Benjamin Dollet

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.

Understanding and predicting viscous, elastic, plastic flows

Abstract.Foams, gels, emulsions, polymer solutions, pastes and even cell assemblies display both liquid and solid mechanical properties. On a local scale, such “soft glassy” systems are disordered assemblies of deformable rearranging units, the complexity of which gives rise to their striking flow behaviour. On a global scale, experiments show that their mechanical behaviour depends on the orientation of their elastic deformation with respect to the flow direction, thus requiring a description by tensorial equations for continuous materials. However, due to their strong non-linearities, the numerous candidate models have not yet been solved in a general multi-dimensional geometry to provide stringent tests of their validity. We compute the first solutions of a continuous model for a discriminant benchmark, namely the flow around an obstacle. We compare it with experiments of a foam flow and find an excellent agreement with the spatial distribution of all important features: we accurately predict the experimental fields of velocity, elastic deformation, and plastic deformation rate in terms of magnitude, direction, and anisotropy. We analyse the role of each parameter, and demonstrate that the yield strain is the main dimensionless parameter required to characterize the materials. We evidence the dominant effect of elasticity, which explains why the stress does not depend simply on the shear rate. Our results demonstrate that the behaviour of soft glassy materials cannot be reduced to an intermediate between that of a solid and that of a liquid: the viscous, the elastic and the plastic contributions to the flow, as well as their couplings, must be treated simultaneously. Our approach opens the way to the realistic multi-dimensional prediction of complex flows encountered in geophysical, industrial and biological applications, and to the understanding of the link between structure and rheology of soft glassy systems.

Nonspherical Oscillations of Ultrasound Contrast Agent Microbubbles

The occurrence of nonspherical oscillations (or surface modes) of coated microbubbles, used as ultrasound contrast agents in medical imaging, is investigated using ultra-high-speed optical imaging. Optical tweezers designed to micromanipulate single bubbles in 3-D are used to trap the bubbles far from any boundary, enabling a controlled study of the nonspherical oscillations of free-floating bubbles. Nonspherical oscillations appear as a parametric instability and display subharmonic behavior: they oscillate at half the forcing frequency, which was fixed at 1.7 MHz in this study. Surface modes are shown to preferentially develop for a bubble radius near the resonance of radial oscillations. In the studied range of acoustic pressures, the growth of surface modes saturates at a level far below bubble breakage. With the definition of a single, dimensionless deformation parameter, the amplitude of nonspherical deformation is quantified as a function of the bubble radius (between 1.5 and 5 microm) and of the acoustic pressure (up to 200 kPa).

Two-dimensional flow of foam around a circular obstacle: local measurements of elasticity, plasticity and flow

We investigate the two-dimensional flow of a liquid foam around a circular obstacle by measuring all the local fields necessary to describe this flow: velocity, pressure, and bubble deformations and rearrangements. We show how our experimental set-up, a quasi-two-dimensional ‘liquid pool’ system, is adapted to the determination of these fields: the velocity and bubble deformations are easy to measure from two-dimensional movies, and the pressure can be measured by exploiting a specific feature of this system, a two-dimensional effective compressibility. To describe accurately neighbour swapping (so-called ‘T1’ processes), we propose a new, tensorial descriptor. All these quantities are evaluated via an averaging procedure that we justify by showing that the fluctuations of the fields are essentially Gaussian. The flow is extensively studied in a reference experimental case; the velocity presents an overshoot in the wake of the obstacle, and the pressure is maximum at the leading side and minimal at the trailing side. The study of the elastic deformations and of the velocity gradients shows that the transition between plug flow and yielded regions is smooth. Our tensorial description of T1s highlights their correlation both with the bubble deformations and the velocity gradients. A salient feature of the flow, notably for the velocity and T1 distribution, is a marked fore–aft asymmetry, the signature of the elastic behaviour of the foam. We show that the results do not change qualitatively when various control parameters (flow rate, bubble area, fluid fraction, bulk viscosity, obstacle size and boundary conditions) vary, identifying a robust quasi-static regime. These results are discussed in the framework of the foam rheology literature. A movie is available with the online version of the paper.

Discrete rearranging disordered patterns, part I: Robust statistical tools in two or three dimensions

Abstract.Discrete rearranging patterns include cellular patterns, for instance liquid foams, biological tissues, grains in polycrystals; assemblies of particles such as beads, granular materials, colloids, molecules, atoms; and interconnected networks. Such a pattern can be described as a list of links between neighbouring sites. Performing statistics on the links between neighbouring sites yields average quantities (hereafter “tools”) as the result of direct measurements on images. These descriptive tools are flexible and suitable for various problems where quantitative measurements are required, whether in two or in three dimensions. Here, we present a coherent set of robust tools, in three steps. First, we revisit the definitions of three existing tools based on the texture matrix. Second, thanks to their more general definition, we embed these three tools in a self-consistent formalism, which includes three additional ones. Third, we show that the six tools together provide a direct correspondence between a small scale, where they quantify the discrete patterns local distortion and rearrangements, and a large scale, where they help describe a material as a continuous medium. This enables to formulate elastic, plastic, fluid behaviours in a common, self-consistent modelling using continuous mechanics. Experiments, simulations and models can be expressed in the same language and directly compared. As an example, a companion paper (P. Marmottant, C. Raufaste, and F. Graner, this issue, 25 (2008) DOI 10.1140/epje/i2007-10300-7) provides an application to foam plasticity.

Nonspherical Shape Oscillations of Coated Microbubbles in Contact With a Wall

In this experimental study, the nonspherical and translational behavior of individual coated microbubbles of different sizes, in contact with a 20-μm thickness cellulose wall, are observed and categorized systematically. Images from two orthogonally positioned microscopes are merged and then recorded with an ultra-fast framing camera. Large nonspherical deformations were found with 2.25 MHz frequency ultrasound pulses having driving pressures from 80 to 325 kPa. A parametric model combining potential flow theory with a viscous boundary layer at the wall is developed and used to calculate stresses from the optically recorded microbubble oscillations. Peak shear stress of up to 300 kPa and normal stresses of up to 1 MPa are estimated when microbubbles are insonifed with a 2.25 MHz pulse at 325 kPa. The clinical relevance of these results is discussed.

Dynamics of coated microbubbles adherent to a wall.

Molecular imaging with ultrasound is a promising noninvasive technique for disease-specific imaging, enabling for instance, the diagnosis of thrombus and inflammation. Selective imaging is performed by using ultrasound contrast agent microbubbles functionalized with ligands, which bind specifically to the target molecules. Here, we investigate in a model system, the influence of adherence at a wall on the dynamics of the microbubbles, in particular, on the frequency of maximum response, by recording the radial response of individual microbubbles as a function of the applied acoustic pressure and frequency. The frequency of maximum response of adherent microbubbles was found to be over 50% lower than for bubbles in the unbounded fluid and over 30% lower than that of a nonadherent bubble in contact with the wall. The change is caused by adhesion of the bubbles to the wall as no influence was found due solely to the presence of the targeting ligands on the bubble dynamics. The shift in the frequency of maximum response may prove to be important for molecular imaging with ultrasound as this application would benefit from an acoustic imaging method to distinguish adherent microbubbles from freely circulating microbubbles.

Microbubble formation and pinch-off scaling exponent in flow-focusing devices

We investigate the gas jet breakup and the resulting microbubble formation in a microfluidic flow-focusing device using ultra high-speed imaging at 1 × 106 frames/s. In recent experiments [Dollet et al., Phys. Rev. Lett. 100, 034504 (2008)], it was found that in the final stage of the collapse the radius of the neck scales with time with a 1/3 power-law exponent, which suggested that gas inertia and the Bernoulli suction effect become important. Here, ultra high-speed imaging was used to capture the complete bubble contour and quantify the gas flow through the neck. The high temporal resolution images enable us to approach the final moment of pinch-off to within 1 μs. It revealed that during the collapse, the flow of gas reverses and accelerates towards its maximum velocity at the moment of pinch-off. However, the resulting decrease in pressure, due to Bernoulli suction, is too low to account for the accelerated collapse. We observe two stages of the collapse process. At first, the neck collapses with a scaling exponent of 1/3 which is explained by a “filling effect.” In the final stage, the collapse is characterized by a scaling exponent of 2/5, which can be derived, based on the observation that during the collapse the neck becomes less slender, due to the driving through liquid inertia. However, surface tension forces are still important until the final microsecond before pinch-off

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.

The role of surface rheology in liquid film formation

The role of surface rheology in fundamental fluid dynamical systems, such as liquid coating flows and soap film formation, is poorly understood. We investigate the role of surface viscosity in the classical film-coating problem. We propose a theoretical model that predicts film thickening based on a purely surface-viscous theory. The theory is supported by a set of new experimental data that demonstrates slight thickening even at very high surfactant concentrations for which Marangoni effects are irrelevant. The model and experiments represent a new regime that has not been identified before.