Secondary Bjerknes Force Research Articles

Non-focal acoustic lens designs for cavitation bubble consolidation

During shockwave lithotripsy, cavitation bubbles form on the surface of urinary stones aiding in the fragmentation process. However, shockwaves can also produce pre-focal bubbles, which may shield or block subsequent shockwaves and potentially induce collateral tissue damage. We have previously shown in-vitro that low amplitude acoustic waves can be applied to actively stimulate bubble coalescence and help alleviate this effect. A traditional elliptical transducer lens design produces the maximum focal gain possible for a given aperture. From experiments and simulation, we have found that this design is not optimal for bubble consolidation as the primary and secondary Bjerknes forces may act against each other and the effective field volume is too small. For this work, we designed and constructed non-focal transducer lenses with complex surface geometries using rapid prototyping stereolithography to produce more effective acoustic fields for bubble consolidation during lithotripsy or ultrasound therapy. We demonstrate a design methodology using an inverse problem technique to map the desired acoustic field back to the surface of the transducer lens to determine the correct phase shift at every point on the lens surface. This method could be applied to other acoustics problems where non-focused acoustic fields are desired.

Influence of acoustic pressure and bubble sizes on the coalescence of two contacting bubbles in an acoustic field

In this study, the coalescence time between two contacting sub-resonance size bubbles was measured experimentally under an acoustic pressure ranging from 10kPa to 120kPa, driven at a frequency of 22.4kHz. The coalescence time obtained under sonication was much longer compared to that calculated by the film drainage theory for a free bubble surface without surfactants. It was found that under the influence of an acoustic field, the coalescence time could be probabilistic in nature, exhibiting upper and lower limits of coalescence times which are prolonged when both the maximum surface approach velocity and secondary Bjerknes force increases. The size of the two contacting bubbles is also important. For a given acoustic pressure, bubbles having a larger average size and size difference were observed to exhibit longer coalescence times. This could be caused by the phase difference between the volume oscillations of the two bubbles, which in turn affects the minimum film thickness reached between the bubbles and the film drainage time. These results will have important implications for developing film drainage theory to account for the effect of bubble translational and volumetric oscillations, bubble surface fluctuations and microstreaming.

Stable tridimensional bubble clusters in multi-bubble sonoluminescence (MBSL)

In the present work, stable clusters made of multiple sonoluminescent bubbles are experimentally and theoretically studied. Argon bubbles were acoustically generated and trapped using bi-frequency driving within a cylindrical chamber filled with a sulfuric acid aqueous solution (SA85w/w). The intensity of the acoustic pressure field was strong enough to sustain, during several minutes, a large number of positionally and spatially fixed (without pseudo-orbits) sonoluminescent bubbles over an ellipsoidally-shaped tridimensional array. The dimensions of the ellipsoids were studied as a function of the amplitude of the applied low-frequency acoustic pressure (PAcLF) and the static pressure in the fluid (P0). In order to explain the size and shape of the bubble clusters, we performed a series of numerical simulations of the hydrodynamic forces acting over the bubbles. In both cases the observed experimental behavior was in excellent agreement with the numerical results. The simulations revealed that the positionally stable region, mainly determined by the null primary Bjerknes force (F→Bj), is defined as the outer perimeter of an axisymmetric ellipsoidal cluster centered in the acoustic field antinode. The role of the high-frequency component of the pressure field and the influence of the secondary Bjerknes force are discussed. We also investigate the effect of a change in the concentration of dissolved gas on the positional and spatial instabilities through the cluster dimensions. The experimental and numerical results presented in this paper are potentially useful for further understanding and modeling numerous current research topics regarding multi-bubble phenomena, e.g. forces acting on the bubbles in multi-frequency acoustic fields, transient acoustic cavitation, bubble interactions, structure formation processes, atomic and molecular emissions of equal bubbles and nonlinear or unsteady acoustic pressure fields in bubbly media.

Experimental and Theoretical Studies on the Movements of Two Bubbles in an Acoustic Standing Wave Field

When subjected to an ultrasonic standing-wave field, cavitation bubbles smaller than the resonance size migrate to the pressure antinodes. As bubbles approach the antinode, they also move toward each other and either form a cluster or coalesce. In this study, the translational trajectory of two bubbles moving toward each other in an ultrasonic standing wave at 22.4 kHz was observed using an imaging system with a high-speed video camera. This allowed the speed of the approaching bubbles to be measured for much closer distances than those reported in the prior literature. The trajectory of two approaching bubbles was modeled using coupled equations of radial and translational motions, showing similar trends with the experimental results. We also indirectly measured the secondary Bjerknes force by monitoring the acceleration when bubbles are close to each other under different acoustic pressure amplitudes. Bubbles begin to accelerate toward each other as the distance between them gets shorter, and this acceleration increases with increasing acoustic pressure. The current study provides experimental data that validates the theory on the movement of bubbles and forces acting between them in an acoustic field that will be useful in understanding bubble coalescence in an acoustic field.

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.

Dynamics of cavitation clouds within a high-intensity focused ultrasonic beam

In this experimental study, we generate a 500 kHz high-intensity focused ultrasonic beam, with pressure amplitude in the focal zone of up to 1.9 MPa, in initially quiescent water. The resulting pressure field and behavior of the cavitation bubbles are measured using high-speed digital in-line holography. Variations in the water density and refractive index are used for determining the spatial distribution of the acoustic pressure nonintrusively. Several cavitation phenomena occur within the acoustic partially standing wave caused by the reflection of sound from the walls of the test chamber. At all sound levels, bubbly layers form in the periphery of the focal zone in the pressure nodes of the partial standing wave. At high sound levels, clouds of vapor microbubbles are generated and migrate in the direction of the acoustic beam. Both the cloud size and velocity vary periodically, with the diameter peaking at the pressure nodes and velocity at the antinodes. A simple model involving linearized bubble dynamics, Bjerknes forces, sound attenuation by the cloud, added mass, and drag is used to predict the periodic velocity of the bubble cloud, as well as qualitatively explain the causes for the variations in the cloud size. The analysis shows that the primary Bjerknes force and drag dominate the cloud motion, and suggests that the secondary Bjerknes force causes the oscillations in the cloud size.

Instability of a bubble chain

Based on the theory of shape instability and diffusive instability for single bubbles, we have studied the instability of an individual bubble in a bubble chain and found that its stable area enlarges the narrower the distance between bubbles. The spatial stability of the bubble chain is due to the secondary Bjerknes force between bubbles. Numerical calculations show the tension of the bubble chain varies with bubble distance and maxima appear at certain distances which could correspond to the stable states of the bubble chain.

Secondary Bjerknes Forces Deform Targeted Microbubbles

In this study, we investigated the effect of secondary Bjerknes forces on targeted microbubbles using high-speed optical imaging. We observed that targeted microbubbles attached to an underlying surface and subject to secondary Bjerknes forces deform in the direction of their neighboring bubble, thereby tending toward a prolate shape. The deformation induces an elastic restoring force, causing the bubbles to recoil back to their equilibrium position; typically within 100 μs after low-intensity ultrasound application. The temporal dynamics of the recoil was modeled as a simple mass-spring system, from which a value for the effective spring constant k of the order 10−3 Nm−1 was obtained. Moreover, the translational dynamics of interacting targeted microbubbles was predicted by a hydrodynamic point particle model, including a value of the spring stiffness k of the very same order as derived experimentally from the recoiling curves. For higher acoustic pressures, secondary Bjerknes forces rupture the molecular adhesion of the bubbles to the surface. We used this mutual attraction to quantify the binding force between a single biotinylated microbubble and an avidin-coated surface, which was found to be between 0.9 and 2 nanonewtons (nN). The observation of patches of lipids left at the initial binding site suggests that lipid anchors are pulled out of the microbubble shell, rather than biotin molecules unbinding from avidin. Understanding the effect of ultrasound application on targeted microbubbles is crucial for further advances in the realm of molecular imaging.

Interaction of a bubble and a bubble cluster in an ultrasonic field

Using an appropriate approximation, we have formulated the interacting equation of multi-bubble motion for a system of a single bubble and a spherical bubble cluster. The behavior of the bubbles is observed in coupled and uncoupled states. The oscillation of bubbles inside the cluster is in a coupled state. The numerical simulation demonstrates that the secondary Bjerknes force can be influenced by the number density, initial radius, distance, driving frequency, and amplitude of ultrasound. However, if a bubble approaches a bubble cluster of the same initial radii, coupled oscillation would be induced and a repulsive force is evoked, which may be the reason why the bubble cluster can exist steadily. With the increment of the number density of the bubble cluster, a secondary Bjerknes force acting on the bubbles inside the cluster decreases due to the strong suppression of the coupled bubbles. It is shown that there may be an optimal number density for a bubble cluster which can generate an optimal cavitation effect in liquid for a stable driving ultrasound.

Effect of ultrasound on adherent microbubble contrast agents

An investigation into the effect of clinical ultrasound exposure on adherent microbubbles is described. A flow phantom was constructed in which targeted microbubbles were attached using biotin–streptavidin linkages. Microbubbles were insonated by broadband imaging pulses (centred at 2.25 MHz) over a range of pressures (peak negative pressure (PNP) = 60–375 kPa). Individual adherent bubbles were observed optically and classified as either being isolated or with a single neighbouring bubble. It is found that bubble detachment and deflation are two significant effects, even during low amplitude ultrasound exposure. Specifically, while at very low acoustic pressure (PNP < 75 kPa) 95% of the bubbles were not affected, at medium pressure (151 kPa < P < 225 kPa) 53% of the bubbles detached and at higher pressures (301 kPa < P < 375 kPa) 96% of the bubbles detached. In addition, more than 50% of the bubbles underwent deflation at pressures between 301 and 375 kPa. At pressures between 226 and 300 kPa, more adherent bubbles detached when there was a neighbouring bubble, suggesting the role of multiple scattering and secondary Bjerknes force on bubble detachment. The flow shear, primary and secondary Bjerknes forces exerted on each bubble were calculated and compared to the estimated forces acting on the bubble due to oscillations. The oscillation force is shown to be much higher than other forces. The mechanisms of bubble detachment are discussed.

Collective bubble dynamics near a surface in a weak acoustic standing wave field

The transport of bubbles to a neighboring surface is very important in surface chemistry, bioengineering, and ultrasonic cleaning, etc. This paper proposes a multi-bubble transport method by using an acoustic standing wave field and establishes a model that explains the multi-bubble translation by expressing the balance between Bjerknes forces and hydrodynamic forces on a bubble in a liquid medium. Results indicated that the influence of primary Bjerknes force, secondary Bjerknes force, and buoyancy force on the bubble translation depends on the position of the target bubble in the acoustic field. Moreover, it was found that increasing the size of a bubble or pressure amplitude can accelerate the bubble motion and enhance the bubble-bubble interaction. The secondary Bjerknes force between two bubbles can switch from an attractive one when they oscillate in phase to a repulsive one when the bubble oscillations are out of phase. These findings provide an insight into the multi-bubble translation near a surface and can be applied to future bubble motion control studies, especially in drug delivery, sonoporation, and ultrasonic cleaning.

Experimental study on inter-particle acoustic forces

A method for the experimental measurement of inter-particle forces (secondary Bjerknes force) generated by the action of an acoustic field in a resonator micro-channel is presented. The acoustic radiation force created by an ultrasonic standing wave moves suspended particles towards the pressure nodes and the acoustic pressure induces particle volume oscillations. Once particles are in the levitation plane, transverse and secondary Bjerknes forces become important. Experiments were carried out in a resonator filled with a suspension composed of water and latex particles of different size (5-15 μm) at different concentrations. Ultrasound was generated by means of a 2.5 MHz nominal frequency transducer. For the first time the acoustic force generated by oscillating particles acting on other particles has been measured, and the critical interaction distance in various cases has been determined. Inter-particle forces on the order of 10(-14) N have been measured by using this method.

Investigation of a mutual interaction force at different pressure amplitudes in sulfuric acid

This paper investigates the secondary Bjerknes force for two oscillating bubbles in various pressure amplitudes in a concentration of 95% sulfuric acid. The equilibrium radii of the bubbles are assumed to be smaller than 10 μm at a frequency of 37 kHz in various strong driving acoustical fields around 2.0 bars (1 bar=105 Pa). The secondary Bjerknes force is investigated in uncoupled and coupled states between the bubbles, with regard to the quasi-adiabatic model for the bubble interior. It finds that the value of the secondary Bjerknes force depends on the driven pressure of sulfuric acid and its amount would be increased by liquid pressure amplitude enhancement. The results show that the repulsion area of the interaction force would be increased by increasing the driven pressure because of nonlinear oscillation of bubbles.

Experimental investigation on reversal of secondary Bjerknes force between two bubbles in ultrasonic standing wave

The direction of the secondary Bjerknes force between a free bubble and an attached bubble was experimentally investigated. The behavior of the two bubbles in an ultrasonic standing wave of 27 kHz was observed using an imaging system with a high-speed video camera. It was demonstrated experimentally that the direction of the force reversed at a specific separation distance between the two bubbles, which was defined as the threshold distance. The threshold distance changed with the radius of the attached bubble. In addition, a theoretical calculation was performed using a previously derived model that coupled the vibrations of two free bubbles [Ida, Phys. Lett. A 297, 210-217 (2002)]. The experiment data for the threshold distance qualitatively agreed with the theoretical predictions, except when the separation distance was very small. Then, it was discovered that the free bubble became trapped near the attached bubble when the separation distance between the two bubbles was very small. This indicated that a stable equilibrium point for the separation distance exists that cannot be predicted by the theoretical model.

Selective particle trapping using an oscillating microbubble

The ability to isolate and sort analytes within complex microfluidic volumes is essential to the success of lab-on-a-chip (LOC) devices. In this study, acoustically-excited oscillating bubbles are used to selectively trap particles, with the selectivity being a function of both particle size and density. The operating principle is based on the interplay between the strong microstreaming-induced drag force and the attractive secondary Bjerknes force. Depending upon the size of the bubble, and thus its resonant frequency, it is possible to cause one force to dominate over the other, resulting in either particle attraction or repulsion. A theoretical analysis reveals the extent of the contribution of each force for a given particle size; in close agreement with experimental findings. Density-based trapping is also demonstrated, highlighting that denser particles experience a larger secondary Bjerknes force resulting in their attraction. This study showcases the excellent applicability and versatility of using oscillating bubbles as a trapping and sorting mechanism within LOC devices.

Dynamics Simulation of Microbubbles in Ultrasonic Wave Field Considering the Secondary Bjerknes Force

In order to control a large number of microbubbles with a pumping ultrasonic wave, it is essential to consider a bubble-to-bubble interaction. This paper proposes a simplified microbubble dynamics simulation method considered with bubble-to-bubble interaction in ultrasonic wave field. The microbubble dynamics is solved with difference method based on the Stokes’s low and the secondary Bjerknes force between microbubbles. The laminar flow is considered in the cylindrical calculation region. The effectiveness of the frequency sweep method in laminar flow is shown with simulation results. And a usefulness of this dynamics simulation is also discussed through the experimental result.

Interaction of two oscillating sonoluminescence bubbles in sulfuric acid

The mutual interaction of two oscillating gas bubbles in different concentrations of sulfuric acid is numerically investigated. A nonlinear oscillation for spherical symmetric bubbles with equilibrium radii smaller than 10 μm at a frequency of 37 kHz in a strong driving acoustical field P(a)=1.8 bar is assumed. The calculations are based on the investigation of the secondary Bjerknes force with regard to adiabatic model for the bubble interior which appears as repulsion or attraction interaction force. In this work the influence of the various concentrations of sulfuric acid in uncoupled and coupled distances between bubbles has been investigated. It is found that the sign and value of the secondary Bjerknes force depend on the sulfuric acid viscosity and its amount would be decreased by liquid viscosity enhancement. The results show that big change in the parameters of produced bubbles occurs in the sulfuric acid with concentrations from 65% to 85%.

History force on coated microbubbles propelled by ultrasound

In this paper the unsteady translation of coated microbubbles propelled by acoustic radiation force is studied experimentally. A system of two pulsating microbubbles of the type used as contrast agent in ultrasound medical imaging is considered, which attract each other as a result of the secondary Bjerknes force. Optical tweezers are used to isolate the bubble pair from neighboring boundaries so that it can be regarded as if in an unbounded fluid and the hydrodynamic forces acting on the system can be identified unambiguously. The radial and translational dynamics, excited by a 2.25 MHz ultrasound wave, is recorded with an ultrahigh speed camera at 15×106 frames/s. The time-resolved measurements reveal a quasisteady component of the translational velocity, at an average translational Reynolds number ⟨Ret⟩≈0.5, and an oscillatory component at the same frequency as the radial pulsations, as predicted by existing models. Since the coating enforces a no-slip boundary condition, an increased viscous dissipation is expected due to the oscillatory component, similar to the case of an oscillating rigid sphere that was first described by Stokes [“On the effect of the internal friction of fluids on the motion of pendulums,” Trans. Cambridge Philos. Soc. 9, 8 (1851)]. A history force term is therefore included in the force balance, in the form originally proposed by Basset and extended to the case of time-dependent radius by Takemura and Magnaudet [“The history force on a rapidly shrinking bubble rising at finite Reynolds number,” Phys. Fluids 16, 3247 (2004)]. The instantaneous values of the hydrodynamic forces extracted from the experimental data confirm that the history force accounts for the largest part of the viscous force. The trajectories of the bubbles predicted by numerically solving the equations of motion are in very good agreement with the experiment.

Interaction between two laser-induced cavitation bubbles in a quasi-two-dimensional geometry

We report on experimental and numerical studies of pairs of cavitation bubbles growing and collapsing close to each other in a narrow gap. The bubbles are generated with a pulsed and focused laser in a liquid-filled gap of 15 μm height; during their lifetime which is shorter than 14 μs they expand to a maximum radius of up to Rmax = 38 μm. Their motion is recorded with high-speed photography at up to 500000 frames s−1. The separation at which equally sized bubbles are created, d, is varied from d = 46–140 μm which results into a non-dimensional stand-off distance, γ = d/(2Rmax), from 0.65 to 2. For large separation the bubbles shrink almost radially symmetric; for smaller separation the bubbles repulse each other during expansion and during collapse move towards each other. At closer distances we find a flattening of the proximal bubbles walls. Interestingly, due to the short lifetime of the bubbles (≤14 μs), the radial and centroidal motion can be modelled successfully with a two-dimensional potential flow ansatz, i.e. neglecting viscosity. We derive the equations for arbitrary configurations of two-dimensional bubbles. The good agreement between model and experiments supports that the fluid dynamics is essentially a potential flow for the experimental conditions of this study. The interaction force (secondary Bjerknes force) is long ranged dropping off only with 1/d as compared to previously studied three-dimensional geometries where the force is proportional to 1/d2.

Microbubble Adhesion to Target Wall by Ultrasonic Wave Frequency Sweep Method

Microbubble cavitation enhances the permeability of cell membranes if the bubbles are in the vicinity of the target. A method aimed at attaching microbubbles to a target wall using the frequency sweep of an ultrasonic pumping wave is proposed. A slow frequency decrease of the pumping ultrasonic wave moves the aggregated bubbles toward the surrounding target wall. The following quick frequency return rearranges the spatial distribution of the bubble aggregations. Repetition of this sequence increases the number of bubbles adhering to the target wall. To analyze the rearrangement by the quick frequency return, Numerical analyses are carried out based on a three-dimensional model in which bubble-to-bubble interaction caused by the secondary Bjerknes force is taken into account. Experiments are carried out using ultrasonic contrast agent microbubbles. As a result, we obtain twice the trapping efficiency of the microbubble with the frequency sweep method from 2.5 to 1.75 MHz in comparison with the fixed frequency method at 2.1 MHz.