visualization. Certain types of bubbles oscillations may be beneficial or undesirable depending on a given application and, hence, the dependence of the regimes of bubbles oscillations on the control parameters is worth studying. We demonstrate that there is a wide variety of types of dynamics in the model by constructing a chart of dynamical regimes in the control parameters space. Here we focus on hyperchaotic attractors characterized by three positive Lyapunov exponents and strange attractors with one or two positive Lyapunov exponents possessing an additional zero Lyapunov exponent, which have not been observed previously in the context of bubbles oscillations. We also believe that we provide a first example of a hyperchaotic attractor with additional zero Lyapunov exponent. Furthermore, the mechanisms of the onset of these types of attractors are still not well studied. We identify two-parametric regions in the control parameter space where these hyperchaotic and chaotic attractors appear and study one-parametric routes leading to them. We associate the appearance of hyperchaotic attractors with three positive Lyapunov exponents with the inclusion of a periodic orbit with a three-dimensional unstable manifold, while the onset of chaotic oscillations with an additional zero Lyapunov exponent is connected to the partial synchronization of bubbles oscillations. We propose several underlying bifurcation mechanisms that explain the emergence of these regimes. We believe that these bifurcation scenarios are universal and can be observed in other systems of coupled oscillators.

Photo‐Induced Electrical Gating with the Inserting of an Interfacial Carrier Blocking Layer for Organic/Graphene Phototransistors

AbstractAlthough ultrahigh photoconductive gain is achieved in organic semiconductor‐sensitized graphene phototransistors, the response speed is limited by the photo‐induced carrier injection at the organic/graphene interface. In this work, an insulating hexagonal boron nitride layer is inserted between the organic heterostructure and graphene to block the interfacial carrier transport. Above the h‐BN layer, the organic heterostructure is adopted as the light‐absorbing layer, and the photoresponse of graphene is realized through electrical gating of the atop h‐BN dielectric layer, by the accumulated carriers in the organic heterostructure. Under this “photo‐induced electrical gating” mechanism, the recombination of nonequilibrium carriers takes place only within the organic heterostructure. Furthermore, monolayer perylene‐3,4,9,10‐tetracarboxylic dianhydride (PTCDA) is deposited on graphene before transfer of the h‐BN, by which the interfacial encapsulated bubbles are effectively reduced. As a step further, the trap states are pre‐filled with background light illumination, then the carrier recombination follows the band‐to‐band pathway. Finally, the response time of the organic/h‐BN/graphene phototransistor is reduced to 7.15 ms, and a high photoresponsivity is achieved even though the carrier multiplication is sacrificed, which is reconciled by the improvement of the device quantum efficiency.

Enhanced removal of toluene in heterogeneous aquifers through injecting encapsulated ozone micro-nano bubble water

Organic contaminants have a tendency to accumulate in low-permeability aquifers, making their removal challenging and creating a bottleneck in groundwater remediation efforts. The use of ozone micro-nano bubbles, due to their smaller size compared to traditional macrobubbles, shows potential for efficient penetration into the low-permeability aquifer and effective oxidization of contaminants. This study conducted batch experiments, column studies, and 2D tank experiments to systematically investigate the remediation efficiency of toluene in a heterogeneous aquifer using ozonated water (OW), ozone micro-bubble water (OMBW), and encapsulated ozone micro-nano bubble water (EOMBW) with rhamnolipid. Experimental results showed that rhamnolipid effectively increased the densities and reduced the sizes of micro-nano bubbles, leading to improved ozone preservation and enhanced toluene degradation. Nanobubbles exhibited higher mobility compared to microbubbles in porous media, while rhamnolipid increased the density of penetrated nanobubbles by 9.6 times. EOMBW demonstrated superior efficiency in oxidizing toluene in low-permeability aquifers, and a numerical model was developed to successfully simulate the ozone and toluene concentration. The model revealed that the increased oxidation rate by EOMBW was attributed to the preservation of ozone in micro-nano bubbles and the enhanced toluene oxidation rate. These findings contribute significantly to the application of EOMBW in heterogeneous aquifer remediation.

Sound scattering and radiation suppression by pressurized spherical shells.

Thin-shell models offer important insights into the complex process of sound-structure interaction but are found to be inconsistent with the rigorous thick-shell theory for fluid-loaded spherical shells. Here, linearized equations of motion of fluid-loaded, thin, spherical shells are re-derived from the first principles. The shell may be prestressed due to the difference in the static pressures in the internal and external fluids. Differences in the fluid-loading terms from previously proposed ad hoc models are identified and their significance is analyzed. Analytic solutions are derived of the problems of spherical sound wave scattering by a fluid-filled, prestressed spherical shell and resonant vibrations of the shell. The results reduce to a number of known exact and asymptotic solutions in appropriate limiting cases. The mathematical model of the shell vibrations is applied to characterize the influence of the shell's material properties and the prestress on passive suppression of low-frequency underwater sound radiation due to diffraction on an acoustically compliant sphere, such as an encapsulated gas bubble. Using soft rubber as the encapsulating membrane is found to preserve the sound suppression qualities of the free gas bubble.

Vibrations of a thin, prestressed, fluid-loaded spherical shell forced by a point sound source

This paper investigates vibrations of and sound scattering by a thin elastic spherical shell when static pressures are different in the fluids inside and outside the shell. The work is motivated by stratospheric balloons being employed to carry acoustic sensors and by the proposed use of large, encapsulated gas bubbles for passive suppression of underwater sound [O. A. Godin and A. B. Baynes, J. Acoust. Soc. Am., 143, EL67–EL73 (2018)]. Linearized equations of motion of thin, prestressed shells are derived from first principles. Differences in the fluid-loading terms from previously proposed ad hoc models are identified and their significance is analyzed. Analytic solutions are derived for vibrations of a spherical shell excited by an incident spherical sound wave and for acoustic fields in the internal and external fluids. The results are verified against exact solutions in several particular cases. The mathematical model of the shell vibrations is applied to characterize the influence of the shell’s material properties on passive suppression of underwater noise and to quantify the effect of wave scattering by the balloon on performance of balloon-borne infrasound sensors in the atmosphere. [Work supported by ONR.]

Describing the dynamics of a nonlinear viscoelastic shelled microbubble with an interface energy model

The present work introduces an interesting revamp to the recently proposed interface energy model [N. Dash and G. Tamadapu, J. Fluid Mech. 932, A26 (2022)] for gas-filled encapsulated bubbles (EBs) suspended in a viscous fluid. Here, the elastic and viscous parts of the viscoelastic shell material are described by the Gent hyperelastic material model and a polymer solute following upper-convected Maxwell (UCM) constitutive relations, respectively. Using the aforementioned framework, the integrodifferential type governing equation has been derived, and the physical features of the radial dynamics of the EB model are studied in detail using numerical simulations. The nonlinear behavior and the underlying implications of the newly introduced interface energy model for EBs are also investigated. It was observed that the interface parameters arising from the interface energy formulation and the Gent material model collectively introduce a stiffening effect into the EB model and the extension limit parameter at its lower values affects the radial dynamics of the bubble. Analysis has been carried out at different relaxation time scales, where the viscoelastic shell material resembles a fluid-like or solid-like behavior. The UCM-type viscous part of the viscoelastic shell material introduces strong nonlinear effects into the bubble model and significantly influences the EB’s behavior. For the present model, a detailed study has been conducted to capture the dynamic behavior of the bubble through the time series curves, phase space analysis, and the nonlinear frequency response of the bubble.

Impact dynamics of air-in-liquid compound droplets

Compound droplets are utilized in applications ranging from preparation of emulsion to biological cell printing and additive manufacturing. Here, we report on the impact dynamics of an air-in-liquid hollow compound droplet on a solid substrate. Contrary to the impact of pure droplets and compound droplets with liquids of similar densities, a compound droplet with an encapsulated air bubble demonstrates the formation of a counterjet in addition to the lamella. We experimentally investigate the influence of the size of the air bubble, liquid viscosity, and height of impact on the evolution of counterjet and the spreading characteristics of the lamella. For a given hollow droplet, the volume of the counterjet is observed to depend on the volume of air and liquid in the droplet and is independent of the viscosity of the liquid and impact velocity of the droplet. We observe that the spread characteristics, counterintuitively, do not vary significantly compared to that of a pure droplet having an identical liquid volume as the hollow droplet. We propose a model to predict the maximum spread during the impact of a hollow droplet based on the energy interaction between the spreading liquid and the liquid in the counterjet during the impact process.

Physico-mathematical model for multiple ultrasound-contrast-agent microbubbles encapsulated by a visco-elastic shell: Effect of shell compressibility on ultrasound attenuation

Owing to its potential for application in ultrasound-based diagnosis and therapy, the dynamics of a microbubble encapsulated by shells has long been theoretically investigated. However, outside of our research group, previous theories have been restricted to the case of single encapsulated bubble, whereas in practical diagnostic scenarios, multiple encapsulated bubbles are used as ultrasound contrast agents. In this study, the most recent theory for a single encapsulated bubble incorporating shell compressibility was extended for multiple encapsulated bubbles. Using the method of multiple scales, weakly nonlinear wave equation for one-dimensional ultrasound in liquids containing multiple encapsulated microbubbles was derived from the set of volumetric averaged equations for bubbly flow. It was found that the shell compressibility significantly increased the advection and dissipation effects of ultrasound. Further, five types of dissipation effects were compared with each other, and showed that the dissipation effects corresponding to shell compressibility were the highest.

Experimental acoustic characterization of an endoskeletal antibubble contrast agent: First results.

PurposeAn antibubble is an encapsulated gas bubble with an incompressible inclusion inside the gas phase. Current‐generation ultrasound contrast agents are bubble‐based: they contain encapsulated gas bubbles with no inclusions. The objective of this work is to determine the linear and nonlinear responses of an antibubble contrast agent in comparison to two bubble‐based ultrasound contrast agents, that is, reference bubbles and SonoVueTM.MethodsSide scatter and attenuation of the three contrast agents were measured, using single‐element ultrasound transducers, operating at 1.0, 2.25, and 3.5 MHz. The scatter measurements were performed at acoustic pressures of 200 and 300 kPa for 1.0 MHz, 300 kPa, and 450 kPa for 2.25 MHz, and 370 and 560 kPa for 3.5 MHz. Attenuation measurements were conducted at pressures of 13, 55, and 50 kPa for 1.0, 2.25, and 3.5 MHz, respectively. In addition, a dynamic contrast‐enhanced ultrasound measurement was performed, imaging the contrast agent flow through a vascular phantom with a commercial diagnostic linear array probe.ResultsAntibubbles generated equivalent or stronger harmonic signal, compared to bubble‐based ultrasound contrast agents. The second harmonic side‐scatter amplitude of the antibubble agent was up to 3 dB greater than that of reference bubble agent and up to 4 dB greater than that of SonoVueTM at the estimated concentration of 8×104 bubbles/mL. For ultrasound with a center transmit frequency of 1.0 MHz, the attenuation coefficient of the antibubble agent was 8.7 dB/cm, whereas the attenuation coefficient of the reference agent was 7.7 and 0.3 dB/cm for SonoVueTM. At 2.25 MHz, the attenuation coefficients were 9.7, 3.0, and 0.6 dB/cm, respectively. For 3.5 MHz, they were 4.4, 1.8, and 1.0 dB/cm, respectively. A dynamic contrast‐enhanced ultrasound recording showed the nonlinear signal of the antibubble agent to be 31% greater than for reference bubbles and 23% lower than SonoVueTM at a high concentration of 2×106 bubbles/mL.ConclusionEndoskeletal antibubbles generate comparable or greater higher harmonics than reference bubbles and SonoVueTM. As a result, antibubbles with liquid therapeutic agents inside the gas phase have high potential to become a traceable therapeutic agent.

Derivation of an effective equation for ultrasound propagation in a liquid containing many encapsulated bubbles

Toward an establishment of basis for medical applications such as ultrasound diagnosis, the mathematical model for the propagation of ultrasound (or pressure waves) in a liquid containing microbubbles encapsulated by a viscoelastic shell is constructed from the theoretical viewpoint. For simplicity, the bubbles do not coalesce, break up, appear, and disappear; the bubbles are spherical, and these oscillations are spherically symmetric; the viscosity of gas inside the bubbles and the thermal conductivities of both phases are neglected: these assumptions are the same as those in our previous studies [e.g., Kanagawa et al., J. Fluid Sci. Technol. (2010); Kanagawa, J. Acoust. Soc. Am. (2015)]. In this paper, the Hoff model (equation of spherical shell bubble) is used for the equation of motion of the bubble in order to theoretically clarify the effect of viscosity and stiffness of the shell, instead of the Rayleigh-Plesset or Keller model. From the method of multiple scales, some linear and /or nonlinear wave equations (e.g., KdV-Burgers equation) as an effective equation are derived from the basic set of governing equations for liquids containing many encapsulated bubbles.

Thermoplastic foaming with thermo-expandable microcapsules: Mathematical modeling and numerical simulation for extrusion process

A numerical simulation for foaming a melted thermoplastic by extrusion process is conducted, where thermo-expandable microcapsules are used as a blowing agent. Firstly, a growth model for an encapsulated bubble is developed to match experimental results. Then, the growth model is used to numerically solve a 2D non-isothermal die-swell problem, making possible to simulate the extrusion foaming process. In particular, the model is able to predict the foaming location. Numerical results are compared with experimental data obtained by optical microscopy observations and density measurements on extruded samples. The effects of thermoplastic melt rheology on foaming and the die swell induced by foaming are also investigated. This simulation helps to estimate process parameters such as extrusion speed and to predict die swell and sample porosity in whole, to finally optimize the industrial process.

Fluidics in an emptying bottle during breaking and making of interacting interfaces

An experimental effort has been reported in the present study analyzing the fluidics during the emptying of a bottle. The viscosity of the containing liquid and the orientation of the bottle while emptying are varied to obtain different emptying modes. Stages during the emptying of a vertical upended and an inclined bottle have been demarcated based on prompt flow features. Fluidic phenomena such as formation and pinch-off of an encapsulated bubble, ejector jet, flooding, and stratification have been observed in a vertically upended bottle. The rise velocity, collapse dynamics, and growth rate of the bubble at the bottle mouth are affected by the angle of inclination and mainly viscosity of the emptying liquid. Two distinct bottle emptying modes have been identified in one of which the discharge rate is increased due to a high-frequency pinch-off of air bubbles inside the bottle and in another mode due to an increase in the volume of the pinched-off bubble at a comparatively lower frequency. The interaction of dominant forces during the emptying process has been established by quantifying Reynolds number, Weber number, and Bond number. For all emptying liquids, bottle emptying time reduces linearly up to a critical angle of inclination, θcrit ∼ 20°, and further follows an asymptote. We hypothesize that the transition between the linear regime and the asymptotic regime is due to the saturation of the voidage of the air at the cylindrical section of the bottle mouth. Furthermore, the geometry of the bottle also facilitates the growth rate of the bubble inside the bottle at θcrit.

3Pb4-1 Basic Theory on Ultrasound Propagation in a Liquid Containing Encapsulated Bubbles toward Medical Application

3Pb4-1 Basic Theory on Ultrasound Propagation in a Liquid Containing Encapsulated Bubbles toward Medical Application

Subwavelength resonances of encapsulated bubbles

The aim of this paper is to derive an original formula for the subwavelength resonance frequency of an encapsulated bubble with arbitrary shape in two dimensions. Using Gohberg-Sigal theory, we derive an asymptotic formula for this resonance frequency, as a perturbation away from the resonance of the uncoated bubble, in terms of the thickness of the coating. The formula is numerically verified in the case of circular bubbles, where the resonance can be efficiently computed using the multipole method. The approach involves the use of a pole-pencil decomposition of the leading order term in the asymptotic expansion of some integral operator in terms of the thickness of the coating, followed by the application of the generalized argument principle to find the characteristic value. This approach is quite general and can be applied to other subwavelength resonators such as coated plasmonic nanoparticles.

Multistability in dynamics of an encapsulated bubble contrast agent: coexistence of three attractors

In this work we discuss complex dynamics arising in a model describing behavior of an encapsulated bubble contrast agent oscillating close to an elastic wall. We demonstrate presence of three coexisting attractors in the system. We propose an efficient numerical procedure based on the continuation method that can be used to locate the area of coexistence of these attractors in the parameters space. We provide area of coexistence of three attractors obtained by means of the proposed procedure.

Deformation of an encapsulated bubble in steady and oscillatory electric fields

In this paper, we investigate the dynamics of an encapsulated bubble in steady and oscillatory electric fields theoretically, based on a leaky dielectric model. On the bubble surface, an applied electric field generates a Maxwell stress, in addition to hydrodynamic traction and membrane mechanical stress. Our model also includes the effect of interfacial charge due to the jump of the current and the stretching of the interface. We focus on the axisymmetric deformation of the encapsulated bubble induced by the electric field and carry out our analysis using Legendre polynomials. In our first example, the encapsulating membrane is modelled as a nearly incompressible interface with bending rigidity. Under a steady uniform electric field, the encapsulated bubble resumes an elongated equilibrium shape, dominated by the second- and fourth-order shape modes. The deformed shape agrees well with experimental observations reported in the literature. Our model reveals that the interfacial charge distribution is determined by the magnitude of the shape modes, as well as the permittivity and conductivity of the external and internal fluids. The effects of the electric field on the natural frequency of the oscillating bubble are also shown. For our second example, we considered a bubble encapsulated with a hyperelastic membrane with bending rigidity, subject to an oscillatory electric field. We show that the bubble can modulate its oscillating frequency and reach a stable shape oscillation at an appreciable amplitude.

Attenuation of low-frequency underwater sound using an array of air-filled balloons and comparison to effective medium theory.

The ultimate goal of this work is to accurately predict the attenuation through a collection of large (on the order of 10-cm-radius) tethered encapsulated bubbles used in underwater noise abatement systems. Measurements of underwater sound attenuation were performed during a set of lake experiments, where a low-frequency compact electromechanical sound source was surrounded by different arrays of encapsulated bubbles with various individual bubbles sizes and void fractions. The measurements were compared with an existing predictive model [Church, J. Acoust. Soc. Am. 97, 1510-1521 (1995)] of the dispersion relation for linear propagation in liquid containing encapsulated bubbles. Although the model was originally intended to describe ultrasound contrast agents, it is evaluated here for large bubbles, and hence low frequencies, as a design tool for future underwater noise abatement systems, and there is good quantitative agreement between the observations and the model.

Oxygen nanobubbles revert hypoxia by methylation programming

Targeting the hypoxic tumor microenvironment has a broad impact in cancer epigenetics and therapeutics. Oxygen encapsulated nanosize carboxymethyl cellulosic nanobubbles were developed for mitigating the hypoxic regions of tumors to weaken the hypoxia-driven pathways and inhibit tumor growth. We show that 5-methylcytosine (5mC) hypomethylation in hypoxic regions of a tumor can be reverted to enhance cancer treatment by epigenetic regulation, using oxygen nanobubbles in the sub-100 nm size range, both, in vitro and in vivo. Oxygen nanobubbles were effective in significantly delaying tumor progression and improving survival rates in mice models. Further, significant hypermethylation was observed in promoter DNA region of BRCA1 due to oxygen nanobubble (ONB) treatment. The nanobubbles can also reprogram several hypoxia associated and tumor suppressor genes such as MAT2A and PDK-1, in addition to serving as an ultrasound contrast agent. Our approach to develop nanosized oxygen encapsulated bubbles as an ultrasound contrast agent for methylation reversal is expected to have a significant impact in epigenetic programming and to serve as an adjuvant to cancer treatment.

Dynamic simulation of a coated microbubble in an unbounded flow: response to a step change in pressure

A numerical method is developed to study the dynamic behaviour of an encapsulated bubble when the viscous forces of the surrounding liquid are accounted for. The continuity and Navier–Stokes equations are solved for the liquid, whereas the coating is described as a viscoelastic shell with bending resistance. The Galerkin Finite Element Methodology is employed for the spatial discretization of the flow domain surrounding the bubble, with the standard staggered grid arrangement that uses biquadratic and bilinear Lagrangian basis functions for the velocity and pressure in the liquid, respectively, coupled with a superparametric scheme with $B$-cubic splines as basis functions pertaining to the location of the interface. The spine method and the elliptic mesh generation technique are used for updating the mesh points in the interior of the flow domain as the shape of the interface evolves with time, with the latter being distinctly superior in capturing severely distorted shapes. The stabilizing effect of the liquid viscosity is demonstrated, as it alters the amplitude of the disturbance for which a bubble deforms and/or collapses. For a step change in the far-field pressure the dynamic evolution of the microbubble is captured until a static equilibrium is achieved. Static shapes that are significantly compressed are captured in the post-buckling regime, leading to symmetric or asymmetric shapes, depending on the relative dilatation to bending stiffness ratio. As the external overpressure increases, shapes corresponding to all the solution families that were captured evolve to exhibit contact as the two poles approach each other. Shell viscosity prevents jet formation by relaxing compressive stresses and bending moments around the indentation generated at the poles due to shell buckling. This behaviour is conjectured to be the inception process leading to static shapes with contact regions.

On the interaction of two encapsulated bubbles in an ultrasound field

We establish a theoretical model for the radial oscillations, translational motions and deformations of two interacting encapsulated bubbles. The flow field outside the bubbles is approximated by a potential flow with a viscous correction. The in-plane stresses and bending moments of the viscoelastic membranes are balanced by the hydrodynamic tractions at the interfaces of the bubbles. Since the material points move along the membranes accompanied by their movements in the radial direction when the encapsulated bubbles undergo deformations, stress balance in both the tangential and normal directions and the no-velocity-jump condition at the bubble surface are applied. The derived expression for the viscous drag includes the quasisteady drag force and the history force, which is validated by the solution of the unsteady Stokes equation. With an appropriate choice of the interface parameters, the present model is suitable for bubbles with free-slip, viscoelastic or no-slip interfaces. The viscous correction and the potential part of our solution are validated, respectively, by comparing them with previous experimental observations. The encapsulated bubble shows more stability in resisting shape oscillation. The attractive or repulsive movements of the two bubbles subjected to a driving frequency are consistent with the prediction by Bjerknes’ theory. For gas bubbles, the drag is mainly from the quasisteady component of the flow. For encapsulated bubbles, the no-velocity-jump condition enhances viscous dissipation, and thus contributes significantly to the history force in the viscous drag, generating more damping in the translational motion.